From Implied to Spot Volatilities

Transcription

1 From Implied to Spot Volatilities Valdo Durrleman A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Operations Research and Financial Engineering June 4

2 c Copyright by Valdo Durrleman, 4. All Rights Reserved

3 Abstract Given the quote price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter to be put into Black-Scholes formula to give the same price as the option quote price. This dissertation is concerned with the link between the implied volatility and the actual volatility of the underlying stock. Such a link is of particular practical interest since it relates the fundamental quantity for pricing financial derivatives the actual volatility of the underlying stock), which is not observable, to directly observable quantities such as implied volatilities. The link that we derive in chapter is a link between the dynamics of the two quantities. So far these quantities were mostly studied at a given time whereas we work at the level of processes. This is the main result of the dissertation. In chapter 1, we shall first review current practical problems in option pricing. Our aim there is twofold. First, we want to show that from a practical point of view, studying dynamics is very natural. Second, we shall identify two practical issues to which we shall propose answers in chapter 3. Although the main motivation of this dissertation comes from contemporary issues in the study of financial markets, chapter also gives a solution to an inverse problem in the mathematical sense. One wishes to recover the structure of a stochastic process from a family of conditional expectations over its distribution. Besides the main result, this dissertation makes the following contributions. It brings new insights about implied volatility dynamics. In particular, it was observed that its motion was extremely rigid in the sense that the motion of a specific point determines the entire surface dynamics. This statement will be made more precise. Second, it provides with simple closed form approximations for implied volatilities. These approximations avoid having to compute expectations to get option prices. Third, this dissertation gives qualitative and quantitative understanding of common models used in practice. iii

4 Acknowledgements I would like to thank my advisor, Professor Carmona. His trust and support helped me a lot, especially during the final steps of completing my work in the Fall of 3. I am greatly indebted to my teachers, Professors Çınlar and El Karoui. To all of them, I express my deepest gratitude. They have taught me what I know about mathematical finance and stochastic processes, and more importantly, shared with me their enthusiasm for mathematics and its applications. I owe many warm thanks to Professor Dayanık for being my dissertation reader and to Professor Sircar for having agreed to be a part of the committee of examiners. iv

5 Contents Abstract Acknowledgements List of Figures iii iv vii 1 Implied volatility: a dynamical point of view The Black, Scholes, and Merton paradigm The smile effect Two practical issues What information does the implied volatility surface contain? Model choice: the Dupire, Derman, and Kani approach The calibration procedure Smile dynamics Static hedging of a down-and-in call option Deterministic models: sticky and floating smiles Hedging a book of options Stochastic models for implied volatility smiles Outline From implied to spot volatilities 1.1 Framework Boundary behavior v

6 .3 Implied volatility dynamics V dynamics X dynamics Recovering the spot volatility Recovering the spot volatility dynamics Concluding remarks about our assumptions From spot to implied volatilities and applications From spot to implied volatilities Black-Scholes deterministic volatility model Heston stochastic volatility model Dupire local volatility model SABR stochastic volatility model Model analysis Stochastic models for implied volatilities Computing implied volatilities First order approximations Higher order approximations Static hedging of a barrier option when the spot is Markov Static hedging with a constant volatility Static hedging in Dupire local volatility model Conclusions 75 A Semimartingales with spatial parameters and generalized Itô formula 77 A.1 Semimartingales with spatial parameters A. Generalized Itô formula A.3 Convergence vi

7 List of Figures 1.1 Calibrated smiles Smile dynamics: CEV vs. Heston Smile dynamics: Quadratic vs. SABR Static hedge ratios for down-and-in call options vii

8 Chapter 1 Implied volatility: a dynamical point of view In this first chapter, we present some of the contemporary issues faced by actors in financial markets. We shall put emphasis on two problems: model calibration to market data on the one hand and risk management of a book of options on the other hand. In so doing, we shall review common practice to tackle these problems as well as their flaws. In the second part of this chapter, we shall advocate a dynamical point of view. Such a point of view is very natural in practice. With this first chapter our hope is to justify from a practical perspective chapters and The Black, Scholes, and Merton paradigm In their seminal works, Black and Scholes 1973) and Merton 1973) show how to price options on a stock. A common option is a European call option; it is defined by a date T called date of maturity, a positive number K called strike or exercise price, and it gives the right to its owner to acquire at time T one unit of the stock at the unit price K. Black and Scholes 1973) came up with the now well known Black-Scholes formula which gives the price of such an option as a function of the stock price, K, 1

