Brownian motion

The Concepts and Practice of Mathematical Finance

by Mark S. Joshi  · 24 Dec 2003

of the BlackScholes model. In Chapter 5, we step up a mathematical gear and introduce the Ito calculus. With this calculus we introduce the geometric Brownian motion model of stock price evolution and deduce the Black-Scholes equation. We then show how the BlackScholes equation can be reduced to the heat equation

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we refer the author to [63]. For a more rigorous treatment, see [118]. We discuss further reading at the end of the chapter. 5.2 Brownian motion One of the fundamental tools in option pricing is the theory of stochastic calculus. This theory allows the manipulation of the random processes described in

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will also affect the value of the option. We can achieve this property by requiring the random increments to come from a Brownian motion, and by requiring that the same Brownian motion drive all the random processes. Recall that Wt+h - Wt = h112N(0, 1), (5.7) in a distributional sense. With this in

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it impossible to simplify via a change of coordinates. Example 5.1 Suppose the stocks Xt and Yt follow geometric Brownian motion with the same underlying Brownian motion. Show that XtYt also follows a geometric Brownian motion and compute its drift and volatility. Solution Write dXt = aXtdt + QXtdWt, dYt = ,BYtdt + vYtdWt. We compute d(XtYt) = XtdYt

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the Black-Scholes analysis to it directly. Note that Xt is described by the process dXt = (µ + d)Xtdt + XtadWt, (5.76) which is just geometric Brownian motion with a different drift. But the drift does not affect the Black-Scholes price. We therefore conclude that the price of the option C satisfies

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- s and mean zero and is independent of Ws. Stochastic calculus generalizes ordinary calculus by letting the derivative have a random component coming from a Brownian motion. Stochastic calculus deals with Ito processes, in which the random variable's derivative has both a deterministic linear part and a random part which is

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the pure mathematician which requires hard work, but is worthwhile for those who are not willing to take results on faith is Karatzas & Shreve's Brownian Motion and Stochastic Calculus, [941. The PDE approach to mathematical finance developed in this chapter has now been superseded by the martingale approach we describe in

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< i 2, with Xo') = X(" show that for t > 0, X(1) < X(2) t t Exercise 5.11 Suppose an asset follows Brownian motion instead of geometric Brownian motion. Find the analogue of the Black-Scholes equation. Exercise 5.12 Suppose we have a call option on the square of the stock price

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instead of the stock. 6.2 Introduction We have presented two different approaches to deriving the Black-Scholes equation. The first approach relied on approximating Brownian motion by a discrete process in which the value at each node of a tree was determined by no-arbitrage arguments. The second approach relied on

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normally distributed with mean 0 and variance t - s. This measure is known as Wiener measure, and ensures that 142 Risk neutrality and martingale measures Brownian motion actually exists in a mathematical sense. One curious aspect of the theory is that it is the existence of the measure that is important rather

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the first stage of martingale pricing - identifying the processes which are martingales. We are interested in the case where the stock, St, is following geometric Brownian motion and the bond, Bt, is continuously compounding at the risk-free rate. Following the procedure in the discrete case, we want StlBt to be a

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to make the discounted prices into martingales. This will, of course, be a change in the measure on the space of paths which underlies the Brownian motion driving the stock-price movements. Such a measure change must preserve probability 1 and probability 0 events. This means that the measure-changed paths will

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measure has absolutely no effect on it. The interesting thing about (6.68) is that it is the evolution equation for a stockprice under geometric Brownian motion with drift the risk-free rate. Our switch to an equivalent martingale measure has the same effect as pretending investors are risk-neutral - the market

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s from below. The crucial point about the martingale representation theorem is that Mt is a martingale with respect to the filtration generated by the Brownian motion. This means that the information which determines the movement of Mt is contained in the behaviour of Wt. It is therefore not so surprising that

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Broadie & Glasserman, [26]. For a discussion of the issues involved in implementing tree models see [37]. 7.10 Exercises Exercise 7.1 Suppose we discretize Brownian motion by taking a trinomial tree. What conditions (if any) on the probabilities and the branches will get the third and fourth moments of one step

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drift. In this section, we combine that result with our results on Girsanov's theorem to derive the joint law for a Brownian motion with drift. Let Wt be a Brownian motion. Let Yt = aWt, and my be the minimum of Yt up to time t. We then have for y < 0 and x

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maximum. Let MZ denote the maximum over the interval [0, t]. Fortunately, the fact that the negative Continuous barrier options 216 of a Brownian motion with drift is a Brownian motion with drift means that the law for the maximum is easily deducible from the law for the minimum. We can write M? = max

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expectations by a random process which is a positive martingale called the Radon-Nikodym derivative. The measure change for changing the drift of a Brownian motion uses geometric Brownian motion as Radon-Nikodym derivative. The formula for a down-and-out call is most easily deduced by dividing the payoff into two pieces and

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corresponding knock-out option is not equal to the price of the corresponding vanilla, construct an arbitrage portfolio. Exercise 8.2 A stock follows geometric Brownian motion with time-dependent volatility. How will the time-dependence affect the price of a down-and-out call? 8.11 Exercises 221 Distinguish the two

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= c22, (9.19) which imply that a21 = X21 all , a22 = (9.20) c22 - a21. We can continue similarly for the remaining rows. In fact for Brownian motion, Cholesky decomposition is nothing new for us: inspecting our incremental path generation, we see that it is equivalent to multiplying by a lower triangular matrix

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we now order the eigenvalues X j so that) j > ,l j+l. We can regard each eigenvector as being a different component of the Brownian motion, then the first eigenvector expresses the largest component. Each successive eigenvalue represents the weighting of higher frequency vibrations. If we take the pseudo-square-root

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substantial increase in the rate of convergence for low-discrepancy-based simulations at little computational cost. It is equivalent to taking the covariance matrix for Brownian motion, rearranging the order of the rows and columns, and then performing a Cholesky decomposition. Whilst the computational burden in computing the spectral square root is

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. In order to understand multi-asset options, we will need to understand multidimensional Brownian motions, how to understand correlations between Brownian motions and how to extend the Ito calculus to higher dimensions. 11.2 Higher-dimensional Brownian motions Recall that a one-dimensional Brownian motion, Xt, is defined so that the distribution of Xt - XS is always a

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of random variables which are each following Ito processes, that is we need a multi-dimensional Ito rule. Thus suppose we have correlated Brownian motions W(j). Associated to each Brownian motion, we have an Ito process X(j), dX(j) = Aj (X(j), t)dt + aj (X(j), t)dW(j). (11.7

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-dimensional Ito calculus: dWrj)dW(k) = pjkdt. To summarize, we have Theorem 11.1 (Multi-dimensional Ito lemma) Let Wtj) be correlated Brownian motions with correlation coefficient pjk between the Brownian motions WU) and Wtk). Let Xj be an Ito process with respect to Wt W. Let f be a smooth function; we then

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also that if P12 = 1 then 6=61+Q2. (11.21) We can interpret (11.20) geometrically. Suppose we regard a Brownian motion as being a vector times a one-dimensional Brownian motion. Perfect correlation means the vectors point the same way, perfect negative correlation means they point the opposite way, and zero correlation

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as vectors in 1R which add according to their directions. Example 11.1 Suppose the stocks Xt and Yt follow correlated geometric Brownian motions. Show that XtYt also follows a geometric Brownian motion and compute its drift and volatility. Solution We write dXt = aXtdt + o XtdWW 1), dYt = ,BYtdt + vYdWt(2), and take the

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we can change. drifts. Once again we cannot do anything else. This second statement has some new aspects however. When we are dealing with correlated Brownian motions, this means that the correlations between them cannot be changed via measure change. In financial terms, this ensures that the correlation between two assets affects

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dBt = rBtdt, dDt = dDtdt, dFt = FtpFdt + Fto-FdWt, (11.51) dMt = MtµMdt + MtoxMdZt (11.54) (11.52) (11.53) where Wt and Zt are Brownian motions correlated with coefficient p. We want to identify the risk-neutral processes associated with taking the sterling money-market account as numeraire. The exchange rate

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lYl + ai2Y2 ), (11.70) where (ai j) is any Cholesky decomposition of the correlation matrix. To see this process is a discretization of two-dimensional Brownian motion, we simply invoke the two-dimensional Central Limit Theorem, just as we did in the one-dimensional case. This technique can be extended easily to

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Ito calculus goes over to higher dimensions with the additional rule dWjdWk = pjkdt where Pjk is the correlation between Wj and Wk. When adding correlation Brownian motions we can find the volatility of the new process by treating the original processes as vectors. We can change the drift of a multi-dimensional

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the forward rate is above H at the start of the FRA. Develop an analytic formula for its price if the forward rate follows geometric Brownian motion. 14 The pricing of exotic interest rate derivatives 14.1 Introduction The critical difference between modelling interest rate derivatives and equity/FX options is that

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+ p12dW2), (14.24) df2 = f2µ2dt + f2Q2dW2, df3 = f3/s3dt + f3a3(p23dW2 + (14.23) 1 - p23dW3), (14.25) where W1, W2 and W3 are uncorrelated Brownian motions. The fact that W1 and W3 are uncorrelated comes from our decorrelation assumption. Note that this is a quite strong interpretation of our condition. It

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more theory. 364 Incomplete markets and junzp-diffusion processes 15.3 Modelling jumps in a continuous framework Suppose we have a stock moving under geometric Brownian motion, with the added possibility of crashes. What properties should crashes have? They should occur instantaneously: the probability of one occurring in a given small time

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regardless of which the way the moves are, and the variance of the path is T. Letting N tend to infinity, the paths converge to Brownian motion and the first variation becomes infinite. Since (almost) every path has infinite first variation, this property must be preserved by changes of measure. How can

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volatility 390 with a a positive real number. We work with the instantaneous variance, V, which is the square of the instantaneous volatility a. The Brownian motions WO and W(2) may be correlated or uncorrelated as we choose. Many of the issues which arise with stochastic-volatility models are similar to

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within the space of continuous martingales will not buy us much. A second classification theorem says that a general martingale is a random-time-changed Brownian motion. This means that in mathematical terms, the introduction of random times is inevitable if we wish to study new processes. The random process modelling information

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fact that we can easily deduce from the characteristic function is that Variance Gamma paths are of finite first variation. This is very different from Brownian motion where the second variation is finite and non-zero, and the first variation is always infinite. We prove that the first variation is finite by

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, and of being independent of previously elapsed time. Variance Gamma paths consist of many small jumps. Variance Gamma paths are of finite first variation whereas Brownian motion paths are of infinite first variation. In passing to an equivalent measure, there are no constraints on the changes in parameter values unlike in the

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out in the context of hedging a vanilla call option. The perfect Black-Scholes world Implement an engine which evolves a stock under a geometric Brownian motion with drift µ, volatility o, in N steps using the solution for the stochastic differential equation. Write a hedging simulator that accounts for the profits and

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possible. There are two ways to do this: the Brownian bridge and spectral decomposition. Our objective is to produce draw a path, Wt, from a Brownian motion and pass back the increments Wt; -Wt1_I . When we use the intuitive method of incremental path-generation, we are effectively drawing n Gaussian variables

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N(µ, a2) distribution has mean µ and variance a2. The normal distribution is important for two reasons; the first is that it underlies the definition of Brownian motion which will be crucial to us in modelling stock price movements. The second is that it is, in a certain sense, the distribution one obtains

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. [93] M. Joshi, J. Theis, Bounding Bermudan swaptions in a swap-rate market model, Quantitative Finance 2, 2002, 370-7. [94] I. Karatzas, E. Shreve, Brownian Motion and Stochastic Calculus, second edition, Springer Verlag, 1997. [95] I. Karatzas, E. Shreve, Methods of Mathematical Finance, Springer Verlag, 1998. [96] S.G. Kou, A

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callable, 301 convertible, 7, 430 corporate, 7 government, 1 premium, 2 riskless, 5, 7 zero-coupon, 5, 24-26, 28, 302, 433 Brownian bridge, 230 Brownian motion, 97-100, 101, 107, 142, 260, 430 correlated, 263 higher-dimensional, 261-263 Buffett, Warren, 2 bushy tree, see tree, non-recombining calibration to vanilla

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-negativity of, 384 Index Gamma distribution, 402 Gamma function, 402 incomplete, 405 Gaussian distribution, 57, 103 Gaussian random variable synthesis of, 191 gearing, 300 geometric Brownian motion, 111, 114 gilt, 314 Girsanov transformation, 214 Girsanov's theorem, 158, 166, 210-213, 368, 390, 431 higher-dimensional, 267-271 Greeks, 77-83, 431

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Ito's Lemma, 106-110 application of, 111-114 multi-dimensional, 264 joint density function, 464 joint law of minimum and terminal value of a Brownian motion with drift, 213 without drift, 208 jump-diffusion model, 87, 364-381 and deterministic future smiles, 244 and replication of American options, 293 price of

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deviation, 463 static replication, see replication, static stepping methods for Monte Carlo, 439 stochastic, 433 stochastic calculus, 97 stochastic differential equation, 105 for square of Brownian motion, 107 stochastic process, 102-106, 141 stochastic volatility, 88, 389 and risk-neutral pricing, 390-393 implied, 400 pricing by Monte Carlo, 391-394 pricing

Mathematical Finance: Theory, Modeling, Implementation

by Christian Fries  · 9 Sep 2007

.2. Special Sections . . . . . . . . . . . . . . . . . . . . . . . . 21 22 22 23 I. 25 Foundations 2. Foundations 2.1. Probability Theory . . . . . . . . . . . . . . . . 2.2. Stochastic Processes . . . . . . . . . . . . . . . 2.3. Filtration . . . . . . . . . . . . . . . . . . . . . 2.4. Brownian Motion . . . . . . . . . . . . . . . . . 2.5. Wiener Measure, Canonical Setup . . . . . . . . 2.6. Itô Calculus . . . . . . . . . . . . . . . . . . . . 2.6.1. Itô Integral . . . . . . . . . . . . . . . . 2.6.2. Itô Process . . . . . . . . . . . . . . . . 2.6.3. Itô

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Lemma and Product Rule . . . . . . . 2.7. Brownian Motion with Instantaneous Correlation 2.8. Martingales . . . . . . . . . . . . . . . . . . . . 2.8.1. Martingale Representation Theorem . . . 2.9. Change of Measure (Girsanov, Cameron, Martin) 2.10. Stochastic Integration . . . . . . . . . . . . . . . 2

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.3.3. Normal Distributed Random Variables . . . . . . . . . . . . . A.3.4. Poisson Distributed Random Variables . . . . . . . . . . . . . A.3.5. Generation of Paths of an n-dimensional Brownian Motion . . A.4. Generation of Correlated Brownian Motion . . . . . . . . . . . . . . A.5. Factor Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6. Optimization (one dimensional): Golden Section Search . . . . . . . A.7. Convolution with the Normal Density . . . . . . . . . . . . . . . . . 387 B.

