Financial Engineering Advanced Background Series Published or forthcoming A Primer for the Mathematics of Financial Engineering, by Dan Stefanica Numerical Linear Algebra Methods for Financial Engineering Applications, by Dan Stefanica A Probability Primer for Mathematical Finance, by.Elena Kosygina Differential Equations with Numerical Methods for Financial Engineering, by Dan Stefanica A PRIMER for the MATHEMATICS of FINANCIAL ENGINEERING DAN STEFANICA Baruch College City University of New York FE PRESS New York FE PRESS New York www.fepress.org Information on this title: www.fepress.org/mathematicaLprimer ©Dan Stefanica 2008 All rights reserved 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, or otherwise, without the prior written permission of the publisher To Miriam First published 2008 to Rianna Printed in the United States of America ISBN-13 978-0-9797576-0-0 ISBN-IO 0-9797576-0-6 and Contents List of Tables xi Preface xiii Acknowledgments xv How to Use This Book xvii O Mathematical preliminaries 0.1 Even and odd functions 0.2 Useful sums with interesting proofs 0.3 Sequences satisfying linear recursions 0.4 The "Big 0" and "little 0" notations 0.5 Exercises Calculus review Options 1.1 Brief review of differentiation 1.2 Brief review of integration 1.3 Differentiating definite integrals 1.4 Limits 1.5 L'Hopital's rule 1.6 Multivariable functions 1.6.1 Functions of two variables Plain vanilla European Call and Put options 1.8 Arbitrage-free pricing 1.9 The Put-Call parity for European options 1.10 Forward and Futures contracts 1.11 References 1.12 Exercises Vll 1 12 15 19 19 21 24 26 28 29 32 34 35 37 38 40 41 viii CONTENTS Numerical integration Interest Rates Bonds 2.1 Double integrals 2.2 Improper integrals 2.3 Differentiating improper integrals 2.4 Midpoint, Trapezoidal, and Simpson's rules 2.5 Convergence of Numerical Integration Methods 2.5.1 Implementation of numerical integration methods 2.5.2 A concrete example 2.6 Interest Rate Curves 2.6.1 Constant interest rates 2.6.2 Forward Rates 2.6.3 Discretely compounded interest 2.7 Bonds Yield, Duration, Convexity 2.7.1 Zero Coupon Bonds 2.8 Numerical implementation of bond mathematics 2.9 References 2.10 Exercises Probability concepts Black-Scholes formula Greeks and Hedging 3.1 Discrete probability concepts 3.2 Continuous probability concepts 3.2.1 Variance, covariance, and correlation 3.3 The standard normal variable 3.4 Normal random variables 3.5 The Black-Scholes formula 3.6 The Greeks of European options 3.6.1 Explaining the magic of Greeks computations 3.6.2 Implied volatility 3.7 The concept of hedging ~- and r-hedging 3.8 Implementation of the Black-Scholes formula 3.9 References 3.10 Exercises 45 45 48 51 52 56 58 62 64 66 66 67 69 72 73 77 78 81 81 83 85 89 91 94 97 99 103 105 108 110 111 Lognormal variables Risk-neutral pricing 117 4.1 Change of probability density for functions of random variables 117 4.2 Lognormal random variables 119 4.3 Independent random variables 121 IX 4.4 4.5 Approximating sums of lognormal variables Power series 4.5.1 Stirling's formula 4.6 A lognormal model for asset prices 4.7 Risk-neutral derivation of Black-Scholes 4.8 Probability that options expire in-the money 4.9 Financial Interpretation of N(d ) and N(d2 ) 4.10 References 4.11 Exercises 126 128 131 132 133 135 137 138 139 Taylor's formula Taylor series 143 5.1 Taylor's Formula for functions of one variable 143 5.2 Taylor's formula for multivariable functions 147 150 5.2.1 Taylor's formula for functions of two variables 5.3 Taylor series expansions 152 155 5.3.1 Examples of Taylor series expansions 158 5.4 Greeks and Taylor's formula 5.5 Black-Scholes formula: ATM approximations 160 5.5.1 Several ATM approximations formulas 160 5.5.2 Deriving the ATM approximations formulas 161 5.5.3 The precision of the ATM approximation of the BlackScholes formula 165 170 5.6 Connections between duration and convexity 5.7 References 172 5.8 Exercises 173 Finite Differences Black-Scholes PDE 6.1 Forward, backward, central finite differences 6.2 Finite difference solutions of ODEs 6.3 Finite difference approximations for Greeks 6.4 The Black-Scholes PDE 6.4.1 Financial interpretation of the Black-Scholes PDE 6.4.2 The Black-Scholes PDE and the Greeks 6.5 References 6.6 Exercises 177 177 180 190 191 193 194 195 196 Multivariable calculus: chain rule, integration by substitution, and extrema 203 7.1 Chain rule for functions of several variables 203 CONTENTS x 7.2 Change of variables for double integrals 7.2.1 Change of Variables to Polar Coordinates 7.3 Relative extrema of multivariable functions 7.4 The Theta of a derivative security 7.5 Integrating the density function of Z 7.6 The Box-Muller method 7.7 The Black-Scholes PDE and the heat equation 7.8 Barrier options 7.9 Optimality of early exercise 7.10 References 7.11 Exercises 205 207 208 216 218 220 221 