discrete time Concepts

Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Nikolai Dokuchaev · 24 Apr 2007

203-96472-9 (Print Edition) (ebk) © 2007 Nikolai Dokuchaev Contents Preface vi 1 Review of probability theory 1 2 Basics of stochastic processes 17 3 Discrete time market models 23 4 Basics of Ito calculus and stochastic analysis 49 5 Continuous time market models 75 6 American options and binomial trees 110

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be used independently as a reference. Chapter 1 can be also considered as a short introduction to measure and integration theory. Further, we describe generic discrete time market models (Chapter 3) and continuous time market models (Chapter 5). For these two types of models, the book provides all basic concepts of

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variables or vectors that depend on time. Definition 2.1 A sequence of random variables ξt, t=0, 1, 2,…, is said to be a discrete time stochastic (or random) process. be given. A mapping ξ:[0,T]×Ω→R is said to be a Definition 2.2 Let continuous time stochastic

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uniquely defined by its initial data. The following definitions give examples that are different. Definition 2.5 Let ξt, t=0, 1, 2,…, be a discrete time random process such that ξt are mutually independent and have the same distribution, and Eξt≡0. Then the process ξt is said to be a

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10 Let ξ(t) be a random process. The filtration defined as the minimal filtration such that ξ(t) is adapted to it. Consider a discrete time random process Example 2.11 Let Ω={ω1,ω2,ω3}, ξt, t=0, 1,… such that Let us find the filtration generated by ξt for

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consist of smaller σ-algebras, and Example 2.12 Let Ω={ω1,ω2,ω3}, ξt, t=0, 1,… such that © 2007 Nikolai Dokuchaev Consider a discrete time random process Basics of Stochastic Processes Let us find again the filtration Let 19 generated by the process ξt, t=0, 1, 2. denote the

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We say that ξ(t) is a martingale if ξ(t) is a martingale with respect to the filtration Problem 2.16 Prove that any discrete time random walk is a martingale. © 2007 Nikolai Dokuchaev Mathematical Finance 20 Problem 2.17 Let ζ be a random variable such that E|ζ|2

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notations of Definition 2.23, for all measurable deterministic functions F such that the corresponding random variables are integrable. Problem 2.25 Prove that a discrete time random walk is a Markov process. Vector processes Let ξ(t)=(ξ1(t),…, ξn(t)) be a vector process such that all its components are

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respect to a filtration, (i) Is 2τ a Markov time? (ii) Is τ/2 a Markov time? © 2007 Nikolai Dokuchaev 3 Discrete time market models In this chapter, we study discrete time mathematical models of markets. These models are relatively simple and straightforward. However, they still allow us to introduce all core definitions of

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are not discussed here in detail, because this course is focused on the problem of pricing (the only exception is Section 3.12). 3.2 Discrete time model with free borrowing We introduce a model of a financial market consisting of the risky stock with price St, t=0, 1, 2,…, where

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at time t=0. For example, for the trivial risk-free strategy, when γt≡0, the corresponding total wealth is Xt≡X0. © 2007 Nikolai Dokuchaev Discrete Time Market Models 25 Note that these definitions present a simplification of the real market situation, because transaction costs, bid and ask gap, possible taxes and

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dividends, interest rate for borrowing, etc., are not taken into account. 3.3 A discrete time bond-stock market model A more realistic model of the market with non-zero interest rate for borrowing can be described via the following bond

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portfolio given C. For simplicity, we assume that ρt≡1. Then Xt+1−Xt=γt(St+1−St)=γtStξt+1=CXtξt+1, © 2007 Nikolai Dokuchaev Discrete Time Market Models 27 i.e., For instance, let (S0, S1, S2, S3,…)=(1, 2, 1, 2, 1,…). It follows that Let X0=1. For

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1, and everywhere in this chapter. After that, one can read this chapter again taking into account the impact of ρt≠1. © 2007 Nikolai Dokuchaev Discrete Time Market Models 29 3.5 Risk-neutral measure Up to this point, we have not needed probability space, and the market model was not a

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for the corresponding wealth. Let P* be a risk-neutral measure, and let E* be the corresponding expectation. Let E*ψ2<+∞. Then © 2007 Nikolai Dokuchaev Discrete Time Market Models Proof. Clearly, XT=ψ iff the process 31 a.s. By Theorem 3.22, it follows that is a martingale under P* with

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The proof of this assertion is beyond the scope of this book. The equivalence relation between the existence of equivalent risk-neutral © 2007 Nikolai Dokuchaev Discrete Time Market Models 33 measure and the absence of (certain types of) arbitrage is called the fundamental theorem of asset pricing. 3.8 A case of