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Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 2.4. BROWNIAN MOTION ω2 ω9 ω8 ω1 ω10 ω3 ω 7 ω6 ω4 ω5 ω1 ω10 Figure 2.4.: Illustration of a filtration and an adapted process that

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and more specific statements about the nature of Z. Compare this to the illustrations in Figure 2.1 and 2.3. C| 2.4. Brownian Motion Definition 23 (Brownian Motion): q n Let W : [0, ∞) × Ω → R denote a stochastic process with the following properties: 37 This work is licensed under a Creative

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with mean 0 and covariance matrix (t − s)In , where In denotes the n × n identity matrix. Then W is called (n-dimensional) P-Brownian motion or a (n-dimensional) P-Wiener process. y We have not yet discussed the question whether a process with such properties exists (it does). The

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see that we have prescribed the distribution of W(t) as well as the distribution of the increments W(t) − W(s). Remark 24 (Brownian Motion): The property 4 is less axiomatic than one might assume: The central requirement is the independence of the increments together with the requirement that increments

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0. That the increments are normal distributed is more a consequence than an requirement, see Theorem 25. This theorem also gives a construction of the Brownian motion. Tip (time discrete realizations): In the following we will often consider the realizations of a stochastic process at discrete times 0 = T 0 < T

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27 (Canonical Setup): The space (C([0, ∞)), B(C([0, ∞)), P∗ ) q (as defined in Theorem 25) is called the canonical setup for a Brownian motion W defined by W(t, ω) := ω(t), ω ∈ C([0, ∞)). y Remark 28: A more detailed discussion of Theorem 25 may be found in

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that occurs from T to T + ∆T ; to be precise W(T ) models the probability distribution of the particle position. The model of a Brownian motion√is that position changes are normal distributed with mean 0 and standard deviation ∆T . Requiring mean 0 corresponds to requiring √ that the position change has

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requirement that position changes are independent from the position and time at which they occur. To motivate the class of Itô processes we consider the Brownian motion at discrete times 0 = T 0 < T 1 < . . . < T N . The random variable W(T i ) (position of the particle) may be expressed through

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http://www.christian-fries.de/finmath/ 2.6. ITÔ CALCULUS W(t,ω) 0 t Figure 2.6.: Paths of a (discretization of a) Brownian motion (for example). Then the increments may be still normal distributed but their standard p deviation no longer will be T − T j . Instead it might

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example we would use σ(T j ) := e−T j . X(t,ω) 0 t Figure 2.7.: Paths of a (discretization of a) Brownian motions with time dependent instantaneous volatility Next we consider the case where the particle has a preference for a certain direction, i.e. a drift. This

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+ σ(t, ω)dW(t, ω). The stochastic integral Z t2 X(t) dY(t) t1 was defined pathwise for integrators Y that are Brownian motions (dY = dW) and for integrands X belonging to the class of integrands of the Itô integral. We extend the definition of the stochastic integral to

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q dY = µdt + σdW 18 The dimension n denotes the dimension of the image space. The factor dimension m denotes the number of (independent) Brownian motions needed to construct the process. 46 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed

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discrete case this is not the case. E.g. consider (X(ti ) + ∆X(ti ))3 in the example above. C| 2.7. Brownian Motion with Instantaneous Correlation In Definition 23 Brownian motion was defined through normal distributed increments W(t) − W(s), t > s having covariance matrix (t − s)In . In other words we

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have that for W = (W1 , . . . , Wn ) the components are one dimensional Brownian motions with pairwise independent increments, i.e. for i , j we have that Wi (t) − Wi (s) and W j (t) − W j (s) are independent

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.en Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 2.7. BROWNIAN MOTION WITH INSTANTANEOUS CORRELATION fi, j (i = 1, . . . , n, j = 1, . . . , m) denote stochastic processes belonging to the class of integrands of the Itô integral

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i.e. that R := FF T is a correlation matrix. By this assumption we ensure that the components of Wi of W are one dimensional Brownian motions in the sense of Definition 23. By means of the factor matrix F we may interpret the implied correlation structure R in a geometrical way

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free). 2.8.1. Martingale Representation Theorem Theorem 48 (Martingale Representation Theorem21 ): Let W(t) = (W1 (t), . . . , Wm (t))T denote an m-dimensional Brownian motion, FRt the corresponding filtration. Let M(t) denote a martingale with respect to Ft with Ω |M(t)|2 dP < ∞ (∀ t ≥ 0). Then there exists

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-Nikodým density and denote it by dQ y dP . Theorem 53 (Change of Measure (Girsanov, Cameron, Martin)): Let W denote a (d-dimensional) P-Brownian motion and {Ft } the filtration generated by W. Let Q denote a measure equivalent to P (w.r.t. {Ft }). 1. Then there exists a

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(integrands) for which we may define a stochastic integral depends on the properties of the integrators (and vice versa). For continuous integrators (as the Brownian motion) the integrands merely have to be adapted processes. For more general integrators one has to restrict to a smaller class of integrands, e.g. previsible

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stochastic process evaluated at time t (≡ random variable) see above X(ω) stochastic process evaluated in state ω Path of X in state ω. W Brownian motion Model for a continuos (random) movement of a particle with independent increments (position changes). F σ-algebra sets) {Ft | t ≥ 0} filtration (set of

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://www.christian-fries.de/finmath/ CHAPTER 4. PRICING OF AN EUROPEAN STOCK OPTION UNDER THE BLACK-SCHOLES MODEL N (where W Q denotes a QN -Brownian motion).2 From the quotient rule 43 we find S (t) d B(t) ! =   N S (t)  S (t)  QN µ (t)dt + σ(t)dW

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denote by ω j . Here ∆W(ti , ω j ) and ∆W(tk , ω j ) (i , k) are independent random numbers, following the definition of the Brownian motion. If we follow this rule to generate paths ω1 , . . . , ωnpaths , where ∆W(ti , ω j ) and ∆W(ti , ωk ) ( j , k) are independent, then

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.christian-fries.de/finmath/ CHAPTER 13. DISCRETIZATION OF TIME AND STATE SPACE ∆B(ti ) such that in the limit ∆ti → 0 we recover a brownian motion, i.e. for X we recover the original time-continuous process. Using the increments ∆B(ti ) the process X from (13.7) may take only

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, (17.1) with initial conditions Li (0) = Li,0 , mit Li,0 ∈ [0, ∞), i = 0, . . . , n − 1, where WiP denote (possibly instantaneously correlated) P-Brownian motions with dWiP (t)dW Pj (t) = ρi, j (t)dt. Let σi : [0, T ] 7→ R and ρi, j : [0, T ] 7→ R be deterministic

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derivatives the time dependency matters. A further degree of freedom that is introduced in (17.1) is the instantaneous correlation ρi, j of the driving Brownian motions. For the value of a caplet the instantaneous correlation is insignificant (indeed, it does not enter in the Black model). For the evaluation of

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see Appendix A.5. The advantage of the factor reduction is that after it only an m-dimensional Brownian motion has to be simulated (and not an n-dimensional Brownian motion). Often 8 An n − 1-dimensional Brownian motion is sufficient here, since we may choose W0 = 0, because the forward rate L0 is not

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.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 18. EXCURSION: INSTANTANEOUS CORRELATION AND TERMINAL CORRELATION and Uk denote independent Brownian motions. Furthermore, let fi,k such that m  X    R := ρi, j (t) =  fi,k f j,k  i, j=1,...,n k=1 is

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T ) · dW P (t) f (0, T ) = f0 (T ) (19.1) for 0 ≤ t < T , where W P = (W1P , . . . , WmP ) is an m-dimensional P-Brownian motion with instantaneously uncorrelated components.2 Furthermore we assume that σ(t, T ) = (σ1 (t, T ), . . . , σm (t, T )) and αP (t, T ) are adapted processes

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choose the volatility structure such that (21.2) corresponds to the process of a LIBOR market model: Let W = (W1 , . . . , Wn ) denote an n-dimensional Brownian motion as given in Section 2.7. dW(t) = F(t) · dU(t), with correlation matrix R := FF T , i.e. dWi (t) = Fi (t) ·

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To this correlation model we apply a factor reduction (Principal Component Analysis), see Appendix A.5. The number of factors is the number of independent Brownian motions (effectively) entering the model, see Definition 45. Upon a factor reduction 315 This work is licensed under a Creative Commons License. http://creativecommons.org/

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curves are parallel or exhibit some regular structure, see Figure 22.2. 22.3. Mean Reversion We consider the example of a simple one factor Brownian motion (ρi, j = 1, i.e. r = 0). Figure 22.1 shows the simulated forward rates for different parameters a in (22.1). 1 We

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. dx(t) = σ(t) · dW Q (t), x(0) = x0 , (23.2) where σ is a deterministic function and W Q denotes a Q-Brownian motion. Without loss of generality we may assume x0 = 0. Equation (23.2) is the most simplest choice of a Markovian driver process. We will consider

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1 , T 2 ] liegt und λ konstant, so ist log(1 − q) λ=− . T2 − T1 A.3.5. Generation of Paths of an n-dimensional Brownian Motion Es sei T 0 < T 1 < . . . < T m eine gegebene Zeitdiskretisierung. Es sollen an diesen Zeitpunkten die Realisierungen einer n-dimensionalen Brownschen Bewegung W :=

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zeitdiskrete Brownsche Bewegung durch den Code in Listing A.1 erzeugt. Listing A.1: Erzeugung einer n-dimensionalen Brownschen Bewegung /** * This class represents a multidimensional brownian motion W = (W(1),...,W(n)) * where W(i),W(j) are uncorrelated for i not equal j. * Here the dimension n is called factors

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since this brownian motion is used to * generate multi-dimensional multi-factor Ito processes and there one might use * a different number of factors to generate Ito processes of

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2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ A.4. GENERATION OF CORRELATED BROWNIAN MOTION A.4. Generation of Correlated Brownian Motion Lemma 227 (Faktorzerlegung): Sei R = (ρi, j )i, j=1...n eine gegebene Korrelationsmatrix. Somit ist R symmetrisch und positiv semi-definit

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List of Figures Illustration of Measurability . . . . . . . . . . . . . . . . . . . . . . Lebesgue integral versus Riemann integral . . . . . . . . . . . . . . Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . Illustration of a filtration and an adapted process . . . . . . . . . . . Brownian motion: Time discretization . . . . . . . . . . . . . . . . . Brownian motion: Paths . . . . . . . . . . . . . . . . . . . . . . . . Brownian motions with time dependent instantaneous volatility . . . Brownian motion with drift . . . . . . . . . . . . . . . . . . . . . . Drift as a consequence of a non-linear function of a stochastic process Integration of stochastic processes . . . . . . . . . . . . . . . . . . . 30 32

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J, M S.: The Concepts and Practice of Mathematical Finance. Cambridge University Press, 2003. ISBN 0-521-82355-2. [18] K, I; S, S E.: Brownian Motion and Stochastic Calculus. Second Edition. Springer Verlag, 1991. ISBN 0-387-97655-8. [19] K, P E.; P, E: Numerical Solution of Stochastic Differential Equations

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.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ INDEX – Implementation . . . . . . . . . . . . . . . . 135 bootstrapping . . . . . . . . . . . . . . . . . . . . 133 Borel σ-algebra . . . . . . . . . . . . . . . . . . . 28 Bouchaud-Sornette Methode . . . . . . . 100 Brownian motion . . . . . . . . . . . . . . . . . . 38 C Cache . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 calibration . . . . . . . . . . . . . . . . . . . . . . . 265 Call Spread . . . . . . . . . . . . . . . . . . 146, 147 cancelation right . . . . . . . . . . . . . . . . . . 165 Canonical Setup . . . . . . . . . . . . . . . . . . . 40 Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Caplet . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Capped . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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. . . . . . . . . . . . . . . . . . . . . . . . . . . 172 static (Java™ Schlüsselwort) . . . 366 statische Methode . . . . . . . . . . . . . . . . 365 Stochastic Integral – Itô Process as Integrator . . . . . . . . . . 46 stochastic integral . . . . . . . . . . . . . . . . . . 57 – semi-martingale as integrator . . . . . 58 – with Brownian motion as integrator 44 stochastic process . . . . . . . . . . . . . . . . . . 35 – lognormal . . . . . . . . . . . . . . . . . . . . . . . 49 – previsible . . . . . . . . . . . . . . . . . . . . . . . 58 – previsible process . . . . . . . . . . . . . . . . 58 Stochastische Differentialgleichung – Diskretisierung . . . . . . . . . . . . . . . . . 177 Stochastischer Prozess . . . . . . . . . . . . . 35 – Gestoppter Prozess . . . . . . . . . . . . . 222 – quadratische

Derivatives Markets

by David Goldenberg  · 2 Mar 2016  · 819pp  · 181,185 words

To Modeling Prices 15.7 Appendix: Essential Martingale Properties CHAPTER 16 OPTION PRICING IN CONTINUOUS TIME 16.1 Arithmetic Brownian Motion (ABM) 16.2 Shifted Arithmetic Brownian Motion 16.3 Pricing European Options under Shifted Arithmetic Brownian Motion with No Drift (Bachelier) 16.3.1 Theory (FTAP1 and FTAP2) 16.3.2 Transition Density Functions 16

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.3.3 Deriving the Bachelier Option Pricing Formula 16.4 Defining and Pricing a Standard Numeraire 16.5 Geometric Brownian Motion (GBM) 16

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.5.1 GBM (Discrete Version) 16.5.2 Geometric Brownian Motion (GBM), Continuous Version 16.6 Itô’s Lemma 16.7 Black–Scholes Option Pricing 16.7.1 Reducing GBM

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for N=3 14.10 Summary of Stock Price Evolution (N-Period Binomial Process) 15.1 Strategy for Example 2 16.1 Non Smoothness of Brownian Motion Paths TABLES 1.1 Forward Mortgage Rates (March 7, 2014) 1.2 Weekly Average Spot 30-Year Fixed Mortgage Rates 1.3 GBP/USD Futures

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martingale measures (EMMs) in the representation of option and stock prices; 30. the efficient market hypothesis (EMH) as a guide to modeling prices; 31. arithmetic Brownian motion (ABM) and the Louis Bachelier model of option prices; 32. easy introduction to the tools of continuous time finance, including Itô’s lemma; 33. Black