225 228 230 231 Lagrange multipliers Newton's method Implied volatility Bootstrapping 8.1 Lagrange multipliers 8.2 Numerical methods for 1-D nonlinear problems 8.2.1 Bisection Method 8.2.2 Newton's Method 8.2.3 Secant Method 8.3 Numerical methods for N-dimensional problems 8.3.1 The N-dimensional Newton's Method 8.3.2 The Approximate Newton's Method 8.4 Optimal investment portfolios 8.5 Computing bond yields 8.6 Implied volatility 8.7 Bootstrapping for finding zero rate curves 8.8 References 8.9 Exercises 235 235 246 246 248 253 255 255 258 260 265 267 270 272 274 Bibliography 279 Index 282 List of Tables 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Pseudocode for Midpoint Rule Pseudocode for Trapezoidal Rule Pseudocode for Simpson's Rule Pseudocode for computing an approximate value of an integral with given tolerance Pseudocode for computing the bond price given the zero rate curve Pseudocode for computing the bond price given the instantaneous interest rate curve Pseudocode for computing the price, duration and convexity of a bond given the yield of the bond 3.1 3.2 Pseudocode for computing the cumulative distribution of Z Pseudocode for Black-Scholes formula 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Pseudocode Pseudocode Pseudocode Pseudocode Pseudocode Pseudocode Pseudocode for for for for for for for the Bisection Method Newton's Method the Secant Method the N-dimensional Newton's Method the N-dimensional Approximate Newton's Method computing a bond yield computing implied volatility Xl 59 59 60 61 74 75 77 109 109 247 250 254 257 259 266 269 Preface The use of quantitative models in trading has grown tremendously in recent years, and seems likely to grow at similar speeds in the future, due to the availability of ever faster and cheaper computing power Although many books are available for anyone interested in learning about the mathematical models used in the financial industry, most of these books target either the finance practitioner, and are lighter on rigorous mathematical fundamentals, or the academic scientist, and use high-level mathematics without a clear presentation of its direct financial applications This book is meant to build the solid mathematical foundation required to understand these quantitative models, while presenting a large number of financial applications Examples range from Put-Call parity, bond duration and convexity, and the Black-Scholes model, to more advanced topics, such as the numerical estimation of the Greeks, implied volatility, and bootstrapping for finding interest rate curves On the mathematical side, useful but sometimes overlooked topics are presented in detail: differentiating integrals with respect to nonconstant integral limits, numerical approximation of definite integrals, convergence of Taylor series, finite difference approximations, Stirling's formula, Lagrange multipliers, polar coordinates, and Newton's method for multidimensional problems The book was designed so that someone with a solid knowledge of Calculus should be able to understand all the topics presented Every chapter concludes with exercises that are a mix of mathematical and financial questions, with comments regarding their relevance to practice and to more advanced topics Many of these exercises are, in fact, questions that are frequently asked in interviews for quantitative jobs in financial institutions, and some are constructed in a sequential fashion, building upon each other, as is often the case at interviews Complete solutions to most of the exercises can be found at http://www.fepress.org/ This book can be used as a companion to any more advanced quantitative finance book It also makes a good reference book for mathematical topics that are frequently assumed to be known in other texts, such as Taylor expansions, Lagrange multipliers, finite difference approximations, and numerical methods for solving nonlinear equations This book should be useful to a large audience: • Prospective students for financial engineering (or mathematical finance) xiii PREFACE xiv programs will find that the knowledge contained in this book is fundamental for their understanding of more advanced courses on numerical methods for finance and stochastic calculus, while some of the exercises will give them a flavor of what interviewing for jobs upon graduation might be like • For finance practitioners, while parts of the book will be light reading, the book will also provide new mathematical connections (or present them in a new light) between financial instruments and models used in practice, and will so in a rigorous and concise manner • For academics teaching financial mathematics courses, and for their students, this is a rigorous reference book for the mathematical topics required in these courses • For professionals interested in a career in finance with emphasis on quantitative