MONTE CARLO SIMULATION

Topics History of Monte Carlo Simulation GBM process How to simulate the Stock Path in Excel, Monte Carlo simulation and VaR

History of the Monte Carlo

Markov Property A Markov process is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future

Continuous-Time Stochastic Process

Wiener Process

Graphically

Generalized Wiener Process dS = a(S,mean change per unit of time is known as drift rate and the variance per unit is called as the variance rate)dt + b(S, t)dz dx = adt + bdz dx = a(S, t )dt + b(S, t)dz

Example Suppose stock price follow the process of dx = adt or dx/dt = a Integrating with respect to time, we get x = x 0 + at - Where x 0 is the value of x at time 0. In a period of time of length T, the variable x increase by an amount of aT - bdz is regarded as noise or variability term added to the path of x - Wiener process has a standard deviation of 1.0. so, b times a Wiener process has a standard deviation of b.

Stock price process: with out volatile

Stock price process with volatile

Change of x at small time changes and in time interval T

Log normal return

Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull The Lognormal Property These assumptions imply ln S T is normally distributed with mean: and standard deviation : Because the logarithm of S T is normal, S T is lognormally distributed

Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull The Lognormal Property continued where  m,s] is a normal distribution with mean m and standard deviation s

Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull The Lognormal Distribution

Monte Carlo Simulation (See Excel)