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’. The first model of prices consistent with the empirical findings was the random walk model or what we will call, in continuous time, the arithmetic Brownian motion model (ABM). We will be discussing this model in detail in Chapter 16. Later, it was found that independence was too strong a condition, because

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that the property E(W1(ω)|W0)=W0 holds. CHAPTER 16 OPTION PRICING IN CONTINUOUS TIME 16.1 Arithmetic Brownian Motion (ABM) 16.2 Shifted Arithmetic Brownian Motion 16.3 Pricing European Options under Shifted Arithmetic Brownian Motion with No Drift (Bachelier) 16.3.1 Theory (FTAP1 and FTAP2) 16.3.2 Transition Density Functions 16

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.3.3 Deriving the Bachelier Option Pricing Formula 16.4 Defining and Pricing a Standard Numeraire 16.5 Geometric Brownian Motion (GBM) 16

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.5.1 GBM (Discrete Version) 16.5.2 Geometric Brownian Motion (GBM), Continuous Version 16.6 Itô’s Lemma 16.7 Black–Scholes Option Pricing 16.7.1 Reducing GBM

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and FTAP2, apply in this continuous-time context as well. We will begin with the prototype of all continuous time models, and that is arithmetic Brownian motion (ABM). ABM is the most basic and important stochastic process in continuous time and continuous space, and it has many desirable properties including the strong

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to partially prepare you for courses in mathematical finance. The details we have to leave out are usually covered in such courses. 16.1 ARITHMETIC BROWNIAN MOTION (ABM) ABM is a stochastic process {Wt(ω)}t≥0 defined on a sample space (Ω,ℑW,℘W). We won’t go into all the

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) for an underlying process that has some flaws (ABM). Showing this connection is one of the main purposes of this chapter. 16.2 SHIFTED ARITHMETIC BROWNIAN MOTION The first step in pricing options is to get a reasonable process for the underlying stock where the obvious, removable flaws have been removed. Denote

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arithmetic Brownian motion, {Wt(ω)}t≥0, by (Wt). We shift the ABM process, (Wt), by adding to it the current stock price S0>0. It doesn’t

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is that, E(Xt(ω)|X0)=S0+W0=X0, which is the martingale requirement for Xt(ω). 16.3 PRICING EUROPEAN OPTIONS UNDER SHIFTED ARITHMETIC BROWNIAN MOTION WITH NO DRIFT (BACHELIER) 16.3.1 Theory (FTAP1 and FTAP2) To price a European option on a shifted ABM, we will use the modern

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. This shows that the normalization works in generating an N(0,1) random variable. In order to derive the transition density function for shifted arithmetic Brownian motion, we just have to normalize the terminal (at time T) stock price distribution by subtracting the conditional mean and dividing by the conditional standard deviation

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time in 1900, to derive the arbitrage-free price of a European call option for an asset (Xt)0≤t≤T following a shifted arithmetic Brownian motion. K is the exercise price of the option, t=0, and the option expires at time T. Given all the work we have already done

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was basically rediscovered around 1965. At that point, it was dismissed for three reasons by Samuelson, who presented three objections to Bachelier. First, shifted arithmetic Brownian motion violates the limited liability property of stock prices, because the process can go negative. This can be resolved by absorbing the ABM at zero, when

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a drift term, μt, to ABM that is proportional to time. Samuelson’s well-known resolution of these issues was to replace shifted arithmetic Brownian motion with geometric Brownian motion (GBM), also called the usual log-normal diffusion process. However, as indicated, it is relatively easy to resolve Samuelson’s three objections within the

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context of shifted arithmetic Brownian motion. 16.4 DEFINING AND PRICING A STANDARD NUMERAIRE In Chapter 4, section 4.2.1, we discussed how to price a zero-coupon bond with

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used, in general, to discount the underlying price process in order to obtain an EMM with which to price the financial derivative. 16.5 GEOMETRIC BROWNIAN MOTION (GBM) 16.5.1 GBM (Discrete Version) In order to price European options on GBM, which is considered the workhorse for underlying processes used to

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a Gaussian distribution with conditional mean μΔt and (conditional) variance σ2Δt, because μΔt+σ*ΔWt(ω) is a constant plus the increment of a scaled Brownian motion, which by assumption is Gaussian (see property 3 in the definition of an ABM). Note that all of these calculations are in discrete time. When

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-time GBM process in (μ equation discrete, risk-adjusted). That process is called the log-normal process or geometric Brownian motion (GBM). The reason for calling it geometric, as opposed to arithmetic Brownian motion is that it expresses the idea of geometric growth, otherwise known as continuous compounding. Over any period of length μΔt

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. In the ABM process one is looking at absolute dollar changes in the asset’s value—as opposed to percentage changes. 16.5.2 Geometric Brownian Motion (GBM), Continuous Version Take (equation μ discrete, risk-adjusted) and wherever you see Δ, replace it by the letter d, Now drop the ω and

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example. However, integration with respect to dWs requires that Ws has sufficiently ‘nice’ paths, which means differentiable paths. One of the main features of the Brownian motion paths (Ws(ω)) is that, while they are almost surely continuous, they are almost surely nowhere differentiable. We can’t prove that here but we

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can give an economic rationale for it in terms of the EMH. FIGURE 16.1 Non Smoothness of Brownian Motion Paths If a Brownian motion path happened to be differentiable at some stock price and the derivative was positive, for example, then it would be locally riskless. This

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looks roughly the same. In order to reduce (GBM SDE) to (Risk-Neutralized GBM SDE), one has to change the Brownian motion measure using Girsanov’s theorem, Wt, to get an equivalent Brownian motion process, . The economics of this important mathematical procedure is that it removes the risk premium in μ by transferring it

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, We will change the notation slightly here to avoid confusion, and call the Risk-Neutralized GBM Yt, and we will drop the ~, remembering that the Brownian motion process is . The SDE for it is simply (Risk-Neutralized GBM SDE), Then Yt is called the risk-neutral GBM and the unique solution to

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a component of some option pricing models. We should probably keep it around for these reasons. ■ KEY CONCEPTS 1. Arithmetic Brownian Motion (ABM). 2. Shifted Arithmetic Brownian Motion. 3. Pricing European Options under Shifted Arithmetic Brownian Motion (Bachelier). 4. Theory (FTAP1 and FTAP2). 5. Transition Density Functions. 6. Deriving the Bachelier Option Pricing Formula. 7. Defining

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and Pricing a Standard Numeraire. 8. Geometric Brownian Motion (GBM). 9. GBM (Discrete Version). 10. Geometric Brownian Motion (GBM), Continuous Version. 11. Itô’s Lemma. 12. Black–Scholes Option Pricing. 13. Reducing GBM to an ABM with Drift. 14

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, 100–1, 103, 167, 172, 188, 192, 208, 260, 292, 450, 474; pricing by 464; risk-free arbitrage 100, 373; risky arbitrage 100–1 arithmetic Brownian motion (ABM) model of prices: equivalent martingale measures (EMMs) 530–1; option pricing in continuous time 540–1 back stub period 294 backwardation, contango and 198

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) 88 block trade eligibility 214, 228 block trade minimum 214, 228 Bond Equation 552, 554–5 boundaries, absorption of 541 Brownian motion paths, non-smoothness of 560; see also arithmetic Brownian motion (ABM); geometric Brownian motion (GBM) Buffett, Warren 252 buyers and sellers, matching of 125, 126–7 buying back stock 339 buying forward 7–8

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arb 190; unwinding arb 190–2 equity in customer’s account 145, 148 equivalent annual rate (EAR) 70 equivalent martingale measures (EMMs) 507–38; arithmetic Brownian motion (ABM) model of prices 530–1; computation of EMMs 529; concept checks: contingent claim pricing, working with 514; martingale condition, calculation of 525; option pricing

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–1 Gaussian distributions 543, 546, 548, 557, 565, 577 general equilibrium (GE) 453; models of, risk-neutral valuation and 615 generalized forward price 402 geometric Brownian motion (GBM) 553–61; continuous version 559–61; discrete version 553–9 Girsanov’s theorem 605 Globex and Globex LOB 134–6 Globex trades, rule for

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–9; trades entry into clearing system 138–9; trading cards, submission of 139 option buyers 328 option pricing in continuous time: absorbing boundaries 541; arithmetic Brownian motion (ABM) model of prices 540–1; Black-Scholes option pricing 566–85, 588–9; from Bachelier 571–83; historical volatility estimator method 583–4; implied

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; risk-neutral transition density functions, generation of unknown from knowns 570–1; volatility estimation in Black-Scholes model 583–5; Bond Equation 552, 554–5; Brownian motion paths, non-smoothness of 560; clustering (persistence), volatility and 585; concept checks: arbitrage opportunity construction 554; Bachelier option pricing formula, derivation of 550; Black-Scholes

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; exercises for learning development of 590–3; fundamental theorems of asset pricing (FTAP1 and FTAP2) 540; Gaussian distributions 543, 546, 548, 557, 565, 577; geometric Brownian motion (GBM) 553–61; continuous version 559–61; discrete version 553–9; Heston volatility model 587–8; Itô’s Lemma 562–6; key concepts 590; Log

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585; leverage effect 586; stochastic volatility (SVOL) models 586–7; numeraire 554; definition and pricing a standard 551–3; pricing European options under shifted arithmetic Brownian motion (ABM) with no drift 542–51; Bachelier option pricing formula, derivation of 547–51; fundamental theorems of asset pricing (FTAP) 542–3; transition density functions

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process SDE 568; risk-neutralized GBM SDE 567, 570; risk premia 554, 558, 561, 567, 588; scaled-by-? increment of ABM process 555; shifted arithmetic Brownian motion (ABM) model of prices 541–2; reduced process 570; stochastic differential equations (SDEs) 553, 559, 562–3, 564, 566, 567–8, 570, 571, 583; stochastic

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integral equations (SIEs) 559, 560, 561, 564, 565–6, 567; stochastic processes 540–1, 543, 562, 587, 588; transition density function for shifted arithmetic Brownian motion 545–6; Wiener measure (and process) 540–1 option sellers 328 option trading strategies 345–67, 415–34; basic (naked) strategies 347–63; ‘calling away

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spot markets 6–7 pricing a swap 294 pricing by arbitrage and FTAP2 597–8 pricing currency forwards 105 pricing European options under shifted arithmetic Brownian motion (ABM) with no drift 542–51; Bachelier option pricing formula, derivation of 547–51; fundamental theorems of asset pricing (FTAP) 542–3; transition density functions

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214, 228, 229, 258–9 settlement variation 146 Sharpe ratio: equivalent martingale measures (EMMs) and 526, 605–6; risk-neutral valuation and 624 shifted arithmetic Brownian motion (ABM) model of prices 541–2; reduced process 570 short a European call option on the underlying 348, 355–7; economic characteristics 357 short a

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period rate 237; intermediate execution, basis risk and 237–8; liquidity advantage in execution 237 transfer of obligations 16 transition density function for shifted arithmetic Brownian motion 545–6 transportation across time, storage as 195 treasury bill synthesis 166–7 trinomial model (three stock outcomes) 464 turning points 22 unallocated foreign exchange

Handbook of Modeling High-Frequency Data in Finance

by Frederi G. Viens, Maria C. Mariani and Ionut Florescu  · 20 Dec 2011  · 443pp  · 51,804 words

the ‘‘symmetric case’’ (which is a reasonable assumption for equity prices), both models require only one additional parameter, κ, compared to the two-parameter geometric Brownian motion (also called the Black–Scholes model). This additional parameter can be interpreted as the percentage excess kurtosis relative to the normal distribution and, hence, this

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Tankov (2011) for further information. Exponential (or Geometric) Lévy models are arguably the most natural generalization of the geometric Brownian motion intrinsic in the Black–Scholes option pricing model. A geometric Brownian motion (also called Black–Scholes model) postulates the following conditions about the price process (St )t≥0 of a risky asset

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of time are independent. Finally, the last condition is tantamount to asking that X has continuous paths. Note that we can represent a general geometric Brownian motion in the form St = S0 eσ Wt +μt , where (Wt )t≥0 is the Wiener process. In the context of the above Black–Scholes model

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time. Of course, continuity is just a convenient limiting abstraction to describe the high trading activity of liquid assets. In spite of its shortcomings, geometric Brownian motion could arguably be a suitable model to describe low frequency returns but not high frequency returns. An ELM attempts to relax the assumptions of the

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correlations and Levy models applied to the study of memory effects in high frequency (tick) data, Physica A 2009;388(8):1659–1664. Osborne MFM. Brownian motion in the stock market, Oper Res 1959;7(2):145–173. Sun W. Relationship between trading volume and security prices and returns, MIT LIDS Technical

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process v → Zv is Gaussian, with Zv having mean 0 and variance 1 for each v. For instance, Zv = a(v)Bb(v) , for a Brownian motion v → Bv , and a(·) and b(·) are suitable functions, which determine the correlation φv,v . In this chapter, for simplicity, we always assume that Zv

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Volatility Indices where Xt = logSt and St is the asset price, r is the short-term risk-free rate of interest, and Wt a standard Brownian motions. ϕt models the stochastic volatility process. It has been proved that for any proxy of the current stochastic volatility distribution at t the option prices

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temporal time series have been of a great importance comparing the financial markets. The first model that described the evolution of option prices was the Brownian motion. This model assumes that the increment of the logarithm of prices follows a diffusive process with Gaussian distribution [12]. However, the empirical study of temporal

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of the Levy distribution is inversely proportional to the Hurst parameter. The Hurst parameter is an indicator of the memory effects coming from the fractional Brownian motion, which has correlated increments. Furthermore, the TLF maintains statistical properties that are indistinguishable from the Levy flights [15]. 6.2.2 RESCALED RANGE ANALYSIS Hurst

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pattern. He measured the trend using an exponent now called the Hurst exponent. Mandelbrot [28,29] later introduced a generalized form of the Brownian motion model, the fractional Brownian motion to model the Hurst effect. The numerical procedure to estimate the Hurst exponent H by using the R/S analysis is presented next (for

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memory increase while during the time of crisis the stocks start behaving randomly and therefore their H estimates are closer to the estimates for a Brownian motion (0.5). Looking at the DFA estimates (Table 6.3), we recognize this behavior in other equity as well. What we also see here is

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; 2008. 27. Hurst HE. Long term storage of reservoirs. Trans Am Soc Civ Eng 1950;116:770–808. 28. Mandelbrot BB, Van Ness JW. Fractional Brownian motions, fractional noises and applications. SIAM Rev 1968;10(4): 422–437. 29. Mandelbrot BB. The fractal geometry of nature. New York: Freeman and Co.; 1982

