hedge with an edge an introduction to the mathematics of finance riaz ahmed & adnan khan lahore...
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Hedge with an EdgeAn Introduction to the Mathematics of Finance
Riaz Ahmed & Adnan KhanLahore Uviersity of Management Sciences
Monte Carlo Methods
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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
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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’
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Some Properties
• Distribution of successes in trials
• Expected Value
• Variance
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Simulation of Binomial
• Generating Bernoulli
• Binomial as the sum of Bernoulli
• Monte Carlo Simulation
• Numerical vs. Exact Mean and Variance
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Simulation of Binomial
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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
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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())
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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
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Rejection Method
• Simulate &
• To Simulate look @
• If accept, else reject
• To Simulate N(0,1) let
• If set
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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
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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)
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Simulating Brownian Motion
• Initialize at 0 as W(0)=0
• Simulate Weiner Increments according to
• The Weiner Process then follows
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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
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Weiner Process
Weiner Process
Tim
e
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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
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Weiner Process 1Weiner Process 2Weiner Process 3Weiner Process 4Weiner Process 5
Tim
e
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Stock Price Model
• Modeled by Geometric Brownian Motion
• Note
• To simulate use the ‘Euler Scheme’
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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
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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
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Mean Reverting Process
• Arithmetic Brownian Motion is mean reverting
• Interest rate models
• The numerical scheme is
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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
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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
0.2
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ExactNumerical
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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.
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Pricing of European Call
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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