More Mathematical Finance

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The long-awaited sequel to the "Concepts and Practice of Mathematical Finance" has now arrived. Taking up where the first volume left off, a range of topics is covered in depth. Extensive sections include portfolio credit derivatives, quasi-Monte Carlo, the calibration and implementation of the LIBOR market model, the acceleration of binomial trees, the Fourier transform in option pricing and much more. Throughout Mark Joshi brings his unique blend of theory, lucidity, practicality and experience to bear on issues relevant to the working quantitative analyst....

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  1. PWP The long-awaited sequel to the "Concepts and Practice of ZRm Mathematical Finance" has now anived. Taking up where the first volume left off, a range of topics is covered in depth More Extensive sections include portfolio credit derivatives. quasi- Monte Ca rlo , the calibration and implemen t.l tion of 由 e L1 BOR market model , the acceleration of binomial trees ,由 e 之 Mathematical Fourier transform in option pricing and much more ω Throughout Mark Joshi brings his new unique blend of 同E 由 eo町, lucidity, practicality and experience to bear on Issues Finance relevant to 由 e working quanti t.ltive analyst. m E 巳ω 2 Praise for the Concepts and Practice of Mathematical Finance: - 响 Mark Joshi 咱 vershadows many 。由 er books available on the same subject" - 言 ω ZentralBlall Math = "Mark Joshi succeeds admirably - an excellent s 臼 rting point for a n numerate person in the field of mathematical finance."·· Risk m Magazine "Very few books provide a balance between financialtheory and ] practice. This book is one of the few books that strikes 由 at OME balance." .. SIAM Review ISBN 978-0-9871228-0-3 90000 > ii - -nr-o- i-‘ i一白 - 一 -c3- 一 - -VA- - 舍 W 、 白 n F EZ-、 、 r d E ­ - 、 一 一 J - - 一 一 •
  2. PILOT WHALE PRESS 如felbourne www.markjosh i. com @Mark Suresh Joshi 2011 Contents This publication is in copyright. Except where permitted by law, Preface no reproduction of any part may take place without the written X lll permission of the copyright holder. 1. , Optionality, convexity and volatility t Chapter I s 1.1. Introduction i t First published 2011 - 1.2. Volatility and convexity 句 1.3. Convexity and optionality 3 1.4. 吁 Is convexity necessary? / 0 0 1. 5. Key points 0 0 0 1. 6. Further reading 0 1.7. Exercises n Chapter 2. Where does the money go? y n National Library of Australia Cataloguing-in-Publication entry: y 2.1. Introduction n υ A 2.2. The money bleed t " i t 2.3. Analyzing the examples i .. Author: Joshi,岛1ark, - S. - 2.4. Volatility convexity and the existence of smiles 句 I 1 1 2.5. Key Points 句 Title: 1 , More mathematical finance / Mark S. Josh i. 1 白 1 2.6. Further reading 付 1 / M 2.7. Exercises 呵 ISBN: , 9780987122803 (hbk.) " 222 Chapter 3. The Bachelier model 4 Notes: 3 Includes bibliographical references and index. 3.1. Introduction 呵 3 3.2. The pricing formula 呵 3 Subjects: Finance-Mathematical models. / O 3.3. Approximations and comparisons Business mathematics. 3.4. Key points 吁 / 叮 3.5. Further reading I Dewey Number: 332.0151 3.6. Exercises 竹 / 今 AYAY Chapter 4. Deriving the Delta , 由 ? 4.1. Introduction " ? " 4.2. The stock measure
  3. V CONTENTS CONTENTS iv 73 4.3. Homogeneity 30 Computing the loss distribution in a single-factor model 8.3. 4 .4. 74 Other cases 31 Turning loss distributions into prices 8.4. 76 4.5. Key points 32 Stochastic recovery rates 8.5. 77 4.6. Further reading 32 The Fourier transform approach 8.6. 78 4.7. Exercises 32 Bucketing 8.7. 79 Key points 8.8. Chapter 5. Volatility derivatives and model-free dynamic replication 33 80 Further reading 8.9. 5.1. Introduction 33 80 Exercises 8.10. 5.2. Variance swaps 34 81 Chapter 9. Implied correlation for portfolio credit derivatives 5.3. Pricing general volatility derivatives 36 81 5 .4. Hedging a volatility derivative 38 9. 1. Introduction 82 5.5. Key points 40 9.2. Implied correlations 85 5.6. Further reading 40 9.3. Base correlation 87 5.7. Exercises 40 9 .4. Mapping methodologies 89 9.5. Hedging and the computation of Greeks Chapter 6. Credit derivatives 41 91 9.6. Key points 6.1. Introduction 41 92 9.7. Further reading 6.2. The basic instruments 42 92 9.8. Exercises 6.3. The philosophy of pricing credit derivatives 46 93 Chapter 10. Alternate models for portfolio credit derivatives 6 .4. Hazard rates 48 93 10.1. Introduction 6.5. Pricing simple credit instruments 50 95 10.2. Random factor loadings 6.6. Key points 50 100 10.3. Elliptic copulas 6.7. Further reading 51 102 10.4. Multiple default processes 6.8. Exercises 51 104 10.5. Intensity Gamma Chapter 7. The Monte Carlo pricing of portfolio credit derivatives 111 53 10.6. Key points 7. 1. Introduction 111 53 10.7. Further reading 7.2. The Li model 111 54 10.8. Exercises 7.3. Importance sampling for basket default swaps 56 113 Chapter 11. The non-commutativity of discretization 7 .4. Tranched CDOs by Monte Carlo 59 113 1 1.1. Introduction 7.5. The default density in the Li Model 62 113 11.2. Discretization and risk-neutrality 7.6. The likelihood ratio method for basket credit derivatives 63 117 11. 3. Discretization and Greeks 7.7. The pathwise method for nth-to-default swaps 65 120 1 1.4. Factor reduction 7.8. Key points 67 122 11. 5. Importance sampling 7.9. Further reading 67 123 11.6. Coordinate changes 7.10. Exercises 68 124 11.7. Calibration 127 Chapter 8. Quasi-analytic methods for pricing portfolio credit derivatives 71 11. 8. Key points 127 8.1. Introduction 71 11.9. Further reading 127 8.2. The loss distribution for independent defaults 72 11. 10. Exercises
  4. CONTENTS Vll CONTENTS VI 0 557 l111 0 15. Quasi Monte Carlo Simulation Chapter Chapter 12. What is a factor? 129 0 0 15. 1. Introduction 12. 1. Introduction 129 0 O 15.2. Choices and more choices 12 .2. Factors for an implementation of the L1\在M A 130 Y 15.3. The proper use of Sobol numbers 12.3. Factor reduction A 132 U 15 .4. Assessing convergence 12 .4. The number of common factors n 136 υ 15.5. Key points A 12.5. The dimension of the space attainable 138 υ A 15.6. Further reading 12.6. Markovian dimension with drifts 142 υ 15.7. Exercises 12.7. Markov functional models 144 12.8. 1\在 atrix separability 146 Chapter 16. Pricing continuous barrier options using a jump-diffusion 12.9. Key points 148 207 model 12.10. Further reading 148 207 16 .1. Introduction 12.11. Exercises 149 209 16.2. The Merton jump-diffusion model 210 16 .3. Importance sampling and stratification Chapter 13. Early exercise and Monte Carlo Simulation 151 211 16 .4. The price conditional on no jumps occurring 13. 1. Introduction 151 212 16.5. The algorithm 13.2. A sketch of the least-squares method 152 213 16.6. Numerical results 13.3. The details of the least-squares algorithm 153 217 16.7. Key points 13 .4. Carrying out the regression 155 218 16.8. Further reading 13.5. Breaking a contract 157 218 16.9. Exercises 13.6. Assessing and extending least-squares 159 219 13.7. Upper bounds and the seller's price 160 Chapter 17. The Fourier-Laplace transform and option pricing 219 13.8. Recharacterising the optimal hedge 163 17.1. Introduction 13.9. Upper 如 ounds for breakable contracts 219 165 17 .2. Definitions and basic results 228 13.10. Never exercise sub-optimally 166 17.3. Working with the log forward 233 13.11. Multiplicative upper bounds 167 17 .4. The Fourier transform in log-strike space 13 .1 2. Key points 239 172 17.5. The time-value approach 241 13.13. Further reading 172 17.6. The probability decomposition approach 242 13.14. Exercises 172 17.7. Working with characteristic functions 244 17.8. Known characteristic functions 247 14. The Brownian bridge Chapter 175 17.9. The Heston characteristic function 249 14.1. Introduction 175 17.10. Numerical implementation 251 14.2. Reducing to the driftless case 175 17.11. Key points 251 The law of the minimum for a Brownian bridge 14.3. 177 17.12. Further reading 251 14.4. The distribution at intervening times 178 17.13. Exercises 14.5. Using the Brownian bridge for path generation 180 253 Chapter 18. The cos method 14.6. The geometric bridge 181 253 18 .1. Introduction 14.7. Key points 183 253 18 .2. Cosine series 14.8. Further reading 183 255 18.3. Cosine series and characteristic functions 14.9. Exercises 183