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) = μ− dt + σ (Yt ) dWt , 2 = α Yt dt + β dBtH , (8.1) where Wt is a standard Brownian motion and BtH is a fractional Brownian motion with Hurst index H ∈ (0, 1]. The fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1] is a Gaussian process with almost surely continuous paths and covariance structure

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given by Cov(BtH , BsH ) =  1  2H |t| + |s|2H − |t − s|2H . 2 For H = 1/2 the process is the well-known standard Brownian motion. Formally, we say that a process exhibits long-range dependence when the series of the autocorrelation function is nonsummable, that is +∞ ρ(n) = +∞. n=1

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8.1 Introduction 221 From the covariance function of the fractional Brownian motion, we can easily deduce that the autocorrelation function of the increments of fBm is of order n2H −2 . This implies that when H > 1/2

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‘‘short’’ memory. The parameter H is the long-memory parameter and is also known as Hurst index. More details regarding the properties of the fractional Brownian motion can be found in the book by Beran (1994). As in the case of a classical stochastic volatility models, the volatility process is not directly

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(t) = d  j σi (t)dW i + bj (t) dt, j = 1, . . . , n, (10.1) i=1 where W = (W 1 , . . . , W d ) are independent Brownian motions on a filtered probability space satisfying the usual conditions and σ∗∗ and b∗ are adapted random processes satisfying  (H) E T   (b (t)) dt < ∞, i

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.12) 251 10.2 Fourier Estimator of Multivariate Spot Volatility where v(t) := σ 2 (t) is the variance process, W and Z are correlated Brownian motions, and σ (t), γ (t) and a(t), b(t) are adapted random processes satisfying  (H ) E[ 2π  (a (t) + b (t)) dt] < ∞, E[ 2

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) = σ (t)dW1 (t) dσ 2 (t) = α(β − σ 2 (t))dt + νσ (t)dW2 (t), (10.30) where W1 and W2 are independent Brownian motions. Moreover, we assume that the logarithmic noises η are i.i.d. Gaussian and independent from p; this is typical of the bid-ask bounce

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assume that the logarithm of the efficient price p(s) evolves as dp(s) = σ (s)dW 1 (s). (10.39) W 1 is a Brownian motion on a filtered probability space(, (Fs )s∈[0,T ] , P) and T σ is a continuous adapted stochastic process such that E[ 0 σ 4

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(s) evolves according to the process df (s) = m(f (s))dt +  v(f (s))dW 2 (s), (10.43) where W 2 is a Brownian motion independent of W 1 . The coefficients al are real numbers, Pl (f (s)) are the eigenfunctions of the infinitesimal generator associated with f (s), and

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(t) = Pi [bi (t)dt + m  σij (t)dWj (t)], i = 1, . . . , m. (11.2) i=1 Here W (·) = (W1 (·), . . . , Wm (·))∗ is an m-dimensional Brownian motion on a complete probability space (, F, P). We shall denote by F = {Ft }{0≤t≤T } the P-augmentation of the filtration generated by W

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(t)π ∗ (t)σ (t)dW0 (t) − γ (t)dC(t), 0 ≤ t ≤ T . (11.12) The process W0 (t) of Equation 11.11 is Brownian motion under the equivalent martingale measure P0 (A)  E[Z0 (T )1A ], A ∈ FT , (11.13) by the Girsanov theorem (Section 3.5 in Karatzas and

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,C (t)π ∗ (t)σ (t)dW0 (t) − γ (t)dC(t), 0 ≤ t < ∞. (11.40) The process W0 (t) of Equation 11.39 is Brownian motion under the equivalent martingale measure P0 (A)  E[Z0 (T )1A ], A ∈ FT (11.41) 307 11.7 Infinite Horizon Case by the Girsanov theorem

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some results. 11.7.5 SOME RESULTS ON OPTIMAL STOPPING FOR ONE-DIMENSIONAL DIFFUSIONS Consider the one-dimensional diffusion process driven by an m-dimensional Brownian motion {Wt } dY (t) = (β − r)Y (t)dt − Y (t)θdWt , Y (0) ∈ (0, ∞) (11.55) and the optimal stopping problem   W (y) = sup Ey

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τx  inf {t ≥ 0 / Y (t) = x}, x > 0, the first hitting time of level x. The reduction of the optimal stopping problem to the Brownian motion case has been studied by Dayanik and Karatzas in the case when the diffusion was driven by one-dimensional

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Brownian motion. Their results hold also for the case in which we have a m-dimensional Brownian motion with the only modification in the equation solved by the Green functions ψ and ϕ. Still, the problem

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is solved explicitly for the Brownian motion case by using a graphical method. By taking advantage of the theory developed in Dayanik and Karatzas (2003), in terms of the existence of an

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one-dimensional diffusions have been established by Karatzas and Dayanik. Note that although their theory requires that the diffusion is driven by a one-dimensional Brownian motion, their results hold for our case and a sufficient assumption for the existence of an optimal stopping time is (Proposition 5.14 Dayanik and Karatzas

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portfolio and consumption decisions for a small investor on a finite time-horizon. SIAM J Control Optim 1987;25:1557–1586. Karatzas I, Shreve SE. Brownian motion and stochastic calculus. 2nd ed. New York: Springer; 1991. Karatzas I, Shreve SE. Methods of mathematical finance. New York: Springer; 1998. Karatzas I, Wang H

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John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc. 327 328 CHAPTER 12 Stochastic Differential Equations and Levy Models where Bt indicates the Brownian motion [1–3]. If m = 2 this equation arises in modeling the growth of a population of size Xt in a stochastic, crowded environment. The constant

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, the underlying security S follows, as in the Black–Scholes model, a stochastic process dSt = μSt dt + σt St dZt , where Z is a standard Brownian motion. Unlike the classical model, the variance v(t) = σ 2 (t) also follows a stochastic process given by √ dvt = κ(θ − v(t))dt + γ

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vt dWt , where W is another standard Brownian motion. The correlation coefficient between W and Z is denoted by ρ: Cov (dZt , dWt ) = ρ dt. This leads to a generalized Black–Scholes equation ∂ 2F

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) Here, M is some diffusion matrix, and u0 is some payoff function. The Black–Scholes models with jumps arise from the fact that the driving Brownian motion is a continuous process, and so there are difficulties fitting the financial data presenting large fluctuations. The necessity of taking into account large market movements

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the volatility is stochastic, we may consider the processes dS = Sσ dZ + Sμdt, dσ = βσ dW + ασ dt, where Z and W are two standard Brownian motions with correlation coefficient ρ. If F (S, σ , t) is the price of an option depending on the price of the asset S, then by

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) as |x| → ∞. 13.4 Integro-Differential Equations in a Lévy Market The Black–Scholes models with jumps arise from the fact that the driving Brownian motion is a continuous process, and so there are difficulties fitting the financial data presenting large fluctuations. The necessity of taking into account the large market

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value in the presence of transaction costs. We outline the steps used in the next section. 14.1.1 OPTION PRICE VALUATION IN THE GEOMETRIC BROWNIAN MOTION CASE WITH TRANSACTION COSTS Let C(S, t) be the value of the option and  be the value of the hedge portfolio. The asset follows

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a geometric Brownian motion. Using discrete time, we assume the underlying asset follows the process √ δS = μSδt + σ S δt, (14.1) where  is drawn from a standard normal

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the underlying security S follows, as in the Black–Scholes model, a stochastic process dS = μS dt + σ S dX1 , where X1 is a standard Brownian motion. Unlike the classical model, the variance v(t) = σ 2 (t) also follows a stochastic process given by dv = κ(θ − v(t)) dt + γ

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√ v dX2 , where X2 is another standard Brownian motion. The correlation coefficient between X1 and X2 is denoted by ρ: Cov(dX1 , dX2 ) = ρ dt. This leads to a generalized Black–Scholes equation. A

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the volatility is stochastic we may consider the process dS = μSdt + σ SdX1 , (14.27) dσ = ασ dt + βσ dX2 , (14.28) where the two Brownian motions X1 and X2 are correlated with correlation coefficient ρ: E[dX1 dX2 ] = ρdt. (14.29) 14.4 Model with Transaction Costs and Stochastic Volatility 401

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will be traded. However, we can compute the expected number of this variable and hence the expected transaction cost. Since X1 and X2 are correlated Brownian motions, we consider Z1 and Z2 two independent normal variables with mean 0 and variance 1 and thus we may write the distribution of X1 , X2

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how many shares will be traded, but we can compute the expected number and hence the expected transaction cost. Since X1 and X2 are correlated Brownian motions, we consider Z1 and Z2 two independent normal variables with mean 0 and variance 1 and thus, we may write the distribution of X1 , X2

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with decision tree learning, 49 as an interpretive tool, 67 Boundary value problem, 319, 320 Bounded parabolic domain, 352, 368 Bozdog, Dragos, xiii, 27, 97 Brownian motion, 78, 120, 220 BSC indicators, 52, 53. See also Balanced scorecards (BSCs) BSC management system, 51–52 Index Calendar time sampling, 9 Call options chains

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, 251, 252 Fourier method high-frequency data using, 243–294 gains yielded by, 290 Fourier transform(s), 122, 246, 335 numerically inverting, 13–14 Fractional Brownian motion (FBM), 125, 220, 221 FRE data series, DFA and Hurst methods applied to, 154 Frequency range, identifying, 22 Frequency sampling, 5 Functional analysis, review of

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, 166–167, 169, 170 Generalized tree process, 354 General semilinear parabolic problem, 355–362 General utility functions, 311 Genetic algorithms, 63, 64 Geometric Brownian motion, 4, 6–7 Geometric Brownian motion case, transaction costs and option price valuation in, 384–386 Geometric Lévy models. See Exponential Lévy models German Society of Financial

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Mortgage-backed securities (MBSs); Subprime MBS portfolios slicing into tranches, 88–89 MBS tranches, 76 MBS units, 79 MBS vehicle, function of, 77 m-dimensional Brownian motion, 311, 312 Mean squared error (MSE), 245, 254–256. See also MSE entries cutting frequency and, 259, 260 Mean–variance mixture definition, 170 Mean-variance

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(s), 406 discrepancies among, 219 in stochastic volatility models, 401 Option price evolution model, 120 Option price formula, 384 Option price valuation, in the geometric Brownian motion case, 384–386 Option pricing algorithm, 226 Options, 348. See also Call options chain; European option entries; European call option; Put options chains; Stock options

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–81 Tranches, of a portfolio, 77 Tranche seniority, 82, 89, 93 Transaction costs, 402–404, 406–407 financial models with, 383–408 in the geometric Brownian motion case, 384–386 Transition level, 89 Truncated Lévy flight (TLF), 120, 122–125, 338 440 Two-dimensional Itö’s formula, 328–329 Two-factor

Monte Carlo Simulation and Finance

by Don L. McLeish  · 1 Apr 2005

an important part of the literature in Physics, Probability and Finance at least since the papers of Bachelier and Einstein, about 100 years ago. A Brownian motion process also has some interesting and remarkable theoretical properties; it is continuous with probability one but the probability that the process has finite 10 68

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CONTINUOUS TIME 69 Now if we increase the sample size and decrease the scale appropriately on both axes, the result is, in the limit, a Brownian motion process. The vertical √ scale is to be decreased by a factor 1/ n and the horizontal scale by a factor n−1 . The theorem concludes

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√ converges to a positive constant, namely E|Xi |, if we multiply by nh |Xi | the limit must be infinite, so the total variation of the Brownian motion process is infinite. 70 CHAPTER 2. SOME BASIC THEORY OF FINANCE Continuous time process are usually built one small increment at a time and defined

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, ti+1 ] must be made at the beginning of this interval, not at the end or in the middle. Second, in the case of a Brownian motion process W (t), it makes a difference where in the interval [ti , ti+1 ] we evaluate the function h to approximate the integral, whereas it

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can assess the size of dW since its standard deviation is (dt)1/2 . Now consider defining a process as a function both of the Brownian motion and of time, say Vt = g(Wt , t). If Wt represented the price of a stock or a bond, Vt might be the price

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Rb The term a dt above is what distinguishes the Ito calculus from the Riemann calculus, and is a consequence of the nature of the Brownian motion process, a continuous function of infinite variation. There is one more property of the stochastic integral that makes it a valuable tool in the construction

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of models in finance, and that is that a stochastic integral with respect to a Brownian motion process is always a martingale. To see this, note that in an approximating sum Z T h(t)dWt ≈ 0 n−1 X h(

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.} − σ Xt dt = Xt 2Xt2 σ2 = (a − )dt + σdWt . 2 74 CHAPTER 2. SOME BASIC THEORY OF FINANCE which is a description of a general Brownian motion process, a process with increments dYt that are normally distributed with mean (a − σ2 2 )dt and with variance σ 2 dt. This process satisfying

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Xt up to time t. Various choices for the functions a(Xt , t), σ(Xt , t) are possible. For the Black-Scholes model or geometric Brownian motion, a(Xt , t) = aXt and σ(Xt , t) = σXt for constant drift and volatility parameters a, σ. The Cox-Ingersoll-Ross model, used to

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a(Xt , t) is vector valued, σ(Xt , t) is replaced by a matrix-valued function and dWt is interpreted as a vector of independent Brownian motion processes. For technical conditions on the coefficients under which a solution to 2.26 is guaranteed to exist and be unique, see Karatzas and Shreve

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a few cases, while in many others, numerical techniques are necessary. One special case of this equation deserves particular attention. In the case of geometric Brownian motion, a(St , t) = µSt and σ(St , t) = σSt for constants µ, σ. Assume that the spot interest rate is a constant rand that a constant

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value of XT given Xt . The second is the random component, and since it is a weighted sum of the normally distributed increments of a Brownian motion with weights that are non-random, it is also a normal random variable. The mean is 0 and the (conditional) variR T 2 (u)

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taking limits that the random variable normal distribution and find its mean and variance. RT 0 g(t)dWt has a 12. Consider two geometric Brownian motion processes Xt and Yt both driven by the same Wiener process dXt = aXt dt + bXt dWt dYt = µYt dt + σYt dWt . Derive a stochastic differential

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the more realistic situation in which (1) dXt = aXt dt + bXt dWt (2) dYt = µYt dt + σYt dWt (1) (2) and Wt , Wt are correlated Brownian motion processes with correlation ρ. 13. Prove the Shannon inequality that H(Q, P ) = X qi log( qi )≥0 pi for any probability distributions P and