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To prove the completeness of the market, it suffices to find an admissible strategy such that Let © 2007 Nikolai Dokuchaev i.e., a.s., and Discrete Time Market Models Clearly, Let 35 and Define the functions Clearly, Yt=Vt(ξ1,…, ξt). By Bayes formula, for any integrable random variable η and for

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random, and, for any t, there exists a.s. It such that −1<d1(t)<0<d2(t)<1 a.s. and follows that the discrete time market model is also complete for this case of conditionally two-point distribution of ξt. Technically, this model is more general than the Cox-RossRubinstein

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model. It appears that it is the most general assumption that still allows a discrete time market to be complete. (For instance, Corollary 3.35 states that three-point distribution for ξt leads to incompleteness.) Remark 3.40 In fact,

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that p=(ρ−d)/(u−d) in Remark 3.40. (ii) Find the (risk-neutral) probability that ST=S0u2d for t=3. © 2007 Nikolai Dokuchaev Discrete Time Market Models 37 3.10 Option pricing Options and their types Let us describe first the most generic options: the European call option and the

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option that has some economical sense. If possible, suggest a pricing method using the approach described below for European and American options. © 2007 Nikolai Dokuchaev Discrete Time Market Models 39 Fair price of an option The key role in mathematical finance belongs to a concept of the ‘fair price’ of options. The

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following definition is a discrete time analogue of the definition introduced by Black and Scholes (1973) for a continuous time market. Definition 3.48 The fair price of an option of

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market situation with some admissible strategies. Let us assume that a probabilistic concept is accepted. This means that the stock price evolves as a random discrete time process, and a probabilistic measure is fixed. We now rewrite Definition 3.48 more formally for European options. Definition 3.49 The fair price of

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of the option is the initial wealth X0 such that E|XT−ψ|2 is minimal over all admissible self-financing strategies. © 2007 Nikolai Dokuchaev Discrete Time Market Models 41 In many cases, this definition leads to the option price calculated as the expectation under a risk-neutral equivalent measure which needs

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non-constant function, such that the is bounded. Let t0=0, tk+1=k∆, k=0,…, T, where Note that tn=τ. Consider discrete time discounted prices Consider the discrete time market model with discounted stock prices and with the self-financing strategy defined by the stock portfolio {γk}, where γk=g(tk), and

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with constraints such as EX1→max, VarX1≤const., or VarX1→min, EX1≥const. Remark 3.59 The solution of the optimal investment problem for a discrete time market with T>1 is much more difficult. For instance, Markovitz’s results for quadratic U were extended for the case of T>1 only

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recently (Li and Ng, 2000). © 2007 Nikolai Dokuchaev Discrete Time Market Models 45 3.13 Possible generalizations The discrete time market model allows some other variants, some of which are described below. • One can consider an additive model for the stock

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N>1 has different properties compared with the case of N=1. For instance, as far as we know, there are no examples of complete discrete time markets with N>1. Some special effects can be found for N→+∞ (such as strategies that converge to arbitrage). Note also that the most widely

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model that takes into account the impact of a large investor’s behaviour, (ρm, Sm) is affected by {γk}k<m. 3.14 Conclusions • A discrete time market model is the most generic one, and it covers any market with time series of prices. Strategies developed for this model can be implemented

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.39). Therefore, pricing is difficult for the general case. Some useful theorems from continuous time setting are not valid for the general discrete time model. Many problems are still unsolved for discrete time market models (including pricing problems and optimal portfolio selection problems). © 2007 Nikolai Dokuchaev Mathematical Finance 46 • The complete Cox-Ross-

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for the constantly rebalanced portfolio. Solve Problems 3.11 and 3.12. Problem 3.63 (Make your own model). Introduce a reasonable version of the discrete time market model that takes into account transaction costs (a brokerage fee), and derive the equation for the wealth evolution for self-financing strategy here. (Hint

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: transaction costs may be per transaction, or may be proportional to the size of transaction or may be of a mixed type.) Discrete time market: arbitrage and completeness Solve Problem 3.29. Problem 3.64 Prove that an equivalent risk-neutral probability measure does not exist for Problem 3

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independent, and let there exist for all t. Explain in which cases the market is arbitrage-free, allows arbitrage, complete or incomplete: © 2007 Nikolai Dokuchaev Discrete Time Market Models 47 (i) ρ=1, a=b=0.1, c=−0.05, (ii) ρ=1.1, a=b=0.15, c=−0.05, (

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processes (constructed similarly to the generalized deterministic functions such as the delta function). This generalized process dw/dt is a continuous time analogue of the discrete time white noise. This approach is used mainly for the case of linear equations with constant b in control system theory. 4.4.4 Examples of