  5. 1X CONTENTS CONT丑 NTS V11l 306 Exercises 2 1. 1 1. 18 .4. European option pricing 256 18.5. Homogeneous models and the cos method 259 307 Chapter 22. Adjoint and automatic Greeks 18.6. Bermudan options 260 307 22. 1. Introduction 18.7. American options 263 308 22.2. Model Deltas using the Giles一Glasserman method 18.8. Key points 264 22.3. Pathwise Vegas in the LMM using the Giles-Glasserman method 311 18.9. Further reading 264 313 22 .4. The adjoint acceleration 18 .1 0. Exercises 264 320 22.5. The LMM as a sequence of vector operations 322 22.6. The limitations of the adjoint method Chapter 19. What are market models? 265 323 22.7. Forwards versus backwards 19. 1. Introduction 265 324 22.8. Key points 19.2. The general set-up 266 324 22.9. Further reading 19.3. Drifts and martingales 267 324 22 .1 0. Exercises 19.4. Calibration 268 19.5. Products 272 327 Chapter 23. Estimating correlation for the LffiOR market model 19.6. Key points 279 327 23.1. Introduction 19.7. Further reading 280 327 23.2. The set-up 19.8. Exercises 280 328 23.3. Time parameterization 329 23 .4. Interactions with boot-strapping Chapter 20. Discounting in market models 281 331 23.5. Factor reduction 20.1. Introduction 281 332 23.6. Other market models 20.2. Possible numeraires 282 332 23.7. Time-series step size 20.3. The most common choices and their consequences 284 333 23.8. Correlation smoothing 20 .4. Using the numeraire to discount 286 337 23.9. Does it really matter? 20.5. Numeraire matching , variance reduction and discretization bias 288 338 23.10. Key points 20.6. Forward discounting in the spot measure 289 338 23 .11. Further reading 20.7. Key points 290 338 23.12. Exercises 20.8. Further reading 291 20.9. Exercises 291 341 Chapter 24. Swap-rate market models 341 24.1. Introduction Chapter 21. Drifts again 293 342 24.2. Deducing the bond-ratios for the co-terminal model 2 1.1. Introduction 293 343 24.3. Cross-variation derivative 21.2. Rapid computation of drifts 293 346 24 .4. Swap-rate drift computations 2 1. 3. Evolving the bond 295 348 24.5. Constant maturity market models 21 .4. Positivity issues with bond evolution 297 350 24.6. Co-initial swap-rates 21.5. Predictor corrector 299 352 24.7. Incremental market models 2 1.6. Stopping predictor corrector 299 356 24.8. Calibrating the co-terminal swap-rate market model 21.7. Pietersz-Pelsser-Regenmortel 301 357 24.9. Evolving swap-rates 21.8. Numerical comparisonsof drift methods 303 358 24 .1 0. LIBOR versus swapωrate market models 2 1.9. Key points 305 359 24 .11. Key points 21.10. Further reading 306
  6. X CONTENTS CONTENTS Xl 24.12. Further reading 360 Chapter 28. The convergence of binomial trees 407 24.13. Exercises 360 28. 1. Introduction 407 28.2. Richardson extrapolation 408 Chapter 25. Calibrating market models 363 28.3. Convergence of simple trees for European options 412 25.1. Introduction 363 28 .4. Convergence theorems 414 25 .2. Understanding pseudo-square roots 365 28.5. Redesigning trees 415 25.3. Decomposing pseudo-roots 367 28.6. The Leisen-Reimer tree 417 25 .4. Time dependence and factor maintenance 368 28.7. Higher order convergence 419 25.5. Mapping between models and swaption approximations 368 28.8. Code for higher order trees 420 25.6. Cascade calibration 371 28.9. More and more trees 422 25.7. Fitting caplets and co-terminal swaptions 374 28 .1 0. Choices for trees 425 25.8. Rescaling and LMM calibration 381 28.1 1. American options 426 25.9. Period mismatch 383 28 .1 2. Assessing accuracy 428 25.10. Global optimization 385 28 .1 3. Truncation choices 429 25.11. Calibration with displacements 386 28 .1 4. Key points 430 25.12. Key points 387 28 .1 5. Further reading 430 25.13. Further reading 388 28 .1 6. Exercises 430 25 .1 4. Exercises 388 Chapter 29. Asymmetry in option pricing 433 Chapter 26. Cross-currency market models 29. 1. Introduction 433 389 26.1. Introduction 29.2. American optionality 434 389 26 .2. Notation 29.3. Incomplete markets 437 390 26.3. Dynamics 29 .4. Transaction costs 439 390 26 .4. Understanding cali 也ration 29.5. Key points 441 393 26.5. Pricing given a calibration 29.6. Further reading 441 395 26.6.Approximation foymulas fof the voiatiiity of the forward FX fate 29.7. Exercises 441 396 26.7. Equity-linked notes 397 Chapter 30. A perfect model? 26.8. Key points 443 398 30.1. Introduction 443 26.9. Further reading 399 30.2. The vanilla options trader 26.10. Exercises 444 399 30.3. Dynamic hedging with a perfect model 445 30 .4. The portfolio Chapter 27. 1\在 ixture models 446 401 30.5. The exotics trader 447 27.1. Introduction 401 30.6. Key points 447 27 .2. Uncertain parameter models 402 30.7. Further reading 448 27.3. As a smoothing methodology 403 30.8. Exercises 448 27 .4. The advantages and disadvantages 403 27.5. Key points 404 Chapter 31. The fundamental theorem of asset pricing. 449 27.6. Further reading 405 3 1.1. Introduction 449 27.7. Exercises 405 31 .2. The easy direction 450
  7. X ll CONTENTS 31 .3. The hard direction in the discrete case 451 31 .4. Attaining the minimal price 454 3 1. 5. Key points 456 31.6. Further reading 456 31.7. Exercises 456 Appendix A. The discrete Fourier transform 457 Preface A.l. Introduction 457 A .2. Roots of unity 457 A.3. The discrete Fourier transform 460 It is now ten years since the first draft of "the Concepts and Practice of 1\在athe­ A .4. The fast Fourier transform 462 matical Finance" was 自 nished. The volume of research published during that time A.5. The discrete Fourier transform and convolutions 463 has been immense. New areas have arisen and many questions have been resolved. A.6. The fast Fourier transform and matrix multiplication 464 Some markets such as portfolio credit derivatives have arisen , boomed and crashed. A.7. Key points 466 "More Mathematical Finance" is therefore a sequel , and it is intended t。如e a sec- A.8. Further reading 466 ond or third book on financial mathematics. In particular, rather than recall basic Bibliography theory, I wi1 I refer tωO 出 丑 467 and maximize the amount of new material. Index 477 This sequel is not intended to be comprehensive. The field is now far too large for such an undertaking to be practical. In any case , I am a firm believer in "write what you know." Most of the topics in the book are related to my own research in one way or another, and I hope to pass on some of the insights I have gained from using and implementing these models.