skills, the book can be used as a stepping stone toward that goal, by building a solid mathematical foundation for further studies, as well as providing a first insight in the world of quantitative finance The material in this book has been used for a mathematics refresher course for students entering the Financial Engineering Masters Program (MFE) at Baruch College, City University of New York Studying this material before entering the program provided the students with a solid background and played an important role in making them successful graduates: over 90 percent of the graduates of the Baruch MFE Program are currently employed in the financial industry The author has been the Director of the Baruch College MFE Program1 since its inception in 2002 This position gave him the privilege to interact with generations of students, who were exceptional not only in terms of knowledge and ability, but foremost as very special friends and colleagues The connection built during their studies has continued over the years, and as alumni of the program their contribution to the continued success of our students has been tremendous This is the first in a series of books containing mathematical background needed for financial engineering applications, to be followed by books in N umerical Linear Algebra, Probability, and Differential Equations Dan Stefanica New York, 2008 Acknow ledgments I have spent several wonderful years at Baruch College, as Director of the Financial Engineering Masters Program Working with so many talented students was a privilege, as well as a learning experience in itself, and seeing a strong community develop around the MFE program was incredibly rewarding This book is by all accounts a direct result of interacting with our students and alumni, and I am truly grateful to all of them for this The strong commitment of the administration of Baruch College to support the MFE program and provide the best educational environment to our students was essential to all aspects of our success, and permeated to creating the opportunity for this book to be written I learned a lot from working alongside my colleagues in the mathematics department and from many conversations with practitioners from the financial industry Special thanks are due to Elena Kosygina and Sherman Wong, as well as to my good friends Peter Carr and Salih Neftci The title of the book was suggested by Emanuel Derman, and is more euphonious than any previously considered alternatives Many students have looked over ever-changing versions of the book, and their help and encouragement were greatly appreciated The knowledgeable comments and suggestions of Robert Spruill are reflected in the final version of the book, as are exercises suggested by Sudhanshu Pardasani Andy Nguyen continued his tremendous support both on QuantNet.org, hosting the problems solutions, and on the fepress.org website The art for the book cover is due to Max Rumyantsev The final effort of proofreading the material was spareheaded by Vadim Nagaev, Muting Ren, Rachit Gupta, Claudia Li, Sunny Lu, Andrey Shvets, Vic Siqiao, and Frank Zheng I would have never gotten past the lecture notes stage without tremendous support and understanding from my family Their smiling presence and unwavering support brightened up my efforts and made them worthwhile This book is dedicated to the two ladies in my life Dan Stefanic a IBaruch MFE Program web page: http://www.baruch.cuny.edu/math/masters.html QuantNetwork student forum web page: http://www.quantnet.org/forum/index.php New York, 2008 xv How to Use This Book While we expect a large audience to find this book useful, the approach to reading the book will be different depending on the background and goals of the reader Prospective students for financial engineering or mathematical finance programs should find the study of this book very rewarding, as it will give them a head start in their studies, and will provide a reference book throughout their course of study Building a solid base for further study is of tremendous importance This book teaches core concepts important for a successful learning experience in financial engineering graduate programs Instructors of quantitative finance courses will find the mathematical topics and their treatment to be of greatest value, and could use the book as a reference text for a more advanced treatment of the mathematical content of the course they are teaching Instructors of financial mathematics courses will find that the exercises in the book provide novel assignment ideas Also, some topics might be nontraditional for such courses, and could be useful to include or mention in the course Finance practitioners should enjoy the rigor of the mathematical presentation, while finding the financial examples interesting, and the exercises a potential source for interview questions The