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t)) for a.e. t a.s. and (β(1)(t), 5.6 Arbitrage possibilities and arbitrage-free markets Similarly to the case of the discrete time market, we define arbitrage as a possibility of a risk-free positive gain. The formal definition is as follows. Definition 5.29 Let T>0

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option seller or for the option buyer. Problem 5.47 Is it possible to prove analogs of Propositions 5.44 and 5.45 for the discrete time market model? 5.9.3 Option pricing for a complete market For a complete market, Definition 5.43 leads to replication. Theorem 5.48

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also arise for the case of random r. 5.11.2 Pricing for an incomplete market Mean-variance hedging Similarly to the case of the discrete time market, Definition 5.43 leads to superreplication for incomplete markets. Clearly, it is not always meaningful. Therefore, there is another popular approach for an

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the risk-free interest rate r is non-random and constant, then the corresponding market model is equivalent to the Cox-Ross-Rubinstein discrete time market model for the discrete time prices The parameters p, u, and d are chosen to match the stock price expectation and volatility. © 2007 Nikolai Dokuchaev American Options and

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1.2 Choice of u, d, p for the case of constant r and σ The most popular binomial tree represents the Cox-Ross-Rubinstein discrete time market model described in Section 3.9. It is known as the Cox-Ross-Rubinstein binomial tree. Let us assume that the continuous time stock

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tree. It is natural to assume that © 2007 Nikolai Dokuchaev American Options and Binomial Trees where 113 are random variables such that We want the discrete time approximation to be close in some sense to the original continuous stock price. There are three unknown parameters (u, d, p), hence three restrictions can

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American option and non-arbitrage prices We describe the American option in the continuous time setting. However, all definitions given below are valid in the discrete time setting as well (see Definition 3.43). Up to the end of this chapter, we assume that the risk-free rate r≥0 is a

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not stationary. In fact, stationarity in a wide sense is sufficient for many applications. Definition 8.3 Let ξt, t=0, 1, 2,…, be a discrete time random process such that ξt are mutually independent and have the same distribution, and Eξt≡0. Then the process ξt is said to be a

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hand, a white noise can be used as a basic construction block for the modelling of random processes. For instance, one can model a stationary discrete time process with given characteristics as the output of an autoregression, where a white noise is used as the input. In contrast to the white noise

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dimensional equation © 2007 Nikolai Dokuchaev Review of Statistical Estimation 141 yt=β0+βxt+εt, t=1, 2,…. (8.1) Here yt and xt represent observable discrete time processes; yt is called the regressand, or dependent variable, xt is called the regressor, or explanatory variable, εt is an unobserved and are parameters that

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1, then {q(tk)} are mutually independent, and R(tk) = βR(tk−1)+β0+εk, where and and is a random discrete time process that represents stochastic changes. It is a discrete time white noise process, Eεk=0, and In the following chapters, we shall study estimation of market parameters from the observation of

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{R(tk)}. Note that the process R(t) can be thought of as the limit of the discrete time process R(tk) as the time interval h becomes very small. Remark 9.3 The stock price in the mean-reverting model has log-normal

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© 2007 Nikolai Dokuchaev Estimation of Models for Stock Prices 173 where β=1 and β0=µh, and {εk} is a random discrete time process that represents stochastic changes. It is a discrete time white noise process, εk=q(tk), Eεk=0, and The LS estimator By Proposition 8.12 and Definition 8.19, it

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Z and h=tk−tk–1, then R(tk)=βR(tk−1)+β0+εk, where and and {εk} is a random discrete time process that represents stochastic changes. It is a discrete time white noise process, Eεk=0, and (see (9.8)). Then estimates and can be found by LS or ML methods, as

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(1979). Option pricing: a simplified approach. J. of Financial Economics 7, 229–263. [5] Föllmer, H. and Schied, A. (2002). Stochastic Finance: An Introduction in Discrete Time 2. De Gruyter Studies in Mathematics, Berlin. [6] Gujarati, D. (1995). Basic Econometrics. New York: McGraw-Hill. [7] Higham, D.J. (2004). An Introduction to

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New York: John Wiley & Sons. [14] Neftci, S. (1996). Mathematics of Financial Derivatives. New York: Academic Press. [15] Pliska, S. (1997). Introduction to Mathematical Finance: Discrete Time Models. Oxford: Blackwell. [16] Shiryaev, A.N. (1999). Essentials of Stochastic Finance. Facts, Models, Theory. Hackensack, NJ: World Scientific. [17] Söderlind, P. (2003). Lecture Notes

Tools for Computational Finance

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

of the (S, t) half strip is semi-discretized in that it is replaced by parallel straight lines with distance ∆t apart, leading to a discrete-time model. The next step of discretization replaces the continuous values Si along the parallel t = ti by discrete values Sji , for all i and appropriate

