Model Calibration in Option Pricing SQU Journal for Science, 17 (1) (2012) 84-102 © 2012 Sultan Qaboos University 84 Model Calibration in Option Pricing Andre Loerx* and Ekkehard W. Sachs** Department of Mathematics, University of Trier, Trier, Germany. *Email: loerx@uni-trier.de and **Email: sachs@uni-trier.de ABSTRACT: We consider calibration problems for models of pricing derivatives which occur in mathematical finance. We discuss various approaches such as using stochastic differential equations or partial differential equations for the modeling process. We discuss the development in the past literature and give an outlook into modern approaches of modelling. Furthermore, we address important numerical issues in the valuation of options and likewise the calibration of these models. This leads to interesting problems in optimization, where, e.g., the use of adjoint equations or the choice of the parametrization for the model parameters play an important role. KEYWORDS: Adjoints, Calibration, Jump models, Local volatility models, Mixed models, Partial differential equation (PDE), Stochastic differential equation (SDE), Stochastic volatility models. معايرة النماذج في تسعير الخيارات أندريه لوركس و إكهارد زاكس فً التموٌل الرٌاضً. نناقش طرق متنوعة كاستخدام التسعٌر التً تظهر اتنفترض مسائل المعاٌرة لنماذج مشتق :خصمل التطورات المنشورة مسبقا ونعطً عرضأو المعادالت التفاضلٌة الجزئٌة لعملٌة النمذجة. نالعشوائٌة المعادالت التفاضلٌة تلك النماذج. ةنعالج مواضٌع عددٌة هامة فً تقٌٌم الخٌارات ومعاٌر ،آفاق حول الطرق الحدٌثه للمعاٌرة. إضافة إلى ذلك ا فً ماها دور تلعب تم معادالت مساعدة أو اختٌار معامالااستخدأن حٌث، مثآل، ٌاتمثلاألهذا ٌؤدي إلى مسائل مهمة فً .نموذج المعامالت 1. Introduction inancial derivatives, like options and futures, have gained considerable importance since the Chicago Board Options Exchange (CBOE), the first exchange to list standardized exchange-traded stock options, was founded in 1973. Starting with 911 contracts on 16 underlying stocks on the first trading day on April 26, 1973, the CBOE reported a total number of over 1.1 billion traded contracts in 2009, which corresponds to an average volume of more than 4.5 million contracts a day. 1 The rapid growth over the last 40 years of financial derivative markets, certainly owes its success to the publication of Black and Scholes (1973) and its extension by Merton (1973), since they laid the foundation of preference-free valuation of contingent claims. Particularly, they developed a simple, but powerful model that governs the price of European-style call and put options over time. The main achievement, however, was not only the derivation of a valuation formula in closed form but also the idea of building a (hedge) portfolio by buying and selling the underlying asset and a risk-free bond in such a 1 Source: http://www.cboe.com/aboutcboe/History.aspx & http://www.cboe.com/data/marketstats-2009.pdf F MODEL CALIBRATION IN OPTION PRICING 85 self-financing way that it perfectly matches the payoff (at maturity) of the option to be priced. Consequently, the amount of initial capital needed for building up the hedge portfolio coincides with the price of the considered European-style option. These publications form the cornerstone of today's financial industry. But not only the total number of contracts, also the variety have grown in a remarkable way. Nowadays, in addition to standard European-style plain vanilla call and put options, exotic derivatives like digital or barrier options of American- or Bermudan-type, Asian-style options like lookbacks, or chooser options, cliquets or any reasonable combination, are frequently traded on financial derivative markets. However, it has been shown (in a variety of publications and text books) that due to its simplicity the classical Black-Scholes model cannot properly capture the real market dynamics. The Black-Scholes model is, unfortunately, not suitable to adequately price and hedge exotic options. In the literature, a multiple of models can be found subsequently relaxing the assumptions of the classical Black-Scholes model, for instance, by adding another degree of freedom to the process of the underlying asset. In order to extract accurate market dynamic information to price and hedge exotic options, practitioners, e.g., traders and risk managers, need to adapt their models to the current market situation, i.e. the models have to be calibrated to a set of liquidly traded standard instruments like plain vanilla options. The pricing of options as well as model calibration are interesting mathematical problems from various points of view. They pose challenges in several areas like mathematical