book was written with the aim of ensuring that anyone thoroughly studying it will have a strong base for further study and full understanding of the mathematical models used in finance A point of caution: there is a significant difference between studying a book and merely reading it To benefit fully from this book, all exercises should be attempted, and the material should be learned as if for an exam Many of the exercises have particular relevance for people who will interview for quantitative jobs, as they have a flavor similar to questions that are currently asked at such interviews The book is sequential in its presentation, with the exception of Chapter 0, which can be skipped over and used as a collection of reference topics XVll xviii HOW TO USE THIS BOOK Chapter Mathematical preliminaries Even and odd functions Useful sums with interesting proofs Sequences satisfying linear recursions The "Big 0" and "little 0" notations This chapter is a collection of topics that are needed later on in the book, and may be skipped over in a first reading It is also the only chapter of the book where no financial applications are presented Nonetheless, some of the topics in this chapter are rather subtle from a mathematical standpoint, and understanding their treatment is instructive In particular, we include a discussion of the "Big 0" and "little 0" notations, i.e., 0(·) and 0('), which are often a source of confusion 0.1 Even and odd functions Even and odd functions are special families of functions whose graphs exhibit special symmetries We present several simple properties of these functions which will be used subsequently Definition 0.1 The function f : ~ -7 ~ f( -x) = f(x), is an even function if and only if Vx E ~ (1) The graph of any even function is symmetric with respect to the y-axis Example: The density function f (x) of the standard normal variable, i.e., MATHEMATICAL PRELIMINARIES is an even function, since 0.1 EVEN AND ODD FUNCTIONS For example, the proof of (4) can be obtained using (2) as follows: f(-x) = - e _(_x)2 I: = f(x); V2ir j(x) dx = lim o see section 3.3 for more properties of this function t-+-oo t f(x) dx = lim (t f(x) dx = t-+oo Hm t-+-oo roo Jo Jo l-t f(x) dx f(x) dx Lemma 0.1 Let f(x) be an integrable even function Then, 1: I: and therefore Moreover, if j( x) dx Jooo f (x) j (x) dx = = 1" 1" D (2) j (x) dx, V a E R., f(-x) j (x), V a (3) E R j(x) dx = and f' f' j(x) dx = 1-a r Ja = f( -v) (-dy) = r Jo j(x) dx = 1" j(x) dx = 1: E JR (7) r Jo + j(x) dx = 0, V a E R (8) exists, then 1: j(x) dx = o (9) Proof Use the substitution x = -y for the integral from (8) The end points x = -a and x = a change into y = a and y = -a, respectively, and dx = -dy Therefore, = f(y) dy, (6) since f( -V) = - f(y); cf (7) Since y is just an integrating variable, we can replace y by x in (10), and obtain that t j(x) dx We conclude that j(x) dx = - f(x), V x a f( -v) dy Then, t Jooo f (x) dx (5) j(x) since f( -V) = f(y); cf (1) Note that y is just an integrating variable Therefore, we can replace y by x in (6) to obtain 1: I: Moreover, if a O f(x) dx JR is an odd function if and only if If we let x = in (7), we find that f(O) = for any odd function f(x) Also, the graph of any odd function is symmetric with respect to the point (0,0) (4) j(x) dx, Proof Use the substitution x = -y for the integral on the left hand side of (2) The end points x = -a and x = change into y = a and y = 0, respectively, and dx = -dy We conclude that -+ Lemma 0.2 Let f(x) be an integrable odd function Then, dx exists, then I: I: Definition 0.2 The function f : JR 1" j(x) dx = 21" j(x) The results (4) and (5) follow from (2) and (3) using the definitions (2.5), (2.6), and (2.7) of improper integrals j(x) dx = - I: t j(x) dx j(x) dx = O The result of (9) follows from (8) and (2.10) D 174 CHAPTER MULTIVARIABLE CALCULUS Let f(x) = ln~x) with f : (0 , 00) • JR Then f'(x) = 弓凶 The function f (x) has one crit iC al point correspondi吨 tox = 飞 IS 1肌reasi吨 on the interval (0 , e) and is decreasing on the interval (e ， ∞) We conclude that f (x) has a global maximum point at x = e, i.e , f (x) < f (e) = ~ for all x > with x 护民 and therefore In other words， ω( x) is a decreasi吨 function on the 凶 i nt怡 er凹 va 址I [归 ，∞) an 丑l therefore ω(0) 三四 (x) for all x 主 Since ω(0) = M , and using (7.23) , it follows that M:: : (M + 1" u(t)v(叫叫- 1" v(t) dt) ° ln(作) 忡忡 -r correspond to for the fu日ction u(x , r) defined as follows: V(8 , t) = exp( 一 αx - br)u(x , r) , where x = In I飞 functionω: [0 ，∞)→ [0 ，∞)as \1111/ /III-\ Z z /III-\ 丑ecall ° 也(x) 三 M 叫1" v(t) dt) , Solution: Define the 1" 口 Problem 3: Let 叩) : [0 ，∞)→ [0 ，∞) be two continuous functions with positive values Assume that there exists a constant M > such that u(x) 三 M + 175 7.3 SOLUTIONS TO SUPPLEMENTAL EXERCISES θν2\ 2) T θu rF = ° CHAPTER MULTIVARIABLE CALCULUS 176 Solution: Using chain