9 T, the short rate of interest, and the volatility of the stock. Only this last parameter is not directly observable but it can be estimated from time series data. European call options as we just described them are now actively traded and their price is set by supply and demand in the marketplace in very much the same way stocks set their prices. At a first glance, the works of Black and Scholes 1973) and Merton 1973) on how to price an option seem now irrelevant: the market in doing it for us! Such is however not the case. First, there exist more complex options, often called exotic options, a simple example that we shall consider later is a barrier option. These are not actively traded and therefore need to be priced. More importantly the work of Black and Scholes 1973) and Merton 1973) is still of utmost importance because of the paradigm they proposed to price options. Their fundamental contribution was to see that the risk contained in an option the uncertainty about stock prices in the future) could be exactly synthesized in a self financing portfolio. In other words, they provide a trick to change risks in the future into portfolio strategies of stocks and bonds. These portfolios have simply to be rebalanced continuously and in a precise way but without injecting cash. This means that the seller of an option will exactly meet his or her obligations at maturity by simply holding at each time a certain quantities of stocks and bonds. This is true in every state of the world, whatever happens to the stock! The fair price of the option ought then to be the initial cost of such a replicating strategy. As we shall see in the next paragraph, the hypotheses under which the Black- Scholes formula was established are wrong. However, the idea of dynamically hedging the risk is still the main methodology to price options The smile effect The Black-Scholes formula in Black and Scholes 1973) and Merton 1973) is obtained under the assumption that the stock can be modeled as a geometric Brownian motion

10 with constant volatility. Since option quote prices are available and the only unknown parameter is the volatility, if the Black-Scholes assumption were true, we could find the volatility of the stock by simply inverting Black-Scholes formula. A volatility thus obtained is called implied. In particular, for options with different maturities and different strikes but written on the same stock, one should find the same implied volatility, i.e., the volatility of the stock which is unique. Such is not the case. At a given maturity, options with different strikes trade at different implied volatilities. When plotted against strikes, implied volatilities exhibit a smile or a skew effect. The Black-Scholes model and its pricing formula are wrong. As often quoted, the implied volatility is the wrong number to put in the wrong formula to obtain the right price. It comes at first as a surprise to see this apparently irrelevant number being constantly used by traders. Why should it deserve so much attention? A first answer is pointed out in Lee ).... ) it is helpful to regard the Black-Scholes implied volatility as a language in which to express an option price. Use of this language does not entail any belief that volatility is actually constant. A relevant analogy is the quotation of a discount bond price by giving its yield to maturity, which is the interest rate such that the observed bond price is recovered by the usual constant interest rate bond pricing formula. In no way does the use or study of bond yields entail a belief that interest rates are actually constant. As yield to maturity is just an alternative way of expressing a bond price, so is implied volatility just an alternative way of expressing an option price. The language of implied volatility is, moreover, a useful alternative to raw prices. It gives a metric by which option prices can be compared across different strikes, maturities, underlyings, and observation times; and by which market prices can be compared to assessments of fair value. It is a standard in industry, to the extent that traders quote option prices in vol points, and exchanges update implied volatility 3

11 indices in real time. In chapter, we shall give another answer to this apparent puzzle by showing that the implied volatility smile dynamics for short maturities and the spot volatility dynamics are intimately related. We shall soon see that spot volatilities are crucial in practice Two practical issues Before we continue the discussion of implied volatilities, we introduce some of the problems we would like to solve. We shall focus on two issues. First we need a method to price exotic options. The simplest example of such options are barrier options that give the same rights as European options under the additional constraint that the spot go/do not go below/above a prespecified barrier level. These options will be priced using the Black-Scholes paradigm, i.e., their price will be the initial value of a self financing portfolio that synthesizes their risks if such a portfolio exists. Following the line of reasoning of Black and Scholes 1973) and Merton 1973), one must start off with a model for the spot process. Whereas Black and Scholes 1973) and Merton 1973) had lots of degrees of freedom to choose their model for the spot price, we are now constrained by the fact that our model candidate must give back the price or, equivalently, the implied volatilities) of actively traded options. It is therefore crucial to understand what the implied volatility smile tells us about the spot process before we can model it any further. Before assessing this question, we would like to mention a second problem faced by traders: how should one manage the risk of a book of options? Dynamically hedging a single option is rather well understood. We are guided here by El Karoui et al. 1998) and Rebonato 1999). By buying a European call option and selling its replicating portfolio, a trader is in fact trading the volatility. He or she enters such a deal only if he or she thinks the European call option is cheap 4

12 or, equivalently, that its implied volatility is low. Indeed, his or her profit at option s maturity precisely depends on the future realization of the implied volatility during the option s life. To see why this is so, one can compute over a small time interval t, the profit or loss of the portfolio option replicating portfolio). It is given by: S P & L = 1 ) ΓS Σ t) S where Σ is the implied volatility at which the option was purchased and S/S) is the stock volatility experienced during the small time interval t. Γ is the Black- Scholes gamma of the option, i.e., the second derivative of the value function with respect to the current value of the spot. In the case of a European call option, it is always positive. One sees on the formula above that the trader wants the stock to vibrate as much as possible, i.e., S/S) to be as large as possible. In particular, if the realized volatility is always above the implied volatility at which the option was purchased, the trader makes a profit in all states of the world. This example sheds some light on terminology like being long/short or trading the volatility, being long/short or trading the gamma, etc. Put it another way, El Karoui et al. 1998) show that using the replicating strategy proposed by Black and Scholes with the implied volatility makes sense even if the spot is not Markov. From this example, one should also keep in mind that traders use nowadays the Black, Scholes, and Merton paradigm to dissociate the risk in the spot delta risk) from that in the volatility vega risk.) Dynamic hedging of a portfolio consisting of several options with different strikes and different maturities is more complicated. In particular, the gamma of such a book does not have a constant sign. As pointed out in Avellaneda et al. 1995) in the case of a call spread option whose payoff has a mixed convexity, no hedge based on a Black-Scholes delta can superhedge the claim even when the unknown spot volatility 5

