stochastic process

The Volatility Smile

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

, interest rates, or whatever 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

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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 stock price turns out to be very difficult. Such attempts typically involve more complex stochastic differential equations with

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up with a value estimate. If, however, you can trade options, and if you know (or, rather, assume that you know) the stochastic process for volatility in addition to the stochastic process for stock prices, then you can hedge an option’s exposure to volatility with another option. By doing this, you can derive

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, one must hedge it at every instant; to hedge it, one must hedge against the instantaneous change in value of the option caused by the stochastic process driving the stock price; but the risk-neutral distribution at expiration tells you nothing about the evolution of the stock price on its way to

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an option, will converge to the BSM formula. We will use these binomial processes, and generalizations of them, as a basis for modeling more general stochastic processes that can perhaps explain the smile. THE BINOMIAL MODEL FOR OPTIONS VALUATION In this section we explain how we can use the binomial model to

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that the expected growth rate of the stock, 𝜇, appears nowhere in the equation. You can derive many of the continuous-time partial differential equations for stochastic processes (the mean time to reach a barrier, for example) as limits obtained from the binomial framework in this way. EXTENDING THE BLACK-SCHOLES-MERTON MODEL

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F = Se(r−b)dt (14.4) which is also the forward price of the stock. Figure 14.1 shows a binomial approximation to the stochastic process over time dt. In our binomial approximation, the forward price is simply the probability-weighted average of the two possible stock prices Su and Sd

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. The local volatility model is therefore in fact a stochastic volatility model in which both the underlier and its volatility are governed by the same stochastic process. But volatility can also change for other reasons, independent of changes in the underlier. In the next chapter, we will formally introduce stochastic volatility models

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addition to shares of stock you can also use other options to hedge the stochastic volatility of the target option, and if you know the stochastic process for option prices (i.e., volatility) as well as stock prices, then you can hedge your option’s exposure to volatility with another option, and

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formula for the option’s value, which we will do in the following chapter. In reality, we understand the stochastic process for option prices and volatility even less well than we understand the stochastic process for stock prices (which is to say, not very well at all). In the next chapter we will nevertheless

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. “A Theory of the Term Structure of Interest Rates.” Econometrica 53:385–407. Cox, John, and Stephen Ross. 1976. “The Valuation of Options for Alternative Stochastic Processes.” Journal of Financial Economics 3 (1–2): 145–166. Cox, John, Stephen Ross, and Mark Rubinstein. 1979. “Option Pricing: A Simplified Approach.” Journal of Financial

Mathematics for Economics and Finance

by Michael Harrison and Patrick Waldron  · 19 Apr 2011  · 153pp  · 12,501 words

that each consumption plan is a random vector. Let us review the associated concepts from basic probability theory: probability space; random variables and vectors; and stochastic processes. Let Ω denote the set of all possible states of the world, called the sample space. A collection of states of the world, A ⊆ Ω

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vector is just a vector of random variables. It can also be thought of as a vector-valued function on the sample space Ω. A stochastic process is a collection of random variables or random vectors indexed by time, e.g. {x̃t : t ∈ T } or just {x̃t } if the time

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, we will assume that the index set consists of just a finite number of times i.e. that we are dealing with discrete time stochastic processes. Then a stochastic process whose elements are N -dimensional random vectors is equivalent to an N |T |-dimensional random vector. The (joint) c.d.f. of a random

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vector or stochastic process is the natural extension of the one-dimensional concept. Random variables can be discrete, continuous or mixed. The expectation (mean, average) of a discrete r

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are denoted by the set Ω. So a consumption plan or lottery is just a collection of |T | k-dimensional random vectors, i.e. a stochastic process. Again to distinguish the certainty and uncertainty cases, we let L denote the collection of lotteries under consideration; X will now denote the set of

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application of Siegel’s n o n oparadox is in the case of currency exchange rates. In that case, F̃t and S̃t are stochastic processes representing forward and spot exhange rates respectively. It seems reasonable to assume that forward exchange rates are good predictors of spot exchange rates in the