rule , it is easy to see that 7.3 SOLUTIONS TO SUPPLEMENTAL EXERCISES which is what we wanted to show 177 己 θVθFInSθF δtθt T θν7 δV 1θF θI T δV T-t θF θu? δS S T θ2V θν? ( (T - t)2 θ2F 一一_ θS2 Problem 7: For the same maturity, options with different strikes are traded simultaneously The goal of this problem is to compute the rate of change of the implied volatility ωa function of the strike of the options In other words , assume that S , T , q and r are given , and let C(K) be the (know叫 value of a call option with maturity T and strike K Assume that options with all strikes K exist Defi 缸 fine 也 t he implied volatility σ 叽inη1以 K) 创 as the unique solution to C(K) = CBs(K， σimp(K)) ， S2 飞 T-t θF\ 一一一一 T2 θy2 T θy) Then , the PDE (7.27) for V(S , I , t) becomes the followi鸣 PDE for F(y , t): o= θVθ V 一一 + ')~')护 VθV 1口 S 一一 +一 σL SL 一一τ +γS一一 ' 2- - 8S θS rV θ t θI θ FInS θF 一一一一一一一十 θt T θy 1δF lnS~~ -T θσimp(K) θu 十 1σ2 i( (T - t)Z 一一一一一一一一一…一一一 θ2F T-t θFV+T-tθF I + r 一一一一一一一一 2\ T2 δy2 T θF σ2(T - t)2 θ2F , 一一+ n ~ ~十 θt' 2T2 where CBs(K， σ 叽仰nη以 阮cl挝 S hoωoles 臼s value of a call 叩 pt挝io ∞ n with strike K on an unde 臼rl勾i坊抖 y 甘切 切 rin 吨 1鸣 ga 部ss附 创t ，f'01 e 扣如 low 叭in 吨 ga log 伊 nω10ωr m 口 口 r丑na 1恼al 叫 model 呐 w it由 h volatility σ叫 (K) Find an implicit differential equ 任 tion satisfied byσ叫 (K) ， i.e ，缸d θy) , • T θu - ~ (σ2\ T-t θF Ir 一 -=::-1 一一一一一 - rF 8γ\ 2) T θu θK 邸saf缸 a un 旧ct钊io ∞丑 of σ 叽imη1以 K) 口 Solution: We first 自nd the partial derivative of the Black-Scholes value CBs(K) of a call option with respect to its strike K Recall that CBs(S , K) = Se- qT N(d ) Problem 6: One way to see that American calls on non-dividend-paying assets are never optimal to exercise is to note that the Black-Scholes value of the European call is always greater than the intrinsic premium S - K , for S>K Show that this argument does not work for dividend-paying assets In other words , prove that the Black-Scholes value of the European call is smaller than S - K for S large enough , if the underlying asset pays dividends continuously at the rate q > (and regardless of how small q is) Solution: We want to show that , if the dividend rate of the underlying asset is q > 0, then CBS(S , K) < S - K for S large enough Note that CBS(S , K) = Se- qT N( d1 ) - K e- rT N( d2) < Se- qT , since N(d ) < and N(d 2) > O If Se- qT < S - K , which is equiv，由lt to S > 亡ιT > since q > 0, it follows that CBs(氏 K) < S - K We conclude that CBS(S , K) < S - K , V S > 4K 川 Ke- rT N(d 2) - Then , θd θCBS 一币 俨 EJd Se一句'(d ) ;~一严 N(d ) - Ke- r :J.'N'(d ) 坛 θK Also , recall that Se- qT N'(d ) Ke- rT N'(d 2); - (7 到 (7.29) d Lemma 3.15 of [2] From (7.28) and (7.29) , we find that 到C 'R.c: #二 Since d ~'T' _ _, • , ~~ rr ~ TI I (8d 8ι\ - e- rT N(d2 ) 十 Ke一叩'(也) \拔一括) 工 d 十 σ叮， _", " (7.30) it follows that θd θd θK θK We co肌lude from (7.30) tl川 θCBS θK - e- rT N(d ) (7.31 ) CHAPTER MULTIVARIABLE CALCULUS 178 We now differentiate with respect to K the formula C(K) CBs(K， σ问 (K)) = θC θK δCBS IθCBS 一一一· θK θσθK (7.32) where θCBS _-aT Lagrange multipliers N- dimensional Newton's method Implied volatility Bootstrapping I T_ '!J 万 =Se-qVV=dse-PNf(do C1 V'T Ke- rT N'(d ) - K产品生 d (η9) We conclude that the implied differential equation (7.32) can be written as IT VK with e +iu r θσimp(K) -e一句(d2 ) + 鸣a(CBs) 叫ffk)? vega(CBs) ch ap 民U which is the de直缸缸 旧 nl让i扰 山tior丑1 of σ 町仰mη1以 K) Note that C (K) is assumed to be known for all K , as it the market prices Using Chain Rule and (7.31) we find that 一兰 θσimp(K) θ-y- = 在 + e一叩 (d ) ， dry = In (去)十 (r - q)T 一切 (K)VT h (j imp(K)VT 2' ~ 8.1 Solutions to Chapter Exercises Problem 1: Find the maximum and mi山num ofthe function f(Xl' X2 ， 句)= 4X2 - 2X3 subject to the constraints 2X1 - X2 - X3 = and xI + x~ = 13 Solution: We reformulate the problem as a constrained optimization problem Let f : JR3 • JR and : JR3 • JR be defined as follows: 削 = 4X2 - 2叫仰) = (专JZ二3) where x = (X l, X2 , X3) We want to 五nd the maximum and minimum of f (x) on JRδsubject to the constraint g(叫 =0 We 自rst check that ra出(飞7 g(x)) = for any x such that g(x) = O Note that \1 g(x) = (2;，二 en It is easy to see that rank( \7 g( x)) = 2, unless Xl = X2 = 0, in which case g(x) 并 O The Lagrangian associated to this problem is F(x ， λ) 4X2 - 2♂3 十 λ1(2x1 - X2 - X3) 十七 (xr + x~ - 13) , whereλ=(λ1 ， λ2)t εJR2 (8.1) is the Lagra口ge multiplier We now find the critical points of F ( x ， λ) Let 均 x 0=(阳 均0 ， ， X协 Zωa Z 缸 a丑 n1 沁0=(λ λ 沁0，1 ， λ 沁O叩 ω ，2公) From (8.1) it follows that \7 (x，) )F(xo， λ0) = is equivalent 179 180 CHAPTER LAGRANGE MULTIPLIERS NEWTON'S JVIETHOD eη 飞八 ZZ 缸， 队，川 -3 "ZZ 在-·龟， ， hm 12132 ZZ mm 一一十= λλJzll 十十 22J4 句B A - - 矶， 认入川， ;「 (ii) Find the asset allocation for a maximum expected return portfolio with standard deviation of the rate of return equal to 24% 0; 0; 0; 13 Solution: For i 二 : , denote by Wi the weight of asset i in the portfolio Recall that the expected value and the variance of the rate of return of a portfolio made of the four assets given above are , respectively, 、 、E E， ， ， f'51 飞 。