13 stays within a band. In that context, one needs a model to compute the risks. Such a model will provide a way of aggregating the risks of options with different strikes and maturities. This second practical issue, like the pricing of exotic options, shows the importance of a model consistent with the observed implied volatility smile. The next paragraph is about its informational content What information does the implied volatility surface contain? If the Black-Scholes model is wrong, how can we assess the main issues faced by the actors in option markets? We now come back to the question of knowing what the smile tells us about the spot process. As we have explained, this is the very first question if one wants to build a model for the spot process. If we keep the Black-Scholes paradigm in mind, the price of an option is the initial cost of a self financing replicating strategy. If such a replication is possible, i.e., if the market is complete, this initial cost can be represented as the expected value of the option payoff under the risk-neutral measure. If we denote the price of European call option with maturity T and strike price K at time by CT, K), this means that CT, K) = E { S T K) +} 1.1) where S is the risk-neutral spot price process. We shall assume for simplicity that interest rates are zero. Breeden and Litzenberger 1978) showed that the knowledge CT, K) for all K at a given T is equivalent to the knowledge of the risk-neutral distribution of S T. This is seen by formally) differentiating the above equality. It gives the cumulative 6

14 risk-neutral distribution function of the spot at time T : P {S T K} = 1 + K CT, K). Observing at a given day option prices of all strikes for a specific maturity date gives the marginal distribution of the spot process for this date. If in addition we can also observe prices for all maturities, the marginal distributions of the process are completely known. However, it is important to notice that this leaves many possibilities about the full risk-neutral distribution Model choice: the Dupire, Derman, and Kani approach In view of the questions of paragraph 1.1., one needs to propose models to explain smiles. Different ideas have been proposed. Instead of a constant volatility, one can posit a volatility that is a function of the spot process itself. This way, the spot process is a Markov process solution of a stochastic differential equation. A more general approach consists in a volatility being a stochastic process on its own. In such a case the spot process alone is no longer Markov and such models are often called fully) stochastic volatility models. Finally, people have further proposed to introduce jumps in the spot process. This last idea can of course be combined with any of the previous ones. In this section, we shall focus on the first idea, i.e, on the case where the spot is a continuous Markov process. A first systematic approach to construct a spot model consistent with the observed implied volatility smile was given by Dupire 1994) and Derman and Kani 1994). A very detailed exposition of this method with emphasis on its numerical implementation can be found in Rebonato 1999). As explained in the previous section, observed prices of European options only give information about the marginal distributions of the spot process. Dupire 1994) 7

15 and Derman and Kani 1994) show that under the assumption that the spot process is continuous and Markov, the spot process distribution is completely specified. More precisely, let us suppose that the volatility of the spot is a deterministic function of the current value of the spot and possibly time, say, σt, S). Dupire 1994) in particular says that we can find back this function if we observe the entire implied volatility surface function of T and K) at a given date. His result reads in terms of prices: T Ct, S) σt, S) = 1.) S KK Ct, S). where C is the function defined in 1.1). This method is very appealing from a theoretical point of view. Let alone its numerical difficulties, it has a main shortcoming. It is its dependence on the date at which the volatility function σ is computed. This calibration procedure, may give very different answers at very close dates. This means that traders must change their model every day. This is enough to loose confidence in such a method. Also, empirical tests show that spot processes are not Markov. We shall further discuss the calibration problem in the next paragraph. One should notice that formula 1.) gives the spot volatility from prices or, equivalently, from implied volatilities.) Chapter is therefore very much in the same spirit as this result The calibration procedure In this paragraph, we shall give an example of calibration. For definiteness, let us assume that the spot process is continuous and Markov and more precisely that it is a CEV process. The volatility of a CEV Constant Elasticity of Variance) process is a simple power law: σt, S) = αs β 1. 8

16 α > and β 1 are fixed parameters. Today s stock price is S = $1. It evolves as ds t = S t σt, S t )dw t where W is Brownian motion under the risk-neutral measure. Again, interest rates are zero for simplicity. Since the volatility is not constant, this model gives rise to a smile as shown by the solid line in Figure 1.1. The parameter α is chosen so that today s spot volatility is % and β =.1. Let us imagine a trader. Of course, he or she does not know the form of the spot volatility, let alone the particular values of α and β. Let us suppose that his or her favorite model is the Heston model. This model is a stochastic volatility model for which the spot volatility is a correlated mean reverting square root process. This means: ds t = σ t S t dw t together with dσ t = RevSpd RevLvl σ t ) dt + Vvolσt dw t where W and W are correlated Brownian motions with correlation ρ. One reason he or she might like this model could be that there are four degrees of freedom the reversion speed RevSpd, the mean reversion level RevLvl, the volatility of the volatility Vvol, and the correlation ρ), allowing to fit many different smile shapes yet being parsimonious. In Figure 1.1, we see that two possible values of the parameters can give equally good fit to the observed market data. There can be more than one answer to the calibration procedure. This is a general fact which is not particular to the Heston model. As we argued in paragraph 1.1.3, we only know the marginal distribution of S from the implied volatility smile. There are many reasonable models that fit any given implied volatility surface as there are many processes with specified marginal distributions. More importantly these processes can 9