Analysis of Financial Time Series

by Ruey S. Tsay  · 14 Oct 2001

Duration Models, 216 6. Continuous-Time Models and Their Applications 6.1 6.2 6.3 6.4 6.5 Options, 222 Some Continuous-Time Stochastic Processes, 222 Ito’s Lemma, 226 Distributions of Stock Prices and Log Returns, 231 Derivation of Black–Scholes Differential Equation, 232 221 ix CONTENTS 6.6

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moments of xt are time-invariant). The ARMA process of Chapter 2 is linear because it has an MA representation in Eq. (4.1). Any stochastic process that does not satisfy the condition of Eq. (4.1) is said to be nonlinear. The prior definition of nonlinearity is for purely stochastic time

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, Inc. ISBN: 0-471-41544-8 CHAPTER 6 Continuous-Time Models and Their Applications Price of a financial asset evolves over time and forms a stochastic process, which is a statistical term used to describe the evolution of a random variable over time. The observed prices are a realization of the underlying

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stochastic process. The theory of stochastic process is the basis on which the observed prices are analyzed and statistical inference is made. There are two types of stochastic process for modeling the price of an asset. The first type is called the

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discrete-time stochastic process, in which the price changes at discrete time points. All the processes discussed in the previous chapters

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belong to this category. For example, the daily closing price of IBM stock on the New York Stock Exchange forms a discrete-time stochastic process. Here the price changes only at the closing of a trading day. Price movements within a trading day are not necessarily relevant to the observed

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daily price. The second type of stochastic process is the continuous-time process, in which the price changes continuously, even though the price is only observed at discrete time points. One can think

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number, whereas a discrete price can only assume a countable number of possible values. Assume that the price of an asset is a continuous-time stochastic process. If the price is a continuous random variable, then we have a continuous-time continuous process. If the price itself is discrete, then we have

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Chapter 5 is an example of discrete-time discrete process. In this chapter, we treat the price of an asset as a continuous-time continuous stochastic process. Our goal is to introduce the statistical theory and tools needed to model financial assets and to price options. We begin the chapter with some

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diversifiability of jump risk or defining a notion of risk and choosing a price and a hedge that minimize this risk. For basic applications of stochastic processes in derivative pricing, see Cox and Rubinstein (1985) and Hull (1997). 6.1 OPTIONS A stock option is a financial contract that gives the holder

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, only in-the-money options are exercised in practice. For more description on options, see Hull (1997). 6.2 SOME CONTINUOUS-TIME STOCHASTIC PROCESSES In mathematical statistics, a continuous-time continuous stochastic process is defined on a probability space (, F, P), where  is a nonempty space, F is a σ -field consisting of subsets

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numbers. For a given η, {x(η, t)} is a time series with values depending on the time t. For simplicity, we 223 STOCHASTIC PROCESSES write a continuous-time stochastic process as {xt } with the understanding that, for a given t, xt is a random variable. In the literature, some authors use x(t

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) instead of xt to emphasize that t is continuous. However, we use the same notation xt , but call it a continuous-time stochastic process. 6.2.1 The Wiener Process In a discrete-time econometric model, we assume that the shocks form a white noise process, which is not

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use a simple approach that focuses on the small change wt = wt+ t − wt associated with a small increment t in time. A continuous-time stochastic process {wt } is a Wiener process if it satisfies 1. 2. √ t, where is a standard normal random variable, and wt = wt is independent of w

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t ∈ [0, ∞). Remark: A formal definition of a Brownian motion wt on a probability space (, F, P) is that it is a real-valued, continuous stochastic process for t ≥ 0 with independent and stationary increments. In other words, wt satisfies 1. continuity: the map from t to wt is continuous almost surely

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with respect to the probability measure P; 2. independent increments: if s ≤ t, wt − ws is independent of wv for all v ≤ s; 225 STOCHASTIC PROCESSES 3. stationary increments: if s ≤ t, wt − ws and wt−s − w0 have the same probability distribution. It can be shown that the probability distribution

