Simulation Techniques in Financial Risk Management
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Simulation Techniques in Financial Risk Management
STATISTICS IN PRACTICE Advisory Editor
Peter Bloomfield North Carolina State University, USA
Founding Editor Vic Barnett Nottingham Trent University, UK
Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study. With sound motivation and many worked practical examples, the books show in downtoearth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title’s special topic area. The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on. The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges. It is our aim to present judiciously chosen and wellwritten workbooks to meet everyday practical needs. Feedback of views from readers will be most valuable to monitor the success of this aim. A complete list of titles in this series appears at the end of the volume.
Simulation Techniques in Financial Risk Management
NGAI HANG CHAN HOI YING WONG The Chinese University of Hong Kong Shatin, Hong Kong
@E;!;iCIENCE A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 02006 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate percopy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 7508400, fax (978) 7504470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 1 1 1 River Street, Hoboken, NJ 07030, (201) 748601 1, fax (201) 7486008, or online at http://www.wiley.condgo/permission. Limit of LiabilityiDisclaimer of Wamnty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or foi technical support, please contact our Customer Care Department within the United States at (800) 7622974, outside the United States at (3 17) 5723993 or fax (3 17) 5724002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our web site at www.wiley.com.
Library of Congress CataloginginPublicationData: Chan, Ngai Hang. Simulation techniques in financial risk management / Ngai Hang Chan, Hoi Ying Wong. p. cm. Includes bibliographical references and index. ISBN13 9780471469872 (cloth) ISBN10 0471469874 (cloth) 1. FinanceSimulation methods. 2. Risk managementSimulation methods. I. Wong, Hoi Ying. 11. Title. HG173.C47 2006 338.54~22 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1
2005054992
To Pat, Calvin, and Dennis N.H. Chan
and
To Mei Choi H.Y. Wong
Contents
List of Figures
xi
List of Tables
xaaa
Preface
...
xv
1 Introduction 1.1 Questions 1.2 Simulation 1.3 Examples 1.3.1 Quadrature 1.3.2 Monte Carlo 1.4 Stochastic Simulations 1.5 Exercises 2 Brownian Motions and It6’s Rule 2.1 Introduction 2.2 Wiener’s and It6’s Processes 2.3 Stock Price 2.4 It6’s Formula
11 11 11 18
22 vii
2.5
Exercises
3 BlackScholes Model and Option Pricing
3.1 3.2 3.3 3.4 3.5
Introduction One Period Binomial Model The BlackScholesMerton Equation BlackScholes Formula Exercises
28 31 31 32 35 41 45
4 Generating Random Variables
49 49 49 50 52 54 54 56 58 64
5 Standard Simulations in Risk Management 5.1 Introduction 5.2 Scenario Analysis 5.2.1 Value at Risk 5.2.2 Heavy Tailed Distribution 5.2.3 Case Study: VaR of Dow Jones 5.3 Standard Monte Carlo 5.3.1 Mean, Variance, and Interval Estimation 5.3.2 Simulating Option Prices 5.3.3 Simulating Option Delta 5.4 Exercises 5.5 Appendix
67 6r 67 69 70 r2
Introduction Random Numbers Discrete Random Variables 4.4 AcceptanceRejection Method 4.5 Continuous Random Variables 4.5.1 Inverse Transform 4.5.2 The Rejection Method 4.5.3 Multivariate Normal 4.6 Exercises 4.1 4.2 4.3
6
Variance Reduction Techniques 6.1 Introduction 6.2 Antithetic Variables 6.3 Stratified Sampling 6.4 Control Variates
rr rr
r9 82 86 87 89 89 89 9r 104
CONTENTS
6.5 6.6
Importance Sampling Exercises
ix
110 118
7 PathDependent Options 7.1 Introduction 7.2 Barrier Option 7.3 Lookback Option 7.4 Asian Option 7.5 American Option 7.5.1 Simulation: Least Squares Approach 7.5.2 Analyzing the Least Squares Approach 7.5.3 AmericanStyle PathDependent Options 7.6 Greek Letters 7.7 Exercises
121 121 121 123 124 128 128 131 136 138 143
8 Multiasset Options 8.1 Introduction 8.2 Simulating European MultiAsset Options 8.3 Case Study: O n Estimating Basket Options 8.4 Dimensional Reduction 8.5 Exercises
145 145 14 6 148 151 155
9 Interest Rate Models 9.1 Introduction 9.2 Discount Factor 9.2.1 Time Varying Interest Rate 9.3 Stochastic Interest Rate Models and Their Simulations 9.4 Options with Stochastic Interest Rate 9.5 Exercises
159 159 159 160
10 Markov Chain Monte Carlo Methods 10.1 Introduction 10.2 Bayesian Inference 10.3 Sirnuslating Posteriors 10.4 Markov Chain Monte Carlo 10.4.1 Gibbs Sampling 10.4.2 Case Study: The Impact of Jumps on Dow Jones
167 167 167 170 170 170
161 164 166
172
x
CONTENTS
10.5 MetropolisHastings Algorithm 10.6 Exercises
181 184
11 Answers to Selected Exercises 11.1 Chapter 1 11.2 Chapter 2 11.3 Chapter 3 11.4 Chapter 4 11.5 Chapter 5 11.6 Chapter 6 11.7 Chapter 7 11.8 Chapter 8 11.9 Chapter 9 11.10 Chapter 10
187 187 188 192 196 198 200 201 202 205 207
References
21 1
Index
21 r
List of Figures
Densities of a lognormal distribution with mean and variance e(e  I), i.e., p = o and u2 = 1 and a standard normal distribution.