17 Figure 1.1: Two calibration results against observed smile for options maturing in 3 months. Spot trades at $1. The observed smile is that of a CEV model with α = 1.6 and β =.1 for which the volatility of the spot is %. The two fitted smiles are obtained with Heston models, one with RevLvl = 5% and RevSpd =.1 the other with RevLvl = 35% and RevSpd =.6. In both cases, Vvol = 1.9% and ρ =.7. have widely different distributions yielding to very different prices for exotic options. In a recent paper Hagan et al. ) show that the dynamics of the smile can also vary widely for different models. To illustrate this last point, let us make the following experiment. Suppose that the stock drops tomorrow to $95. Since the spot is Markov, tomorrow s spot volatility can be computed and from the parameters we find.9%. Let us assume that the trader can effectively estimate this value and use it together with the estimated parameters in Figure 1.1 to price options. Depending of the model choice that he or she has made he or she would get two different answers as shown in Figure 1.. In one case, the smile stays rather close to the market data but in the other case it moves upwards. This means that two equally good fits can produce different 1

18 Figure 1.: The three smiles of Figure 1.1 after the spot dropped to $95. The volatility of the spot is now.9%. dynamical answers. If the trader had chosen the bad set of parameters, he or she would have to recalibrate his or her model to the market data. This should be sufficient to loose confidence in a model. To compute the implied volatility smiles of this section, we used the formulas derived in paragraph Smile dynamics As shown in Figure 1.1 and 1., smiles can have different dynamics. This fact will be further evidenced in Figure 1.3. In this section, we shall show that the smile dynamics is very natural to a trader in the context of the two questions raised in paragraph

19 1..1 Static hedging of a down-and-in call option This example is inspired by a case study in Rebonato 1999) pp We simplify the situation quite a lot by looking at a single barrier option instead of a double barrier option. Static hedging of barrier option was introduced by Bowie and Carr 1994); see also Carr et al. 1998). By contrast to dynamic hedging where continuous trading in the stock and the risk-free bank account replicates the contingent claim payoff, static hedging accomplishes the same task with a portfolio that is discretely rebalanced. In fact, in the particular example of this paragraph, it is rebalanced once or never. Let us consider the case of a down-and-in call option. The payoff of such an option is the usual call payoff S T K) + under the condition that the stock process goes below the barrier H at some point before maturity T. To make our discussion as simple as possible, we shall assume that the option is regular, i.e., H < K. Let us first recall the idea of static hedging when volatility is constant. This idea can be put on a firm mathematical ground by applying the strong Markov property see, for instance, paragraph 3.5.1). Assume that the stock reaches the barrier H at T H < T. At this very moment, the barrier option is precisely worth a call option with strike K, i.e., it is worth Call spot = H, strike = K, mat = T ). Since volatility is constant, the price is given by Black-Scholes formula. Moreover, a simple computation shows that, Call spot = H, strike = K, mat = T ) = K ) H Put spot = H, strike = H K, mat = T. Therefore, at T H, the option is worth K/H put options with strike H /K and maturity T. The advantage of this representation is that such put options are worthless if the stock stays above H since their strike is H /K < H. Whether or not the spot has reached the barrier during the option s life, the value of the barrier option is worth 1

20 K/H put options with strike H /K and maturity T. This gives the price of the barrier option as well as its replicating strategy: do nothing until the spot hits the barrier, then convert put options into the call option. Obligations are met at maturity in all cases. In the presence of smile, the same reasoning can be made. There is however a major difficulty: how do we know the option prices when the spot hits the barrier? From today s implied volatility smile we only know the distribution of the spot process at a fix date conditionally on having its value today. As shown by the example in Figure 1.1 and 1., this leaves many possibilities for its distribution at maturity conditionally on being at the barrier, let alone the law of the hitting time itself. Part of the information needed to price and hedge barrier option is exactly that of the shape of the smile when the spot moves to the barrier. We need to know the dynamics of the smile. 1.. Deterministic models: sticky and floating smiles The dynamics of the smile is so crucial in the context of pricing and hedging barrier options that traders are ready to abandon the route sketched in paragraph Although this route is mathematically more rigorous, traders often prefer to specify directly the shape of the smile when the spot hits the barrier independently of any given model. There are two common practice in that respect: the sticky smile or the floating smile as explained, for instance, in Rebonato 1999). Sticky smiles are smiles that do not move if the underlying moves. Floating smiles on contrary move exactly with the spot. To illustrate this point, let us go back to the implied volatility smile in Figure 1.1. We wish to compute the implied volatility of a call option with maturity 3 months and strike $1 when the spot trades at $9. In a sticky smile world the implied volatility for this option is again 18.4%. In a floating smile world it would be 19.%, which is implied volatility for a call option with strike 13