τb me that is the essence of the book, my objective is to give the reader my own personal perspectives on how one should view various issues. Thus whilst most of the mathematics and models here pre- sented can be found somewhere in the literature , the perspectives I present often cannot. Much of the book focuses on numerical methods. A pricing model is not much use unless it can be implemented and calibrated. The ability to compute Greeks is another essential. My objective is therefore to show how the mathematics can be translated into an implementable, usable model. However, this is not a recipe book. Although I present algorithms , my objective is to give the reader an understanding that makes the algorithms clear, rather than to present a piece of pseudo啕 code to copy out. I will , however, occasionally point to where the relevant code can be found in the QuantLib open source library. I largely avoid presenting purely numerical techniques which are well known outside finance. For example, I leave the details of how to carry out Gaussian integration to other texts. However, I do present extensive discussion of how to use Sobol numbers for quasi-Monte Carlo simulation , since this seems to be a much misunderstood topic. X111
  8. XIV PREFACE PREFACE XV I have restricted this text to mathematical finance in the sense of derivatives possible. However, there are inevitably some dependencies and I now discuss these. pricing. A more specific but rather unwieldy title might have been "how to think First, the credit chapters should be read in order. Second, the introduction to market about some numerical techniques for pricing derivative contracts." models should be read before the other market models chapters; these are , however, At some point, one must call a halt to writing , and many topics t妇 at were largely independent of each other. The cross-currency market model chapter (26) does , of course , assume familiarity with market models. Third , the cos chapter (18) considered have not made it in to this book.Tf1ese indude Levy processes , OIS relies on the Fourier transform chapter (1 7). The quasi-Monte Carlo chapter (1 5) discounting, SABR, asymptotic expansion approximations , solving SDEs , numer- ical methods for solving PDEs , short rate models , the HJM model, commodities吨 and the importance sampling chapter (16) both depend on the Brownian bridge chapter (14). The remaining chapters are largely stand alone and can be read in power derivatives , CGMYSV, GPUs , proxy methods for Greek computation, in; terPolation methodologies fof interest rates , local volatility, aYITI-value modeh. anyorder. VAR, CES , mean-variance theory, CAPM , utility theory, APT ... The list is end- The website for this book is less. Even 户 ally, when I again feel 出 atIhaveeno 让 i t will be time tωo write "Even More Mathematical 日 Finance." visit there for updates , questions , new editions , typos and news. Please use the So what topics are covered? I spend four chapters on portfolio credit deriva- forum there to ask questions about the text and to inform me of typos. tIves since it is an area that has gone from obscurity to fame to notoriety in the I have included end of chapter exercises. These take a variety offorms ranging last few years-I iook at binomial trees in depth since it is a topic which is much from simple computations to complicated proofs. Many of them are more com- misunderstood:we wili see that there are at least twenty difemIlt ways to place puter projects than exercises since ultimately this book is about modelling. I have the nodes of a tree , and that each of these can be implemented in at least sixteen not included solutions , but you are encouraged to discuss the problems on the digerent ways.Monte Carlo techniques are examined in depth with cf1apters on the book's website. Brownian bridge, quasi-Monte cario , the early exercise problem and stratiacation- Market modeis for pricing exotic interest rate derivatives have been my prin 翻 Various versions of the manuscript have been read by a rather large number of people and I thank them all for their comments. The readers include Barbara La cipai researcf1interest for many years.This is reaected by chapters on their applim Scala, Will Wright , Chris Beveridge, Jiun Hong Chan , Stephen Chin , Nick Den- cability, drift computation and aPPI-oxiInation , correlation estimation, swap-rate son , Andrew Downes , Robert Tang , Chao Yang , Ferdinando Ametrano , Paulius mafket models , calibration, discounting and cross-currency market models.The cha户户s on disc削 zation , factor reduction , ql邸 Jakubenas , Harry Lo , Lew Burton , Agustin Lebron , Oh Kang Kwon , Alan Lewis , Nagulan Saravanamuttu, Graeme West , Dherminder Kainth and Lorenzo Lies饨, 臼 四 tiv itie s S ens 山 叽 创 with adjoint m 妇 出 whilst written in a more genera1 context are SI 丑 lod s led 剖 剖 a Isωo directly relevant to market models. as well as many others. I also include a few chapters on more philosophical questions. These include Mark Joshi chapters on asymmetry, evaluating a perfect model , the fundamental theorem of asset pricing , convexity, mixture models and the money bleed. Melbourne , August 2011 Certain chapters have been included simply because I think the results and/or the mathematics are neat and I want them to share them with the reader. These in- clude a chapter on how to differentiate the Black-Scholes formula ,
  9. CHAPTER 1 Optionality, convexity and volatility 1.1. Introduction For certain contracts , increasing volatility will always increase prices. For ex- ample, in the Black-Scholes model the Vega of a call option is always positive so price must be an increasing function of volatility. In the Merton jump-diffusion model , price is an increasing function of jump intensity so again increasing uncer- tainty leads to an increase in price. However, there are certainly plenty of contracts with positive pay-offs for which increasing volatility can reduce price. For example, if a digital call option is in-the-money, zero volatility will clearly result in a maximal price. It is also well-known that for barrier options , an increase in volatility can result in a higher probability of knock-o时, and therefore a decrease in value. Similarly, in a jump- diffusion model increasingjump intensity can result in either a decrease or increase in price for a digital call option , see Chapter 15 of [105]. Of course , one cannot expect general monotonicity results to hold for digital options , as a digital call plus a digital put of the same strike is simply a zero-coupon bond , which has a price independent of volatility or any other uncertainty. We shall see that convexity is the important property of a call option that makes monotonicity results work, and that in a sense convexity and optionality are the same concep t. 1.2. Volatility and convexity We want a more general concept of increasing uncertainty that encompasses both increasing volatility in the Black-Scholes model and increasing intensity in a jump-diffusion model. We shall say that a positive random variable X is more uncertain than a pos- itive random variable Y if they can be defined on the same probability space in
