Hedge with an EdgeAn Introduction to the Mathematics of Finance

Riaz Ahmed & Adnan KhanLahore Uviersity of Management Sciences

Monte Carlo Methods

Topics

• Simulating Bernoulli Random Variable • Generating Random Variables – Inverse Transform Method– Box Muller Method– Rejection Method

• Simulate a 1-D random Walk– Calculate the mean– Calculate the Variance

• Simulating Brownian Motion • Geometric Brownian Motion• Arithmetic Brownian Motion• Variance Reduction Techniques

Simulating a Binomially Distributed Random Variable

• Note sum of Bernoulli trials is a binomial

• Let X i be a Bernoulli trial with probability ‘p’ of success

• is binomial ‘n’, ‘p’

Some Properties

• Distribution of successes in trials

• Expected Value

• Variance

Simulation of Binomial

• Generating Bernoulli

• Binomial as the sum of Bernoulli

• Monte Carlo Simulation

• Numerical vs. Exact Mean and Variance

Simulation of Binomial

0 1 2 3 4 5 6 7 8 9 100

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hist

hist

Continuous Random Variables

• Inverse Transform Method– Suppose a random variable has cdf ‘F(x)’– Then Y=F-1(U) also had the same cdf

• Generating the exponential

• Generate the exponential, compare with exact cdf

• Generate a r.v. with cdf

Simulating the Exponential

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Simulating Normal using Inverse Transform

• Cannot get a closed form in terms of elementary functions

• Excel has built in command normsinv()

• Use normsinv(rand())

Simulation of Normal

-3-2.76

-2.52-2.28

-2.04-1.8

-1.56-1.32

-1.08

-0.8399999999...

-0.5999999999...

-0.3599999999...

-0.1199999999...

0.1200000000...

0.3600000000...

0.6000000000...

0.8400000000...1.08

1.321.56 1.8

2.042.28

2.522.76 3

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Series1Series2Series3

Rejection Method

• Simulate &

• To Simulate look @

• If accept, else reject

• To Simulate N(0,1) let

• If set

Box Muller Method• Recall the cdf for the standard normal is

• We saw one way was to invert this• Another technique is to generate

• Then and where

Simulation

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Weiner Process

• W(t) CT-CS process is a Weiner Process if W(t) depends continuously on t and the following holda)b) are independentc)

Simulating Brownian Motion

• Initialize at 0 as W(0)=0

• Simulate Weiner Increments according to

• The Weiner Process then follows

Simulation

00.15 0.3

0.45 0.60.75 0.9

1.05 1.21.35 1.5

1.65 1.81.95 2.1

2.25 2.42.55 2.7

2.85 33.15 3.3

3.45 3.6

3.74999999999999

3.89999999999999

4.04999999999999

4.19999999999999

4.34999999999999

4.49999999999999

4.64999999999999

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Weiner Process

Weiner Process

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Simulation

00.15 0.3

0.45 0.60.75 0.9

1.05 1.21.35 1.5

1.65 1.81.95 2.1

2.25 2.42.55 2.7

2.85 33.15 3.3

3.45 3.6

3.74999999999999

3.89999999999999

4.04999999999999

4.19999999999999

4.34999999999999

4.49999999999999

4.64999999999999

-0.4

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Weiner Process 1Weiner Process 2Weiner Process 3Weiner Process 4Weiner Process 5

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Stock Price Model

• Modeled by Geometric Brownian Motion

• Note

• To simulate use the ‘Euler Scheme’

Simulating GBM

00.15 0.3

0.45 0.60.75 0.9

1.05 1.21.35 1.5

1.65 1.81.95 2.1

2.25 2.42.55 2.7

2.85 33.15 3.3

3.45 3.6

3.74999999999999

3.89999999999999

4.04999999999999

4.19999999999999

4.34999999999999

4.49999999999999

4.649999999999990

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GBM1GBM2Mean

Simulating GBM

00.15 0.3

0.45 0.60.75 0.9

1.05 1.21.35 1.5

1.65 1.81.95 2.1

2.25 2.42.55 2.7

2.85 33.15 3.3

3.45 3.6

3.74999999999999

3.89999999999999

4.04999999999999

4.19999999999999

4.34999999999999

4.49999999999999

4.649999999999990

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Series1exact

Mean Reverting Process

• Arithmetic Brownian Motion is mean reverting

• Interest rate models

• The numerical scheme is

Simulating ABM

00.15 0.3

0.45 0.60.75 0.9

1.05 1.21.35 1.5

1.65 1.81.95 2.1

2.25 2.42.55 2.7

2.85 33.15 3.3

3.45 3.6

3.74999999999999

3.89999999999999

4.04999999999999

4.19999999999999

4.34999999999999

4.49999999999999

4.64999999999999

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Arithmetic Brownian Motion

Simulating ABM

00.15 0.3

0.45 0.60.75 0.9

1.05 1.21.35 1.5

1.65 1.81.95 2.1

2.25 2.42.55 2.7

2.85 33.15 3.3

3.45 3.6

3.74999999999999

3.89999999999999

4.04999999999999

4.19999999999999

4.34999999999999

4.49999999999999

4.649999999999990

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ExactNumerical

Option Pricing using Monte Carlo

• Generate several risk-neutral random walks for the asset starting at the asset price today and going on till expiry.

• For each path generated calculate the payoff.• Calculate average the average of all the

payoffs• Take the present value of this average to get

the option value today.

Pricing of European Call

Challenge Problem

Simulate using Monte Carlo techniques the price of a European call option where the underlying with volatility 0.5 interest rate 3% exercise price 100 and currently underlying at 90