21 $18 today. In other words, a trader who believes in a sticky smile world would label the x-axis in Figure 1.1 with strikes and use the same graph whatever the spot value is. On the contrary, a trader who believes in a floating smile world would label the x-axis with moneyness i.e., with the ratio K/S) and use the same graph whatever the spot value is. This explains why floating smiles are also called sticky delta smiles since the Black-Scholes delta is a function of moneyness only Hedging a book of options Let us go back for a moment to the example of paragraph If the trader had chosen the bad set of parameters, he or she would have to recalibrate his or her model to the market data. As explained in Hagan et al. ), in a case a trader is dynamically hedging a book of options, a recalibration procedure leads to a recomputation of hedges. Non stable hedges can prove extremely costly in practice. With the formulas derived in paragraph 3.4., we reproduce in this paragraph the dramatic example of Hagan et al. ). Let us consider the following two models. The first one is termed quadratic Markov: σ t = γ + δ S t S ). This model may explode in finite time because the function S γ + δ S S ) does not satisfy the usual linear growth constraint on the coefficients of a stochastic differential equation. This should not be of much concern since we can always modify this function for large values of S to make it grow linearly. We are looking at short maturities and such a modification should not change the conclusions of this qualitative discussion. The second model is the SABR model of Hagan et al. ). It is a lognormal 14

22 stochastic volatility model: ds t = σ t S t dw t together with σ t = α t S β 1 t dα t = να t dw t where W and W are correlated Brownian motions with correlation ρ. The dramatic difference between these two models is shown in Figure 1.3. With an initial spot value of $1, the two models produces similar smiles. They differ somewhat for large strikes. However, when the spot drops from $1 to $95 the two smiles move in opposite directions. Figure 1.3: Two different smile behaviors for options maturing in 3 months when the spot drops from $1 to $95. The quadratic smile was obtained with S = 15, γ =.14 and δ = 1 4 for which the volatility of the spot is %. The SABR smile was obtained with α =.8, β =.7, ν = 64% and ρ =.7 for which the volatility of the spot is also %. The two smiles move in opposite directions. 15

23 The value of a portfolio of options depends explicitly on the values of the underlying spot process and of different points of the implied volatility surface, one for each option. Viewed this way, the problem of hedging a portfolio of options is similar to that of a portfolio of bonds in fixed income markets. Each bond in the portfolio involves one or more points of the yield curve. To hedge such a portfolio, a trader would buy additional instruments so that the overall portfolio is unsensitive to moves of the yield curve. The most important move to be hedged is typically a parallel shift of the yield curve. Other important moves can be found by performing a principal component analysis of the yield curve. In practice to find the hedge ratios one computes the value of today s portfolio and then its value after bumping the whole yield curve by a small amount. One buys enough bonds to get the same value before and after bumping the yield curve. One would like to do the same in the case of the implied volatility surface. Cont and Da Fonseca 1) have performed principal component analysis for implied volatility surfaces. However, how the methodology for the yield curve must be translated for the implied volatility surface is not straightforward. Indeed, Heath et al. 199) give a way of justifying the methodology for interest rate by providing an arbitrage free framework for the dynamics of the yield curve. But such an approach has not been carried out yet in the case of implied volatilities. We will review attempts in this direction in the next paragraph. The practical issue can be summarized as follows. Suppose for instance as in Figure 1.1 that the at-the-money implied volatility i.e., strike $1) is % and that the implied volatility with strike $11 is 1%. We bump the at-the-money implied volatility by 1 pt. By how much should one bump the other implied volatility to respect no arbitrage conditions? 16

24 1..4 Stochastic models for implied volatility smiles As explained in the previous paragraph, an appealing idea consists in modeling implied volatilities directly. This approach is very much in the same spirit as the Heath et al. 199) approach. It was initiated in the present context by Dupire 1993), Derman and Kani 1998), Zhu and Avellaneda 1998), Ledoit and Santa-Clara 1999), Schönbucher 1999), Carr ), Brace et al. 1). Whether they look at a single implied volatility or at the entire surface, their idea is to model implied volatilities directly. There are many reasons why this approach has not been successful yet. In the case of implied volatilities there are structural constraints on the surface shape. Any model must ensure that these constraints are satisfied. More precisely, any smooth function T ft ) can be a valid initial forward curve for any Heath-Jarrow-Morton model HJM for short). On the contrary, a smooth option price surface T, K) CT, K) must be such that T CT, K), K CT, K), and KKCT, K). These constraints come directly from 1.1) and they of course translate into constraints on the volatility surface. Note that these necessary conditions are not far from being sufficient in the following sense. In case they hold, there is a spot volatility process that produces that precise implied volatility smile. A simple candidate is the Markov spot process of Dupire 1994) and Derman and Kani 1994) of paragraph Not only are the admissible surface shapes more complicated but their dynamics are also poorly understood. The HJM methodology gives us risk free dynamics for the forward curve f without any reference to the short rate of interest r. In case one wishes to model the implied volatility directly there seems to be no simple way of 17