modeling, stochastic processes, partial differential equations, optimization and numerical analysis. In Section 2, we briefly review the fundamentals of smile-consistent option pricing and its numerical pricing techniques like bi- and trinomial trees, Monte Carlo methods, and PDE pricing, for the case where no closed form solution is available. We will focus mostly on European-style call options and briefly discuss some pros and cons of the main classes of smile-consistent volatility models proposed in the literature. More precisely, we consider stochastic volatility models, local volatility models, jump models, as well as mixed volatility models and emphasize their relevance in practice. Section 3 gives an exhaustive survey of publications on the calibration of financial market models. Although several references on Monte Carlo calibration are given, we focus mostly on literature concerning the reconstruction of the local volatility function. We distinguish parametric and non-parametric approaches and briefly illustrate three categories of calibration procedures proposed in the literature. In doing so, we closely follow the distinction of Bouchouev and Isakov (1999), i.e. optimization-based algorithms, extra- and interpolation schemes, and iterative methods. 2. Option pricing Starting with the Black-Scholes model, today's price of a European-style call (put) option with maturity T and strike K under some risk-neutral measure Q is defined as 0Call: ( , 0) = (max( , 0)), rT Q TC S e S K   (1) 0Put: ( , 0) = (max( , 0)), rT Q TP S e K S   (2) where TS denotes the asset price at maturity T given the asset price process 0( )t t TS   as a solution of the Black-Scholes stochastic differential equation (SDE) = ( )t t t tdS r d S dt S dW  (3) with 0 (0, )S   and [0, ].t T The (constant) instantaneous drift term consists of the risk-less interest rate r and the dividend yield d of the underlying. 2 Furthermore,  denotes the (constant) instantaneous volatility function and 0( )t t TW   represents a Brownian motion (or Wiener process) defined on a probability space ( , , )Q with  -algebra over the set =  and Q the unique risk-neutral measure (or martingale measure). The Brownian motion 0( )t t TW   is adapted to an adequate filtration 0( ) ,t t T  where the filtration 0( )t t T  2 For simplicity, we assume constant interest rates and dividend yields and further omit equity premiums. ANDRE LOERX and EKKEHARD W. SACHS 86 satisfies some 'technical' conditions (see (Karatzas and Shreve, 2008) for details). As already mentioned, one of the key achievements of Black and Scholes (1973) was to provide an explicit valuation formula (known as the Black-Scholes formula 3 ) for European-style call and put options, namely ( ) ( ) 1 2( , ; , ; , , ) = ( ) ( ) , BS d T t r T t t tC S t K T r d e S e K          (4) ( ) ( ) 2 1( , ; , ; , , ) = ( ) ( ) , BS r T t d T t t tP S t K T r d e K e S            (5) where 2 1 2 1 1 ln( / ) ( )( ) 2= , = tS K r d T t T t T t              with ( )x the cumulative distribution function of the standard normal distribution, i.e. 2 2 1 ( ) = . 2 y x x e dy    Hence, the Black-Scholes option price at time [0, ]t T depends on the current value of the underlying ,tS i.e. the spot price, the time the option expires T, i.e. the maturity, the exercise price or strike price K, the interest rate r, dividend yield d and finally the (constant) volatility . A well-known model-free relationship between calls and puts on the same underlying asset, with equal strike and maturity, is the put-call parity ( ) ( ) ( , ; ) ( , ; ) = . BS BS d T t r T t t t tC S t P S t e S e K       It is easy to show that given a fixed Black-Scholes price C satisfying reasonable non-arbitrage conditions, i.e. ( ) ( ) ( ) max( , 0) , d T t r T t d T t t te S Ke C e S          the mapping ( )C C   has a unique root , impl  called implied volatility. Conversely, in the classical Black-Scholes model the option price ( , ; , ; , , ) BS impl tC S t K T r d  depends uniquely on its implied volatility , impl  where impl is assumed to be constant in time t, stock price ,tS strike K and maturity T. This assumption, however, cannot be observed on the market. If one plots the observed market implied volatility against the strike K, the resulting graph will usually be downward sloping in equity markets, while it is typically valley-shaped in currency markets or for equity index options. This behavior is referred to as 'volatility skew' or 'volatility smile', respectively. Furthermore, it can be observed that the volatility skew or smile usually flattens for long term maturities. The change of implied volatilities with respect to