  10. 2 1. OPTIONALITY, CONVEXITY AND VOLATILITY 1.3. CONVEXITY AND OPTIONALITY 3 such a way that X , defining the final stock price in A is more uncertain than the random variable, X=YZ , Y , for B. We need to show that with Y and Z independen t. We also require Z to be positive. JE(f(X)) 注 JE(f(Y)). Wehave By definition and risk-neutrality, we can write X = Y Z. We write JEy for + log Z , logX = logY expectation over Y and similarly for Z. We then have so JE(f(X)) =JE(f(YZ)) , Var(logX) = Var(logY) 十 Var (l og Z). (1. 2.1) =JEy(JEz(f(YZ))). (1. 2.2) Note that if Z is not constant this immediately implies that log X has higher vari- ance than iog Y.This deanMOIlincludes increasing volatility in the BiacKJcholes Observe that f (Y s) is a convex function of s for each value of Y , so applying world since we have for σ1 注 σ2 , that a normal of standard deviation 0" 1 is dis Jensen's inequality to the expectation in Z , we have that 甘 ibuted as a normal of standard deviation ofσ2plus m independentomdsun- dard deviation 飞/付 -4. JE(f(X)) 注 JEy(f(YEz(Z))). (1. 2.3) For the jump-diffusion case , increasing jump intensity can be seen in terms of However, we know that the expectation of Z is one and we conclude muitiplying by a random vafiable genefated by the jumps arising fkom a process JE(f(X)) 注 JEy(f(Y)) , with intensity equal to the difference. (1. 2 .4) Note that when PI-icing derivatives , the increase in uncertainty Will not afeet th which is what we wanted to prove. e mean since the price ofabrward contract must be invariant.This implies that Note that we have shown that the Black-Scholes price of any derivative with the variable Z must have mean equalt01and we impose this restriction for the a convex pay-off is an increasing function of volatility. We have also shown that rest of this section. the same holds in the jump-diffusion model as a function of each of volatility and We will show that for European options with convex pay-offs , price is an in- jump intensity. creasing function of uncertainty.Recall that a function , f , on an inteIVal U is convex if for all 叽 Y E U and () E [0 , 1] , we have that 1.3. Convexity and optionality + (1- ())f(y). f(()x 十 (1 - ())y) ζ ()f(x) In the previous section , by invoking Jensen's inequality we were able to show Geometrically, the chord between any two points on the graph lies on or above the that the prices of European derivatives with convex pay-offs increase with uncer- graph. tainty. In this section , we explore the finance behind thisTesul t. For what follows , YτT、挝 ……st important result 台rompr ability 出 ∞ 吨a灿 h err1ωO 览创 1 此凯 f f 严汕 灿 t heorη')γy it is important that we are careful to make the distinction between an option and a Jensenγ's inequali ty. 让 is derivative. An option is a choice between one of two , or possibly more , portfolios , whereas a derivative is the right and obligation to receive a function of the stock THEOREM 1.1. 扩 f isa convexβmction, and X 归 α random ν'ariable such that JE(/X/) is finite, and JE(lf(X) I) is finite then price at maturity. All options can be viewed as derivatives simply by taking the function to be the maximum of the portfolio values. JE(f(X)) 注 f(JE(X)). For example , a call option is the right to choose between the empty portfolio and a portfolio consisting of a unit of stock and - K riskless bonds. A put is the For a proof see, for example , [197]. right to choose between the empty portfolio and a portfolio of K bonds and minus Now suppose we have a derivative C paying a convex function f at time T. one stock. Note that we require the composition of each portfolio to be fixed at the Suppose we have two risk---neutral models A and B such that the mndom variable, star t.
  11. 4 OPTIONALITY, CONVEXITY AND VOLATILITY 1. CONVEXITY AND OPTIONALITY 1.3. 5 Intuitiveiy, as the holder ofarloption has choice, we would expect more uncer- tainty to increase the price of an option;the right to be immune against some risk 。 .9 should be worth momas the risk gets bigger.In a simple market consisting ofa non- dividend paying stock and a riskless bond, a portfolio corresponds to a derivative that pays a iiIlear function of stock.Options therefore correspond to defivatives that pay the maximum of a collection of linear functions.Linear functions are , of 、、 course, trivially convex. 、\ =Ixl 一似) \、 、、 . .. tangent one We show that the supremum of a collection of convex functions is also convex \\ It is then immediate from 趾 results of the p削 ous sec阳 tl川heprice ofL - .- tangent tw。 option increases with uncertainty. Let feo αε A , be a collection of convex functions with the same domain U. \、 \、 Let \\ -0.3 、\ f(x) = s 叩 fα (x) , \ \、 αεA \、 for x ε U. For x , y E U, and 0 ε(0 , 1) , we have , by definition , for each α fα (Ox + (1 - O)y) ζ °iα (x) + (1 - O)iα (y). The function I 叫 has multiple ta吨e附 at O. FIGURE 1. 3. 1. Taking the supremum of both sides , our result is immediate. We have proven be the slope of the chord from x to y. If x < z < 机 the chord from x to z must lie THEOREM 1.2. Any European option can be represented by a derivatiν'e pay- on or below the one from x to y , so we have that ~ (民 y) is an inc 阴阳 g function ex function of the stock price. For any option, the price increas ω with of y for y > 叽 (and so will decrease as y • x + .) It must also be equal to or uncertainty in the risk-neutral measure. greater than ~ (叽 x) for any w < x as x must lie on or below the chord from ω toy. An obvious question now arises: are there convex functions that do not corre spond to 0严ions? We show 阳 the answer is negative. In P ar t 巾 叩 盯此 阳阳削削 It must therefore conve脐部 U → x +. Call this limit D+(x) , the 咆ht-hand aco盯ex function is the maximum of a countable number of linear fur削 0 瓜 We derivative. Similarly, ~(叽 Z) mt C on rge to a va Iu e D_(x) as ω → Z 一 .We 吟 削 ∞ 附耶 昏 剖阴 do this by showing 出 at at every point on tl问 raph ,阳'e is a (non-u 向 ue) 胁 also have that D 一位 )ζ D + (x). They will be equal if and only if f is differentiable gent iine which only lIItefsects the graph at that point.We also shw that ail convex at x. (This is actually the definition of differentiablity.) functions are co的1 阳IS which means 削 it is s 耐阳It to 毗e the ta耶…iies Any line with slope s through (凯 f(x)) such that D_(x) ζ8ζ D+(x) , will at rational points. lie on or below the graph. To see this observe that if it intersects at t > x , then we We follow the proof of Jemen's inequality in[197]·The key to the proof is must have in the existence and monotonicity of one国 sided derivatives for convex derivatives. t) = s , S ζ D+(x) ζ ~(x , r) ζ ~(x , Let f be aconvex function and suppose Z < U-Note tMt f is convex ifand only if g deamd viag(s)=f(-s)is.This means that wecan invoke symmetry to carry for all r ε (x , t) , which means that the graph is a straight line between x and t. By results for right derivatives to left derivatives symmetry, the same holds on the left. Let Taking the collection of lines of the form , 一 f(y) - f(x) A(ZJ)-7 v-x + f(x) , yx(z) = D+(x)(z - x)