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(t) = √ 2c2 t c 2πt3 (3.21) for parameters θ > 0, c > 0. This is the distribution of a first passage time for Brownian motion. In particular consider a Brownian motion process B(t) having drift 1 and diffusion coefficient c. Such a process is the solution to the stochastic differential equation dB(t

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) = dt + cdW (t), B(0) = 0. Then the first passage of the Brownian motion to the level θ is T = inf(t; B(t) = θ} and this random variable has probability density function (3.21). The mean 160 CHAPTER

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−θ √ c θ approaches the standard normal or, more loosely, the distribution (3.21) approaches Normal(θ, θc2 ). Lemma 30 Suppose X(t) is a Brownian motion process with drift β and diffusion coefficient 1, hence satisfying dXt = βdt + dWt , X(0) = µ. Suppose a random variable T has probability density function (3

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.21) and is independent of Xt . Then the probability density function of the randomly stopped Brownian motion process is given by p p K1 (α δ 2 + (x − µ)2 ) αδ 2 2 (3.22) f (x; α, β, δ, µ) = exp(δ

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(Xt , t)dt + σ(Xt , t)dWt (3.36) with some initial value X0 for Xt at t = 0. Here Wt is a driving standard Brownian motion process. Solving deterministic differential equations can sometimes provide a solution to a specific problem such as finding the arbitrage-free price of a derivative. In

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(∆t)3/2 2 (3.38) which allow an explicit representation of the increment in the process X in terms of the increment of a Brownian motion process ∆Wt ∼ N (0, ∆t). The approximation (3.38) is called the Milstein approximation, a refinement of the first, the Euler approximation. It is the

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price under the risk-neutral measure Q follows a constant elasticity of variance (CEV) process dSt = rSt dt + σStγ dWt (3.40) for a standard Brownian motion process Wt . A down-and-out call option with exercise price K provides the usual payment (ST − K)+ of a European call option on maturity

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, time to maturity T , out barrier b < S0 and constant interest rate r and compare with the Black-Scholes formula as b → 0. A geometric Brownian motion is most easily simulated by taking logarithms. 194 CHAPTER 3. BASIC MONTE CARLO METHODS For example if St satisfies the risk-neutral specification dSt = rSt

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dt + σSt dWt (3.41) dYt = (r − σ2 /2)dt + σdWt . (3.42) then Yt = log(St ) satisfies This is a Brownian motion and is simulated with a normal random walk. Independent normal increments are generated ∆Yt ∼ N ((r − σ2 /2)∆t, σ2 ∆t) and their partial sums

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, the European option is valued closer to $1.28 so the increase must be due to the differences betwen the CEV process and the geometric Brownian motion. We can confirm this by simulating the value of a barrier option in the Black_Scholes model later on. Problems 1. Consider the mixed generator

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X = (µ1 , ..., µN ) + ZA has the desired distribution. 23. (Euler vs. Milstein Approximation) Use the Milstein approximation with step size .001 to simulate a geometric Brownian motion of the form dSt = .07St dt + .2St dWt Compare both the Euler and the Milstein approximations using different step sizes, say ∆t = 0.01, 0

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depends on the frequency of sampling (e.g. if T = .25 (years) and sampling is weekly, then k = 13. If S(t) follows a geometric Brownian motion, then S̄k is the sum of lognormally distributed random variables and the distribution of the sum or average of lognormal random variables is very

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fact that V2 has a known lognormal distribution, the prospects of this are excellent. Since S(t) = S0 eY (t) where Y (t) is a Brownian motion with Y (0) = 0, drift r − σ 2 /2 and diffusion σ, it follows that S̃k has the same distribution as does S0 exp

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at time T. In particular we may stratify the generation of St = S0 exp(Zt ) where Zt can be written in terms of a standard Brownian motion Zt = µt + σWt , with µ = r − σ 2 /2. To stratify into K strata of equal probability for ST we may generate ZT using ZT =

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of the path interpolating the value of S0 and ST using Brownian Bridge interpolation. To do this we use the fact that for a standard Brownian motion and s < t < T we have that the conditional distribution of Wt given Ws , WT is normal with mean a weighted average of the

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stratify the sample; i.e. sample more often when S(T ) is large or to use importance sampling and generate S(T ) from a geometric Brownian motion with drift larger than rSt so that it is more likely that S(T )˙ > K. As before if we do this we need to then

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model drt = µ(rt )dt + σdWt (5.13) for some choice of function µ(rt ). Then according to Girsanov’s theorem, we may simulate rt under the Brownian motion model drt = µ0 dt + σdWt (having the same initial value r0 as in our original simulation) and then average the values of Z T Z

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conditioning, although there are many other potential variance reduction tools here. Suppose the asset price, (under the risk-neutral probability measure, say) follows a geometric Brownian motion model of the form (1) (5.15) dSt = rt St dt + σSt dWt where rt is the spot interest rate. We assume rt is stochastic

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and follows the Brennan-Schwartz model, (2) drt = a(b − rt )dt + σ0 rt dWt (1) (5.16) (2) where Wt , Wt are both Brownian motion processes and usually assumed to be correlated with correlation coefficient ρ. The parameter b in (5.16) can be understood to be the long run

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exp{− 0 rt dt}. However, in this case we can exploit the assumption that ρ = 0 so that interest rates are (1) independent of the Brownian motion process Wt which drives the asset price process. For example, suppose that the interest rate function rt were known (equivalently: condition on the value of

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exiting the histogram. This is above the value d2m−u and dm is between du and d2m−u . A similar result is available for Brownian motion and Geometric Brownian motion. A justification of these results can be made by taking a limit in the discrete case as the time steps and the distances dj

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− dj−1 all approach zero. If we do this, the parameter θ is analogous to the drift of the Brownian motion. The result for Brownian motion is as follows: Theorem 44 Suppose St is a Brownian motion process dSt = µdt + σdWt , S0 = 0, ST = C, H = max{St ; 0 · t · T } and L = min{St ; 0

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∆ → c. Then this ratio approaches f0 (2h − c) f0 (c) where f0 is the probability density function of C under µ = 0. This implies for a Brownian motion process, P [H ≥ h|C = c] = f0 (2h − c) for h ≥ c. f0 (c) (5.25) If we temporarily denote the cumulative distribution function

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(x) and uniformly distributed 278 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS Figure 5.2: Generating H for a fixed value of C for a Brownian motion. on {(c, y); 0 · y · f0 (c)}. This point is shown in Figure ??. We regard the y−coordinate of this point as the generated

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move horizontally to the left. There is a similar argument for generating the high under a geometric Brownian motion as well, since the logarithm of a geometric Brownian motion is a Brownian motion process. Corollary 45 For a Geometric Brownian motion process dSt = µSt dt + σSt dWt , S0 = O and ST = C SIMULATING BARRIER AND LOOKBACK OPTIONS with

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. This is an example of an knock-out barrier but other types are similarly handled. Once again we assume the simplest form of the geometric Brownian motion d ln(St ) = σdWt and assume that the upper barrier is at the point Oeb so that the payoff from the option on maturity

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Redekop (1995) to test the local Brownian nature of various financial time series. These are easily seen in Figure 5.5. For example, for a Brownian motion process with sero drift, suppose we condition on the value of 2H − O − C. Then the point PH must lie (uniformly distributed) on the line

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the curve labelled C2 whose y-coordinate is uniformly distributed showing that C−O ∼ U [0, 1]. 2H − O − C Redekop shows that for a Brownian motion process, the statistic H−O 2H − O − C (5.29) is supposed to be uniformly [0, 1] distributed but when evaluated using real financial data

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the high results in an interval of values for 284 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS Figure 5.5: Some uniformly distributed statistics for Brownian Motion (2H − C) of width −∆y/φ0 (2y − x) where φ0 is the derivative of the standard normal probability density function (the minus sign is to

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µ (x) f0 (x) and this gives the more general result in the table below. The table below summarizes many of our distributional results for a Brownian motion process with drift on the interval [0, 1], dSt = µdt + σdWt , with S0 = O. SIMULATING BARRIER AND LOOKBACK OPTIONS 285 Figure 5.6: Confirmation of

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, O TABLE 5.2: Some distributional results for High, Close and Low. We now consider briefly the case of non-zero drift for a geometric Brownian motion. Fortunately, all that needs to be changed in the results above is the marginal distribution of ln(C) since all conditional distributions given the value

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is an extremely low probability that the high and low of the process will both lie in the same short increment. For example for a Brownian motion with the time interval partitioned into 5 equal subintervals, the probability that the high and low both occur in the same increment is less than

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|O = 0] A first step in this direction is the the following result, obtained from the reflection principle with two barriers. Theorem 47 For a Brownian motion process dSt = µdt + dWt , S0 = 0 defined on [0, 1] and for −a < u < b, P (L < −a or H > b|C = u) = ∞ 1

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event A, denote by P (A|u) probability of the event conditional on C = u. Then according to the reflection principal the probability that the Brownian motion leaves the interval [−a, b] is given from an inclusion-exclusion argument by P (A+1 |u) − P (A+2 |u) + P (A+3 |

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= ln(56.25/40) = 34%. Is this difference significant and are these returns reasonably accurate in view of the survivorship bias? We assume a geometric Brownian motion for both portfolios, (5.34) dSt = µSt dt + σSt dWt , and define O = S(0), C = S(T ), H = max S(t), 0 t

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this is essentially the same as a problem already discussed, the valuation of a barrier option. According to (5.27), the probability that a given Brownian motion process having open 0 and close c strikes a barrier placed at l < min(0, c) is exp{−2 zl } σ2 T with zl =

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l(l − c). Converting this statement to the Geometric Brownian motion (5.34), the probability that a geometric Brownian motion process with open O and close c SURVIVORSHIP BIAS 293 breaches a lower barrier at l is P [L · l|O, C

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of the performance of portfolio 1. 296 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS Figure 5.9: The Effect of Surivorship bias for a Brownian Motion For a Brownian motion process it is easy to demonstrate graphically the nature of the surivorship bias. In Figure 5.9, the points under the graph of the

The Misbehavior of Markets: A Fractal View of Financial Turbulence

by Benoit Mandelbrot and Richard L. Hudson  · 7 Mar 2006  · 364pp  · 101,286 words

time. He also did pioneering work in many now-well-trodden avenues of economics. From 1965 he was publishing on what he soon called fractional Brownian motion and on the underlying concept of fractional integration, which has recently become a widespread econometric technique. In 1972, he published a multifractal model that incorporates

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to express the predictions of my models in this unique graphical form, a kind of forgery of reality. Here, the underlying model is called fractional Brownian motion in multifractal time. Though the name is forbidding, later chapters will elaborate and show the model to be extremely parsimonious. The “daily changes” in the

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motion, observed that it is not a manifestation of life but a physical phenomenon, and received (possibly inflated) credit for the discovery through the term “Brownian motion.” In 1905, Albert Einstein developed for it equations very similar to Bachelier’s own equations of bond-price probability—though Einstein never knew that. Regardless

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. Samuelson. Bachelier’s idea of a “fair game” caught on; and economists recognized the practical virtues of describing markets by the laws of chance and Brownian motion. They were, in the 1960s and 1970s, put into a broader theoretical framework by Eugene F. Fama. As a student at the University of Chicago

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anomaly best ignored, is an essential ingredient of markets that helps set finance apart from the natural sciences. 4. Assumption: Price changes follow a Brownian motion. Theory: Brownian motion, again, is a term borrowed from physics for the motion of a molecule in a uniformly warm medium. Bachelier had suggested that this process can

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charts use the same drawing methods we applied to the Dow—but the picture is very different. These are price charts according to the Bachelier Brownian motion model. As discussed earlier, this is in the catechism of orthodox financial theory. It assumes each day’s price change is independent of the last

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repeating this process over and over, a jagged, complex chart gradually appeared. By careful design, the specific kind of chart shown before was of a Brownian motion—the standard model underlying conventional financial theory. What made it so was the specific shape of the generator: Starting at the point (0, 0), it

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width is the square of each height. It is a nice, tidy relationship—just the kind of thing you would expect from a well-mannered Brownian motion. A cartoon of discontinuity. There are many ways to illustrate the crucial concepts of fat tails and discontinuity—and this one employs the kind of

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range. Statisticians would have expected the “detrended” range to increase as the square root of the length of the time interval, as it does in Brownian motion. In fact, Hurst found the range grew faster than that. This striking anomaly made no sense to statisticians. But Hurst defended it doggedly. It might

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expectation of tomorrow’s news. A Random Run A nice idea, long memory. But what do you do with it? Go back to the original Brownian motion, of individual particles in water. How far will a molecule get from its starting position in two nanoseconds, or two hours? The square-root rule

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, in any given holding period, an asset’s price may rise or fall and how much it is likely to fluctuate within that broad band. Brownian motion is a bank economist’s best friend. When asked by his boss to predict the dollar-sterling rate a year from now, he can smartly

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rider and gallop off into the dark fields to left or right. The diagrams following illustrate the point. They show, not the position of the Brownian motion, but the changes or steps up or down from one instant to the next as time proceeds. The bottom one shows the persistent case, when

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action is furious but still constrained. Because of the fractional values that H can adopt, I denoted the sums of those interdependent increments as “fractional Brownian motions.” Spot the trends. The standard financial models assume each price change is independent of the last. And if that is wrong? These diagrams, drawn by

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fall, appear and disappear. They could vanish at any instant. They have no real permanence. They cannot be predicted. Look again at the persistent fractional Brownian motion charts, the top and bottom ones. You can spot intervals in which the motion appears to trend upwards, or slide downwards. Mere chance, of course

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he distinguish the graphics of real hydrological data from the fake ones? And of the forgeries, could he tell which were based on my fractional Brownian motion calculations, and which were drawn using the conventional hydrology models? The latter were, for all practical purposes, identical to those of the “modern” theory of

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than any one simplistic test could resolve. A long dependence fully described by a single H is a very special case, that of the fractional Brownian motions shown in the charts on page 188. It is also possible to have a multitude of distinct H-like exponents. For instance, in dollar-Deutschemark

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. As shown in prior chapters, fractal geometry allows for synthesis that starts from some simple ideas and generates complex structures. We began with an approximate Brownian-motion diagram, and later obtained a fat-tailed, discontinuous price chart. Now, the same process can be used to illustrate the theme of the current chapter

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, the width equals the height to the power of one half. Call that power H. No coincidence there. As described earlier in this chapter, a Brownian motion has no dependence—each increment is unaffected by past or future changes—and the H exponent describing its behavior is exactly one half. So what

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my preferred model, yet able to be tuned to capture the essence of every type of market effect, from Bachelier’s original idea of a Brownian motion, and then to Noah, Joseph, and both together. First, let us pull together all the strands of the prior cartoons—a recapitulation of old themes