25 disentangling implied volatilities from the corresponding spot volatility. To be more precise, let us sketch the HJM methodology. Start with a short rate process r. Zero coupon bonds are securities that pay $1 at maturity T and their price at time t is denoted by Bt, T ). Risk neutral valuation gives Bt, T ) = E {e T t r sds } F t. Then, we define the continuously compounded forward rate at time t for maturity T, ft, T ), by Bt, T ) = e T t ft,s)ds. 1.3) Then, Heath et al. 199) derive that f satisfies a stochastic differential equation for each T : T ) dft, T ) = γt, T ) γt, s)ds dt γt, T )dw t 1.4) t { where γt, T ) is related to the martingale E e } T rsds Ft in the following way. If we are working in a Brownian filtration, this positive martingale has the martingale representation property so that we can write it as E {e } T rsds Ft = E {e } T rsds + Bs, T )Γs, T )dw s for some adapted processes Γs, T ). Finally, γt, T ) = T Γt, T ). What is important to notice is that r has disappeared in 1.4). It can be recovered by proving that in fact ft, t) = r t. This remark allows us to use 1.4) for modeling purposes by specifying a volatility function γt, T ) for the forward rate. The dynamics are completely specified and arbitrage free. Let us now try to mimic this reasoning in case of implied volatilities. We start off 18

26 by writing the price of call options as C t T, K) = E { S T K) + Ft } where S is the spot process. We denote its volatility by σ. Then, we define the implied volatility Σ t T, K) by inversion of the Black-Scholes formula that plays the same role as the exponential function in the interest rate case at 1.3). The problem is that the stochastic differential equation for Σ t T, K) is plagued by S and its volatility σ see, for instance, Proposition.3.1 below or Brace et al. 1).) There seems to be no simple way of getting rid of σ. This annoying fact prevents us from describing the dynamics of the implied volatility surface intrinsically as it was possible for f. However, we shall show how the Σ t T, K) dynamics relates to that of σ t in chapter. In fact, we shall see that Σ t t, S t ) = σ t. This very much in the same spirit as the equality ft, t) = r t in the case of interest rates. 1.3 Outline We close this first chapter by an outline of the next two chapters. Chapter aims at answering the following question. How much do we know about the spot volatility if we can observe sufficiently many prices of options written on that spot? This is precisely the question of paragraph We shall tackle this problem from a different angle in the sense that we will study dynamics instead of focusing on data at a precise date. We shall take the point of view sketched in the previous section. We are considering a market where the primary securities are the spot and liquid options. As already 19

27 pointed out, there is a lot of information about the spot process in option prices. In fact, we shall see that under some regularity conditions, we can find back the spot volatility dynamics. We want to make very few assumptions on the spot dynamics in order to have the most general understanding. On the other hand, our results will only be local. Unlike in the case of Dupire s result, in the Markov case we will not be able to recover entirely the deterministic volatility function as in 1.). Instead, we shall get its local shape, namely its first and second order derivatives in the space variable at the current value of the spot and its first derivative in the time variable at the current time. In chapter 3, we shall present applications of the results in chapter. We shall see that they lead to a very simple understanding of dynamics of volatility surfaces. We shall also derive closed form approximations for implied volatilities for several popular spot volatility models. This provides some insights and some answers to the stochastic modeling of implied volatility surfaces. Finally, we shall describe a static hedging procedure for barrier options in presence of smile. The two issues raised in paragraph 1.1. will therefore find partial answers in this last chapter. Whether they are satisfactory from a practical point of view is left for future work.

28 Chapter From implied to spot volatilities The main results of this chapter are Theorems.4.4 and.5.1. The first theorem gives back the value of the current spot volatility from the observed implied volatility smile. The second theorem gives its dynamics across time via a stochastic differential equation. These theorems are of the inverse type. Indeed, we start off with a spot volatility process and we are interested in recovering it from actually observable quantities such as implied volatilities. We obtained these theorems by a careful study of option prices near expiry. The main finding of this chapter is that asymptotic behavior of option prices near the money give all information about the spot process 1. The idea of the proof is to observe that implied volatilities are constrained by no arbitrage arguments to have a very special behavior near expiry. Namely, when multiplied by the square root of the remaining time to maturity, they go to zero as the time to maturity approaches zero. We then draw the consequences of that particular behavior on the dynamics of the implied volatility surface. This dynamics is given by an application of Itô formula. 1 In a recent paper, Carr and Wu 3) build an econometric test to tell from data of option prices near expiry whether the underlying spot process is a purely continuous process, a pure jump process or a combination of both. Although we restrict ourselves to the continuous case, it should be noted that on a general level our idea bears some similarities with theirs. 1

29 To study these questions we found the language of semimartingales with spatial parameters particularly appropriate. We follow closely the definitions and theorems of Kunita 199). In order to have a self-contained dissertation, these results are recalled in appendix A. The other techniques are basic real analysis techniques and standard stochastic calculus results for finite dimensional Brownian motion..1 Framework Let Ω, H, P) be a probability space with a n-dimensional Wiener process W t ) t on it. We shall use boldface letters for vectors; W i will denote the i-th component of W. The filtration generated by the Wiener process has been augmented as usual and is denoted by F t ) t. All processes considered in this chapter are adapted with respect to that filtration. We shall assume that probability measure P is risk-neutral, that is, discounted price processes are localmartingales if there is no arbitrage in the market. We first define the risk-neutral stock process S. As is well known, there is no loss of generality in assuming that interest rates are zero. We are given a spot volatility process σ, which is jointly measurable, adapted to the filtration F t ) t, and satisfies the following integrability condition: t σ sds < a.s..1) Since σ satisfies.1), we can define S to be the following stochastic exponential S t = S exp σ s dws 1 1 ) σsds. In other words, S is a typical positive localmartingale in a Brownian filtration. We shall say that it is a global localmartingale because it is well defined for all t.