5
2.1
Sample paths of the process S[,t] for diflerent n and the same sequence of ei.
13
2.2
Sample paths of Brownian motions o n [O,l].
15
2.3
Geometric Brownian motion.
20
3.1
One period binomial tree.
32
4.1
Sample paths of the portfolio.
63
4.2 5.1
Sample paths of the assets and the portfolio.
64
The shape of GED density function.
71
5.2
Left tail of GED.
72
5.3
QQ plot of normal quantiles against dailg Dow Jones returns.
73
5.4 5.5
Determine the maximum of 2f (y)eg graphically.
74
QQ plot GED(l.21) quantiles against Dow Jones return quantiles.
76
1.1
e0.5
Xi
xii
LlST OF FIGURES
5.6
Simulations of the call price against the size.
84
5.7
The log likelihood against a), where X has p.d.f. f and a is a given constant. Let I ( X > a) = 1 if X > a and 0 otherwise. Then
P(X > a ) = E f ( l ( X>a))
Take g(x) = AeXx, x > 0 , an exponential density with parameter A. Then the above derivation shows
P ( X > a ) = E,[eXXf ( X ) l X > a]e’”/A. Using the socalled “memoryless property” , i.e., P ( X > s+tlX > s) = P ( X > t ) , of an exponential distribution, it can be easily seen that the conditional distribution of an exponential distribution conditioned on { X > a } has the
same distribution as a
+ X . Therefore, eAa
P ( X > a ) = E,[eX(X+a) A =
f( X + a)]
1
xE,[eXXf ( X f a)].
We can now estimate Q by generating X I , . . . ,X , according to an exponential distribution with parameter X and using
..
Q =
11
_ _ An
ce””. f + (Xi
a).
i=l
Example 6.13 Suppose we are interested in Q = P ( X > a ) , where X is standard normal. Then f i s the normal density. Let g be an exponential
116
VARIANCE REDUCTION TECHNIQUES
density with X
= a.
Then
P(X > a)
=
1 Eg[eax f ( X a
+ a)]
W e can therefore estimate 0 by generating X , an exponential distribution with rate a , and then using
to estimate 0. To compute the variance of 6, we need to compute quantities and E,[eXZ]. These can be computed numerically and can be shown to be Eg[eX2/2]= aeaz/2&(1
(s)’ 

@(U)),
Eg[exz] = ~ e ~ ’ / ~ f i @ ( l( a / & ) ) .
For example, i f a = 3 and n = 1, then Var(ex2/2) = 0.0201 and Var(9) = x 0.0201 4.38 x 1OP8. O n the other hand, a standard estimator has 0 vanance O(1  0) = 0.00134. Consider simulating a vanilla European call option price again, using the importance sampling technique. Suppose that we evaluate the value of a deep outofmoney (So where Xi = log Si  log Si1. With a fixed At, a discrete approximation to the dynamics (10.7) is
AlogS=pAt+aAWt+YANt.
(10.8)
When At is sufficiently small, A N , is either 1, with probability AAt, or 0, with probability 1  AAt.
Example 10.3 Simulate 100 sample paths from the asset price dynamics of (10.7) with parameters: p = 0.08, CJ = 0.4, A = 3.5, s = 0.3, and k = 0. Each sample path replicates daily logreturns of a stock over a oneyear horizon. Based on these 100 paths, estimate the values of p, CJ, A, s, and k with the Gibbs sampling. Compare the results with input values.
Simulating paths Sample paths are simulated by assuming n = 250 trading days a year and so the discretization (10.8) has At = 1/250. On each path, 1 ogasset price at each time point is generated as follows, log Si+l  log s,
=
p At
+
C
J
~
C
,
p A t + k + v ’ W c ,
if if
U>AAt UIAAt ’
where E N ( 0 , l ) and U N U ( 0 , l ) are independent random variables. To simplify notations, we denote xi = log Si+1 log Si. The corresponding SPLUS code and a graph of three sample paths are given below. N
### generate observation of Y ####
MUY