  12. 6 OPTIONALITY. CONVEXITY AND VOLATILITY 1. IS CONVEXITY NECESSARY? 1.4. 7 we have 1.4. Is convexity necessary? f(s) = sup Yx(S) , We have seen that convexity is equivalent to optionality, and that the price of for all s. a derivative with a convex pay-off always increases with uncertainty. This raises the obvious question of whether convex pay-offs are the widest class of pay-offs for which uncertainty always leads to extra value. We show in this section that for any non-convex pay-off, uncertainty can decrease value. 0.25 We proceed constructively; i. e. , given a non-convex function , we construct an example where uncertainty decreases value. Suppose f is a non-convex function 0.15 /./// on JR+ , then there exists points x < y and 0 E (0 , 1) such that f(Ox + (1- O)y) > Of(x) + (1- O)f(y). chord 100.5 For our construction , we take zero interest rates for simplicity (but the argu- 吨A 』/ /m -0.6 0.2 0 .4 ment is easily adjustable.) We therefore suppose the existence of a riskless bond o 2i /5 of constant value 1. Let 8 0 = Ox + (1 - O)y. jymAJtll l4lIl -lf In the model 儿'h , let 8T take value 8 0 with probability one , i. e. , the stock is / -0.15 / simply 80 riskless bonds. The value of a derivative paying f is therefore trivially // f(8 0 ). In the model M 2 , let 8T take value x with probability 0 and y otherwise. We then have A parabola with the chords through the points at (士 0.5 , 0.25). FIGURE 1. 3. 2 . lE (8T ) = 80 , and It remains to prove that f is continuous. We fix a point x , and prove continuity lE (f(8T )) = Of(x) 十 (1 - O)f(y). there. Observe that for x < z < 弘 we have that f (z) lies on or below the chord from x to y; this guarantees that f is not too big as z → Z 十. Extending the We clearly have that M2 is more uncertain that Ml but the value of a deriva- chord , this also shows that f is not too small as z → Z 一. By symmetry, we also tive paying f is lower by our original hypothesis on f , and we are done. have that f is not too small as z → z 十, and not too big as z → Z 一. We have out side 臼 the Black丽…δ. This example moves shown that the graph of f is constrained to lie between a pair of lines intersecting non- convex pay-o仔 have aBlack Scholes 心 一 price thati s strictIy increasing in volatil- 创始 抄 - at (x , f(x)) , and continuity is now clear. (See Figure 1.3 .2.) 拄 i ty? 矶恒 由 wes howthat 扰 由瓜 i t can. We now have We take r = d = 0 for simplicity. Let 80 = 1 and T = 1. Suppose the s 叩 yq(s) , f(s) = de巾 ative pays 181 -11 for 81 手 1 and -1 otherwise. If volatility is zero the p 出 e qEQ is -1 and otherwise it is positive and equal to the price of a derivative paying showing that f can be written as a supremum of a countable set of lines. To see 181-11 which is convex and positive. τT、hepr eiS there"fi岛陀 挝 柱 严巾 妇趴 川由 orepos it ivean d increasin 时 this relation , observe that both sides define a continuous function and that they 伽 fOr positive 叫 a til ity. We conclude that the price is strictly increasing in volatility, 町 vola 蚓 且 印 凶汹 创 agree on the rationals which is a dense se t. despite the pay-off's being non-convex. This example is a little contrived in that We conclude that any derivative with a convex pay-off is equivalent to an op- it relies on the fact that the value of a pay-off at a point is irrelevant for non-zero tion to choose between a countable set of simple portfolios. volatility but not for zero volatility.
  13. 8 1. OPTIONALITY, CONVEXITY AND VOLATILITY 1.5. Key points We have examined convexity, volatility and optionality in this chapter, and seen that convexity and optionality are equivalent concepts. CHAPTER 2 • Increasing uncertainty is a more general concept than each of increasing volatility and increasing jump intensity. Where does the money go? • Not all derivatives have prices that increase with uncertainty. • The price of a derivative with a convex pay-off increases with uncer- tainty. 2.1. Introduction • Convex functions are continuous. • The pay-off of a true option is always a convex function. Before reading any further, attempt to answer the following problem. If you • Any European derivative on a stock price with a convex pay-off can al- find it hard , you need to read this chapter. ways be represented as a choice between a countable number of simple portfolios. EXAMPLE 2. 1. In the Black-Scholes model, r = d = 0 , σ= 0.1 , T = 1 , and So = 100. A trader sells a call option struck at 100 for the Black-Scholes price. 1.6. Further reading He continuously Delta-hedges ω maturity using an implied volatility ofO.1 at all times. Black-Scholes assumptions are correct except as stated below. What can Some investigation of the relationships between convexity and volatility can you say about his final pro..卢 t or loss for the following volatiliη scenarios? be found in [17]. • instantaneous volatility is a constant 0.1 , 1.7. Exercises • instantaneous volatility is a constant 0.08 , • instantaneous volatility is initially high then low with the final root-mean- 1.1. Show that the relationship X is more uncertain than Y defines EXERCISE squ α re volatility equal to 0. 1. a partial ordering on random να riα bles. , • instantaneous νolatility is initially low then high with the final root-mem• 1.2. Give an example of a convα function which is not di庐 ren­ square volatility equal to 0. 1. EXERCISE tiα ble. o/spot 80 , 100 and 120. Discuss each of these for the final ν'alues 1.3. Show that α function with positive second derivative is convex. EXERCISE The answer to the first volatility scenario should be easy. All the assumptions EXERCISE 1.4. Show that the price of a call option in the ~旬 riance Gamma of the Black-Scholes model hold and the volatility forecast is correc t. Given these model increases with σ. facts , we know that the replication cost of the call option is precisely its Black- EXERCISE 1. 5. R吃formulate and prove the results of this chapter for options Scholes price; the terminal value of spot is irrelevan t. Thus the trader、 final po- on rates that take all ν'alues in IR instead of being positive. sition is zero. However, this begs another question? Where does the money go? Suppose the final spot is less than 100. The option's pay-off is then zero. Consider EXERCISE 1.6. Suppose a pay 叫ff is continuous and the Black-Schol ω price the trader's replicating portfolio. He initially invested the premium in a mix of is strictly increasing as a function of volatility. Must the pay-off be con νex 扩 stock and bonds. He continuously rebalanced it to keep his position delta响 neutral. ·的 is holds for all valuω ofSo? The final value of the replicating portfolio is zero. The entire option premium has • this holds for one value of So? evaporated from the Delta-hedging. What is it about Delta-hedging that makes the trader lose money so that all the premium has bled away? 9