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of Wtj and (Wtj+1 − Wtj ), but not of Wtj+1 and (Wtj+1 − Wtj ). Wiener processes are examples of martingales — there is no drift. Discrete-Time Model Let ∆t > 0 be a constant time increment. For the discrete instances tj := j∆t the value Wt can be written as a sum

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number of units of the asset held in a portfolio at time t. We start with the simplifying assumption that trading is only possible at discrete time instances tj , which deﬁne a partition of the interval 0 ≤ t ≤ T . Then the trading strategy b is piecewise constant, b(t) = b(tj−1

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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 1.3: References on speciﬁc numerical methods are given where appropriate. As computational ﬁnance is concerned, most quotations

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. A Bermudan option restricts early exercise to speciﬁed dates during its life. In our context, the “dates” are created artiﬁcially by a ﬁnite set of discrete time instances ti . This resembles the time discretization of the binomial method of Section 1.4, see Figure 1.8. In this semi-discretized setting the

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). Decide whether an exact calculation of the hitting point makes sense. (Run experiments comparing such a strategy to implementing the hitting time restricted to the discrete time grid.) Think about how to implement upper bounds. Chapter 4 Standard Methods for Standard Options We now enter the part of the book that is

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.1 The Payoﬀ There are several ways how an average of past values of St can be formed. If the price St is observed at discrete time instances ti , say equidistantly with time interval h := T /n, one obtains a times series St1 , St2 , . . . , Stn . An obvious choice of average is the

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Discrete Monitoring Instead of deﬁning a continuous averaging as in (6.3), a realistic scenario is to assume that the average is monitored only at discrete time instances t1 , t2 , . . . , tM . These time instances are not to be confused with the grid times of the numerical discretization. The discretely sampled arithmetic average

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) riskless. As will be shown next, there is a simple analytic formula for ∆ in case of European options. (In reality, hedging must be done in discrete time.) The Black-Scholes equation has a closed-form solution. For a European call with continuous dividend yield δ as in (4.1) (in Section 4

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] [BBG97] [BTT00] [Br91] [BrS77] [BrS02] [Br94] [BrD97] [BrG97] [BrG04] [BH98] [BuJ92] [CaMO97] [CaF95] [CaM99] [Cash84] [CDG00] G.I. Bischi, V. Valori: Nonlinear eﬀects in a discrete-time dynamic model of a stock market. Chaos, Solitons and Fractals 11 (2000) 21032121. F. Black, M. Scholes: The pricing of options and corporate liabilities. J

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). [Fisz63] M. Fisz: Probability Theory and Mathematical Statistics. John Wiley, New York (1963). [FöS02] H. Föllmer, A. Schied: Stochastic Finance: An Introduction to Discrete Time. de Gruyter, Berlin (2002). [FV02] P.A. Forsyth, K.R. Vetzal: Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comp

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York (1983). E. Platen: An introduction to numerical methods for stochastic diﬀerential equations. Acta Numerica (1999) 197-246. S.R. Pliska: Introduction to Mathematical Finance. Discrete Time Models. Blackwell, Malden (1997). D. Pooley, P.A. Forsyth, K. Vetzal, R.B. Simpson: Unstructured meshing for two asset barrier options. Appl. Mathematical Finance 7

Mathematical Finance: Theory, Modeling, Implementation

by Christian Fries · 9 Sep 2007

four Prices . . . 6.2.2. Example (2): Interpolation of two Prices . . . . 6.3. Arbitrage Free Interpolation of European Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Hedging in Continuous and Discrete Time and the Greeks English version of this part not available 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Deriving the Replications Strategy from Pricing Theory . . . . . . . . 7.2.1

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PDE under a BlackScholes Model . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Greeks einer europäischen Call-Option unter dem BlackScholes Modell . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Hedging in Discrete Time: Delta- and Delta-Gamma-Hedging . . . . 7.4.1. Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3. Delta-Gamma Hedging . . . . . . . . . . . . . . . . . . . . . 12 This work is

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Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ Contents 7.4.4. Vega Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Hedging in discrete Time: Minimizing the Residual Error (BouchaudSornette Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. Minimizing the Residual Error at Maturity T . . . . . . . . . 7.5.2. Minimizing the Residual Error in each

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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 1 < . . . < T N only (e.g. this will be the case when we consider the implementation). If we need only the

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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 the increments ∆W(T

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particle movement: First we assume that the particle may lose energy over time 11 In a (one dimensional) random walk a particle changes position at discrete time steps by a (constant) distance (say 1) in either direction with equal probability. In other words we have binomial distributed Yi in Theorem 25. 40

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7 Hedging in Continuous and Discrete Time and the Greeks An English version of this part is currently not available. What follows it the German version. 7.1. Introduction Die vorstehende Bewertungstheorie