，血 。δ 才t牛 nuO& 一­ 飞八 门4 一一 巧t 人 吼叫 一 一一 nu 4t4 and XO ,l = -2; XO ,2 = 3; XO ,3 = -7;λ。， = -2;λ0， = -1 (8.3) For the 在rst solution (8.2) , we compute the Hessian D 月 (x) of 凡 (x) = f(x) + λ切 (x) ， i.e , of 凡 (x) - 4x2-2x3-2(2x1-x2-x3)+xi+x~-13 xi+x~-4x1 +6x2-13 and obtain I 0 飞 I \ 0 I which is (semi)positive definite for a町 Z εJR3 This allows us to conclude directly that the point (2 ，一 ， 7) is a minimum point for f(x) Note that f(2 , -3 , 7) = -26 Similarly, for the second solution (8.3) , we 自nd that 月 (x) - xi - x~ - 4X1 = I -2 0 飞 。 一-2 I 飞 o 0 I I which is (semi)negative de主nite for any x 巴 JR3 We conclude that the point (-2 ， ，一 7) is a maximum point for f (x ) Note that f (… , 3, -7) = 26 口 Problem 2: Assume that you can trade four assets (and that it is also possible to short the assets) The expected values , standard deviatio风 and correlations of the rates of return of the assets are: μ1 μ2 μ3 = μ4 = 0.08;σ1 0.12;σ2 0.16;σ3 0.05;σ4 - 0.25; P1 ,2 = - 0.25; - 0.25; P2 ,3 - - 0.25; - 0.30;ρ1 工 0.25; - 0.20; PiA - 0, V i = : (8.4) (8.5) 十2 (ω1ω伊1σ2P1 ，2 十 ω2ω3σ2{J3P2 ，3 十 ωlω3σlσ3P1 ，3) ， = for i = : We not require the weightsωi to be positive , i e , we allow taking short positions on each one of the assets However , the following relationship between the weights must hold true: si口ce Pi ,4 ω1 十 ω2 十 ω3+ω4 = (8.6) (i) We are looki吨 for a portfolio with given expected rate of return E[R] = 0.12 and minimal variance of the rate of return Using (8 4-8.6) , we obtain that this problem can be written as the following constrained optimization problem: 且ndω° such that gErof(ω) = f(ω0) ， + 6X2 + 13 and D 几位 ) E[R] = ω1μl 十 ω2μ2+ω3μ3 十 ω4μ4; var(R) = ω?σ? 十 ω2σ~+ω;σ; 十 ωiσ~ I D 凡 (x) 工 181 (i) Find the asset allocation for a mi山nal variance portfolio with 12% expected rate of return; 000020·7 。; hL ，，" ， 一­ 4EA diU2' 们向 Cο O EflJll σba WUW 1μe 4ιu·τ···4 ρUQU ωt f w u H M 4L m h z-7 'b s QU m m E S 8.1 SOLUTIONS TO CHAPTER EXERCISES 呐咄 w t扭ler陀e ω (8.7) =( ωi) 山 )i= 吐= f(ω 叫) = 0.0625ωi + 0.0625ω;+0Oω9ω:+O 04tωρi 一 0.03125w1W2 - 0.0375ω2ω3 十 0.0375ω1ω3; (8.8) 飞 /ωl 十 ω2 十 ω3 十 ω4- g(ω) = \ 伽1 十 O 山2 + 0.16ω3+ 伽4 一- 0.12 ) (8.9) It is easy to see that rank( \7 g(ω)) = for 缸lY ωεJR ， since The Lagrange multipliers method can therefore be used variance portfolio to 丑nd the minimum 182 CHAPTER LAGRANGE MULTIPLIERS NEWTON'S METHOD 8.1 SOLUTIONS TO CHAPTER EXERCISES Denote by λ1 and λ2 the Lagra且ge multipliers From (8.8) and (8.9) , we obtain that the Lagrangian associated to this problem is Note that the D Fo(w) is equal to twice the covariance matrix of the rates of return of the four assets , i.e , F(ω7 人) = 0.0625ωi + 0.0625ω2+O 09吟 +0.04ω2 - 0.03125ωlω2 一 0.0375ω2ω3 +λ1 (ω1 十 ω2 十 ω3+ω4 +λ2 (0.08ω1 The gradient of the Lagra吨ian \7(叫) F(ω?λ)= + (8.10) D2 0.0375ω1 W3 - 1) 0.08ω4+λ1 十 0.05λ2 ω1 十 ω2 十 ω3 十 ω4- 0.08ω1 十 0.12ω2 + 0.16ω3 十 0.05ω4 - 0.12 To 自nd the critical points of F(叽 λ) ， we solve \7 (w ，λ) F(ω?λ) = , which can be written as a linear system as follows: 0.125 一 0.03125 0.0375 0.125 -0.0375 。 。 l 0.12 0.08 0.0375 O -0.0375 。 0.18 O O 0.08 l 0.16 1 1 0.08 0.12 0.16 0.05 。 。 0.05 。 也)1 000σ~ J We co叫ude that D 2Fo (ω) is a positive definite matrix Therefore , the associated quadratic form q( v) = v t D Fo(wo)v is positive definite , and so will be the reduced quadratic form corresponding to the linear constraints 飞7g(ω。 )υ=0 We conclude that the point ωo = (0.1586 4143 0.3295 0.0976) is a constrained minimum for f (ω) given the constraints g(ω) = O The portfolio with 12% expected rate of return and minimal variance is invested 15.86% in the first asset , 43% in the second asset , 32.95% in the third asset , and 9.76% in the fourth asset The minimal variance portfolio has a standard deviation of the expected rate of return equal to 13.13% 包)3 W4 /σ?σ1 0"2P1 ，2 σ1σ3P1 ，3 0\ |σlσ3P1 ，3 σ2σ3P2，3 O~ M=Iσ向P1，2 λ1 λ2 一 (8.11 ) / 0.1586\ I 0.4 143 I 0:3295 I;λω= 0.0112;λ0.2 =一 0.3810 \0.0976 / , σ§ \000σ1 μtω; E[R] var(R) ω tMω7 J J (8.12) (8.13) -iqAnOA \飞 -litz-1/ 也 μ'μ'μ'μ' μ' //IIll 飞 where 一一 凡 (ω) = 0.0625叫 + 0.0625w~ + 0.09咛 + 0.04wl - 0.03125w1 ω2 - 0.0375w2W3 + 0.0375w1W3 + 0.0112 (ωl 十 ω2+ω3 十 ω4 - 1) 一 0.3810 (0.08ω1 + 0.12ω2 + 0.16ω3 + 0.05ω4 - 0.12) , O"~σ2σ向3 ~ I the covariance matrix of the rates of return of the four assets Let σp = 0.24 be the required standard deviation of the rate of return of the portfolio If Wi denotes the weight of the asset i in the portfolio , i = : , it follows from (8 4) and (8.5) that Let 月 (ω) = F(ω7λ0，1， λ0.2) ， i.e , and compute its Hessian σ:3 0\ ~ I U I (ii) Denote by W2 The solution of the linear system (8.11) is ωo = iσlσ3ρ1 ， 3σ2σ3P2 ， + 0.12ω2 + 0.16ω3 + 0.05ω4 - 0.12) 0.125w1 - 0.03125ω2 + 0.0375ω3+λ1 十 0.08λ2 、 t 0.125ω2 - 0.03125ω1 - 0.0375ω3+λ1 + 0.12λ2 0.18ω3 十 0.0375ω1 - 0.0375ω2 十沁十 0.16λ2 一 0.03125 / σ? σ1σ2ρ口内 σ3P1 ，3 月 (ω ) = 21σ1σ2P1， 作 σ2匀2，3 飞 is the following (row) vector: 183 is the vector of the expected values of the rates of return of the four assets The problem of 五日ding a portfolio with m以imum expected rate of return and standard deviation of the rate of return equal to σp can be formulated as a constrained optimization problem as follows: find Wo such that ~~n_ f(ω) = f(ω。)