  14. l nu 2. WHERE DOES THE MONEY GO? 11 2.2. THE MONEY BLEED Our objective in this chapter is to study the process of Delta-hedging more and putting the balance (possibly negative) in the bond , that is closely to develop an intuition of how profit and loss fluctuate when dynamically 向 (St , t) = JV(d 1 (St , t)) (2.2.1) hedging. units of St and Before reading the rest of the chapter, consider another problem. f3t(St ,t) = C(St ,t) - JV(d 1 )St (2.2.2) 2.2. We are in the Black-Schol ω model as before. The real-world EXAMPLE units of the riskless bond. Taking interest rates to be zero , at the end of the step , drift is μ > r = O. Suppose we run a Monte Carlo simulation ofthe cα II option our portfolio is worth price. How does the discounted average price 仰 ssuming no hedging) compare between the real• vorld measure and the risk-neutral one? Now suppose we intro- + C(St , t). JV(d 1 (St , t))(St+dt - St) duce discrete Delta-hedging so that the hedge is rebalanced afier e ν'ery ~t units Now consider a second step , we obtain the same change in value across the step of time with ~t > O. How do the mean and variance chα nge in each measure? replacing t by t + dt. So our new value is What happens as ~t tends to zero? + dt))(St+2dt - + JV(d 1 (St , t))(St 十 dt - + C(St , t). JV(d 1 (St+dt , t St+dt) St) Suppose that in the first step we went up by a quantity x , and on the second , we 2.2. The money bleed went down by the same amoun t. After two steps , we have therefore returned to our In this section , we examine the question of why carrying out a Delta四 hedging initial poin t. Our pro自 t and loss expression becomes strategy costs money. In particular, we focus first on the case where we are replicat- + x , t + dt)) + JV(d 1 (St , t))) x (-JV(d 1 (St ing a long position in a call option which is out ofthe money for the entimiifetime We can neglect the dt in time since over a short time step x will be of the magnitude of the contrac t. Note that replicating a long position is equivalent to hedging a ofV玩 and so much larger. Since we assumed that x was positive and Delta is an short position , and it will be useful to consider both viewpoints in this section. increasing function , the Delta after the first step is higher than its initial value and The Black-Scholes Delta-hedging argument says that if we take the option so value at time zero and trade correctly, then at the end we have the option's pay-off. -JV(dl(St + x , t)) + JV(d 1 (St , t)) < O. In the case above, this means that we start with a positive amount of money and Thus we lose money across the step. What happens if we go down and and then end up with nothing! If we are Delta-hedged then we are supposed to be immune up? We then have approximately to stock price moves so why are we losing money?! + JV (d 1 ( St , t)) > 0 一 JV(d 1 (St 十叽 t)) First, it is important to make the distinction between the value of the replicat- ing portfolio which we will denote P and the value of the hedged position that but we are now multiplying a negative quantity and again we lose money. Summa- is P minus the option value , C. The latter is worth zero at all times if we are rizing , if across two steps we return to our initial point , we lose money. continuously Delta hedging. The value of P will move around , however. This is correct! 0 旧 replicating portfolio is supposed to be worth C(St , t 十 2dt) In practice, truly continuous Delta hedging is not possible either in the markets and the value of a call option is an increasing function of time to maturity for fixed or in a computer simulation. We can model by dividing time into many small steps St. The time to maturity has decreased so the value has gone down. and axing the Delta at the beginning of each step-An alternate approach, which What property of the call option is causing us to lose money? The essential we will not explore although it is closer to what a trader would actually do , is to point in the above argument is that the Delta goes up when the stock price goes up , consider rehedging every time the Delta changes more than a fixed small amoun t. and goes down when it goes down. In other words , Delta is an increasing function Consider P. We examine behaviour across a time step from t to t+dt assuming of St. Since Delta is the first derivative ofthe price , this is the same as saying that perfect replication up to time t. There will , of course, be a small error due to the second derivative , or Gamma, is positive. Positivity of the second derivative the discrete nature of the hedging but if dt is very small , this will be vanishingly is equivalent to convexity so the crucial point is that the Black-Scholes value is small-We set up our replication Portolio by purchasing Delta units of the stock convex.
  15. WHERE DOES THE MONEY GO? 12 2. THE MONEY BLEED 2.2. 13 …………........……….....................……....….....................................................…….........…....._...............~................................~...................................……"………·‘ Change in value when Delta-hedging Difference to approximation 0.006 'T"'川...................… O 刀 5152 nu-nu" ω = -〉 a mZ ω ÷川川 Value change i u m a g ω ω』 ω ω 缸 ----… Parabolic g 巾 。 m approximation f E -0.25 U -0.3 -0.003 Change in stock price Change in stock price FIGURE 2.2. 1. The change in value of a Delta-hedged position FIGURE 2.2.2. The difference between a parabolic approximation across one day as a function of the change in stock price. Black- and the Black- Scholes value for the change in value of a Delta- Scholes value and parabolic approximation shown. Hedged position across one day. Of course, our up and down moves will generally not cancel in such a precise the Gamma is multiplied by (St+dt - St)2; this will scale with time like dW; fashion.We now look at thchange in value of tkhedged short position across a which means its size is like dt. Since dt is like time divided by the number of step as a function of St+dt - St. We display one such case in Figure 2.2.1. Spot is steps , summing across all steps yields a term that does not disappear in the small- 90 , strike is 100 , expiry is 1, r = d = 0 , and volatility is 0.2. The time-step size step-size limi t. is 1 day that is 1/365. We make money if the stock price has not moved because We make a small profit if the magnitude of the stock price move is less than the only effect is that the value of C becomes smaller and we are short C. approximately one and make a loss which increases with magnitude otherwise. We are Delta-hedged so the derivative of our position's value with respect to Our expected change in net value across the step will be zero , since interest rates 8 t is zero which gives the horizontal tangent at 0. The next dominant effect is the are zero and we are working in the martingale measure , so the value of a self- Gamma , indeed we can write financing portfolio is a martingale. θC ♂C How often do we make money? In Table 2.2.1 , the changes in the value of S as C(St+dt , t 十 dt) = C(St , t) + o; (St ,t)dt + ~瓦 (St , t)(St+dt - St) v a function of the underlying standard normal increment are displayed together with 102 C the cumulative probabilities. We see that that profits in this case correspond to the 十一一τ (St , t)(St+dt - St)2 + small error normal increment being less than one standard deviation and we therefore make a 2 oSt~ profit about 70% of the time. The other 30% of the time we make a loss. Note th创 Since we are Delta-hedged this results in the change in value , ~ V , across a small the loss will generally be bigger than the profi t. Indeed , since the expected profit step being approximately is zero , the losses must be on average 7/3 times as big as the profi t. θC /~,. 102 C ~V = 一万 (Stl t)dt - ~日($川 )(St+dt - St)2 Note the psychological implications here , if you like to make a little money (2.2 .3) every day but sometime lose a lot, go long Theta 1 and short Gamma. If that is too The Theta is negative so the 岳 rst term is positive, and the second term is negative since the Gamma is positive.We plot this approximation on Figure tfle 1We are long Theta if t忧 Theta of our portfolio's value is positive. 1f it is negative then we are difference between the approximation and the true price in Figure 2.2.2. Note that short Theta.