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baby. The Baby Theorem. This diagram shows how two generators can pass on traits to a third. The mother generator at top right is a Brownian motion, in conventional clock time—as apparent from the chart of its increments shown above the generator. The father, at bottom right, transforms clock time into

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of the price-generating process. On the left sidewall is the fractal chart that the mother generator produces. It is a variant cartoon of the Brownian motion model of how prices happen—in fact, a cartoon pared to the essentials of our original, up-down-up generator without any random shuffling. This

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fact, just a different way of representing what was shown in the prior, Baby Theorem diagram. The left wall is a non-randomized cartoon of Brownian motion—a variant of the mother fractal. The jagged path along the floor is the father; it shows clock time getting deformed, in fits and starts

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lunch and simplification has a cost. Instead, it is best to go beyond cartoons. My current best model of how a market works is fractional Brownian motion of multifractal time. It has been called the Multifractal Model of Asset Returns. The basic ideas are similar to the cartoon versions above—though far

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more intricate, mathematically. The cartoon of Brownian motion gets replaced by an equation that a computer can calculate. The trading-time process is expressed by another mathematical function, called f(α), that can

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time. It compresses it in some places, stretches it out in others. The result appears very wild, very random. The two functions, of time and Brownian motion, work together in what mathematicians call a compound manner: Price is a function of trading time, which in turn is a function of clock time

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, warranted or not. We see patterns where there are none. Between the wars, Evgeny Slutzky, a Soviet statistician, showed how even the record of a Brownian motion—accumulation of a coin-toss game—can appear deliberate and ordered. The eye spontaneously decomposes it into up and down cycles, and then into smaller

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deceptive is long-term dependence that it has found a place in the toolkit of our age’s ultimate fabulists: Hollywood. I have devised fractional Brownian motions “forgeries” that yield quick, realistic-looking landscapes. A demonstration follows. The illustra-tion looks, for all the world, like a relief map of the Himalayas

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some information about it in a 1998 scholarly paper; they called it “tail chiseling”: Under conventional portfolio theory, based on all the old assumptions of Brownian motion in prices, you build a portfolio by laboriously calculating how all the assets in a portfolio vary against each other; good diversification would mean some

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-step formula. It starts with an equation to deform time, to make it jump ahead randomly before slowing again. It follows with a type of Brownian motion to generate a price. There are many others—and so far, no consensus in the industry about which work best. In the absence of clear

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arguing over new rules. The old methods are inadequate, they agree. So what should replace them? One of the standard methods relies on—guess what?—Brownian motion. The same false assumptions that underestimate stock-market risk, mis-price options, build bad portfolios, and generally misconstrue the financial world are also built into

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beyond those works of Pearson and Rayleigh, the year 1905 remains marked by three papers by Albert Einstein, one of which concerns Brownian motion in statistical physics. Random walk and Brownian motion instantly became a core topic of science. In due time, random walk in the plane found a long-forgotten precursor in John

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apply in economics. 209 “as early as 1975” The first multifractal models of price variation were the cartoons to be discussed momentarily and the fractional Brownian motions in multifractal trading time to be discussed starting on page 127. They are closely related and were first presented in Mandelbrot 1997a; see also Mandelbrot

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and Equilibrium. Edited by Doyne Farmer and John Geanakoplos. Oxford, UK: The University Press, 2004. Mandelbrot, Benoit B. 2001c. Scaling in financial prices, III: Cartoon Brownian motions in multifractal time. Quantitative Finance 1: 427-440. Mandelbrot, Benoit B. 2001d. Scaling in financial prices, IV: Multifractal concentration. Quantitative Finance 1: 641-649. Mandelbrot

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of stock price differences. Operations Research 15: 1057-1062. • Reprint: Chapter E21 of Mandelbrot 1997a. Mandelbrot, Benoit B. and J.W. Van Ness. 1968. Fractional Brownian motions, fractional noises and applications. SIAM Review 10: 422-437. • Reprint: Chapter H11 of Mandelbrot 2002. Mandelbrot, Benoit B. and James R. Wallis 1968. Noah, Joseph

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of exchange rates: measurement and forecasting. Journal of International Financial Markets, Institutions and Money 10: 163-180. Rogers, L.C.G. 1997. Arbitrage with fractional Brownian motion. Mathematical Finance 7 (1): 95-105. Roll, Richard. 1970. The Behavior of Interest Rates: An Application of the Efficient Market Model to U.S. Treasury

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Book-to-market Bouchaud, Jean-Philippe Bourbaki Boussinesq, Joseph Box-counting dimension fractals with Brahe, Tycho Bridge range in fractal geometry Bronchia fractals Brown, Robert Brownian motion charts with computer-simulated chart of dependence with Dow price movement compared to financial modeling with multifractal model with Nile river flooding with ordered appearance

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score with pictorial essay with portfolio theory with stocks of turbulent trading with EBITDA École Polytechnique Econometrica Economics astronomy v. Bachelier’s insights in behavioral Brownian motion price assumption of closed shop of computers for continuous price change assumption of cotton prices example in cycle in equilibrium in exogenous effects in faddishness

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in Nile flooding with research need for Federal Reserve Board Feller, Willy Fever charts FIGARCH Filter method Finance analyzing investments in Black-Scholes influence on Brownian motion price assumption of CAPM in case against modern theory of continuous price change assumption of fractals in global crises in investors all alike assumption of

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roughness in trend following in turbulence in volatility in Financial modeling alien visitor in Bachelier in Brownian dependence in development of fractal cartoon in fractional Brownian motion in importance of irregular trends observed in multifractal model in Oanda as random walk model in volatile volatility in Fisher, Adlai Foreign exchange dependence in

The Volatility Smile

by Emanuel Derman,Michael B.Miller  · 6 Sep 2016

before its existence had no plausible or defensible theoretical value at all. In the Platonic world of BSM—a world with normally distributed returns, geometric Brownian motion for stock prices, unlimited liquidity, continuous hedging, and no transaction costs—their model provides a method of dynamically synthesizing an option. It’s a

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to approximate the messiness of actual markets. Other BSM assumptions are violated in more significant ways. For example, stock prices don’t actually follow geometric Brownian motion. They can jump, their distributions have fat tails, and their volatility varies unpredictably. Adjusting for these more significant violations is not always easy. We

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else your theory uses as constituents. Here, unfortunately, be dragons. Financial engineering rests upon the mathematical fields of calculus, probability theory, stochastic processes, simulation, and Brownian motion. These fields can capture some of the essential features of the uncertainty we deal with in markets, but they don’t accurately describe the characteristic

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richer your intuition will be. Models advance by leapfrogging from a simple, intuitive mental concept (e.g., volatility) to the mathematics that describes it (geometric Brownian motion and the BSM model), to a richer concept (the volatility smile), to experience-based intuition (the variation in the shape of the smile), and,

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motion, F = ma, and a particular force law, the gravitational inverse-square law of attraction, which allow one to calculate any planetary trajectory. Geometric Brownian motion and other more elaborate hypotheses for the movement of primitive assets (stocks, commodities, etc.) look like models of absolute valuation, but in fact they are

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we need to make several assumptions, namely:      The movement of the underlying stock price is continuous, with constant volatility and no jumps (one-factor geometric Brownian motion). Traders can hedge continuously by taking on arbitrarily large long or short positions. No bid-ask spreads. No transaction costs. No forced unwinding of positions

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when you take the expected value over the lognormal distribution of the stock price at expiration. We conclude that—provided that the stock undergoes geometric Brownian motion with drift r, irrespective of what hedge ratio Δ is used, no matter what hedging formula you use for delta, and even if you

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HEDGING STRATEGIES IN THE BSM WORLD We now analyze the P&L that results from hedging an option according to the BSM formula, assuming geometric Brownian motion for the stock price. 94 THE VOLATILITY SMILE In the previous section, we rehedged the option at each intermediate time ti between inception and

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certain threshold. In what follows we will discuss only hedging at regular time intervals, and again assume that the underlying stock price evolves with geometric Brownian motion, with constant volatility and no jumps. A Simulation Approach We begin our investigation using Monte Carlo simulation to replicate an option according to the

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that in the absence of transaction costs the implied and realized volatility are identical and equal to 20%, that the SPX evolves according to geometric Brownian motion, and that interest rates and dividends are zero. Calculate the price of a three-month at-the-money European call option using the BSM

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valuation, in both theory and practice. In the BSM model, the volatility is the constant future volatility of a stock assumed to be undergoing geometric Brownian motion. In the BSM model, therefore, a stock must have a definite volatility. If the model accurately describes stocks and the options written on them,

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-money volatility. It measures the number of lognormal standard deviations between the forward price and the strike, a natural viewpoint if the stock undergoes geometric Brownian motion. Once you are familiar with the BSM formula, you quickly notice that the BSM Δ is a function √ of the variable d1 , which depends

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equities, foreign exchange rates, bonds, etc.) have smiles with different characteristic shapes. In each case, these differences hint at the difference between our idealized geometric Brownian motion with constant volatility and the actual behavior of these securities in markets, differences that need to be accounted for if we are going to value

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in an attempt to produce models whose BSM implied volatilities are consistent with the smile. The first strategy is to move away from traditional geometric Brownian motion for the evolution of the underlying asset. The second directly models the movements of the BSM implied volatility surface 𝛴(S, t, K, T) rather

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but also the most ambitious. It attempts to explicitly model the stochastic evolution of the stock price S via a more general process than geometric Brownian motion. The advantage of this approach is that arbitrage violations are more easily avoided, but finding a stochastic process that accurately describes the evolution of a

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of BSM (Merton 1976). These models sensibly allow the stock to make an arbitrary number of jumps in addition to undergoing the diffusion described by Brownian motion. The introduction of jumps allows us to capture the fear of stock market crashes that was responsible for the initial appearance of the smile.

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suggests a method of static replication in the BSM framework, and perhaps even more generally. Valuing a Down-and-Out-Barrier Option under Geometric Brownian Motion with a Zero Riskless Rate and Zero Dividend Yield Consider a European down-and-out call option with strike K and barrier B (now not

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To illustrate it, we initially make one more temporary simplification and consider a stock that undergoes arithmetic Brownian motion. The Method of Images for Arithmetic Brownian Motion Consider a stock S that undergoes arithmetic (rather than geometric) Brownian motion with constant volatility and zero interest rates in a risk-neutral world. Now consider its mirror image

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distribution of a stock starting at S′ (the dashed trajectory). Because S and S′ are symmetrically situated about B, as a consequence of arithmetic Brownian motion with zero rates and dividend yields, both probability distributions have the same value at any time 𝜏 on the boundary B. The Black-Scholes partial differential

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Replication discounted integral of the call’s payoff at expiration over this distribution is the correct price for the down-and-out call, assuming arithmetic Brownian motion. One can understand this pictorially, too: Any gray dashed path in Figure 12.2 that emerges from the barrier and ends up in-the

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image S′ below the barrier that cancels the contribution of those paths arising from S that end up above the barrier. Thus, for arithmetic Brownian motion, we can find the correct riskneutral probability distribution for the terminal stock price of a barrier option by subtracting the distribution of the reflected image

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satisfies the PDE and has the correct boundary conditions, Equation 12.6 is the correct solution. Valuing a Down-and-Out-Barrier Option under Geometric Brownian Motion with a Nonzero Riskless Rate When the riskless rate is nonzero, the similarity of the probabilities of reaching B from both S and S′

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for the implied volatility smile. We begin our search in the framework of the binomial model because it provides a clear way to extend geometric Brownian motion to more general processes. We are preparing for the next chapter, where we will extend the binomial model to accommodate local volatility and the

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Cox-Ross-Rubinstein and the Jarrow-Rudd conventions describe the same continuous process in Equation 13.1. In both cases we are modeling purely geometric Brownian motion, which, when we use it to value an option, will converge to the BSM formula. We will use these binomial processes, and generalizations of

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are contributions to the option payoff from paths that go above the strike and below the current price, but, because of the nature of geometric Brownian motion, these paths have lower risk-neutral probabilities than the more direct paths. In a subsequent chapter, we will discover a better averaging approximation. The

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expiration, and that somewhere between the current stock price and the strike of the call option the local volatility becomes zero. The stock, undergoing geometric Brownian motion, will be unable to move beyond the point where the local volatility first equals zero, and the call option should therefore be worthless. If

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question then becomes how to introduce the second factor. There are two general approaches: 1. Extended Black-Scholes-Merton. This approach begins with the geometric Brownian motion that underlies the Black-Scholes-Merton (BSM) model, which has no implied volatility skew. We then allow the volatility of the stock to itself

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Volatility We begin this section by exploring how we can model the evolution of volatility. Just as with stock price evolution, we often use geometric Brownian motion to model the evolution of volatility. The Hull-White stochastic volatility model (Hull and White 1987) is one of the simplest and earliest models.