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Beide Modellierungen (kontinuierliche oder diskrete Replikation) entsprechen nicht der Realität: Zum einen ist es nicht möglich Finanzprodukte kontinuierlich und in

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS 7.2.1. Deriving the Replication Strategy under the Assumption of a Locally Riskless Product Der obige Ansatz V(t) = V(t

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Dies und dS i dS j = 0 falls i = 0 oder j = 0 eingesetzt in (7.4) liefert n ∂V(t)n

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Für die Funktion V(t, n, s) = sΦ(d+ ) − n K Φ(d− ) N(T ) mit σ̄2 (T − t) 1

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Die Definition von Rho wird lediglich für Aktienderivate verwendet. Für Zinsderivate, wie wir sie in den folgenden Kapitel betrachten werden

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Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING 7.4. Hedging in Discrete Time: Delta- and Delta-Gamma-Hedging Wird ein Delta-Hedge kontinuierlich angewendet, so liefert er eine exakte Replikation. Das Portfolio ist gegenu

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS zum Beispiel durch die Wahl von φ0 1 φ0 (tk ) = φ0 (tk−1 ) − S 0 (tk )  n  X  ·  φi (tk ) − φi (tk

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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING 7.4.2. Error Propagation Die Wahl von φ0 bestimmt sich zum Zeitpunkt t = 0 aus der Anfangsbedingung Π(0

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Wir benutzen diese Wahl um an diskreten Zeitpunkten T i , i = 0, 1, . . . in einem Replikationsportfolio zu handeln. Die Größe der

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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING Hedge Portfolio versus Payoff 1,0 Wöchentlicher Delta Hedge Wöchentlicher Delta Hedge mit falscher Volatilität Portfolio Value 0,8 0

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Der durch die nur lineare Approximation des Derivates durch das Replikationsportfolio gemachte Fehler lässt sich Reduzieren, in dem Produkte in das

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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING so bleibt ein Restrisiko |∆V(t) − ∆Π(t)| = O(|∆t|2 , |∆t∆S i |, |∆S i |3 ). Um das Gamma

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Hedge Portfolio versus Payoff 1,0 Monatlicher Delta Hedge Monatlicher Delta-Gamma Hedge Portfolio Value 0,8 0,5 0,2 0

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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING Hedge Portfolio versus Payoff 1,0 Portfolio Value 0,8 Wöchentlicher Delta Hedge mit falscher Zinsrate Wöchentlicher Delta-Gamma Hedge

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Hedge Portfolio versus Payoff 1,0 Statischer Delta Hedge Statischer Delta-Gamma Hedge Portfolio Value 0,8 0,5 0,2 0

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Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.5. HEDGING IN DISCRETE TIME: MINIMIZING THE RESIDUAL ERROR (BOUCHAUD-SORNETTE METHOD) 7.5. Hedging in discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method) Der Delta-Hedge überträgt die optimale Handelstrategie für kontinuierliches Handeln

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Proof: Der Beweis ist elementar. Mit E(X · Y|C) = Y · E(X|C) und E(Y 2 |C) = Y 2 folgt

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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.5. HEDGING IN DISCRETE TIME: MINIMIZING THE RESIDUAL ERROR (BOUCHAUD-SORNETTE METHOD) gilt – man beachte, dass φ0 in der letzten Summe nicht mehr vorkommt. Somit lässt sich mit φ0

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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Handelsstrategie entsteht aus der Forderung im Zeitschritt T k das Residualrisiko zur Zeit T k+1 durch Wahl des Portfolios φ(T

Artificial Intelligence: A Modern Approach

by Stuart Russell and Peter Norvig · 14 Jul 2019 · 2,466pp · 668,761 words

sense of a spoken or written sequence of words. How can dynamic situations like these be modeled? 14.1.1States and observations This chapter discusses discrete-time models, in which the world is viewed as a series of snapshots or time slices.1 We’ll just number the time slices 0, 1

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. On the other hand, in modeling continental drift over geological time, an interval of a million years might be fine. Each time slice in a discrete-time probability model contains a set of random variables, some observable and some not. For simplicity, we will assume that the same subset of variables is

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al. (2008). Significant classified work on filtering was done during World War II by Wiener (1942) for continuous-time processes and by Kolmogorov (1941) for discrete-time processes. Although this work led to important technological developments over the next 20 years, its use of a frequency-domain representation made many calculations quite

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et al., 2006). 1Uncertainty over continuous time can be modeled by stochastic differential equations (SDEs). The models studied in this chapter can be viewed as discrete-time approximations to SDEs. 2The term “filtering” refers to the roots of this problem in early work on signal processing, where the problem is to filter