different maturities is called 'term structure' of the implied volatility surface. Finally, the implied volatility surface also changes dynamically over time. A more detailed introduction to this topic is given, e.g., in (Hull, 2011). Although, the classical Black-Scholes model lacks on realism, implied volatility serves as a standardized (or normalized) value (usually quoted in %) of market volatility. In sticky-moneyness markets 4 , the implied volatility provides more stability than the Black-Scholes option price. Practitioners use implied volatility as a language, rather than as a model. A lot of research has been done over the last 40 years, trying to explain this strike deviation from the Black-Scholes constant volatility assumption. Many factors have been investigated as being possibly responsible for the smile and term structure of the implied volatility surface. They range from the existence of transaction costs or liquidity constraints, to stochastic volatility and jump processes for the underlying asset price process. In the following we focus on the latter ones. 3 In fact, Black and Scholes (1973) derived their valuation formula by solving the Black-Scholes partial differential equation, which will be introduced later on. 4 Moneyness is defined as the quotient between stock price tS and strike price K. MODEL CALIBRATION IN OPTION PRICING 87 2.1 Smile-consistent pricing models The idea behind the development of new pricing models, so called 'smile-consistent pricing models', is to directly extract information about the asset price and volatility dynamics from frequently traded standardized plain vanilla options in order to price and hedge exotic options. This is done by assuming the coefficients of (3) to be some deterministic function of the spot price tS and the time t, by adding new sources of randomness or by adding all of it. Since the sources of randomness are usually added to the volatility [cf. Gatheral (2006)] the generalized framework (or extension) of the Black-Scholes model is given by replacing (3) by = ( , ) ( , )t t t t t tdS a S t S dt b S t S dW (6) with 0 (0, ).S   The asset price 0( )t t TS   is, therefore, modeled by a 0( )t t T  adapted stochastic process, driven by the SDE (6), where ( , )ta S t and ( , )tb S t are the instantaneous drift and the volatility, respectively. Fengler (2005) assumed that the instantaneous volatility ( , , )t tb S t  follows some 0( )t t T  adapted arbitrarily depending stochastic process, where the ' t -dependence' simply emphasizes that ( , , )t tb S t  may also depend either on the history of ,tS i.e. 1 = , , ,t t tN S S for 0 0 and drift compensation k. The jump sizes in stock price and volatility, i.e. s nZ and , v nZ respectively, for = 1, , ,tn N are assumed to be i.i.d. (independent and identically distributed). Finally, := lim t tnn S S  denotes the stock price at n right before the jump occurs. Thus, (8) denotes a stochastic volatility model with simultaneous jumps in stock price and volatility (SVJJ). Note that Heston's stochastic volatility model is a jump-free special case, i.e. = = 0 s v n nZ Z for all n, of an AJD process and as such has at least a quasi-closed form solution as mentioned before. 5 The Merton and Heston approaches were combined by Bates (1996), who proposed a model with stochastic volatility and jumps (SVJ). Bates' model is also incorporated in (8) as a special case, where = 0 v nZ for all n. Gatheral (2006) shows that SVJ models perform empirically as well as SVJJ, but they have less parameters. Therefore, SVJ models like Heston's model are frequently used in practice. An extensive discussion about jump diffusion models can be found in (Cont and Tankov, 2004). We now turn our focus onto 'local volatility (LV) models'. They became quite popular in the past due to their simplicity, however, they have also gained a lot of criticism in financial literature (cf., e.g., (Ayache et al., 2004; Hagan et al., 2002)). 