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• http://www.youtube.com/watch?v=ioVccVC_Smg

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• Suppose a variable follow a Markov stochastic process and its

current value is 10. Suppose further that its value during 1 year is

• What is the probability of the change in the value during 2 year?......Ans. Because of the Markov process (independent distribution), the distribution

• What is the prob. of change during 6 months?….• Generally, the change during a very short time period ∆t) but note

that the variances of changes are additive but the standard deviations are not additive. Variance in 2, 3 years are 2 and 3 but the standard deviation are 2 and 3

𝑋 ∅ (0 ,1)

𝑋 ∅ (0 ,2)

Wiener Process• Wiener process is a particular type of Markov process. In

physics, it is called as Brownian motion• If a variable z follows wiener process it must follow two

properties• Property 1. The change in ∆z during a small time ∆t is

t

Where has a standardized normal distribution

• Property 2. The value of ∆z for any two different short intervals of time ∆t are independent, thus

• Mean of ∆z = 0,• Standard deviation of ∆z = • Variance of ∆z = t

The second property implies that z follows Markov process

Graphically

∆ 𝑧 1 ∆ 𝑧  2 ∆ 𝑧 3 ∆ 𝑧 5∆ 𝑧 4

t ttt t

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 + bdzdx = 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 = x0 + at

- Where x0 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

If the volatility of stock price is zero, thenWhen

Or,

Integrating between 0 and time T, we get

Meaning that the stock price grow at a continuous compound rate of

Stock price process with volatile

+

Or,

+

For the discrete time,

+

Return of stock price is normal distributed as;

(

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

• has a standard normal distribution. has a normal distribution with

mean of =

variance of =

standard deviation of = • So, change in value of x in any time interval T

mean of =

variance of = standard deviation of =

Log normal return• When a log of any variable distribute as normal, we call it

as lognormal distribute. We can show that the Log of returns is normally distributed as

• Or

Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull 11.14

The Lognormal Property

• These assumptions imply ln ST is normally distributed with mean:

and standard deviation:

• Because the logarithm of ST is normal, ST is lognormally distributed

TS )2/(ln 20

T

TS )2/(ln 20

Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull 11.15

The Lognormal Propertycontinued

where m,s] is a normal distribution with mean m and standard deviation s

TTS

S

TTSS

T

T

,)2(ln

or

,)2(lnln

2

0

20

Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull 11.16

The Lognormal Distribution

E S S e

S S e e

TT

TT T

( )

( ) ( )

0

02 2 2

1

var

Monte Carlo Simulation (See Excel)• Suppose X follow the Wiener Process

Suppose Find the path of X using Excel• To generate random variables using Excel

Normsinv (rand())

* Note that rand() function generate the variables drawn from the uniform distribution, but to keep it simple just use it to generate ‘Z’.

Monte Carlo Simulation (See Excel)

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