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. Bertsekas, D. and Tsitsiklis, J. N. (2008). Introduction to Probability (2nd edition). Athena Scientific. Bertsekas, D. and Shreve, S. E. (2007). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific. Bertsimas, D., Delarue, A., and Martin, S. (2019). Optimizing schools’ start time and bus routes. PNAS, 116 13, 5943–5948. Bertsimas, D

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the magnetospheric multiscale mission and SEXTANT pulsar navigation demonstration. Tech. rep., NASA Goddard Space Flight Center. Witten, I. H. (1977). An adaptive optimal controller for discrete-time Markov environments. Information and Control, 34, 286–295. Witten, I. H. and Bell, T. C. (1991). The zero-frequency problem: Estimating the probabilities of novel

Market Risk Analysis, Quantitative Methods in Finance

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

.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 Return on a Linear Portfolio I.1.4.9 Sources of Returns I.1.5 Functions of

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130 130 131 133 x Contents I.3.7 Stochastic Processes in Discrete and Continuous Time I.3.7.1 Stationary and Integrated Processes in Discrete Time I.3.7.2 Mean Reverting Processes and Random Walks in Continuous Time I.3.7.3 Stochastic Models for Asset Prices and Returns I

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particular. They are usually written at a higher mathematical level than the present text but have fewer numerical and empirical examples. • Those which focus on discrete time mathematics, including statistics, linear algebra and linear regression. Among these books are Watsham and Parramore (1996) and Teall and Hasan (2002), which are written at

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a lower mathematical level and are less comprehensive than the present text. Continuous time finance and discrete time finance are subjects that have evolved separately, even though they approach similar problems. As a result two different types of notation are used for the

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in two different ways. One of the features that makes this book so different from many others is that I focus on both continuous and discrete time finance, and explain how the two areas meet. Although the four volumes of Market Risk Analysis are very much interlinked, each book is self-contained

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matrices, and their eigenvectors and eigenvalues in particular, since these lay the foundations for principal component analysis (PCA). PCA is very widely used, mainly in discrete time finance, and particularly to orthogonalize and reduce the dimensions of the risk factor space for interest rate sensitive instruments and options portfolios. A case study

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financial risk management. The sections on statistical inference and maximum likelihood lay the foundations for Chapter 4. Finally, we focus on the continuous time and discrete time statistical models for the evolution of financial asset prices and returns, which are further developed in Volume III. xxvi Preface Much of the material in

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in continuous time, and assuming we know the value of an instrument at any point in time. But in reality we only accrue interest over discrete time intervals and we only observe prices at discrete points in time. The definition a return requires the distinction between discrete and continuous time modelling. This

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is a source of confusion for many students. The continuous and discrete time approaches to financial models were developed independently, and as a result different 14 The P&L on an asset is the change in value, and

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denote the value of an investment, e.g. the price of a single asset or the value of a portfolio at time t. But in discrete time the notations Pt or pt are standard. In each case we mean that the asset price or portfolio value is a function of time. And

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price or value at time t is a random variable. The use of the notation in discrete and continuous time must also be distinguished. In discrete time it denotes the first difference operator, i.e. the difference between two consecutive values, but in continuous time it is used to denote an increment

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holdings imply variable weights and constant weights imply variable holdings over time. Basic Calculus for Finance I.1.4.3 19 Profit and Loss Discrete Time Consider first the discrete time case, letting Pt denote the value of a portfolio at time t. In a long-short portfolio this can be positive, negative or

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of $1 million some years ago. For this reason we often prefer to analyse returns, if possible. I.1.4.4 Percentage and Log Returns Discrete Time We shall phrase our discussion in terms of backward-looking returns but the main results are the same for forward-looking returns. Suppose that: • • the

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dSt/St has a normal distribution with mean dt and variance 2 dt.20 I.1.4.6 Discrete and Continuous Compounding in Discrete Time Another way of expressing the return (I.1.27) is: 1 + Rt = Pt Pt−1 (I.1.33) On the left-hand side we have

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discretely compounded return. Other examples of discrete and continuous compounding are given in Section III.1.2. I.1.4.7 Period Log Returns in Discrete Time Using (I.1.9) we can write the log return over one period as Pt − Pt−1 P − Pt−1 Pt = ln +1 ≈ t ln

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with the profit and loss instead, which can always be defined. Returns and P&L can be defined in both discrete and continuous time. In discrete time we distinguish between discrete compounding and continuous compounding of returns. The continuously compounded return is called the log return and in

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discrete time the log return is approximately equal to the return over very short time intervals. Profit and loss, returns and log returns can be forwardlooking, as