  16. 2. WHERE DOES THE MONEY GO? 14 ANALYZING THE EXAMPLES 2.3. 15 N(Z) ~S Z • the option was sold with the same wrong implied volatility, σ7 -3 -2.78734553 0.001349898 • the instantaneous volatility is always lower than σ. -2 -1.869566241 0.022750132 This will work regardless of the final value of the stock price. -0.942128734 0.158655254 。 -0.004931372 0.5 Now we consider the final two cases. If volatility was a deterministic function 0.942128554 0.841344746 of time known in advance in each of these , and we hedged using the residual root- 1. 899154832 2 0.977249868 mean-square volatility then the replication would be perfect and there would be 3 2.866252342 0.998650102 nothing to say. However, instead we are hedging with the initial implied volatility at all times and there are some not so obvious effects. TABLE 2.2.1. The changes in value of S as a function of a standard To see this , suppose we mark our books always using the initial volatility σso normal increment for our model case. the value of the call option is the Black-Scholes price as a function of the current spot , current time and σ. We have seen that our approximate profit and loss is given by the negative of the Theta of the option minus a half of the Gamma times much for your nerves do the opposite: lose a little every day and once in a while the spot-move squared , (recall equation (2.2.3).) Since we are marking our books make a big profi t. using 矶 this will still hold using σin the formulas for Gamma and Theta. The Note that whilst our profit and loss is symmetric a也 out ~S = 0 , there is a change will be in the size of the spot-move. hidden effec t. The sign of the move will affect whether both option and hedge are The typical size of the spot-move will vary according to the current instan- getting bigger or smaller. Thus when the stock price increases the option value taneous volatility. The subtlety lies in the fact that Gamma will also vary along increases and so does the hedge value. If the option stays out of the money, then the path. Its value will also depend on where the stock price lies. For example , both go to zero across time and we end up with nothing. suppose the volatility is decreasing and the stock price finishes close to the strike. To answer the original question: where does the money go? we have seen that Even with one day to go the option will have non-trivial value (0.21 for the option it dissipates day to day by buying the stock when it moves up and selling when it at the start ofthe chapter with T = 1/365 , if St = K = 100.), this money still has moves down. Note also the crucial points that the sum of squares of stock price to be realized (or dissipated) by Delta-hedging so the hedging will go on through- moves does not go to zero as the step size go to zero; it is the quadratic variation out the final day and the exercise will take place as late as possible on the day. The of Brownian motion that causes the losses. The continual up and down moves of Gamma on the 在 nal day will be very large in this case (see Figure 4.3 in [105].) the stock price are the source of the money bleed. The volatility is low so the product of Gamma times spot-moved-squared will be smaller than expected and the hedging will cost less money. The hedger makes a profi t. 2.3. Analyzing the examples Conversely, if the volatility is high at the end and the spot move finishes at Returning to Example 2.1 , what can we now say? If the hedge is systematically the strike , then we willlose more money than expected and our final position will correct , and we Delta-hedge continuously with the correct volatility everything be a loss. The crucial factor to consider in each case is the interaction between balances perfectly, and we finish with the right replication value regardless of the instantaneous volatility and Gamma: if these are simultaneously high then we lose final value of spot. money, if not we make it. If the actual realized volatility is 0.08 then the magnitude of the daily move is In particular, a burst of volatility at the end is harmless if the stock is far from smaller, and so we are in the profit area more often and the loss area less often. We the money. If deeply in the money, the Delta stays close to 1 and will not va therefore make money! Whilst the amount of money is random , as the step-size goes to zero it will be positive. Thus hedging with the wrong implied volatility will make a profit provided
  17. 16 2. WHERE DOES THE MONEY GO? 2.4. VOLATILITY CONVEXITY AND THE EXISTENCE OF SMILES 17 Value out of the money Value at the money 9876543210 2.5 2 ω u℃ .ZB 1.5 ZL & 0.5 O O 0.05 0.15 0.1 0.2 0.25 0.05 。 0.2 0.1 0.15 0.25 Volatility Volatility FIGURE 2.4. 1. The Black-Scholes value of an out-of-the-money FIGURE 2 -4 .2.The Black-Scholes value of an at-the-money call call option as a function ofvolatility. Spot is 100 and strike is 120. option as a function of volatility. Spot is 100 and strike is 100. Expiry is one year. The interest rate is zero. Expiry is one year. The interest rate is zero. Value in the money 21 .4 21.2 21 Wh at happens in Example 2.2? If we are working in the martingale measure , 20.8 ω the mean will not change whatever hedging we do. This is the essential feature of .~ 20.6 a. 20 .4 that measure: adding a self-financing portfolio of initial value zero cannot make 20.2 any difference to expectations since such a portfolio's value will be zero. The vari- 20 ance will change , however, the more frequently we Delta-hedge the greater the 19.8 effect, and if we hedge continuously the variance goes to zero. Given that the real- o 0.05 。 .2 0.1 0.15 0.25 world is not a martingale measure why is this relevant? It is an effective methodol- Volatility ogy for variance reduction in Monte Carlo simulations; if we have a good approx- imation for the Delta we can use that to hedge along paths and thereby achieve FIGURE 2 -4 .3. The Black-Scholes value of an in-the-money call variance reduction without introducing bias. We will rarely have the true Delta in option as a function of volatility. Spot is 100 and strike is 80. Ex- a pricing simulation , since if we knew that we would probably already know the piry is one year. The interest rate is zero. pnce. What about the real-world measure? The unhedged call will have greater ex- 2 .4. Volatility convexity and the existence of smiles pectation than the Black-Scholes since the stock will 自 nish higher on every path. Once we start hedging this effect disappears. Each increase in rehedging frequency We now examine smiles in a stochastic volatility model from this point of view. will result in a reduction in variance since we are getting closer to the continuously Suppose we are Delta and Vega hedging an option not struck at the money using Delta-hedged case. It will also bring us closer to the point where the value is the the Black-Scholes model. Suppose the stock is following a stochastic volatility Black-Scholes price on every path , and so trivially the mean will also converge to process and implied volatilities are moving according to their prices in the same the Black-Scholes price. stochastic volatility model.