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𝛼 of the mean reversion increases, the range gets smaller. Figure 19.4 shows the region encompassing ±1 standard deviation for an Ornstein-Uhlenbeck process and Brownian motion. 329 Introducing Stochastic Volatility Models ± FIGURE 19.4 ± 2 Schematic Illustration of the Standard Deviation of Yt SAMPLE PROBLEM Question: Assume that volatility can

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equivalent in a mean reversion model. Introducing Stochastic Volatility Models 331 A Survey of Some Stochastic Volatility Models Most stochastic volatility models assume traditional geometric Brownian motion for the stock price: dS = 𝜇dt + 𝜎dZ S (19.23) If the volatility term 𝜎 is constant, then there will be no smile. The simplest

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% correlation between S and 𝜎. With stochastic volatility, S and 𝜎 can be more flexibly 332 THE VOLATILITY SMILE correlated. We can introduce this correlation through the Brownian motion terms, expressing the correlation 𝜌 between dZ and dW through dZdW = 𝜌dt (19.28) where 𝜌 is assumed to be constant in almost all stochastic volatility

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, 𝜎, t)dt + q(S, 𝜎, t)dZ (20.16) dWdZ = 𝜌dt where p(S, 𝜎, t) and q(S, 𝜎, t) are functions that can accommodate geometric Brownian motion, mean reversion, or more general behaviors. S is the underlying stock price and 𝜎 is its volatility. Approximate Solutions to Some Stochastic Volatility Models 345 Now

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’s look at a more realistic continuous distribution of stochastic instantaneous volatilities of the stock price. We now assume that the volatility itself undergoes geometric Brownian motion (GBM) according to d𝜎 = a𝜎 dt + b𝜎 dZ (21.23) 363 Stochastic Volatility Models: The Smile for Zero Correlation 20.125% 18 yr 19

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formula. The path variance is an arithmetic average of the instantaneous variances to time T, but the instantaneous variance 𝜎 2 itself evolves according to geometric Brownian motion. As a result, there is no closedform expression for the path variance. Nevertheless, one can √ show that the arithmetic average has approximately 1/2

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the path volatility to time 𝜏 is: var [𝜎] ̄ ≈ b2 2 𝜎 𝜏 3 (21.30) which grows linearly with time to expiration because of this standard property of Brownian motion. Substituting these results into Equation 21.24 and keeping terms to second order in 𝜏, we obtain 366 THE VOLATILITY SMILE ( 𝛴atm ) 1 ≈ 𝜎̄ 1 − var [𝜎] ̄ 𝜏

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of the stochastic volatility model with zero correlation. In the two-state model, the range of volatility was constant over time. In the geometric Brownian motion model, the range of volatility grew without bound. In this section we examine a more realistic in-between case in which volatility is mean reverting

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volatility evolves, to understand how the smiles in stochastic volatility models behave for very short and very long expirations. Volatility versus Path Volatility In standard Brownian motion, the diffusion process causes the variance or range of possible outcomes to increase without bound over time. The range of the average value along

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will have the same path volatility so the long-term skew becomes flat. Mean reversion describes a more realistic evolution of volatility than ordinary geometric Brownian motion, and also restores the decreasing curvature of the smile that we often see in actual markets. Figure 22.2 shows the results of a

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and backward approaches do not converge to the same result. In fact, there is no unique continuous result to converge to. In the case of Brownian motion, the forward numerical integration converges to an Itô integral, the backward approach converges to a backward Itô integral, and the midpoint approach converges to

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& Sons. Index Page numbers followed by f refer to figures Absolute valuation, 11–12 Analytical approximation: of jump-diffusion, 410–416 of smile for geometric Brownian motion stochastic volatility with zero correlation, 363–368 of transaction costs, 123–124 Anderson, Leif, 409 Andreasen, Jesper, 409 Apple Inc., 20 Approximate static hedge,

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42–44 Arbitrage opportunity, 14 Arbitrage pricing theory, 33–34 Arithmetic Brownian motion, method of images for, 208–209 Arrow-Debreu securities, 176, 332–334. See also State-contingent securities Asian financial crisis of 1997, 150 Asian options

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(FX) options Gains, from convexity, 52. See also Profit and loss (P&L) Gamma (Γ), 46 Gatheral, Jim, 316 Generalized payoffs, 40–42 Geometric Brownian motion (GBM): assumed, in Black-Scholes-Merton model, 133 for interest rates, 151 method of images for, 209–211 in stochastic volatility models, 331, 332, 342

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168, 395 Merton inequalities, for European option prices, 154–158 Merton’s jump-diffusion model, 395–398, 414 Method of images: for arithmetic Brownian motion, 208–209 INDEX for geometric Brownian motion, 209–211 Mexican peso (MXN), 150f Mixing theorem: in jump-diffusion models, 404–408, 411 and path variance, 364 in stochastic volatility

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, 308 Nonzero correlation, 375–376, 377f–378f Nonzero riskless rate: in Black-Scholes-Merton model, 238–240 valuing down-and-out barrier option under geometric Brownian motion with, 211–212 Normal distributions, 22, 23 Notation, for implied variables, 51–52 Notional variance, 61 Notional vega, of volatility swaps, 61 OAS (option

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Merton model, 321–325 characteristic solution to, 351–352 extending Black-Scholes-Merton model to, 344–350 extending local volatility models to, 337–344 geometric Brownian motion stochastic volatility with zero correlation, 362–368 hedge ratios in, 379 mean-reverting volatility with zero correlation, 369–375 nonzero correlation in, 375–376,

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moneyness, 353–356 as symmetric, 356–360 Zero dividend yield: in Black-Scholes-Merton model, 237–238 valuing down-and-out barrier option under geometric Brownian motion with zero riskless rate and, 207–211 Zero-interest-rate policy (ZIRP), 154n.1 Zero riskless rate: in Black-Scholes-Merton model, 237–238

Tools for Computational Finance

by Rüdiger Seydel  · 2 Jan 2002  · 313pp  · 34,042 words

computational method the simple but powerful binomial method is derived. The following parts of Chapter 1 are devoted to basic elements of stochastic analysis, including Brownian motion, stochastic integrals and Itô processes. The material is discussed only to an extent such that the remaining parts of the book can be understood. Neither

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, l, m, n, M, N, ν various variables: Xt , X, X(t) Wt y(x, τ ) w h ϕ ψ 1D random variable Wiener process, Brownian motion (Definition 1.7) solution of a partial differential equation for (x, τ ) approximation of y discretization grid size basis function (Chapter 5) test function (Chapter

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.5.1 Dow Jones Industrial Average Forward Time Backward Space, see Section 6.5.1 Forward Time Centered Space, see Section 6.4.2 Geometric Brownian Motion, see (1.33) Monte Carlo Ordinary Differential Equation Over The Counter Partial Differential Equation Partial Integro-Differential Equation Projected Successive Overrelaxation Quasi Monte Carlo Stochastic

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take any real number. Further, individual trading will not influence the price. (b) There are no arbitrage opportunities. (c) The asset price follows a geometric Brownian motion. (This stochastic motion will be discussed in Sections 1.6–1.8.) (d) Technical assumptions (some are preliminary): r and σ are constant for 0

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be adapted dynamically. The general definition is ∂V (S, t) ; ∆ = ∆(S, t) = ∂S the expression (1.16) is a discretized version. 1.6 Stochastic Processes Brownian motion originally meant the erratic motion of a particle (pollen) on the surface of a fluid, caused by tiny impulses of molecules. Wiener suggested a mathematical

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model for this motion, the Wiener process. But earlier Bachelier had applied Brownian motion to model the motion of stock prices, which instantly respond to the numerous upcoming informations similar as pollen react to the impacts of molecules. The

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Dow at 500 trading days from September 8, 1997 through August 31, 1999 1.6.1 Wiener Process Definition 1.7 (Wiener process, Brownian motion) A Wiener process (or Brownian motion; notation Wt or W ) is a timecontinuous process with the properties (a) W0 = 0 (with probability one) (b) Wt ∼ N (0, t) for

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. 1.16. Numerically approximated trajectory of Example 1.12 with a = 0.05Xt , b = 0.3Xt , ∆t = 1/300, X0 = 50 Model 1.13 (geometric Brownian motion, GBM) dSt = µSt dt + σSt dWt . (1.33, GBM) This SDE is linear in Xt = St , a(St , t) = µSt is the drift rate with

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the expected rate of return µ, b(St , t) = σSt , σ is the volatility. (Compare Example 1.12 and Figure 1.16.) The geometric Brownian motion of (1.33) is the reference model on which the Black-Scholes-Merton approach is based. According to Assumption 1.2 we assume that µ and

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. A class of models is given by the SDE drt = α(R − rt )dt + σr rtβ dWt , α > 0. (1.40) Wt is again a Brownian motion. The drift term in (1.40) is positive for rt < R and negative for rt > R, which causes a pull to R. This effect is

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. Taking correct limits (similar as in Lemma 1.9) one obtains (1.43). Consequences for Stocks and Options Suppose the stock price follows a geometric Brownian motion, hence Xt = St , a = µSt , b = σSt . The value Vt of an option depends on St , Vt = V (St , t). Assuming C 2 -smoothness of

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and (1.47) This is the density of the lognormal distribution. The stock price St is lognormally distributed under the basic assumption of a geometric Brownian motion (1.33). The distribution is skewed, see Figure 1.20. Now the skewed behavior coming out of the experiment reported in Figure 1.18 is

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)+ f (ST ; T, S0 , r, σ) dST . V (S0 , 0) = e 0 It is inspiring to test the idealized Model 1.13 of a geometric Brownian motion against actual empirical data. Suppose the time series S1 , ..., SM represents consecutive quotations of a stock price. To test the data, histograms of the returns

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Gaussian density. The symmetry may be perturbed, and in particular the tails of the data are not well modeled by the hypothesis of a geometric Brownian motion: The exponential decay expressed by (1.47) amounts to thin tails. This underestimates extreme events and hence does not match reality. It is questionable whether

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plain geometric Brownian motion is suitable to model risks. We conclude this section with deriving the analytical solution of the basic linear constant-coeffficient SDE (1.33) dSt = µSt

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). As a consequence we note that St > 0 for all t, provided S0 > 0. 1.9 Jump Processes 45 1.9 Jump Processes The geometric Brownian motion Model 1.13 has continuous paths St . As noted before, the continuity is at variance with those rapid asset price movements that can be considered

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is the so-called Black-Scholes equation (1.2) with its analytic solution formula (A4.10). For the Assumption 1.2(c) of a geometric Brownian motion see also the notes and comments on Sections 1.7/1.8. For reference on discrete-time models, see [Pli97], [FöS02]. on Section 1

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], [Shi99], [Sato99]. The requirement (a) of Definition 1.7 (W0 = 0) is merely a convention of technical relevance; it serves as normalization. This Brownian motion ist called standard Brownian motion. For a proof of the nondifferentiability of Wiener processes, see [HuK00]. For more hints on martingales, see Appendix B2. In contrast to the results

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is taken from [HPS92]. [KP92] in Section 4.4 gives a list of SDEs that are analytically solvable or reducible. The model of a geometric Brownian motion of equation (1.33) is the classical model describing the dynamics of stock prices. It goes back to Samuelson (1965; Nobel Prize in economics in

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1970). Already in 1900 Bachelier had suggested to model stock prices with Brownian motion. Bachelier used the arithmetic version, which can be characterized by replacing the lefthand side of (1.33) by the absolute change dS. This amounts to

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the process St = S0 + µt + σWt . Here the stock price can become negative. Main advantages of the geometric Brownian motion are the success of the approaches of Black, Merton and Scholes, which is based on that motion, and the existence of moments (as the expectation

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]. on Section 1.9: Section 1.9 concentrates on jump-diffusion processes. This class of processes extends to the large class of Lévy processes. Brownian motion and Poisson jumps are just simple special cases, combined to the jump-diffusion model. A second category of Lévy processes consists of models with

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, increasing Lévy processes are introduced. Call such a subordinator process τ (t), then the model of a financial process is Wτ (t) , with a Brownian motion Wt . Changing the clock in such a way has been successfully applied to match empirical data. Again we refer to [ConT04]. Exercises Exercise 1.1

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of clarity we describe the approach in the one-dimensional context. From Section 1.7.2 we take the one-factor model of a geometric Brownian motion of the asset price St , dS = µ dt + σ dW. S Here µ is the expected growth rate. When options are to be priced we assume a

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constant δ ≥ 0. This continuous dividend model can be easily built into the Black-Scholes framework. To this end the standard model of a geometric Brownian motion represented by the SDE (1.33) is generalized to dS = (µ − δ)dt + σdW. S The corresponding Black-Scholes equation for V (S, t) is ∂V

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relevant PDEs and boundary conditions are required. The equations are of the Black-Scholes type. To extend the onefactor model, an appropriate generalization of geometric Brownian motion is needed. We begin with the two-factor model, with the prices of the two assets S1 and S2 . The assumption of a constant-coefficient

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Black-Scholes Equation The Classical Equation This appendix applies Itô’s lemma to derive the Black-Scholes equation. The first basic assumption is a geometric Brownian motion of the stock price. According to Model 1.13 the price S obeys the linear stochastic differential equation (1.33) dS = µS dt + σS dW

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b1 )dW . (B2.2) Application: dS = rSdt + σSdŴ ⇒ d(e−rt S) = e−rt σSdŴ (B2.3) for any Wiener process Ŵ . Filtration of a Brownian motion FtW := σ{Ws | 0 ≤ s ≤ t} (B2.4) Here σ{.} denotes the smallest σ-algebra containing the sets put in braces. FtW is a model

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’s Theorem Suppose a process γ is such that Z γ is a martingale. Then  t γs ds Wtγ := Wt + (B2.8) 0 is a Brownian motion and martingale under Q(γ). B3 State-Price Process Normalizing A fundamental result of Harrison and Pliska [HP81] states that the existence of a martingale

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µX − rX = σ X γ tr (B3.2) holds. This is a system of n equations for the m components of γ. Special case geometric Brownian motion: For scalar X = S and W , µX = µS, σ X = σS, (B3.2) reduces to µ − r = σγ . Given µ, σ, r, the equation (B3.2) determines

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. Finance & Stochastics 3 (1999) 391-412. [FHH04] J. Franke, W. Härdle, C.M. Hafner: Statistics of Financial Markets. Springer, Berlin (2004). [Fr71] D. Freedman: Brownian Motion and Diffusion. Holden Day, San Francisco (1971). [Fu01] M.C. Fu (et al): Pricing American options: a comparison of Monte Carlo simulation approaches. J. Computational

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] [Lyuu02] [MaM86] [MRGS00] [Man99] [MCFR00] L.W. Kantorovich, G.P. Akilov: Functional Analysis in Normed Spaces. Pergamon Press, Elmsford (1964). I. Karatzas, S.E. Shreve: Brownian Motion and Stochastic Calculus. Second Edition. Springer Graduate Texts, New York (1991). I. Karatzas, S.E. Shreve: Methods of Mathematical Finance. Springer, New York (1998). H

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-Rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-Rate Options. John Wiley & Sons, Chichester (1996). D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. Springer, Berlin 1991. C. Ribeiro, N. Webber: Valuing path dependent options in the VarianceGamma model by Monte Carlo with a gamma bridge. Working paper, City

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, 7–8, 59, 113–115, 121, 139–141, 167–168, 277–279 Box–Muller method 72–75, 85, 117 Bridge 102, 116, 118, 121, 236 Brownian motion 25–26, see Wiener process Bubnov 186 Business time 52 Calculus of variations 200 Calibration 38, 52, 54 Call, see Option Cancellation 54 Cauchy convergence

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240–242 Galerkin 183, 186, 188, 192, 206 GARCH 52 Gaussian elimination 268–270 Gaussian process 25, 51 Gauß–Seidel method 179, 271–272 Geometric Brownian motion (GBM) 9, 34, 36, 41–43, 45, 47, 49, 51, 102, 121, 124, 138, 212, 215, 244, 262 Gerschgorin 137, 268 Girsanov 260 Godunov 236

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, 118 Weak derivative 273–274 Weak solution 92, 120, 195, 199–202 Weighted residuals 183–187 Weighting function 186 White noise 32, 51 Wiener process (Brownian motion) Wt 25–34, 36, 38–44, 47, 50–52, 56, 57, 91–93, 96–102, 105, 119–121, 212, 215, 257–261 Writer 1–2

