Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models Andrey Itkin New York University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Control Number: 2019053436 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FITTING LOCAL VOLATILITY Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-121-276-5 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11623#t=suppl Desk Editors: Herbert Moses/Shreya Gopi Typeset by Stallion Press Email: [email protected] Printed in Singapore Herbert Moses - 11623 - Fitting Local Volatility.indd 1 29-11-19 5:01:26 PM January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pagev Foreword – By Alex Lipton Co-Founder and CTO, Silamoney, Partner, Numeraire Financial, Connection Science Fellow, Massachusetts Institute of Technology, Cambridge, USA Prof. Andrey Itkin, a well-known and highly respected financial engineer, gives a tour de force performance in this short, insightful book. To under- stand his contribution properly, we need to step back in history. Derivatives, including options, forwards, and futures, have been around at least since early modern times. While forwards and futures are linear instruments,whicharerelativelysimpletohandle,optionsarenonlinearin nature and are more difficult to deal with. They come in all kind of flavors and include vanilla calls and puts with hockey-stick payoffs, European op- tions with more general payoffs, American, Bermudan and Asian options, as well as barrier and other exotic options. The corresponding underliers include stocks, bonds, currencies, commodities, etc. The original scientific approaches to options valuation were developed byBachelier(1900),Bronzin(1908),Boness(1964),Samuelson(1965),and others. Yet, explosive development of the field is due to the seminal work of Black–Scholes–Merton (BSM), Black and Scholes (1973), and Merton (1973), who developed a novel theory of rational option pricing, which gained universal acceptance and eventually became an indispensable tool of modern financial engineering. The main advantage of their theory com- pared to its predecessors is its ability to tie together pricing and hedging. By necessity, BSM theory is based on several idealized assumptions, which donotholdintherealworld.Theseassumptionsareasfollows:(A)markets are efficient and frictionless, i.e., there are no transaction costs and taxes; (B) there are no restrictions on short sales, frequency of trading, or the amountofsharesboughtandsold;(C)thestockpricesaregovernedbylog- normal stochastic processes with time-independent parameters. Provided v January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pagevi vi Foreword that these assumptions hold, one can construct a perfect hedge for a con- tingentclaimandpriceitunderarisk-neutralmeasure.IntheBSMframe- work every stock is characterized by a single number σ, which is called its volatility,whileitsphysicaldriftµisunimportant,andcanberiskneutral- ized, so that µ → r−q, where r is the risk-free interest rate, and q is the dividend rate. Thus, in order to price an option, we can assume that the correspondingstockpriceisgovernedbythefollowingstochasticdifferential equation (SDE): dS(t) =(r−q)dt+σdW(t), (1) S(t) S(0)=S , 0 where W is a standard Wiener process. ForaEuropeanoption,thecorrespondingBSMpartialdifferentialequa- tion (PDE) and the terminal condition have the form 1 V (t,S)+ σ2S2V (t,S)+(r−q)SV (t,S)−rV (t,S)=0, (2) t 2 SS S V (T,S)=v(S), where T is the option maturity, and v(S) is its payoff. For calls and puts, we have v(S)=(ω(S−K)) , (3) + where (cid:26) x,x≥0, (x) = (4) + 0 x<0, and ω is the call/put indicator, ω = 1 for calls, and ω = −1 for puts. It is easy to show that BSM prices of vanilla calls and puts with strike K and maturity T have the form (cid:20) (cid:18) ln(F (t,T)/K)+ 1σ2τ(cid:19) V (t,S,T,K;r,d,σ)=ωe−rτ F (t,T)N ω √ 2 (5) σ τ (cid:18) ln(F (t,T)/K)− 1σ2τ(cid:19)(cid:21) −KN ω √ 2 , σ τ whereN(.)isthecumulativenormaldistribution,τ =T−t,andF (t,T)= exp((r−q)τ)S(t)istheforwardprice.Pricesofmorecomplicatedoptions can be computed either analytically or numerically. However, in real markets, idealized BSM assumptions do not hold. Market prices are such that different vanilla options on a given underlier January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pagevii Foreword vii must be priced using different implied volatilities σ (t,S,T,K) in Eq. (3), I so that every underlier has its implied volatility surface, rather than a sin- gle volatility. For a fixed maturity T, this fact results in the volatility skew or smile; for a fixed strike K it manifests itself as the term structure of volatility. As a result, considerable modeling insight is required for pricing vanillaand,especially,exoticoptionsconsistentlywiththemarketandwith each other. Local volatility models assume that the stock price is governed by Eq. (1), with the constant σ replaced by the so-called local volatility σ (t,S).ThesemodelsextendthestandardBSMframeworkinarelatively L mild fashion, so that they are complete and allow for a perfect hedge. In general, local volatility models cannot generate proper implied volatility dynamics and tend to misprice exotic options, while at the same time pric- ing vanillas perfectly. The variance gamma model of Madan and Seneta (1990) can be generalized to the local variance gamma model of Carr and Nadtochiy (2017). Jump diffusion models, first proposed by Merton (1976), augment the regular diffusion with Poissonian jumps. These models are necessarily incompleteanddon’tallowforaperfecthedge,sincesmallandlargemove- ments of the underlier cannot be handled simultaneously. For many un- derliers, local volatility models augmented with jumps tend to agree with market prices well. Stochastic volatility models are useful in many situations. The most popularmodelofthiskindisduetoHeston(1993).Themodelassumesthat volatility is driven by a mean-reverting square-root process and generates