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. This is the purpose of discussing Ito’s calculus in the next section. 6.2.2 Generalized Wiener Processes The Wiener process is a special stochastic process with zero drift and variance proportional to the length of time interval. This means that the rate of change in expectation is zero and the

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rate of change in variance is 1. In practice, the mean and variance of a stochastic process can evolve over time in a more complicated manner. Hence, further generalization of stochastic process is needed. To this end, we consider the generalized Wiener process in which the expectation has a drift

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a generalized Wiener process are timeinvariant. If one further extends the model by allowing µ and σ to be functions 226 CONTINUOUS - TIME MODELS of the stochastic process xt , then we have an Ito’s process. Specifically, a process xt is an Ito’s process if it satisfies d xt = µ(xt , t) dt

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possible confusion in the future, we restate the lemma as follows. 228 CONTINUOUS - TIME MODELS Ito’s Lemma Assume that xt is a continuous-time stochastic process satisfying d xt = µ(xt , t) dt + σ (xt , t) dwt , where wt is a Wiener process. Furthermore, G(xt , t) is a differentiable function of

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these securities are driven by multiple factors, the price of the derivative is a function of several stochastic processes. The two-factor model for the term structure of interest rate is an example of two stochastic processes. In this section, we briefly discuss the extension of Ito’s lemma to the case of several

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stochastic processes. 241 Value of a call 30 35 0 5 10 15 20 25 AN EXTENSION OF ITO ’ S

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CONTINUOUS - TIME MODELS Consider a k-dimensional continuous-time process xt = (x1t , . . . , xkt ) , where k is a positive integer and xit is a continuous-time stochastic process satisfying d xit = µi (xt )dt + σi (xt ) dwit , i = 1, . . . , k, (6.21) where wit is a Wiener process. It is understood that the

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i and j defined by wit = i t and w jt = j t. Assume that G t = G(xt , t) be a function of the stochastic processes xit and time t. The Taylor expansion gives Gt = k
∂G t i=1 + xit + ∂ xit k ∂2Gt 1
2 i=1 ∂ xit ∂t ∂G

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=1 + k
∂G t i=1 ∂ xit σi (xt ) dwit . (6.24) This is a generalization of Ito’s lemma to the case of multiple stochastic processes. 6.8 STOCHASTIC INTEGRAL We briefly discuss stochastic integration so that the price of an asset can be obtained under the assumption that it follows

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integration of a deterministic function, integration is the opposite side of differentiation so that  t d xs = xt − x0 0 continues to hold for a stochastic process xt . In particular, for the Wiener process wt , t t we have 0 dws = wt because w0 = 0. Next, consider the integration 0 ws dws

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. We discuss briefly some statistics that are useful for checking these three features. These statistics are based on some basic statistical theory concerning distributions and stochastic processes. Exceedance Rate A fundamental property of univariate Poisson processes is that the time durations between two consecutive events are independent and exponentially distributed. To exploit

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., Resnick, I. S., Rootzén, and De Vries, C. G. (1989), “Extremal behavior of solutions to a stochastic difference equation with applications to ARCH process,” Stochastic Processes and Their Applications, 32, 213–224. Dekkers, A. L. M., and De Haan, L. (1989), “On the estimation of extreme value index and large quantile

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to most, if not all, of the studies in financial econometrics. We begin the chapter by reviewing the concept of a Markov process. Consider a stochastic process {X t }, where each X t assumes a value in the space Θ. The process {X t } is a Markov process if it has the

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conditional distribution function satisfies P(X h | X s , s ≤ t) = P(X h | X t ), h > t. If {X t } is a discrete-time stochastic process, then the prior property becomes P(X h | X t , X t−1 , . . .) = P(X h | X t ), h > t. 395 396 MCMC METHODS Let

Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Nikolai Dokuchaev  · 24 Apr 2007

(pbk) ISBN13: 978-0-203-96472-9 (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