2.1.3 Local volatility models Following Fengler (2005), 'local variance' may be defined as the risk-neutral expectation of the instantaneous variance conditional on =TS K and time filtration ,t i.e. 2 2 ,ˆ ( , ) := ( ( , , )| = , ) , Q K T t T T T tS t b S T S K  where ( , , )t tb S t  is as before. Then, 'local volatility' (also called 'forward volatility') is given as the square root of local variance. The main advantage of this definition of local volatilities is that it naturally implies the purely deterministic case, but also offers some insights into the concept of stochastic volatility. Within this framework of local volatilities, for some market level = tK S at = ,T t the instantaneous volatility is given by ,ˆ( , ) = ( , ) ,t S t tt S t S t  such that (with ( , ) := ,ta S t r d for simplicity) ,ˆ= ( ) ( , )t t S t t t tt dS r d S dt S t S dW  (9) 5 In fact, the Heston's formula is given as a linear combination of two integrals of real-valued functions. MODEL CALIBRATION IN OPTION PRICING 89 defines the stock price process, which generalizes the classical Black-Scholes theory as desired. The intrinsic stochasticity is integrated out and we are left with a one-factor diffusion process. However, if by assumptions the instantaneous volatility is deterministic in spot and time, i.e. ( , , ) = ( , ),t t tb S t S t  both concepts of instantaneous and local variance coincide, since 2 2 ,ˆ ( , ) := ( ( , , )| = , ) Q K T t T T T tS t b S T S K  2 2 = ( ( , )| = , ) = ( , ) . Q T T tS T S K K T  The local volatility assumption is the easiest way of relaxing the constant volatility case and it introduces much more flexibility. In contrast to stochastic volatility models, the concept of local volatility preserves the assumption of market completeness. 6 Originally, Dupire (1994) and Derman and Kani (1994a) 7 have shown that given the distribution of the final stock price TS for each time T conditional on some starting price 0 ,S there exists a unique risk-neutral diffusion process (9) consistent with these distributions. The reason is that there exists a 'dual' or 'adjoint' PDE to the classical Black-Scholes PDE (cf. Section 2). The remarkable observation that local volatility can be seen as the market expectation of future volatility, known as 'Markovian projection', was independently derived by Dupire (1996) and by Derman and Kani (1998). Different assumptions on the special shape of the local volatility function have been made in the literature. They are either motivated by model calibration in order to reduce the number of unknowns (see Section 3.3.2), or by empirical observations (see, e.g., (Dumas et al., 1998; Coleman et al., 2001)) in order to properly capture the dynamics of the underlying asset. A prominent example is the 'constant elasticity of variance model' (CEV) introduced by Cox and Ross (1976), where 1 ( , ) =t tS t S     with , > 0.  The CEV model attempts to heuristically capture the stochastic volatility, where  controls the relationship between volatility and price. When < 1, commonly observed in equity markets, the volatility of the underlying increases as its price falls. Conversely, in commodity markets, the volatility of the underlying tends to increase as its price increases. Note that for = 1 we obtain the Black and Scholes case. Ingersoll (1997) and Rady (1997) introduced the class of bounded quadratic diffusion models, i.e. ( , )tS t which is considered to be a bounded quadratic function in asset price and/or time. Zühlsdorff (2001) has proven the existence and uniqueness of the solution of the underlying SDE and provided explicit formulas for call options assuming that the deterministic local volatility function can be split in a strictly positive and bounded function  and a quadratic polynomial p such that ( , ) = ( ) ( ).t tS t p S t  Option pricing in the quadratic volatility model is a rather delicate issue, since it touches the limits of no-arbitrage theory. Andersen (2011) clarified some confusion in literature and further extended the range of existing pricing formulas. Coleman et al. (2001) published empirical evidence that a spline representation can provide a more accurate representation in terms of hedging compared to the quadratic model considered by Dumas et al. (1998). Although the deterministic local volatility function may look very complicated, considering local volatilities can be a questionable model simplification. Ayache et al. (2004) and Hagan et al. (2002) doubt that a one-factor diffusion model delivers an adequate description of the asset price behavior. Hagan et al. (2002) illustrated that the model delta of deterministic local volatilities is wrong or at best very misleading. This, however, is a crucial issue in terms of the dynamic hedging performance of the model. Furthermore, another undesirable feature of the local volatility model is that it predicts flat future smiles, such that forward-start options or cliquets are likely to be mispriced. Beside these pricing and hedging problems, Ayache et al. (2004) criticized that local volatility models reveal no reasonable explanation for the existing smile phenomenon. Despite all criticism, local volatility models are widely used in practice. Common problems arising from using complex models like 'jump diffusion models' or even 'mixed volatility models' are the additional 