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translation between discrete and continuous time, and the relationship between the continuous time representation and the discrete time representation of a stochastic process, is very important indeed. The theory of finance requires an understanding of both discrete time and continuous time stochastic processes. Section I.3.8 summarizes and concludes. Some prior knowledge of

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are continuous, indeed they are often assumed to be normal, but the sequence may be over continuous or discrete time. That is, we consider continuous state processes in both continuous and discrete time. • The study of discrete time stochastic processes is called time series analysis. In the time domain the simplest time series models are based

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follows that their returns follow a stationary process. I.3.7.1 Stationary and Integrated Processes in Discrete Time This section introduces the time series models that are used to model stationary and integrated processes in discrete time. Then we explain why prices of financial assets are usually integrated of order 1 and the

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associated returns are therefore stationary. A more formal and detailed treatment of these concepts is given in Section II.5.2. A discrete time stochastic process is a sequence of random variables from the same distribution family and we denote the process, indexed by time t, as {X1 , X2

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= 1 the AR(1) model (I.3.138) becomes Xt = + Xt−1 + $t $t ∼ iid 0 2 (I.3.140) This is the discrete time random walk model with drift and volatility . When the drift is positive the process has an upward trend and when it is negative there is

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) 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 dZt is a standard normal independent process: i.e. as we move from continuous to

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∼ NID0 1 Probability and Statistics 137 Also the increment dX t in continuous time becomes a first difference Xt = Xt − Xt−1 in discrete time.56 Using these discrete time equivalents gives the following discretization of (I.3.141): Xt = + $t where $t = Zt so $t ∼ NID 0 2 But this is the

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same as Xt = + Xt−1 + $t , which is the discrete time random walk model (I.3.140). So (I.3.141) is non-stationary. However a stationary continuous time process can be defined by introducing a

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rate of mean reversion and the parameter is called the long term value of X. In Section II.5.3.7 we prove that the discrete time version of (I.3.142) is a stationary AR(1) model. I.3.7.3 Stochastic Models for Asset Prices and Returns Time series of

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for financial asset prices, the 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

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) looks like. The change in the log price is the log return, so using the standard discrete time notation Pt for a price at time t we have d ln St → ln Pt Hence the discrete time equivalent of (I.3.145) is ln Pt = + $t where = prices, i.e. − 1 2 2

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$t ∼ NID 0 2 (I.3.146) . This is equivalent to a discrete time 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

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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 and the Poisson Process A Poisson process, introduced in Section I.3.3.2, is a stochastic process governing the

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likelihood estimation is used to estimate the parameters of regression models. Finally, we provided an informal introduction to stochastic processes in discrete and continuous time. Discrete time processes, which are also called time series, are often represented by regression models. Here we introduced the simplest first order autoregression model for a stationary

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test 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

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and continuous time models. This lays the foundation for continuous time option pricing models, which are covered in Chapters III.3 and III.5, and discrete time series analysis, which is the subject of Chapter II.5. I.4 Introduction to Linear Regression I.4.1 INTRODUCTION A regression model is a

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of the book by Campbell et al. (1997) and the many references therein. More generally, a considerable body of financial econometrics research has focused on discrete time models for the theory of asset pricing which depends on the assumptions of no arbitrage, single agent optimality and market equilibrium. Indeed, two out of

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only a numerical method will provide a solution. Note that we tend to use the terminology estimation when we are finding the parameters of a discrete time model and calibration when we are finding the parameters of a continuous time model. For instance, by finding the best fit to historical time series

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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 time step t. No arbitrage pricing requires the underlying asset to return the risk free rate, r. Therefore, the log return ln St+t /St will

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our question, we seek p u and d so that (I.5.37) holds. Taking the expectation and variance from the binomial lattice over a discrete time step t, we have ESt+t = St pu + 1 − p d (I.5.38) VSt+t = S2t+t pu2 + 1 − p d2 − pu

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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 is the first

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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 19–20 random walk model 135 stationary/integrated process 134–6

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–1 Finite difference approximation 186, 206–10, 223 first/second order 206–7 the Greeks 207–8 partial differential equations 208–10 First difference operator, discrete time 17 First order autocorrelation 178 Forecasting 182, 254 Forward difference operator, returns 19, 22 Index Fréchet distribution 103 F test 127 FTSE 100 index 204

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118–20 variance 126–7 Inflexion points 14, 35 Information matrix 133, 203 Information ratio 257, 259 Instability, finite difference approximation 209–10 Integrated process, discrete time 134–6 Integration 3, 15–16, 35 Intensity, Poisson distribution 88 Interest rate 34, 171–3 Interest rate sensitivity 34 Interpolation 186, 193–200, 223

