Post on 12-Nov-2014
A U T U M N 2 0 0 9
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)
M S
c
C
O M
P U
T A
T I O
N A
L
F I N
A N
C E
©Finbarr Murphy 2007
Lecture Objectives
Variance Reduction Techniques Antithetic Variates Delta Control Variates
M S
c
C
O M
P U
T A
T I O
N A
L
F I N
A N
C E
Agenda
Page
©Finbarr Murphy 2007
Control Variates
Antithetic Variable Techniques 2
7
M S
c
C
O M
P U
T A
T I O
N A
L
F I N
A N
C E
3
©Finbarr Murphy 2007
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
©Finbarr Murphy 2007
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?)
Antithetic Variables
5
©Finbarr Murphy 2007
See euroMC2.m
Antithetic Variables
6
©Finbarr Murphy 2007
Performance on a Dell Laptop
Antithetic Variables
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%
7
Agenda
Page
©Finbarr Murphy 2007
Control Variates
Antithetic Variable Techniques 2
7
M S
c
C
O M
P U
T A
T I O
N A
L
F I N
A N
C E
8
©Finbarr Murphy 2007
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
©Finbarr Murphy 2007
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
10
©Finbarr Murphy 2007
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
11
©Finbarr Murphy 2007
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
12
©Finbarr Murphy 2007
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
13
©Finbarr Murphy 2007
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
14
©Finbarr Murphy 2007
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
15
©Finbarr Murphy 2007
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
16
©Finbarr Murphy 2007
Here’s the MatLab Code MCEuorCallWithDeltaControlVariate.m
Control Variates
17
©Finbarr Murphy 2007
Some quick calculations without the control variate
Now with
Control Variates
18
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
©Finbarr Murphy 2007
Recall that we said the hedge was the control variate
Extracting this
Giving
Where β1 = -1 (in this case)
Control Variates
19
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
©Finbarr Murphy 2007
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
20
M
kkkT
tTrt CVCeC
1
)( 0
0
©Finbarr Murphy 2007
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
21
©Finbarr Murphy 2007
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
22M S
c
C
O M
P U
T A
T I O
N A
L
F I N
A N
C E