， g(ω)=0 (8.14) 184 CHAPTER LAGRANGE MULTIPLIERS NEWTON'S METHOD 8.1 where ω = (Wi)仨吐 However , it is easy to see that f(ω) (ωtkLz)7 σp (8.16) We can now proceed with finding the portfolio with maximum expected return using the Lagrange multipliers method Denote by λ1 and λ2 the Lagrange ml削pliers From (8.15) and (8.16) , we obtain that the Lagrangian associated to this problem is where 0) F(ω7λ) =μtω+λ1 (Itω-1) 十 λ2(ωtMω - (J'~) G(ω?λ1 ， λ2) C σ2 P ) It (川 飞7(叫) G(ω?λ1， λ2) 2(Mω )t ω wt !VIω - (J'~ \飞 III-f/ l t M- 1 (μ+λ11 + 2川) Itω-1 , This is done using a six dimensional Newton's method; note that the gradient of G(叽 λ1 ， λ2) is the followi吨 x matrix: M00 -inunu We find that the Lagrangian (8.21) has exactly one critical point given by (8.20) From (8.19) and (8.20) , we find that , ifthere exists ωεJR4 such that g(ω) = and ra出 (\7g(ω)) = 1, then 11111/ C \飞 C 孚 = ~1~ω= 2 JR6 is given by 。白 乎 ~~ ω 守J C nL 。 r 飞八 1if σ各=一 ωι1 ω 中=丰〉 ?b wtMω=σ主 (8.19) ， 1=;山一11; 、‘ A气 ， {=中 μ' =1 一一 (8.18) • From (8.16) it follows that , if g(ω) = , then l tw = and wtMw 工作­ Usi吨 (8.18)， we find that l tw 飞八 where G : JR6 ;M九 2-UM" M2p 111 \7 (ω，，\) F(ω7λ) = , G(ω?λ1 ， λ2) In order to use the Lagrange multipliers method for solvi吨 problem (8.14) , we first show that the matrix \7 g(ω) has rank for any ωsuch that g(ω) = O Note that ra此 (\7g(ω)) = if and only if there exists a constant C εR such that 2!VIw = C Using the fact that the covariance matrix !VI of the assets considered here is nonsingular , we obtain that w /1 、、 飞， I/ /'41 、、 。" '?b \l/ 直缸缸时 r1 \飞 III/ !ω rM F ω To /III-\ ω (8.17) 飞A 2(Ax)t = it is easy to see that \7g(ω) /III-11\ 节V (去无) (8.21) The gradient of the Lagrangian is 十 Recall that , if the function h : JRn • JR is given by h(x) = xtAx , where A is an n xηsymmetric square matrix , then the gradient of h( x) is Usi吨 (8.17) ， 185 山飞= 80.01 并 17.36 二 jf (8.15) g(ω) Dh(x) SOLUTIONS TO CHAPTER EXERCISES Wo (om) 0.6450 0.6946 J' -0.3503 λ0 ， = -0.0738;λ0.2 = -0.8510 Let 月 (ω) = F(ω7λ0 ， ， λ0.2) ， i.e , 几 (ω) μtω - 0.0738(l t ω - 1) - 0.8510(ωtMω - (J'~) 186 CHAPTER LAGRANGE 1VIULTIPLIERS NEWTON'S METHOD 8.1 SOLUTIONS TO CHAPTER EXERCISES The Hessian of Fa (ω) is where B = 100 +去 and y = 0.03340 We obtain that the duration of the bond is 4.642735 and the convexity of the bond is 22.573118 口 D 凡 (ω ) = - 0.8510· 2JvI 187 = - 7019M Since the covariance matrix M of the rates of return of the four assets is a positive definite matrix , it follows that D Fo (ω) is a negative defi时te matrix for any ω Therefore , the associated quadratic form q( v) = v t D Fo (ωo)v is negative definite , and so will be the reduced quadratic form corresponding to the linear constraints \7 g(ω。 ) v = O We conclude that the point ωa = (0.0107 0.6450 0.6946 - 0.3503) is a constrained maximum for f (ω) given the constraints g(ω) = O The portfolio with 24% standard deviation of the rate of return and maximal expected return 1.07% in the first asset , 64.50% in the second asset , 69 46% in the third asset , while shorting an amount of asset four equal to 35.03% of the value of the portfolio For example , if the value of the portfolio is $1 ,000 ,000 , then $350 ,285 of asset is shorted (borrowed and sold for cash) , $10 ,715 is invested in asset , $644 ,965 is invested in asset , and $694 ,604 is invested in asset This portfolio has an expected rate of return equal to 17.19% 口 Problem 4: Recall that 缸ding the implied volatility from the given price of a call option is equivalent to solving the nonlinear problem f (x) = , where f(x) = Se- qT N(d (x)) - Ke- rT N(d (x)) - C and d1 (x) 一叫圣) (叫+手)T - -\1' / +xJr"':l r ,d (x) = 叫到十(叫一手)T xJT \1'>./ (i) Show that limx~∞ d (x) = ∞ and li皿z→∞ d (x) = 一∞， and conclude that lim f(x) = Se 一qT _ C (ii) Show that {一∞， ItIr! d (x) = ItIr! d (x) = { 、。 叭。 