Mathematical and Quantitative Finance Chapter 1 Theory PDF

Mathematical and Quantitative Finance: Chapter 1 Theory 1/41 Contents Chapter 1: Wiener Process and Itô’s lemma ▶ Definition of the Wiener Process. ▶ Generalized Wiener Process and its behaviour for different values of drift and dif...

Mathematical and Quantitative Finance: Chapter 1 Theory 1/41 Contents Chapter 1: Wiener Process and Itô’s lemma ▶ Definition of the Wiener Process. ▶ Generalized Wiener Process and its behaviour for different values of drift and diffusion. Computer simulation and visualization. ▶ Itô Process and Itô’s Lemma. ▶ Stochastic differential equations. ▶ Geometric Brownian Motion - process for a stock price. ▶ α-stable Lévy Motion. Computer simulation and visualization. ▶ Poisson Process, compound PP, compensated PP. Computer simulation and visualization. Reference(s): Chapter 14 of Options, Futures and Other Derivatives, 9th edition by J. Hull 2/41 Objectives After completing this chapter you should be able to: ▶ Define Wiener Process. Explain the impact of the drift and diffusion coefficients on the behaviour of trajectories (also known as paths or realizations) of the Generalized Wiener Process. ▶ Manipulate stochastic integrals. ▶ Apply Itô’s Lemma. ▶ Remember the stochastic differential equation for the Geometric Brownian Motion. ▶ Simulate solutions to stochastic differential equations driven by Wiener Process/Standard Brownian Motion, α-stable Lévy Motion, Compensated Poisson Process. 3/41 Continuous-time stochastic process Stochastic process in continuous time {Xt , t ∈ [0, ∞)} is a collection of random variables indexed by t. Note: ▶ We sometimes write X (t) instead of Xt. ▶ We often use {Xt } instead of {Xt , t ∈ [0, ∞)}. ▶ Xt (no braces) represents a random variable, while {Xt } (with braces) represents a stochastic process (many random variables). Why are we interested in/talk about continuous-time stochastic processes? ▶ We need a theoretical framework/model to represent real-life phenomena whose behaviour over time is random, e.g., behaviour of stock prices. 4/41 Wiener Process/Standard Brownian Motion Wiener Process or Standard Brownian Motion {Zt , t ∈ [0, ∞)} is a stochastic process that satisfies: ▶ The change Zt − Zs for a time increment t − s, 0 ≤ s < t, follows a√normal distribution with mean 0 and standard deviation t − s, that is, √ Zt − Zs ∼ N(0, ( t − s)2 ) ▶ {Zt } has independent increments, that is, if 0 = t0 < t1 <... < tn , then Z (tj ) − Z (tj−1 ), j = 1, 2,... , n, are independent ▶ Z0 = 0 (with probability 1) ▶ {Zt } is a continuous function of time (with probability 1) 5/41 Wiener Process/Standard Brownian Motion cont. √ Note: Since Zt − Zs ∼ N(0, ( t − s)2 ), then in particular, the change in {Zt } over the interval [0, T ], ZT − Z0 , satisfies: ▶ E[ZT − Z0 ] = 0 ▶ V[ZT − Z0 ] = T √ ▶ SD[ZT − Z0 ] = T Think of ZT − Z0 as a sum of small changes in {Zt } in many small time intervals between times 0 and T. 6/41 Wiener Process/Standard Brownian Motion cont. Note: If ϵ ∼ N(0, 1) then Zt − Zs has the same distribution as √ t − sϵ. We will sometimes use the following short-hand √ Zt − Zs = t − sN(0, 1) Note: When we write N(mean, variance) the second spot is reserved for the variance as in the 9th edition of our textbook. Note also that N(...) is often used to represent the cdf of a standard normal distribution. 7/41 Wiener Process/Standard Brownian Motion cont. Pseudo-code for simulation of a path of {Zt , t ∈ [0, T ]} ▶ Pick a natural number N, set ∆t = TN−0 and create a grid of equispaced points ti = i · ∆t = iT N for i = 0, 1,... , N on [0, T ]. Zt ∆t ∆t 0 t1 t2 t5 tT =N t ▶ Generate a sequence of N iid standard normal rvs ϵ1 , ϵ2 ,... , ϵN. ▶ Set Z (0) = 0 and then compute sequentially √ Z (ti ) = Z (ti−1 ) + ∆t · ϵi , i = 1, 2,... , N. i √ X  √ X i Note that Z (ti ) = ∆t · ϵj = ∆t ϵj. j=1 j=1 8/41 Wiener Process/Standard Brownian Motion cont. Rcode for simulation of a path of {Zt , t ∈ [0, T ]} #Rvariables TT,dt are already declared/defined tt