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defined by the set of probability distributions of its components. Probability distributions on infinite dimensional spaces are commonly used in the studied in theory of stochastic processes. For example, the Wiener process Chapter 4 is a random (infinity-dimensional) vector with values at the space C(0,T) of continuous functions f

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the vector (1, 0) has the uniform distribution on [0, π). Find the probability that the set © 2007 Nikolai Dokuchaev is finite. 2 Basics of stochastic processes In this chapter, some basic facts and definitions from the theory of stochastic (random) processes are given, including filtrations, martingales, Markov times, and Markov processes

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. 2.1 Definitions of stochastic processes Sometimes it is necessary to consider random variables or vectors that depend on time. Definition 2.1 A sequence of random variables ξt, t=0

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a discrete time white noise, and let t=0, 1, 2,…. Then the process ηt is said to be a random walk. The theory of stochastic processes studies their pathwise properties (or properties of trajectories ξ(t, ω) for given ω, as well as the evolution of the probability distributions. © 2007 Nikolai

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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 σ-algebra generated by the random

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process ξ(t), it is true under some additional conditions; it suffices to require that ξ(t) is pathwise continuous). © 2007 Nikolai Dokuchaev Basics of Stochastic Processes 21 2.4 Markov processes Definition 2.23 Let ξ(t) be a process, and let say that ξ(t) is a Markov (Markovian) process

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of parabolic equations: it is the so-called fundamental solution of the heat equation. The representation of functions of the stochastic processes via solution of parabolic partial differential equations (PDEs) helps to study stochastic processes: one can use numerical methods developed for PDEs (i.e., finite differences, fundamental solutions, etc.). On the other hand

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continuous time market models. 8.1 Some basic facts about discrete time random processes In this section, several additional definitions and facts about discrete time stochastic processes are given. Definition 8.1 A process ξt is said to be stationary (or strict-sense stationary), if the does not depend on m for

Derivatives Markets

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

two results, E(W1(ω)|W0)=W0 and, E(W2(ω)|W1)=W1. 15.3.2 Definition of a Discrete-Time Martingale A discrete-time stochastic process (Xn(ω))n=0,1,2,3,.. is called a martingale if, 1. E(Xn)<∞ and for all n and, 2. E(Xn+1(ω

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prices under risk neutrality are not martingales. However they aren’t very far from martingales. Definition of a Sub (Super) Martingale 1. A discrete-time stochastic process (Xn(ω))n=0,1,2,3,… is called a sub-martingale if E(Xn)<∞, and E(Xn+1(ω)|Xn)>Xn for all n

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=0,1,2,3,… 2. A discrete-time stochastic process (Xn(ω))n=0,1,2,3,… is called a super-martingale if E(Xn)<∞, , and E(Xn+1(ω)|Xn)<Xn for all n

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choose N=2. No attempt at mathematical rigor is claimed. The intuition behind these results is the primary concern. We start with a discrete-time stochastic process (Xn(ω)n=0,1,2,3,… with finite first and second moments E(Xn)<∞ and for all n=0,1,2,3,… and the

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. 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 Markov property, the martingale property, independent increments, normality, and continuous

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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 details as to exactly what (Ω

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zero, because the stock price has effectively ceased to exist. In other words, zero is an absorbing boundary. However, ABM is still the most basic stochastic process of its kind, so not too much should initially be made of this defect. In particular, it is possible to generate an option pricing formula

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density function of XT(ω), given X0. 16.3.2 Transition Density Functions In general, the transition density function describes the probabilistic evolution of a stochastic process from a known position x=X0 assumed at time t to random positions y=yT(ω)=XT(ω) assumed at time T. Let τ denote

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f(x,t) with respect to t. So far so good. Now for a huge jump. How do we take derivatives of smooth functions of stochastic processes, say F(Xt,t), such as (GBM SDE) where the process is the solution of a stochastic differential equation dXt=μXtdt+σXtdWt with initial value

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X0? We start with the observation that we can expect to end up with another stochastic process that is also the solution to another stochastic differential equation. This new stochastic differential equation for the total differential of F(Xt,t) will have