  18. WHERE DOES THE MONEY GO? 18 2. VOLATILITY CONVEXITY AND THE EXISTENCE OF SMILES 2 .4. 19 Volga out of the money Volga at the money 300 O 豆7 ……………………………………………一一 250 -0 .5 200 言" 巳 而ω -。 4 •.• ‘ 100 〉 -Volga 由翩翩翩 Volga 50 -1 .5 o O 0.05 0.1 0.15 0.2 幽2 Volatility Volatility FIGURE 2 -4 .4. The Black-Scholes Volga of an out of the money FIGURE 2 -4 .5. The Black-Scholes Volga of an out of the money call option as a function ofvolatility. Spot is 100 and strike is 120. call option as a function of volatility. Spot is 100 and strike is 100. Expiry is one year. The interest rate is zero. Expiry is one year. The interest rate is zero. For the Vega翻 hedging we use a call option struck at the money, and the Delta Volga in the money hedging is carried out using the stock as usua l. We have seen that for the stock, nUnUnUnUnU nur3nur3 q i!ii.ii:i.!;.Bi!iiii-.E.E- 4 the convexity of the option price as a function of the stock price leads to hedging 4 · · · · costs and this is , in fact , what determines the Black-Scholes price. We can also · expect Vega hedging to have an effect on the price. The nature of the effect will 帽 ω … 咱 4 be determined by the convexity of the price with respect to volatility. We therefore 。 〉 plot in Figures 2 .4.1 , 2 .4.2 , and 2 .4 .3 the value of a call option as a function of 幽幽幽 Volga the implied volatility, for an out-of-the-money option , an at-the-money one and an in-the-money one. 0.05 0.1 0.15 0.2 We see that the at-the-money call option is close to linear in volatility. This Volatility is not surprising if we recall from [105] , Section 3.7.2 , that the value of an at-the- money call option is roughly -LσSVT FIGURE 2 -4 .6. The Black-Scholes Volga of an out of the money 飞/2 1r call option as a function of volatility. Spot is 100 and strike is 80. Expiry is one year. The interest rate is zero. However, the other two graphs are clearly convex. We plot the Volga 2 , that is the second derivatives with respect to volatility, in Figures 2 .4.4, 2 .4 .5 , and 2 .4.6. We see that away from the money these are at-the-money option therefore gives us almost linear exposure to volatility whilst the other two options display convexity. positive. At the money, the Volga is actually negative , however, its magnitude is very small in comparison which reflects the almost linear nature of the price. The τhe process of Vega hedging a short position (i. e. replicating a long position) in an away-from-the-money option therefore leads to losing money on the Volga 2The Volga is sometimes called the Vomma. just as Delta hedging leads to a loss of money via the Gamma. In other words , it
  19. 20 2. WHERE DOES THE MONEY GO? 21 2.7. EXERCISES costs extra money to Delta and Vega-hedge an away-from-the-money option in a 2.5. Key Points stochastic volatility model. Since we need the Vega-hedging to ensure replication , In this chapter, we have looked at the interaction between the Gamma and we conclude that the prices , and therefore the implied volatilities , of these options volatility when hedging. We have seen that it is theconvexity of the option price will be higher. In other words , we expect smile-shaped smiles! that leads to hedging costs in the Black-Scholes model. Note the corollary that for at-the-money options , we do not lose or gain from the volatility's stochasticity: the expected root-mean-square volatility over the life- • When hedging an option , we are long Theta and short Gamma. time of the option is enough. • We lose money from rehedging when short Gamma because we have to buy when the stock goes up and sell when the stock goes down. Of course , we already knew that stochastic volatility models led to smiles with • A small stock move across a discrete time step results in a profit when an increase in the implied volatility level away from the money: just price the op- hedging a short Gamma position , but a large one results in a loss. tions using a Heston model and plot the implied volatilities. However, by examin- • When hedging a short option , we make a small amount of profit many ing the problem from this alternative viewpoint, we have seen how the stochasticity days , and make a larger loss on a few days. directly translates into extra trading costs and so leads to the existence of smiles • When instantaneous volatility is lower than the implied volatility we gain simply because traders would have noticed the extra costs. money from the fact that the stock moves are smaller than expected. How do we reconcile our two viewpoints? If we work in a martingale measure • When instantaneous volatility is bigger than the implied volatility we for the stochastic volatility model , then we know the value of any trading strategy lose money from the fact that the stock moves are bigger than expected. is a martingale. In particula巳 the value of a portfolio replicating a long call-option • Hedging in the martingale measure will reduce variance but will never position is a martingale and there will be no systematic effects , and if we continu- affect mean. ously rehedge we will perfectly replicate. If we discretely rehedge, we will finish • Hedging in the real-world measure will bring the mean closer to the risk- with some variance but the mean will not change. neutral mean as well as reducing variance. However, if we price and compute Greeks using the Black-Scholes model • Introducing stochastic volatility has a bigger effect when the price of a then there are real effects. To use Black- Scholes , we need an implied volatility so contract is convex or concave with respect to implied volatility. suppose we use the expected root-mean-square volatility over the remaining life of the option. The crucial point is that this (discounted) option price will NOT be 2.6. Further reading a martingale in the martingale measure. It is not the price process of a traded asset Euan Sinclair's book "Volatility Trading" , [181] , is highly recommended. The but merely the value of an arbitrary function of market-observed quantities. The intricacies of trading volatility are discussed there in detail at a reasonable level. Volga can therefore cause systematic effects which will drive the price up or down Wilmo 扰, [198] , contains a detailed analysis of discrete Delta hedging with adjust- according to its sign. ment terms to cater for the biases when working in the real-world measure. Thus the advantage of using a stochastic volatility model is that it gives us a price which automatically takes account of Volga effects. We can expect its price 2.7. Exercises to be substantially different from that of the Black-Scholes model when the Volga is large and to be similar to it when the Volga is small. This means that we can use 2.1. Run simulations for the two examples at the start of the chapte r. EXERCISE the Volga to assess whether a stochastic volatility model is required when pricing 2.2. Plot Figure 2.2.1 jar many different values of S , and T. an exot EXERCISE EXERCISE 2.3. Simulate the delta hedging of an out-of-the-money call option in a stochastic volatility (SV) model with volatility parameters computed using the Black-Scholes model. Repeat with vo /,α tility parameters computed using the same SV model. Examine the biases and see how να riα nce changes with step size. ,
  20. 22 2. WHERE DOES THE MONEY GO? EXERCISE 2 .4. Simulate the delta-vega hedging of an out-oFthe-money call option in a stochastic volatility (SV) model with νolatility parameters computed us- ing the Black-Scholes mode l. Use at翩 the-money call options to get the vega hedge. Carry out the same experiment using inside the model hedging with respect to CHAPTER 3 spot and the level of the instantaneous νolatility. Examine the biases and see how ν'ariance changes with step size. The Bachelier model 3.1. Introduction In 1900; Louis Bachelier published his doctoral thesis on the topic of bond options. His model for the movements of underlying bonds was essentially Gauss- ian. Whilst his work was not rigorous by modern standards , he anticipated many modern ideas. In particular, his study of Brownian motion predates Einstein's. In this chapter, our objective is to develop the necessary mathematics to price when the underlying asset is assumed to have a volatility that is independent of St rather than the usuallog-normalσ St. In effect, we are assuming that the process is an ordinary Browian motion rather than a geometric one. We give a modern treatment rather than trying to reproduce Bachelier's work. The results are of more than historial interest. First , in some markets , move- ments are more normal than log-normal , and thus such a model can provide a better fit to market-smile dynamics. Secondly, when using simulation to price we obtain a normally distributed estimate of a price rather than the true price. Thus a price estimate is the true value plus an error which is normal with mean zero. When such estimates are used within other simulations to make decisions , we of- ten effectively obtain a bias which is the positive part of this error. We can regard the bias as a call option the error and therefore quantify its size if we can price in a normal model , and once the bias has been estimated it can be removed. 3.2. The pricing formula Suppose dXt = σ dWt (3.2 .1) in the pricing measure. We have a contract that pays C(XT , T) = max(XT K , O) - 23



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