6 Note that volatility is not a tradeable asset, which implies that the completeness of the market, i.e. the ability to hedge options with the underlying asset only, is lost. 7 While Dupire (1994) developed a continuous time theory, Derman and Kani (1994a) used a discrete binomial tree approach. ANDRE LOERX and EKKEHARD W. SACHS 90 computational effort, the high implementation costs, the loss of intuition, and a potential decrease in calibration stability. Hence, 'practitioners may, and in fact often do, favor a simple and intuitive model', see (Coleman et al., 2011). Furthermore, it is most likely that Dupire (1994) and Derman and Kani (1994a) did not introduce the local volatilities as a model of its own, but instead they intended to propose an intuitive way to price exotic derivatives under certain market circumstances. 2.1.4 Mixed or hybrid volatility models As jump processes have been added to stochastic volatility models to provide a better fit of model implied volatilities to market implied volatilities (especially for short term maturities), the local volatility framework has been applied to stochastic volatility models. So-called 'stochastic local volatility (SLV) models' were proposed by Blacher (2001) and Lipton (2002) and were studied further, e.g., in (Ren et al., 2007; Piterbarg, 2007; Alexander and Nogueira, 2008; Henry-Labordère, 2009). As an example, the governing SDEs for a 'Heston-type stochastic local volatility model' are: = ( ) ( , ) = ( ) s t t LV t t t t v t t t t dS r d S dt S t v S dW dv v v dt v dW        (10) with 0 0, (0, )S v   and , = , s v t tdW dW  where again r, d, , , ,v and  are as in (7) and LV denotes the local volatility function. Then, in the framework of local variance, the instantaneous 'hybrid' variance takes the form: 2 2 ,ˆ ( , ) = ( , ) ( | = , ) . Q K T t LV T T tS t K T v S K   (11) Because of this particular form of (11), it is not possible to separate the influence of the stochastic component from the local component in an intuitive manner. Thus, Tavella et al. (2005) prefer to define the instantaneous hybrid volatility as a weighted sum of a stochastic component and a local component. It is worth mentioning that Lipton (2002) and Lipton and McGhee (2002) further extended (10) by adding jumps to the stock price process .tS Among others, this extension of (10), called the 'universal model', was strongly criticized by Ayache et al. (2004). It is argued that, roughly speaking, there is no chance to reveal the true market smile dynamics, since the freedom of the local volatility function can nearly compensate every dynamics introduced by the stochastic or jump component. In practice, this problem is usually addressed by separately calibrating the model parameters to extract plausible dynamics from the market. 2.2 Numerical evaluation of smile-consistent pricing models Fast model evaluation is a crucial issue in practice. In order to be competitive with other market participants, very complex derivatives need to be priced nearly on-the-fly. Additionally, a fast and stable pricing scheme is essential when calibrating a financial market model to a large number of market data. Therefore, it is not surprising that models, which provide a closed or quasi-closed form solution, have become popular in practice. A survey of most of the existing market models with closed (or quasi-closed) form solutions has been given , e.g., in (Kolb and Overdahl, 2010, Chap. 27; Hull, 2011; Andersen, 2011), - especially for unbounded quadratic local volatility models. In the early years, bi- or trinomial trees have been a typical approach to price path-independent and path-dependent options in consistence with the prevailing volatility smile. This valuation method, which can be seen as a discrete version of Black-Scholes pricing PDE, was pioneered by Cox et al. (1979) (CRR). 8 A very natural way to price complex derivatives are Monte Carlo or quasi-Monte Carlo methods. They are based on the continuous time models, i.e. the fundamental pricing formula (1) or (2) and the considered market model: 8 More precisely, it can be easily shown that, for instance, the trinomial method is an example of an explicit finite difference scheme of Black- Scholes pricing PDE and therefore it inherits certain stability properties of finite difference methods, cf. (Duffy, 2006, Chap. 13) MODEL CALIBRATION IN OPTION PRICING 91  0 0 ( , 0) = ( , ) s.t. = ( , ) ( , , ) , (0, ) , rT Q T T t t t t t t t C S e S dS a S t S dt b S t S dW S         (12) where ( , )T TS  denotes the payoff function, depending on the asset price TS at maturity T and possibly on some history 1 = , ,T t tN S S (0