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problems that K. Itô solved in his famous formula called Itô’s lemma. To understand Itô’s lemma, keep in mind that there are two stochastic processes involved. The first is the underlying process (think of it as the stock). The second process is the derived process, which is a sufficiently smooth

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, which assumes that the stock price process St resembles a standard GBM, with the twist that σt is not only time-dependent, but is a stochastic process itself. The risk-neutral form of the Heston model is, where the variance, is itself a process defined by, Note that Zt is a new

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the constant σ assumption, except to throw out Black–Scholes’ assumption of a stationary log-normal diffusion, and search for a viable (smile-consistent) underlying stochastic process among the vast set of alternatives, many of which will lead to incomplete markets. Black–Scholes and its modifications, however, still have tremendous appeal, especially

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differential equations (SDEs) 553, 559, 562–3, 564, 566, 567–8, 570, 571, 583; stochastic 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

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differential equations (SDEs) 553, 559, 562–3, 564, 566, 567–8, 570, 571, 583 stochastic integral equations (SIEs) 559, 560, 561, 564, 565–6, 567 stochastic processes 540–1, 543, 562, 587, 588 stock forwards when stock pays dividends 88–90 stock index futures 225–30; commentary 230; futures contracts, introduction of

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

by Alain Ruttiens  · 24 Apr 2013  · 447pp  · 104,258 words

ON CURRENCIES 7.7 FUTURES ON (NON-FINANCIAL) COMMODITIES FURTHER READING Part II: The Probabilistic Environment Chapter 8: The basis of stochastic calculus 8.1 STOCHASTIC PROCESSES 8.2 THE STANDARD WIENER PROCESS, OR BROWNIAN MOTION 8.3 THE GENERAL WIENER PROCESS 8.4 THE ITÔ PROCESS 8.5 APPLICATION OF THE

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M month or million, depending on context MD modified duration MtM “Marked to Market” (= valued to the observed current market price) μ drift of a stochastic process N total number of a series (integer number), or nominal (notional) amount (depends on the context) (.) Gaussian (normal) density distribution function N(.) Gaussian (normal) cumulative

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of (.) skew skewness S spot price of an asset (equity, currency, etc.), as specified by the context STD(.) standard deviation of (.) σ volatility of a stochastic process t current time, or time in general (depends on the context) t0 initial time T maturity time τ tenor, that is, time interval between current

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time t and maturity T V(.) variance of (.) (.) stochastic process of (.) stochastic variable zt “zero” or 0-coupon rate of maturity t Z standard Wiener process (Brownian motion, white noise) Introduction The world is the

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as it becomes convenient to buy forward at a reduced (market) cost. Part II The Probabilistic Environment 8 The basis of stochastic calculus 8.1 STOCHASTIC PROCESSES Stochastic is equivalent to random, hence the stochastic calculus develops rules of calculus to be applied if the problems to be handled are of a

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) = 0.7734. Provided F(x) is continuously differentiable, we can determine the corresponding density function f(x) associated to the random variable X as Stochastic Processes A stochastic process can be defined as a collection of random variables defined on the same probability space (Ω, , P) and “indexed” by a set of parameter T

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, we will only consider a one-dimension state space, namely the set of real numbers , that refers to T, and random variables Xt involved in stochastic processes {Xt, t ∈ T} will be denoted by where “∼” indicates its random nature over time t; these random variables will be such as a price, a

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, as follows: discrete parameter set: continuous parameter set: discrete state space: discrete parameter chain continuous parameter chain continuous state space: random sequence random function, or stochastic process In our time series, t may be considered as discrete or continuous. Most liquid financial instruments may look to be traded continuously in time, but

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a continuous time framework, the random variable of prices or returns will change continuously as well. So that the processes we will consider relate to “stochastic processes” properly said. Stationary or Non-Stationary Processes A random process may also be considered as stationary or non-stationary. Broadly speaking, a process is stationary

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what extent can we consider that these values represent the mean and variance of the distribution F(x) of the random variables of a given stochastic process? For sake of simplicity, let us further consider that it is the case. In particular, stationarity implies that the probability P that a random variable

