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M O N T E C A R L O S I M U L A T I O N S

Numerical Methods in Finance (Implementing Market Models)

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©Finbarr Murphy 2007

Lecture Objectives

Variance Reduction Techniques Antithetic Variates Delta Control Variates

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Agenda

Page


Control Variates

Antithetic Variable Techniques 2

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Antithetic variance reduction is a very simple concept

It is based on the construction of an asset with an exact negative correlation with the underlying asset

There are now two SDE’s

tttt

tttt

dzSdtrSds

dzSdtrSds

,2,2,2

,1,1,1

Antithetic Variables

4


A portfolio containing S1 and S2 will have a much lower variance as one asset rises, the other falls

Taking both paths to maturity and averaging the payoff’s

The average of the pair is more accurate than two different simulations (why?)

And is computationally more efficient (why?)


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See euroMC2.m


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Performance on a Dell Laptop


Monte Carlo With Antithetic Reduction

No Simulations

Option Price Std Dev Std Err

Time (secs)

Option Price Std Dev Std Err

Time (Secs)

Difference in StdErr

Difference in Time

100 2.6857 3.27430.327

4 0.015 2.6944 2.5779 0.2578 0.016 21% -7%

500 2.6734 3.42990.153

4 0 2.6569 2.4912 0.1114 0.109 27% N/A

1000 2.7804 3.37690.106

8 0.031 2.8887 2.6943 0.0852 0.031 20% 0%

2500 2.5077 3.18660.063

7 0.063 2.7297 2.6712 0.0534 0.047 16% 25%

5000 2.7118 3.33050.047

1 0.125 2.6881 2.6754 0.04 0.094 20% 25%

10000 2.6672 3.2033 0.032 0.234 2.6944 2.6979 0.027 0.203 16% 13%

20000 2.6585 3.29490.023

3 0.5 2.6977 2.6926 0.019 0.421 18% 16%

50000 2.6903 3.28620.014

7 1.141 2.6975 2.6658 0.0119 1.063 19% 7%

75000 2.6821 3.2758 0.012 1.797 2.7101 2.681 0.0098 1.719 18% 4%

100000 2.69 3.27250.010

3 2.39 2.6685 2.6595 0.0084 2.172 18% 9%

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Agenda

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Control Variates

Antithetic Variable Techniques 2

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Control variates are useful when we have two similar derivatives, H and I

Derivative H is the security being valued, derivative I is a similar security with an analytically available solution

Using normal Monte Carlo techniques, we estimate the value of H = H*

Using normal Monte Carlo techniques, we estimate the value of I = I*

Control Variates

9


A better value for Hb is now calculated asHb = H* - I* + I

Or Hb = I + (H* - I*)

The variance of Hb is given byvar Hb = var H* + var I* -2cov(H*, I*)

The standard deviation is given by

Control Variates

),cov(varvar **** IHIHbH2

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Now…

if

So this variance reduction technique relies on finding a control variate that is highly correlated with the variable being simulated

Control Variates

*HH b

*

*

** ,H

IIH

2

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Now, moving to a specific example

Consider a short position in one option and a long position in ∂C/∂S shares

At maturity, this portfolio will have a value

Control Variates

0

TTT

T CSS

C

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One step before maturity, assuming delta hedging, this portfolio will have a value

Rearranging, we can see that

In other words, the current (T-1) value of the option is related to the changing option deltas and the terminal option prices

Control Variates

0111

1

TT

TT CS

S

C

TTrT

TT

TrT

T SS

CeS

S

CCeC

1

11

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In other words, if we simulate the payoff In other words, if we simulate the payoff And the hedge

In other words, if we simulate the payoff And the hedge We can estimate the current option value

inflated to maturity date at the riskless rate

More formally and using summations over the life of the option:

Eta (η) refers to a hedging error

In other words, if we simulate the payoff

Control Variates

1

0

)()( 1

1

0

0

N

i

tTrtt

tT

tTrt

i

ii

i eSESS

CCeC

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So we need to simulate the payoff, and the hedge and compute the average of the difference

In this case, the hedge is regarded as the control variate

Time to look at a practical example

Control Variates

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We consider the following: noSimulations = 100; noStepsPerSimulation = 10; T = 1; % one year maturity startStockPrice = 100; strike = 100; div = 0.03; sigma = 0.20; dt = T/noStepsPerSimulation; intRate = 0.06; beta1 = -1;

Control Variates

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Here’s the MatLab Code MCEuorCallWithDeltaControlVariate.m

Control Variates

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Some quick calculations without the control variate

Now with

Control Variates

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No simulations Option Price

Std Dev Std Err

100 9.3542 2.2644 0.2264

1000 9.1639 2.1427 0.0678

100000 9.1364 2.0426 0.0065

No simulations Option Price

Std Dev Std Err

100 9.1325 0.5267 0.0527

1000 9.1181 0.4567 0.0144

10000 9.1357 0.4793 0.0048


Recall that we said the hedge was the control variate

Extracting this

Giving

Where β1 = -1 (in this case)

Control Variates

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1

0

)()( 1

1

0

0

N

i

tTrtt

tT

tTrt

i

ii

i eSESS

CCeC

1

0

)(1

1

1

N

i

tTrtt

t i

ii

i eSESS

CCV

11

)( 0

0CVCeC T

tTrt


Assuming that we use multiple control variates

A second, often used control variate is Gamma

The Bi’s can be calculated using a least squares regression technique

Control Variates

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M

kkkT

tTrt CVCeC

1

)( 0

0


Control Variate techniques can and should be combined with Antithetic techniques

The following slide shows how the inclusion of these techniques can be used to increase computation time

Control Variates

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Recommended Texts

Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing

derivative models, 1st ed., John Wiley and Sons Ltd.—Chapter 4

Additional/Useful Brandimarte, P. (2006) Numerical methods in finance

and economics: A matlab-based introduction, 2nd Revised ed., John Wiley & Sons Inc.

Hull, J. (2005) Options, futures and other derivatives, 6th ed., Prentice Hall

—Chapters 17, P417-419

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