I 0, ∞， if Se(r-q)T < K; if Se(r-q厅 K; if Se(r-q)T > K (Recall that F = Se(r-q)T is the forward price.) Conclude that Problem 3: Use Newton's method to find the yield of a five year semiannual coupon bond with 3.375% coupon rate and price 100 古 What are the duration and convexity of the bond? Solution: Nine $1 6875 coupo丑 payments are made every six months , and a final payment of $101.6875 is made after years By writing the value of the bond in terms of its yield , we obtain that /IIiI\ , 时i nu 42·A 吃$ "U 1-9" exp 十 寸t4 noQO \飞 III/' 9ZM vhυ 巧t 1-m 一- 寸t4 + nunu nhuQO vhu exp Uu vhu (8.22) We solve the nonlinear equation (8.22) for y using Newton's method With initial guess Xo = 0.1 , Newton's method converges in four iterations to the solution y = 0.03340 We conclude that the yield of the bond is 3.3401 % The duration and convexity of the bond are given by D ~ (飞J、 Z16吨 exp B ·2 (iii) Show that if Se(r-q)T -C 三 K; > K f (x) is a strictly increasi吨 function and -C < f(x) < Se- qT - C , if Se(r-q)T 三 K; Se- qT - Ke- rT - C < f(x) < Se- qT - C, if Se(叫)T > K (扣) For what range of call option values does the problem f(x) = have a positive solution? Compare your result to the range Se- qT - Ke- rT < C < Se- qT required for obtaining a positive implied volatility for a value C of the call option Solution: (i) Note that = ;(主 1ω巾 C = ( 1Se- qT - K产- C, if Se(r-q)T lV(叫- d1 ‘ ltt/\ 飞Z \/飞)、 + 10 6875· 25 exp(一切) )、IJ‘ , (x) Z 出 (x) = In (丢) + (γ - q)T , x vIT ZZ飞IT 十 In (圣) + (r - q)T x vIT x vIT (8.23) (8.24) 188 CHAPTER LAGRANGE MULTIPLIERS NEWTON'S METHOD 8.1 SOLUTIONS TO CHAPTER EXERCISES It is easy to see that (iii) Differentiati 吨 f (x) with respect to x is the same as computing the derivative of the Black-Scholes value of a European call option with respect to the 飞Tolatility 矶 which is equal to the vega of the call In other words , l豆豆 d1(x) = ∞ J豆豆 d (x) = 一∞? and and therefore f' (x) We conclude that lim f(x) - In (去) xVT = 一节言十一一; xvT I Se- qT - C ' 也 (x) In (去) L/Tr -ln( 是) + (r+手)T where d l = d d z ) - d Thus , f'(x) > , \:j x > 队 and f (x) is strictly increasi吨­ Recall that limx -t∞ f(x) = Se- q'1' - C and (ii) Let F = Se(r-q)T be the forward price From (8.23) and (8.24) it follows that d1 (x) and d2 (x) can be written as d1(x) =仇一听「Le一手 飞/ J马 N(d (x)) = 出 N(ddz))=laad = vega(C) 189 xVT xVT • If F < K , then In (去) < and r 炖忡忡 Since f (x) is strictly increasi吗， - we co叫ude C < f(x) < < K: F 二 K7 if F if C, that Se- qT - C , if F 三 K; C, if F > K (iv) If F 三 K ， the problem f(x) = has a solution x > if and only if Ther陀 e£扣 ore 且 I im♂\oN(何 出l(归 d x)) 工 1且im 八 也2(μ d x)) 工 Oa丑 x \ oN(伺 :炖咆 f 仲(归x)尸巳一 C -C K;~rT -C < f(x) < Se- qT Se- qT - Ke- rT - :坦白(z)zESd2(z)= 一∞ Se- qT - 。< C < Se一ι(8.25) If F > K , the problem f(x) = has a solution x > if and only if Se- qT - Ke- rT • If F = K , 出nd1(x)= 孚 and 也 (x) 口 一千， 缸aI丑侃圳 l叫d时 山阳re伽 缸 b f扣 ωr O臼 < C < Se-qT (8.26) Note that :炖巧 d出1叫巾(归 仰 Z) = :炖坷 d 也2纣 州(x)忏 =0 Se- qT - Ke- rT Thus ,1im x\ a N(d1(x)) = lim八a N(d (x)) = ~ and 一俨r w(Z)=;(SfqT-ke-TT)-czET(F-K)-c = e-rT(Se(叫)T _ K) From (8.25) and (8.26) , we conclude that the problem f(x) = has a positive solution if and only if C belongs to the following range of values: r max (Se一 qT 一 Ke-气 O吗) = -C = e-rT(F - K) < C < Se- qT • If F > K , then In (去) > and ~t~ d1 (x) = l~~ d2 (x) = ∞- z \v X \.U Therefore li叫\\\、、、oN \V(付 出l(仪 d x)川) = 且 I im♂八\oN(何 也2(归 d x)) = a丑 :炖巧 f 仲 (μ x)忏 =S 仇 仇 s f旨e一-qT 一 Ke巳f ♂一刊 T叮 T 一 C Problem 5: A three months at-the-money call on an underlying asset with spot price 30 paying dividends continuously at a 2% rate is worth $2.5 Assume that the risk free interest rate is constant at 6% (i) Compute the implied volatility with six decimal digits accuracy, using the bisection method on the i口terval [0.0001 , 1]' the secant method with initial guess 0.5 , and Newton's method with initial guess 0.5 190 CHAPTER LAGRANGE (ii) Let MULTIPLIER丘 NEWTON'S METHOD σ问 be the implied volatility previously computed method Use the formula σ usi吨 Newtor内 r-.JV2在 C 一与庄s r-.J百万 imp ,approx to compute an approximate value compute the relative error σimp， apprωfor the implied volatility, and |σzmp， αpprox 一 σ vzmp σimp Solution: (i) Both the secant method with ♂ -1 = 0.6 and Xo = 0.5 and Newton's method with initial guess Xo = 0.5 converge in three iterations to an implied volatility of 39.7048% The approximate values obtained at each iteration are given below: k Secant :Nlethod Newton's Method 。 0.5 0.3969005134 0.3970483533 0.3970481868 0.5 0.3969152615 0.3970481867 0.3970481868 [0.375063 , 0.5]; [0.375063 , 406309]; zmp，α，pprox 一 σ| v zmp I 巳 0.000948 = 0.0948% 口 σzmp ( xf + 2XlX2 十 zi-2223 十 9\ ( 2XI + 2XlX~ 十 zizi-ziz3 一 \ Z问仇 +zi-23-zd-4 ' J/ f \ and x' = for Xo = I I 飞 J (-n 扣r Xo 工 0) The iteration counts are given in the table below: Iteration Count Iteration Count Iteration Cou日t Approximate 时ewton Approximate Newton Newton's Method Central Differences Forward Differences ( 1\ {~ } 9 40 65 43 \ Problem 6: Let F : :lR3 • :lR3 given by F(x) - , I 9831072429 I 飞 -0.8845580785 J 旷工 Xo 39.6672% If σimp = 0.3970481868 is the implied volatility obtained using Newton's method , then |σ Fi(x + hej) 2h - -Fi(x - ' J/ hej) , j = :风 £\ ~£\~ Solution: We use Newton's method and the approximate Newton's method both with forward difference approximations and with cen~ral difference approximations with toLconsec = 10- and toLapp r