30 Let us stress that, whereas S is only driven by the first noise, σ is adapted to the entire filtration F t ) t generated by W t ) t. In financial terms, we do not assume completeness of the market. We make the following basic assumption: Assumption 1. S t is integrable for all t and σ t ω) >, Leb P a.e. t, ω). Our main and fundamental assumption is that liquid options are marked-tomarket. By liquid options, we mean options with short maturities and strikes near the money. More precisely, let x > and ȳ > be fixed throughout the rest of the chapter, at any given time t, we are only considering the continuum of options whose strike prices K lie in the open interval S t e ȳ, S t eȳ) and whose maturities T lie in the open interval t, t + x). For T, K), x) S e ȳ, S eȳ), i.e., for an option with strike K and maturity T which is liquid today t = ), let us define τt, K) = inf { t > : ) ln St ȳ} T. K τt, K) is the time at which this option stops being liquid because either it has expired or the value of the stock has gone too far away from its strike. Lemma.1.1. τt, K) is an accessible lower semicontinuous stopping time. Proof. Indeed, for T, K) fixed, τt, K) is a stopping time. Next, for a.e. ω, T, K) τt, K) is a positive lower semicontinuous function. Finally, the sequence of random fields τ n T, K) = inf { t > : ) ln St ȳ1 )} n T K is such that τ n T, K) < τt, K) because S has continuous paths, and such that τ n T, K) τt, K) as n goes to infinity. Let T, K) fixed, on {t < τt, K)}, C t T, K) denotes the price at time t of the option struck at K with maturity T. Since we assume it is marked-to-market, no 3

31 arbitrage arguments yields that it has to be a localmartingale. We know its terminal value and this gives the well known risk neutral representation: C t T, K) = E { S T K) + F t }..) C t T, K) has monotonic and convex properties as a function of T and K. These properties are well known and easy to establish from.). We recall them in the following lemma. Lemma.1.. Let T 1 T and K be fixed. On {t < τt 1, K)} {t < τt, K)} = {t < τt 1, K)}, we have C t T 1, K) C t T, K) a.s. Let K 1 K and T be fixed. On {t < τt, K 1 )} {t < τt, K )}, we have C t T, K 1 ) C t T, K ) a.s. Let λ 1, K 1, K and T be fixed. On {t < τt, K 1 )} {t < τt, K )}, we have C t T, λk λ)k ) λc t T, K 1 ) + 1 λ)c t T, K ) a.s. We wish to study C t T, K) as a function of T, K) for every t and almost every ω. Since C t T, K) is given to us as a conditional expectation, it is only defined off an exceptional set depending on T, K). We need to construct a modification of C t T, K) that is continuous in T, K) for every t and almost every ω. Lemma.1.3. There exists a modification of CT, K) also denoted CT, K)) as a continuous process such that C t T, K) is continuous in T, K) for every t and almost every ω. 4

32 Proof. Let t. Let T n ) n and K m ) m be two enumerations of the positive rational numbers. C t T n, K m ) is defined off a set of measure zero that depends neither on n nor on m for t < T n < t + x and S t e ȳ < K m < S t eȳ. Thanks to the elementary inequality a + b + a b, we have C t T n, K m ) C t T n, K m ) E { STn S Tn Ft } and C t T n, K m ) C t T n, K m ) K m K m. Therefore, C t T n, K m ) C t T n, K m ) E { STn S Tn Ft } + Km K m. Let us now extend C t, ) at an arbitrary point T, K) of t, t + x) S t e ȳ, S t eȳ). Take two sequences of rational numbers such that T n T and K m K. Let ε >, by Lebesgue dominated convergence theorem, there is an integer N, such that E { S Tn S T F t } < ε/3 whenever n N. Obviously, there is also an integer M such that K m K m < ε/3 whenever m, m M. This shows that n, n N, m, m M = C t T n, K m ) C t T n, K m ) ε. The sequence C t T n, K m )) n,m is Cauchy and it therefore converges to a limit that we call C t T, K). One checks as usual that this limit does not depend on the chosen sequences and that the resulting function defined on t, t+ x) S t e ȳ, S t eȳ) is indeed continuous. We shall make a further assumption about the regularity of C t T, K) as a function of T, K). We need a couple of definitions. 5

33 We first define seminorms and function spaces. Let O be a domain i.e., an open and connected set) in R d for some d. Let m be a positive integer; we denote by C m O; R e ), for some e, the set of all maps f : O R e that are m-times continuously differentiable on O. Let K O; we define the seminorms f m:k for m and f C m O; R e ) by f m:k = x K α m sup x α fx). denotes the Euclidean norm in R d or in R e and α the usual multi-indexed differential operator. As usual, for a multi-index α = α 1,..., α k ), α = α α k. If m =, C O; R e ) denotes the set of all continuous maps f : O R e with the corresponding family of seminorms. Let m be a positive integer and δ, 1]; we denote by C m+δ O; R e ), the set of all maps f : O R e which are m-times continuously differentiable and whose m-th derivatives are Hölder continuous of order δ. Let K O; we define the seminorms f m+δ:k for m, δ, 1] and f C m+δ O; R e ) by f m+δ:k = x K α m sup x α fx) + sup x x α =m K α x fx) α x fx ) x x δ. The families of seminorms m:k and m+δ:k where K ranges over the compact sets in O make the sets C m O; R e ) and C m+δ O; R e ) into Fréchet spaces. These spaces are easily seen to be separable. As topological spaces they are also measurable spaces when endowed with their Borel σ-algebras. Let {F x, t); x O, t } be a real valued random field with double parameter x O and t, i.e., a random variable taking values in R O R + when this set is endowed with the smallest σ-algebra such that the coordinate mappings are measurable. If F x, t, ω) is a continuous function of x for almost every ω and all t, we can 6