closed-form expressions for call and put prices. For certain underliers, for example, liquid currency pairs, stochastic volatility models are adequate, though never perfect. Finally, universal volatility models proposed by Lipton (2002) combine the best features of local, jump diffusion and stochastic volatility models; as a result, they produce reliable prices and hedges not only for calls and puts but also for many exotic options. A large portion of Itkin’s book is devoted to the calibration of local volatility models. The instrument of choice for calibrating these models to the market is the celebrated forward equation of option pricing due to Dupire (1994). An alternative but equivalent formulation is given by Derman and Kani (1994). The Dupire equation allows one to price calls and puts with different strikes and maturities at once. Depending on the context,itcanheusedtoexpressσ (t,S)intermsofσ (t,S),or,conversely, I L January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pageviii viii Foreword toexpressσ (t,S)intermsofσ (t,S).Specifically,writtenintermsofcall L I prices C(t,S,T,K), the Dupire equation reads 1 C (T,K)− σ2 (T,K)K2C (T,K) T 2 L KK +(r−q)KC (T,K)+qC(T,K)=0, K C(0,K)=(S−K) . (6) + Foragivenσ ,Eq.(6)canbesolvedforwardtogeneratethecorresponding L C(T,K), and then σ via Eq. (5). Alternatively, and more importantly, I for a given price surface C(T,K), Eq. (6) can be used to calibrate the corresponding local volatility model and get C (T,K)+(r−q)KC (T,K)+qC(T,K) σ2 (T,K)= T K . (7) L 1K2C (T,K) 2 KK Once σ is known, one can use an appropriately modified BSM pricing L problem to find prices of more complicated options. Equation(7)isbeautifulintheory,butratherdifficulttouseinpractice for a variety of reasons, first and foremost due to the fact that in real markets C(T,K) is known for a rather sparse set of discrete pairs Ξ = {(T ,K )}, i = 1,...,N. This book is dedicated to developing novel and i i technicallyadvancedmethodsforsolvingthecalibrationprobleminearnest. InPartI,theauthordiscussesthegeneralconceptoflocalvolatilityand introduces the all-important no-arbitrage conditions. He develops various forms of no-arbitrage interpolation which are used in the rest of the book. The advantage of these interpolations is that, in addition to preserving no-arbitrage conditions by construction, they allow closed-form solutions for various special cases of the general volatility problem. In addition, the authordiscussesarecentworkofCarrandPelts(2015)onnoarbitragefor the implied volatility surface, and puts it into the general framework. In Part II, the calibration problem for a discrete set of maturities and strikes is analyzed in great detail. The author extends the work of Andreasen and Huge (2011), and, more closely, Lipton and Sepp (2011), to produceacontinuouspiecewiselinearlocalvolatilitysurfaceσ (t,S)inthe L spirit of Itkin and Lipton (2018). He also uses modern regression methods to perform parameter calibration, starting with the Stochastic Volatility Inspired (SVI) local volatility parameterization, Gatheral (2006), and pro- ceeding to his own parameterization, Itkin (2015), which he developed a decade ago while working as a market maker. The author convincingly demonstrates the advantages of this parameterization by comparing the fit with representative market data as well as with SVI. January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pageix Foreword ix Finally,inPartIII,theauthorexpandsthelocalvariancegammamodel ofCarrandNadtochiy(2017)inthespiritofCarrandItkin(2018a,2018b). He adds a drift and makes the stochastic driver geometric rather than arithmeticbyusingaclevertimechange.Afterthat,theauthorshowshow to calibrate the model to the market in an efficient manner by exploiting various new no-arbitrage interpolations introduced in Part I. Thisbookexpandsthepreviousworkdonebytheauthor,somesoloand someinco-authorshipwithPeterCarrandAlexanderLipton,byproviding additional material, which makes the exposition clear and uniform. One of the key advantages of the book is its coverage of the most recent findings in the general area of volatility calibration. The other one is that it is a treasuretroveofnumericalrecipesmakingthevolatilitysurfacecalibration both fast and accurate. Itkin’s book can be used as a foundation for an advanced master or PhD course in financial engineering and mathematical finance. It can also beusedbypractitionersandacademicswhowanttolearnthemostmodern and efficient approaches to building local and implied volatility surfaces in a fast and accurate way. I recommend it wholeheartedly. References [1] Andreasen,J.andHuge,B.(2011).Volatilityinterpolation,Risk Magazine, pp. 76–79. [2] Bachelier, L. (1900). Theorie de la speculation, Annales de l’Ecole Normale Superieure, 17, pp. 21–86. [3] Black, F. and Scholes, M. (1973). The pricing of options and corporate lia- bilities, Journal of Political Economy, 81, pp. 637–659. [4] Boness, A.J. (1964). Elements of a theory of a stock option value, Journal of Political Economy, 72, pp. 163–675. [5] Bronzin, V. (1908). Theorie der Pra¨miengesch¨afte; Franz Deuticke. [6] Carr, P. and Itkin, A. (2018a). An expanded local variance gamma model, Available at https://arxiv.org/pdf/1802.09611.pdf. [7] Carr, P. and Itkin, A. (2018b). Geometric local variance gamma model, Available at https://arxiv.org/abs/1809.07727. [8] Carr, P. and Nadtochiy, S. (2017). Local Variance Gamma and explicit cal- ibration to option prices, Mathematical Finance, 27(1), pp. 151–193. [9] Carr, P. and Pelts, G. (2015). Duality, deltas, and derivatives pricing, in SteveShreve’s65thBirthdayConference,http://www.math.cmu.edu/CCF/ CCFevents/shreve/abstracts/P.Carr.pdf. [10] Derman, E. and Kani, I. (1994a). Riding on a smile, RISK, pp. 32–39. [11] Dupire, B. (1994). Pricing with a smile, Risk, 7, pp. 18–20. [12] Gatheral, J. (2006). The volatility surface: A practitionals guide. Wiley, Hoboken.