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.16, using the risk neutral probability measure, is called a semimartingale, That is, a variant of a “martingale”. A martingale is a Markovian (memory-less) stochastic process such as, at t, the conditional expected value of St+1 is St. In our case, we talk of a semimartingale, that is, a martingale

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in the further reading at the end of the chapter). 9 Other financial models: from ARMA to the GARCH family The previous chapter dealt with stochastic processes, which consist of (returns) models involving a mixture of deterministic and stochastic components. By contrast, the models developed here present three major differences: These models

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-deterministic variable such as a return, the difference between the model output and the actual observed value is a probabilistic error term. By contrast with stochastic processes described by differential equations, these models are built in discrete time, in practice, the periodicity of the modeled return (daily, for example). By contrast with

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usual Markovian stochastic processes, these models incorporate in the general case a limited number of previous return values, so that they are not Markovian. For a time series of

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moves appears generally more frequent than implied by the normal distribution, what is called a “fat tails” problem. Due to the poor performance of alternative stochastic processes developed with a non-normal distribution (cf. Chapter 15, Section 15.1), the market practice generally prefers to keep the Gaussian hypothesis but to adjust

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SABR models, let us also mention the one8 consisting – instead of starting from Eq. 12.3 – in considering the following relationship: that creates a third stochastic process Z3 that is independent (uncorrelated) with Z1. Provided some hypothesis can be reasonably made about ρ1, 2, presumably as a function of σt, the model

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4, Section 4.3.7). Finally, diffusion processes that involve jump discontinuities can be generalized by use of the more general class of continuous-time stochastic processes called Lévy processes, of which the Wiener, the Poisson and the gamma processes are particular cases (cf. Further Reading). 15.1.3 Other alternative processes

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spot instruments swaptions see also market prices; option pricing price of time, CAPM price-weighted indexes pricing sensitivities see sensitivities probability risk neutral see also stochastic processes Proportion of failures (POF) test putable bonds put options call-put parity see also options PV see present value quanto swaps randomness random numbers random

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-term rates volatility spreads SPVs see special purpose vehicles standardized futures contracts standard Wiener process see also dZ; general Wiener process stationarity stationary Markovian processes stochastic processes basis of Brownian motion definition of process diffusion processes discrete/continuous variables general Wiener process Markovian processes martingales probability reminders risk neutral probability standard Wiener

Mathematics for Finance: An Introduction to Financial Engineering

by Marek Capinski and Tomasz Zastawniak  · 6 Jul 2003

. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z. Brzeźniak and T. Zastawniak Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and J.M. Jones Elements of Abstract Analysis

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respect to the risk-neutral probability gives the initial bond price 100.05489, so the floor is worth 0.05489. Bibliography Background Reading: Probability and Stochastic Processes Ash, R. B. (1970), Basic Probability Theory, John Wiley & Sons, New York. Brzeźniak, Z. and Zastawniak, T. (1999), Basic

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Stochastic Processes, Springer Undergraduate Mathematics Series, Springer-Verlag, London. Capiński, M. and Kopp, P. E. (1999), Measure, Integral and Probability, Springer Undergraduate Mathematics Series, Springer-Verlag,

Market Risk Analysis, Quantitative Methods in Finance

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

109 110 111 114 115 116 117 118 118 120 122 123 124 125 126 127 130 130 131 133 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

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Pt  Pt−1 See Section I.3.3.4 for an introduction to the normal distribution. See Section I.3.7 for further details on stochastic processes in discrete and continuous time. (I.1.38) Basic Calculus for Finance 23 On the left-hand side we have the continuous compounding factor, so

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will need to understand maximum likelihood estimation. Section I.3.7 shows how to model the evolution of financial asset prices and returns using a stochastic process in both discrete and continuous time. The translation between discrete and continuous time, and the relationship between the continuous time representation and the discrete time

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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 basic calculus and elementary linear

