www.ssoar.info Spectral methods for volatility derivatives Mijatovic, Aleksandar; Albanese, Claudio; Lo, Harry Postprint / Postprint Zeitschriftenartikel / journal article Zur Verfügung gestellt in Kooperation mit / provided in cooperation with: www.peerproject.eu Empfohlene Zitierung / Suggested Citation: Mijatovic, A., Albanese, C., & Lo, H. (2009). Spectral methods for volatility derivatives. Quantitative Finance, 9(6), 663-692. https://doi.org/10.1080/14697680902773603 Nutzungsbedingungen: Terms of use: Dieser Text wird unter dem "PEER Licence Agreement zur This document is made available under the "PEER Licence Verfügung" gestellt. Nähere Auskünfte zum PEER-Projekt finden Agreement ". 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Diese Version ist zitierbar unter / This version is citable under: https://nbn-resolving.org/urn:nbn:de:0168-ssoar-221488 Quantitative Finance SPECTRAL METHODS FOR VOLATILITY DERIVATIVES For Peer Review Only Journal: Quantitative Finance Manuscript ID: RQUF-2006-0098.R2 Manuscript Category: Research Paper Date Submitted by the 29-May-2008 Author: Complete List of Authors: Mijatovic, Aleksandar; Imperial College London, Department of Mathematics albanese, claudio; Independent Consultant Lo, Harry; Imperial College London, Mathematics Volatility Modelling, Volatility Smile Fitting, Volatility Surfaces, Keywords: Stochastic Volatility, Quantitative Finance, Equity Options, Markov Processes, Exotic Options JEL Code: Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. VarSwap.zip E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Page 2 of 41 Quantitative Finance 1 2 3 4 5 6 SPECTRAL METHODS FOR VOLATILITY DERIVATIVES 7 8 ´ 9 CLAUDIO ALBANESE, HARRY LO, AND ALEKSANDAR MIJATOVIC 10 11 Abstract. In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as 12 one of the listed products, options on its implied volatility index (VIX). This created the challenge 13 14 of developing a pricing framework that can simultaneously handle European options, forward-starts, 15 options on the realized variance and options on the VIX. In this paper we propose a new approach 16 to this problemFor using spectralPeer methods. Review We use a regime switching Only model with jumps and local 17 volatility defined in [1] and calibrate it to the European options on the S&P 500 for abroadrange 18 of strikes and maturities. The main idea of this paper is to “lift” (i.e. extend) the generator of the 19 20 underlying process to keep track of the relevant path information, namely the realized variance. The 21 lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty 22 by applying a new semi-analytic algorithm for block-diagonalization. This method enables us to 23 evaluate numerically the joint distribution between the underlying stock price and the realized 24 25 variance, which in turn gives us a way of pricing consistently European options, general accrued 26 variance payoffs and forward-starting and VIX options. 27 28 29 30 31 1. Introduction 32 33 In recent years there has been much interest in trading derivative products whose underlying is 34 a realized variance of some liquid financial instrument (e.g. S&P 500) over the life of the contract. 35 1 36 The most popular payoff functions are linear, leading to variance swaps, square root, yielding 37 volatility swaps, and the usual put and call payoffs defining variance swaptions. 38 39 It is clear that the plethora of possible derivatives on the realized variance is closely related 40 to the standard volatility-sensitive instruments like vanilla options, which are also exposed to 41 other market risks, and the forward-starting options which are almost pure vega bets and are 42 43 mainly exposed to the movements of the forward smile. Recently Chicago Board Options Exchange 44 (CBOE) introduced options on the volatility index2 (VIX) which are also important predictors 45 46 for the future behaviour of implied volatility. The main purpose of this paper is to introduce 47 a framework in which all of the above financial instruments (i.e. the derivatives on the realized 48 49 variance as well as the instruments depending on the implied volatility) can be priced and hedged 50 consistently and efficiently. 51 52 Our central idea is very simple and can be described as follows. We use a model for the underlying 53 that includes local volatility, stochastic volatility (i.e. regime switching) and jumps and can be 54 calibrated to the implied volatility surface for a wide variety of strikes and maturities (for the case 55 56 of European options on the S&P 500 see figure 1). The stochastic process for the underlying is 57 a continuous-time Markov chain, as defined in [1], where it was used for modelling the foreign 58 59 60 The authors would like to thank the referee for lucid and constructive comments that lead to a simplification of the model. AM would also like thank Jim Gatheral for useful conversations. 1For the precise definition of these products see appendices B.1 and B.2. 2For a brief description of the these securities see appendix B.3. For the definition of VIX see [12]. 1 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Quantitative Finance Page 3 of 41 1 2 CLAUDIO ALBANESE, HARRY LO, AND ALEKSANDAR MIJATOVIC´ 2 3 exchange rates. The underlying process is stationary as can be inferred from the fact that the 4 5 implied forward volatility smile behaves in a consistent way (see figures 7, 8, 9 and 10). 6 There are two features of this model that make it possible to obtain the distributions of the 7 8 future behaviour of the implied volatility and the realized variance of the underlying. The first 9 feature is the complete numerical solubility of the model. In other words spectral theory provides a 10 11 simple and efficient algorithm (see [1], section 3) for obtaining a conditional probability distribution 12 function for the underlying between any given pair of times in the future. This property is sufficient 13 to determine completely the forward volatility smile and the distribution of VIX for any maturity. 14 15 The second feature of the framework is that it allows us to describe the realized variance of the 16 underlying riskyFor security as Peer a continuous-time Review Markov chain. ThisOnly makes it possible to deal with 17 18 the realized variance by extending the generator of the original process and thus obtaining a new 19 continuous-time Markov chain which keeps track simultaneously of the realized variance and the 20 21 underlying forward rate. 22 The key idea of the paper is to assume that the state-space of the realized variance process lies on a 23 24 circle. This allows us to apply a block-diagonalization algorithm (described in detail in appendix A) 25 and the standard methods from spectral theory to obtain a joint probability distribution function 26 for the underlying process and its realized variance or volatility at any time in the future (e.g. for 27 28 time horizons of 6 months and 1 year see figure 15). This joint pdf is precisely what is neededto 29 price a completely general payoff that depends on the realized variance and on the underlying. 30 31 There are two natural and useful consequences of this approach. One is that we do not need to 32 specify exogenously the process for the variance and then try to find an arbitrage-free dynamics for 33 34 the underlying, but instead imply such a process from the observed vanilla market via the model for 35 the underlying (a term-structure of the fair values of variance swaps as implied by the vanilla market 36 data and our model is shown in figure 18). The second consequence is that this approach bypasses 37 38 the use of Monte Carlo techniques and therefore yields sharp and easily computable sensitivities of 39 the volatility derivatives to the market parameters. This is because the pricing algorithm yields, 40 41 as a by-product, all the necessary information for finding the required hedge ratios. 42 There is a rapidly growing interest in trading volatility derivatives in financial markets, mainly 43 44 a consequence of the following two factors. On the one hand, pure volatility instruments are used 45 to hedge the implicit vega exposure of the portfolios of market participants, thus eliminating the 46 47 need to trade frequently in the vanilla options market. This in itself is advantageous because of the 48 relatively large bid-offer spreads prevailing in that market. On the other hand, volatility derivatives 49 are a useful tool for speculating on future volatility levels and for trading the difference between 50 51 realized and implied volatility.