Market Risk Analysis, Quantitative Methods in Finance

by Carol Alexander  · 2 Jan 2007  · 320pp  · 33,385 words

Portfolio Holdings and Portfolio Weights I.1.4.3 Profit and Loss I.1.4.4 Percentage and Log Returns I.1.4.5 Geometric Brownian Motion I.1.4.6 Discrete and Continuous Compounding in Discrete Time I.1.4.7 Period Log Returns in Discrete Time I.1.4.8

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A binomial lattice Computing the price of European and American puts Simulating from a standard normal distribution Possible paths for an asset price following geometric Brownian motion A set of three independent standard normal simulations A set of three correlated normal simulations Convex, concave and linear utility functions The effect of correlation

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concepts to define the return on a portfolio, in both discrete and continuous time, discrete and continuous compounding of the return on an investment, geometric Brownian motion and the ‘Greeks’ of an option. The last section focuses on Taylor expansion, since this is used so widely in continuous time finance and all

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are applied to price European and American options consistently with the Black–Scholes–Merton model, and Monte Carlo simulation is applied to simulate correlated geometric Brownian motions, amongst other illustrative examples. As usual, all of these are contained in an Excel workbook for the chapter on the CD-ROM, more specific details

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log price, d ln St. Hence the log return is the increment in the log price of the asset. I.1.4.5 Geometric Brownian Motion We use the differential operator to describe the evolution of prices of financial assets or interest rates in continuous time. Let St denote the

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differential term dWt to (I.1.29) or to (I.1.30). The process Wt is called a Wiener process, also called a Brownian motion. It is a continuous process that has independent increments dWt and each increment has a normal distribution with mean 0 and variance dt.19

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in the price at time t, rather than the absolute change, we call (I.1.32) geometric Brownian motion. If the left-hand side variable were dSt instead, the process would be called arithmetic Brownian motion. The diffusion coefficient is the coefficient of dWt, which is a constant  in the case of

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geometric Brownian motion. This constant is called the volatility of the process. By definition dWt has a normal distribution

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II.5.2. • Continuous time stochastic processes are represented as stochastic differential equations (SDEs). The most famous example of an SDE in finance is geometric Brownian motion. This is introduced below, but its application to option pricing is not discussed until Chapter III.3. The first two subsections define what is meant

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parts to this representation: the first term defines the deterministic part and the second term defines the stochastic part. A Wiener process, also called a Brownian motion, describes the stochastic part when the process does not jump. Thus a Wiener process is a continuous process with stationary, independent normally distributed increments. The

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.3.141) is called the process volatility. The where  is called the drift of the process and model (I.3.141) is called arithmetic Brownian motion. Arithmetic Brownian motion is the continuous time version of a random walk. To see this, we first note that the discrete time equivalent of the Brownian increment dZ

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negative, whilst the prices of tradable assets are never negative. Thus, to represent the dynamics of an asset price we very often use a geometric Brownian motion which is specified by the following SDE: dSt = dt + dZt (I.3.143) St The standard assumption, made in the seminal papers

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Black–Scholes–Merton framework still remains a basic standard against which all other models are gauged. Now we derive a discrete time equivalent of geometric Brownian motion, and to do this it will help to use a result from stochastic calculus that is a bit like the chain rule of ordinary calculus

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Itô’s lemma with f = ln S shows that a continuous time representation of geometric Brownian motion that is equivalent to the geometric Brownian motion (I.3.143) but is translated into a process for log prices is the arithmetic Brownian motion, d ln St = − 21 2 dt + dWt  (I.3.145) We already

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random walk model for the log ln Pt =  + ln Pt−1 + $t  $t ∼ NID 0 2  (I.3.147) To summarize, the assumption of geometric Brownian motion for prices in continuous time is equivalent to the assumption of a random walk for log prices in discrete time. I.3.7.4 Jumps

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for stationarity. By applying Itô’s lemma we demonstrated the equivalence between the continuous time geometric Brownian motion model for asset prices and the discrete time random walk model for log prices. We also introduced arithmetic Brownian motion and mean reverting models for financial asset prices, and jumps that are governed by the Poisson

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III.3.7, but these only apply to standard European call and put options under the assumption that the underlying asset price follows a geometric Brownian motion. In general there are no analytic formulae for the Greeks of an option, or of a portfolio of options, and they are estimated using finite

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-hedged portfolio. Then we can derive a PDE which describes the evolution of the option price.18 More specifically, if the underlying follows a geometric Brownian motion with constant drift  and volatility , then the evolution of the option price f will follow the well-known heat equation, ft + r − ySfS + 21

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measure that is equivalent to the measure defined by the price process (e.g. a lognormal measure when the price process is a standard geometric Brownian motion). Moreover, this measure is unique if the market is complete.23 I.5.6.3 Pricing European Options The pay-off at expiry depends on

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Asset Price Distributions The Black–Scholes–Merton pricing formula for European options is based on the assumption that the asset price process follows a geometric Brownian motion with constant volatility .26 How should the binomial lattice be discretized to be consistent with this model? Consider a risk neutral setting over a discrete

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 d2 (I.5.40) Hence, the discretization in a binomial lattice will be consistent with the assumption that the asset price process follows a geometric Brownian motion provided that conditions (I.5.39) and (I.5.40) hold. Since we have only two conditions on four parameters there are many ways in

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.40) can hold; in other words, there are many different parameterizations of a binomial tree that are consistent with an asset price following a geometric Brownian motion. We mention just two. The discretization  √  exprt − d (I.5.41)  u = d−1 = exp  t  p= u−d due to Cox, Ross

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volatility tends to 0 the lattice does not converge to an exponential trend, which is the case when the underlying asset price follows a geometric Brownian motion. The discretization   √ √  √  p = 1/2 u = exp m +  t  d = exp m −  t  m = r − 1/2 t (I.5.42) (Jarrow and Rudd, 1982

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: Generating Time Series of Lognormal Asset Prices In this subsection we describe how to simulate a time series of asset prices that follow a geometric Brownian motion, dSt = dt + dWt St 30 For instance, suppose the empirical returns are in cells A1:A1000. Use the command RANDBETWEEN(1,1000

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for the specification of the standard normal distribution. 31 32 Numerical Methods in Finance 219 Geometric Brownian motion was introduced in Section I.1.4.5, and we derived the discrete time equivalent of geometric Brownian motion in Section I.3.7.3. Using Itô’s lemma we showed that the log return, which

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St−1 (I.5.48) So this is how we simulate prices that follow a geometric Brownian motion. To illustrate (I.5.48) we generate some possible price paths for an asset that follows a geometric Brownian motion with drift 5% and volatility 20%. Suppose we generate the paths in daily increments over 1

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.18 I.5.7.4 31 61 91 121 151 181 211 241 271 301 331 361 Possible paths for an asset price following geometric Brownian motion Simulations on a System of Two Correlated Normal Returns Correlated simulations are necessary for computing the Monte Carlo VaR of a portfolio, and we shall

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difference 206–10, 223 Taylor expansion 31–4, 36 Arbitrage no arbitrage 2, 179–80, 211–13 pricing theory 257 statistical strategy 182–3 Arithmetic Brownian motion 22, 136, 138–9 Arrival rate, Poisson distribution 87–9 Ask price 2 Asset management, global 225 Asset prices binomial theorem 85–7 lognormal distribution

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–16 lognormal distribution 94 numerical method 185 Taylor expansion 2–3 BLUE (best linear unbiased estimator) 157, 175 Bonds 1–2, 37, 191 Bootstrap 218 Brownian motion 136 arithmetic 22, 136, 139 geometric 21–2, 134, 138, 212, 213–14, 218–19 Calculus 1–36 differentiation 10–15 equations and roots 3

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concave/convex function 13–14 definition 10–11 monotonic function 13–14 rule 11–12 stationary point 14–15 stochastic differential term 22 Diffusion process, Brownian motion 22 Discontinuity 5 Discrete compounding, return 22–3 Discrete time 134–9 log return 19–20 notation 16–17 P&L 19 percentage return 19

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–3 Generalized least squares (GLS) 178–9 Generalized Pareto distribution 101, 103–5 Generalized Sharpe ratio 262–3 General linear model, regression 161–2 Geometric Brownian motion 21–2 lognormal asset price distribution 213–14 SDE 134 stochastic process 141 time series of asset prices 218–20 GEV (generalized extreme value) distribution

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, 256 CAPM 253–4 compounding 22–3 continuous time 16–17 correlated simulations 220 discrete time 16–17, 22–5 equity index 96–7 geometric Brownian motion 21–2 linear portfolio 25, 56–8 log returns 16, 19–25 long-short portfolio 20–1 multivariate normal distribution 115–16 normal probability 91

Frequently Asked Questions in Quantitative Finance

by Paul Wilmott  · 3 Jan 2007  · 345pp  · 86,394 words

the Most Useful Performance Measures? References and Further Reading What is a Utility Function and How is it Used? References and Further Reading What is Brownian Motion and What are its Uses in Finance? References and Further Reading What is Jensen’s Inequality and What is its Role in Finance? References and

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? 83 18. What are the most useful performance measures? 87 19. What is a utility function and how is it used? 90 20. What is Brownian Motion and what are its uses in finance? 94 21. What is Jensen’s Inequality and what is its role in finance? 97 22. What is

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many scientific fields and is commonly used as the model mechanism behind a variety of unpredictable continuous-time processes. The lognormal random walk based on Brownian motion is the classical paradigm for the stock market. See Brown (1827). 1900 Bachelier Louis Bachelier was the first to quantify the concept of

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Brownian motion. He developed a mathematical theory for random walks, a theory rediscovered later by Einstein. He proposed a model for equity prices, a simple normal distribution,

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great success and, naturally, Bachelier’s work was not appreciated in his lifetime. See Bachelier (1995). 1905 Einstein Albert Einstein proposed a scientific foundation for Brownian motion in 1905. He did some other clever stuff as well. See Stachel (1990). 1911 Richardson Most option models result in diffusion-type equations. And often

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weather forecasting. See Richardson (1922). Richardson later worked on the mathematics for the causes of war. 1923 Wiener Norbert Wiener developed a rigorous theory for Brownian motion, the mathematics of which was to become a necessary modelling device for quantitative finance decades later. The starting point for almost all financial models, the

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Exchange. Scholes and Merton won the Nobel Prize for Economics in 1997. Black had died in 1995. The Black-Scholes model is based on geometric Brownian motion for the asset price S The Black-Scholes partial differential equation for the value V of an option is then 1974 Merton, again In 1974

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, LF 1922 Weather Prediction by Numerical Process. Cambridge University Press Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818 Samuelson, P 1955 Brownian motion in the stock market. Unpublished Schönbucher, PJ 2003 Credit Derivatives Pricing Models. John Wiley & Sons Sharpe, WF 1985 Investments. Prentice-Hall Sloan, IH & Walsh, L

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in the text. References and Further Reading Ingersoll, JE Jr 1987 Theory of Financial Decision Making. Rowman & Littlefield What is Brownian Motion and What are its Uses in Finance? Short Answer Brownian Motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. It is the most

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in finance. Example Pollen in water, smoke in a room, pollution in a river, are all examples of Brownian motion. And this is the common model for stock prices as well. Long Answer Brownian motion (BM) is named after the Scottish botanist who first described the random motions of pollen grains suspended in water

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not possible in practice. Second, it hinges on the accuracy of the model. The underlying has to be consistent with certain assumptions, such as being Brownian motion without any jumps, and with known volatility. One of the most important side effects of risk-neutral pricing is that we can value derivatives by

…

’s theorem is the formal concept underlying the change of measure from the real world to the risk-neutral world. We can change from a Brownian motion with one drift to a Brownian motion with another. Example The classical example is to start withdS = µS dt + σ S dWt with W being

…

-world measure) and converting it to under a different, the risk-neutral, measure. Long Answer First a statement of the theorem. Let Wt be a Brownian motion with measure and sample space Ω. If γt is a previsible process satisfying the constraint then there exists an equivalent measure on Ω such that

…

is a Brownian motion. It will be helpful if we explain some of the more technical terms in this theorem. Sample space: All possible future states or outcomes. (Probability

…

-world processes, dS = µS dt + σ S dX1 anddσ = a(S, σ , t)dt + b(S, σ , t)dWX2 , where dX1 and dX2 are correlated Brownian motions with correlation ρ(S, σ , t). Using Girsanov you can get the governing equation in three steps:1. Under a pricing measure ,Girsanov plus the

…

edition. John Wiley & Sons What is a Jump-Diffusion Model and How Does It Affect Option Values? Short Answer Jump-diffusion models combine the continuous Brownian motion seen in Black-Scholes models (the diffusion) with prices that are allowed to jump discontinuously. The timing of the jump is usually random, and this

…

an interest rate. Although there have been research papers on pure jump processes as financial models it is more usual to combine jumps with classical Brownian motion. The model for equities, for example, is often taken to bedS = µS dt + σ S dX + (J − 1)S dq. dq is as defined above

…

n is modelled by where w0 is the long-term variance, α and β are positive parameters, with α + β < 1, and Bn are independent Brownian motions, that is, random numbers drawn from a normal distribution. The latest variance, vn, can therefore be thought of as a weighted average of the most

…

. It has two parameters: a > 0, location; b > 0 scale. Its probability density function is given by This distribution models the time taken by a Brownian Motion to cover a certain distance. Inverse normal Mean a Variance Gamma Bounded below, unbounded above. It has three parameters: a, location; b > 0 scale; c

…

methodology was formalized by Harrison and Kreps (1979) and Harrison and Pliska (1981).6 We start again withdSt = µS dt + σS dWt The Wt is Brownian motion with measure Now introduce a new equivalent martingale measure such that where η = (µ − r)/σ. Under we have Introduce The quantity er(T−t)Gt

…

total slippage will depend on how often the stock crosses this point. Herein lies the rub. This happens an infinite number of times in continuous Brownian motion. If U(∈) is the number of times the forward price moves from K to K + ∈, which will be finite since ∈ is finite, then the financing

…

Equation The penultimate derivation of the Black-Scholes partial differential equation is rather unusual in that it uses just pure thought about the nature of Brownian motion and a couple of trivial observations. It also has a very neat punchline that makes the derivation helpful in other modelling situations. It goes like

…

this. Stock prices can be modelled as Brownian motion, the stock price plays the role of the position of the ‘pollen particle’ and time is time. In mathematical terms

…

Brownian motion is just an example of a diffusion equation. So let’s write down a diffusion equation for the value of an option as a function

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