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vector 39 Orthonormal matrix 53 Orthonormal vector 53 Out-of-sample testing 183 P&L (profit and loss) 3, 19 backtesting 183 continuous time 19 discrete time 19 financial returns 16, 19 volatility 57–8 Pairs trading 183 Parabola 4 Parameter notation 79–80 Pareto distribution 101, 103–5 Parsimonious regression model

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–4 multivariate distributions 107–18 stochastic processes 134–9 univariate distribution 85–107 Profit and loss (P&L) 3, 19 backtesting 183 continuous time 19 discrete time 19 financial returns 16, 19 volatility 57–8 Prompt futures 194 Pseudo-random numbers 217 Put option 1, 212–13, 215–16 Quadratic convergence 188

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165–7 Returns 2–3, 16–26 absolute 58 active 92, 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

Monte Carlo Simulation and Finance

by Don L. McLeish · 1 Apr 2005

{− p(1 − p)}Φ(σ−1 ln(η/l) + σ(p − )) 2 2 p where Φ is the standard normal cumulative distribution function. Application: A Discrete Time Black-Scholes Model Suppose that a stock price St , t = 1, 2, 3, ... is generated from an independent sequence of returns Z1 , Z2 over non

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by simulation. One of the simplest methods of simulating such a process is motivated through a crude interpretation of the above equation in terms of discrete time steps, that is that a small increment Xt+h − Xt in the process is approximately normally distributed with mean given by a(Xt , t)hand

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generate these increments sequentially, beginning with an assumed value for X0 , and then adding to obtain an approximation to the value of the process at discrete times t = 0, h, 2h, 3h, . . .. Between these discrete points, we can linearly interpolate the values. Approximating the process by assuming that the conditional distribution of

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examples are simulations of processes such as networks or queues. Simulation models in which the process is characterized by a state, with changes only at discrete time points are DES. In modeling an inventory system, for example, the arrival of a batch of raw materials can be considered as an event which

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precipitates a sudden change in the state of the system, followed by a demand some discrete time later when the state of the system changes again. A system driven by diﬀerential equations in continuous time is an example of a DES because

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probability in the case of a discrete sample space, for example a simple random walk that jumps up or down by a fixed amount in discrete time steps. This result for Brownian motion obtains if we take a limit over a sequence of simple random walks approaching a Brownian motion process.) Note

Data Mining: Concepts and Techniques: Concepts and Techniques

by Jiawei Han, Micheline Kamber and Jian Pei · 21 Jun 2011

if a certain disease is geographically colocated with certain objects like a well, a hospital, or a river. In time-series data analysis, researchers have discretized time-series values into multiple intervals (or levels) so that tiny fluctuations and value differences can be ignored. The data can then be summarized into sequential

The Concepts and Practice of Mathematical Finance

by Mark S. Joshi · 24 Dec 2003

, we step up another mathematical gear and this is the most mathematically demanding chapter. We introduce the concept of a martingale in both continuous and discrete time, and use martingales to examine the concept of riskneutral pricing. We commence by showing that option prices determine synthetic probabilities in the context of a

Data Mining: Concepts, Models, Methods, and Algorithms

by Mehmed Kantardzić · 2 Jan 2003 · 721pp · 197,134 words

, shown in Figure 7.1, constituting the only computational node of the network. Neuron k is driven by input vector X(n), where n denotes discrete time, or, more precisely, the time step of the iterative process involved in adjusting the input weights wki. Every data sample for ANN training (learning) consists

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as a filter to perform three basic information-processing tasks: 1. Filtering. This task refers to the extraction of information about a particular quantity at discrete time n by using data measured up to and including time n. 2. Smoothing. This task differs from filtering in that data need not be available

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time n can also be used to obtain the required information. This means that in smoothing there is a delay in producing the result at discrete time n. 3. Prediction. The task of prediction is to forecast data in the future. The aim is to derive information about what the quantity of

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a word C″ is represented as C″ = (bcccccbaaaaabbccccbb). The main advantage of the SAX method is that 100 different discrete numerical values in an initial discrete time series C is first reduced to 20 different (average) values using PAA, and then they are transformed into only three different categorical values using SAX

Why Stock Markets Crash: Critical Events in Complex Financial Systems

by Didier Sornette · 18 Nov 2002 · 442pp · 39,064 words

a ﬁnite-time singularity. Suppose, for instance, that the growth rate of the hazard rate doubles when the hazard rate doubles. For simplicity, we consider discrete-time intervals as follows. Starting with a hazard rate of 1 per unit time, we assume it grows at a constant rate of 1% per day

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with the size of economic factors. Suppose, for instance, that the growth rate of the population doubles when the population doubles. For simplicity, we consider discrete time 2 05 0: the end of t h e g r o w t h e r a? 365 intervals as follows. Starting with a

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