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and covariance as a measure of dependency. The very least of our assumptions is that the underlying data are generated by a set of stationary stochastic processes.31 So we do not take the covariance 31 See Section I.3.7 for further information about stationary processes. 112 Quantitative Methods in Finance

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have ˆ (I.3.135) estse X = √ n and ˆ2 (I.3.136) estse ˆ 2 = √  2n I.3.7 STOCHASTIC PROCESSES IN DISCRETE AND CONTINUOUS TIME A stochastic process is a sequence of identically distributed random variables. For most of our purposes random variables are continuous, indeed they are often assumed

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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 on regression analysis, which is introduced in the next

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. We shall introduce the general autoregressive moving average time series models for stationary processes and analyse their properties in Section 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

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application to option pricing is not discussed until Chapter III.3. The first two subsections define what is meant by a stationary or ‘mean-reverting’ stochastic process in discrete and continuous time. We contrast this with a particular type of nonstationary process which is called a ‘random walk’. Then Section I.3

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the 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 ,    , XT } or

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root tests is given in Section II.5.3. I.3.7.2 Mean Reverting Processes and Random Walks in Continuous Time A continuous time stochastic process has dynamics that are represented by a stochastic differential equation. There are two parts to this representation: the first term defines the deterministic part and

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dZt, with EdW = 0 and VdW = dt Now we are ready to write the equation for the dynamics of a continuous time stochastic process X t as the following SDE: dXt = dt + dZt (I.3.141) is called the process volatility. The where  is called the drift

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, even in the presence of market frictions and inefficiencies, prices and log prices of tradable assets are integrated stochastic processes. These are fundamentally different from the associated returns, which are generated by stationary stochastic processes. Figures I.3.28 and I.3.29 illustrate the fact that prices and returns are generated by very

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different types of stochastic process. Figure I.3.28 shows time series of daily prices (lefthand scale) and log prices (right-hand scale) of the Dow Jones Industrial Average (DJIA)

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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 occurrences of events through time. For our purposes the event will be a jump in a price, or a jump in another financial

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provide many more empirical examples where maximum 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

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CAPM and the cross-section of expected returns. Journal of Finance 51, 3–53. Jarrow, R. and Rudd, A. (1982) Approximate option valuation for arbitrary stochastic processes. Journal of Financial Economics 10, 347–369. Jensen, M., (1969) Risk, the pricing of capital assets, and the evaluation of investment portfolios. Journal of Business

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price 2 Asset management, global 225 Asset prices binomial theorem 85–7 lognormal distribution 213–14 pricing theory 179–80, 250–55 regression 179–80 stochastic process 137–8 Assets, tradable 1 Asymptotic mean integrated square error 107 Asymptotic properties of OLS estimators 156 Autocorrelation 175–9, 184, 259–62 Autocorrelation adjusted

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, 35 Continuous time 134–9 long-short portfolio 21 mean reverting process 136–7 Index notation 16–17 P&L 19 random walks 136–7 stochastic process 134–9 Convergence, iteration 188–9 Convex function 13–14, 35 Copula 109–10 Correlation 111–14 beta value 147–8 simulation 220–2 Correlation

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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 stochastic process 134–9 Discretization of space 209–10 Discriminant 5 Distribution function 75–7 Diversifiable risk 181 DJIA (Dow Jones Industrial Average) index 137–8 Dot

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–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 101–3 Global asset management 225 Global minimum variance portfolio 244, 246

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. (independent and identically distributed) variables central limit theorem 121 error process 148 financial modelling 186 GEV distribution 101 regression 148, 157, 175 stable distribution 106 stochastic process 134–5 Implicit function 185 Implied volatility 194, 196, 200–1 Implied volatility surface 200–1 Incremental change 31 Indefinite integral 15 Independent events 74

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distributed (i.i.d.) variables central limit theorem 121 error process 148 financial modelling 186 GEV distribution 101 regression 148, 157, 175 stable distribution 106 stochastic process 134–5 284 Index Independent variable 72, 143 random 109–10, 115, 140 Index tracking regression model 182–3 Indicator function 6 Indices, laws 8