34 regard F, t) as a stochastic process with values in C O; R) or a C -valued process. If F x, t, ω) is a m-times continuously differentiable in x and all its m-th derivatives are Hölder continuous of order δ for almost every ω and all t, we can regard F, t) as a stochastic process with values in C m+δ O; R) or a C m+δ -valued process. In case where F x, t) is a continuous process with values in C m+δ it is called a continuous C m+δ -process. We now give the definition of a C m+δ -localmartingale. Definition.1.4 Cf. Theorem 3.1. p. 75 of Kunita 199)). Let m 1 and < δ 1. Let Mx, t), x O be a family of continuous localmartingales with joint quadratic variation Ax, y, t) = Mx, t); My, t) such that Mx, ) is C m+δ. It is called a C m+δ -localmartingale if it is a continuous C m+δ -process for each α with α m, α x Mx, t), x O is a family of continuous localmartingales with joint quadratic variation α x α y Ax, y, t). To prove that a family of continuous localmartingales is a C m+δ -localmartingale we shall use Theorem A.1.1 in appendix A. Before we can proceed any further, we need to define local C m+δ -localmartingales. Let F t, x), x O and t [, τx)) where τ is an accessible lower semicontinuous stopping time in the sense of Lemma.1.1. Assume that F is continuous in x, t). We set O t ω) = {x : τx) > t}. It is an open subset of O a.s. since τ is lower semicontinuous a.s. Then F x, t) defines a mapping from O t ω) into R for each ω. F is said to be a local C m+δ -process if for almost every ω, the map F, t, ω) : O t ω) R is a C m+δ -function for every t. Let us now fix u. Then, for all t u, O t ω) O u ω) a.s. Let F be a local C m+δ -process. If for every u and almost every ω, the map t F, t, ω) 7

35 from [, u] into C m+δ O u ω); R) is continuous, then F is said to be a continuous local C m+δ -process. We now take the sequence of stopping times associated with τ as in Lemma.1.1 and look at the stopped process F τnx) x, t) = F x, t τ n x)), x O and all t. These are global processes. A continuous local C m+δ -process is called a local C m+δ -localmartingale if for every n and each α with α m, the stopped processes x α F ) τnx) x, t), x O is a family of continuous localmartingales. Let us now formulate our assumption on C. We take O =, x) S e ȳ, S eȳ). Assumption. C is a local C 4+δ -localmartingale on [, τ) for some < δ 1. Moreover, on {t < τt, K)}, C t T, K) > S t K) + Leb P a.e. Let us stress that C t T, K) > S t K) +, i.e., we assume that liquid options have strictly positive time values. The first part of Assumption implies in particular that C t T, K) is a continuous local C 4+δ -process. Given the price of an option i.e., C t T, K)), the Black-Scholes implied volatility Σ t T, K) is by definition the unique volatility parameter for which the Black-Scholes formula recovers the option price. In fact, we found it easier to study the implied variance V t T, K). It is simply the usual Black-Scholes implied volatility squared time-to-maturity, i.e, V t T, K) = Σ t T, K) T t). Hence, V t T, K) is the unique solution to the following equation on {t < τt, K)} KBS S t /K, ) V t T, K) = C t T, K)..3) 8

36 BS is the normalized Black-Scholes formula ln u BSu, v) = uφ v + v ) ln u Φ v v ) It is defined for u, v) R + R +. It is a C function on R + R + and, for every u R +, it is a C -diffeomorphism from R + onto itself. We can and will extend it as a continuous function from R + R + onto R + by setting BSu, ) = u 1) +. This extension is still denoted by BS. By the implicit mapping theorem, we can solve equation.3) on any neighborhood of T, K), where t < T < t + x and S t e ȳ < K < S t eȳ. V t is also a local C 4+δ -process by the same theorem. Moreover it inherits from C t the a.s. continuity in t as a process with values in C 4+δ and the a.s. continuity in t, T, K). Dynamics for C and V will be further studied in Section.3.. Boundary behavior In this section, we study the behavior of implied volatilities just before maturity. This behavior is intimately related to that of the corresponding option prices. In absence of arbitrage, option prices are continuous in maturity. Mathematically, this is just Lebesgue dominated convergence theorem applied to the representation of option prices in terms of risk-neutral expectations. Indeed, since S T is integrable and continuous as a function of T, we have on {t < τt, K)}, lim E { } S T T K) + F t = St K) + t a.s. We use the notation ϕx) and Φx) for the density and the cumulative distribution function of the standard Gaussian distribution, i.e., ϕx) = 1 π e x / and Φx) = 1 π x e u / du. 9