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asset price evolution 87 bid price 2 equity 172 generating time series 218–20 lognormal asset prices 213–14 market microstructure 180 offer price 2 stochastic process 137–9 Pricing arbitrage pricing theory 257 asset pricing theory 179–80, 250–55 European option 212–13 no arbitrage 211–13 Principal cofactors, determinants

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74 Probability and statistics 71–141 basic concepts 72–85 inference 118–29 laws of probability 73–5 MLE 130–4 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

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–2 P&L 19 percentage 16, 19–20, 59–61 period log 23–5 portfolio holdings/weights 17–18 risk free 2 sources 25–6 stochastic process 137–9 Ridge estimator, OLS 171 Risk active risk 256 diversifiable risk 181 portfolio 56–7 systematic risk 181, 250, 252 Risk adjusted performance measure

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distribution 99–100 Standard normal distribution 90, 218–19 Standard normal transformation 90 Standard uniform distribution 89 Stationary point 14–15, 28–31, 35 Stationary stochastic process 111–12, 134–6 Stationary time series 64–5 Statistical arbitrage strategy 182–3 Statistical bootstrap 218 Statistics and probability 71–141 basic concepts 72

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–85 inference 118–29 law of probability 73–5 MLE 130–4 multivariate distribution 107–18 stochastic process 134–9 univariate distribution 85–107 Step length 192 Stochastic differential equation (SDE) 22, 134, 136 Stochastic dominance 227, 258–9

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Stochastic process 72, 134–9, 141 asset price/returns 137–9 integrated 134–6 mean reverting 136–7 Poisson process 139 random walks 136–7 stationary 111–

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-tic data 180 Time series asset prices/returns 137–9, 218–20 lognormal asset prices 218–20 PCA 64–5 Poisson process 88 regression 144 stochastic process 134–9 Tobin’s separation theorem 250 Tolerance levels, iteration 188 Tolerance of risk 230–1, 237 Total derivative 31 Total sum of square (TSS

Frequently Asked Questions in Quantitative Finance

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

multiperiod securities markets. Journal of Economic Theory 20 381-408 Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11 215-260 Haselgrove, CB 1961 A method for numerical integration. Mathematics of Computation 15 323-337 Heath, D, Jarrow, R & Morton

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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 common stochastic building block for random walks in

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an option-pricing context, and by Einstein. The mathematics of BM is also that of heat conduction and diffusion. Mathematically, BM is a continuous, stationary, stochastic process with independent normally distributed increments. If Wt is the BM at time t then for every t, τ ≥ 0, Wt+τ − Wt is independent of

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the Forward and Backward Equations? Short Answer Forward and backward equations usually refer to the differential equations governing the transition probability density function for a stochastic process. They are diffusion equations and must therefore be solved in the appropriate direction in time, hence the names. Example An exchange rate is currently 1

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that other models, such as stochastic volatility, have difficulties in doing. References and Further Reading Cox, J & Ross, S 1976 Valuation of Options for Alternative Stochastic Processes. Journal of Financial Econometrics 3 Kingman, JFC 1995 Poisson Processes. Oxford Science Publications Lewis, A Series of articles in Wilmott magazine September 2002 to August

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models, in which volatility is a function of asset and time, σ(S, t), and stochastic volatility models, in which we represent volatility by another stochastic process. The latter models require a knowledge or specification of risk preferences since volatility risk cannot be hedged just with the underlying asset. If the variance

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multiperiod securities markets. Journal of Economic Theory 20 381-408 Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11 215-260 Joshi, M 2003 The Concepts and Practice of Mathematical Finance. CUP Rubinstein, M 1976 The valuation of uncertain income

Why Stock Markets Crash: Critical Events in Complex Financial Systems

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

discounted value of its future dividends dt  dt+1  dt+2     (which are supposed to be random variables generated according to any general (but known) stochastic process): pt = dt + 1 dt+1 + 1 2 dt+2 + 1 2 3 dt+3 + · · ·  (3) where the factors i = 1 − r < 1, which can fluctuate

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