Heston Model: The Variance Swap Calibration

Heston Model: the Variance Swap Calibration Florence Guillaume∗† Wim Schoutens ‡ April 23, 2013 Abstract This paper features a market implied methodology to infer adequate starting values for the spot and long run variances and for the mean reversion rate of a calibration exercise under the Heston model. More particularly, these initial parameters are obtained by matching the term structure of the future expected total variance, inferred from the volatility surface, with the model’s term structure. In the numerical study, we compare the goodness of fit and the parameter stability of the Heston model calibrated by using either plausible random or market implied starting values for a one-year sample period including the recent credit crunch. In particular, we show that the proposed methodology avoids getting stuck in one “bad” local minimum and stabilizes the calibrated parameters through time. Keywords: Heston model, Starting values, Variance term structure matching 1 Introduction The Heston stochastic volatility model is widely used among practitioners, especially to price path dependent derivatives, such as barrier or cliquet options. To price such typically over-the-counter (OTC) exotic instruments, the first step consists in calibrating the chosen model, i.e. in finding the parameter set which is compatible with the observed market price of liquidly traded (vanilla) derivatives. Typically, a perfect match is not plausible and one looks for an “optimal” match. The most popular methodology consists in solving the so-called inverse problem, where the optimal parameter set is the parameter set which minimizes some distance between the market and model prices of a set of benchmark instruments. In the equity market, the most natural choice of calibration instruments consists of European vanilla options. Most commonly, practitioners are minimizing the root mean square error (RMSE), leading to a least-squares problem ; but there exist other alternatives just as suitable such as the minimization of the average absolute error as a percentage of the mean price (APE) or of the average relative percentage error (ARPE). It is well established that the solution of the inverse problem might turn out to be instable with respect to small changes in the option prices observed in the market. Hence, small variations in ∗ K.U.Leuven, Department of Mathematics, Celestijnenlaan 200 B, B-3001 Leuven, Belgium. E-mail: florence.guillaume@wis.kuleuven.be † Florence Guillaume is a postdoctoral fellow of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.). ‡ K.U.Leuven, Department of Mathematics, Celestijnenlaan 200 B, B-3001 Leuven, Belgium. E-mail: Wim@Schoutens.be 1 market option prices might lead to large changes in the optimal parameter set and, consequently, in the value of exotic and structured products (see, for instance, [1]). Moreover, the root mean square error is typically a non-convex function of the model parameters and can thus have several local minima. This makes the solution of the least-squares calibration problem dependent on the initial parameter set, which is taken as starting value of the optimization algorithm, and on the sophistication of the numerical search performed (see, for instance, [2]). Hence, deriving plausible starting values for the model parameters is a key step to obtain the global minimum (or a “good” local minimum) and to reduce the computation time of the calibration exercise. In the financial literature, several approaches have been proposed to infer adequate starting values for the parameters of the Heston stochastic volatility model. One possible method consists in considering asymptotic formulas for the implied volatility at large maturities or at extreme strikes (see, for instance, [3], [4] or [5]). The moment formula of Lee for implied volatility at extreme strikes can be, for instance, used to determine two of the model parameters from the tail slope of the volatility smile at one particular time horizon (see [5]). An alternative way to determine an initial guess for the parameters could be to use the asymptotic formula of Forde et al. for the Heston implied variance for long term options (see [3]). Although appealing for their analytical approach, these asymptotic methodologies only use a portion of the option surface to infer the initial parameter set. This paper proposes an alternative methodology to derive starting values for three parameters of the Heston model, namely the spot variance v0 , the long run variance η and the mean reverting rate κ ; and this from the whole set of liquidly traded options. More precisely, we take as initial guess for these three parameters the values that replicate as best as possible the term structure of variance swap prices (or more exactly of the expected future total variance). This term structure is inferred from the option price surface by using the spanning option payoff formula of Breeden and Litzenberger (see [6]). Indeed, the expected future total variance term structure under the Heston model can be derived in closed-form by differentiating the integrated CIR (Cox-IngersollRoss) characteristic function, which governs the cumulated variance of the asset log-return under the Heston model. The in this way obtained analytical expression turns out to be independent of the volatility of variance λ and of the correlation between stock and variance returns ρ ; it only depends on the parameters v0 , η and κ. Adequate market implied starting values for these three parameters can thus be obtained in a straightforward way by matching as best as possible the expected variance term structure that we observe in the market. This term structure can be obtained by following a similar methodology as the one adopted by the Chicago Board Options Exchange (CBOE) to compute the VIX volatility index (see [7]). As preliminary study, we will compare the Heston cumulated variance term structure with the one inferred from the market. This comparison will provide evidence that the Heston model is able to fit the expected variance term structure observed in the market in most cases and under different volatility regimes. We will then show the improvement, in terms of both the computation time and the goodness of fit, when adequate market implied starting values for the model parameters are plugged into the optimization algorithm. This paper is organized as follows. Section 2 recalls the Heston stochastic volatility model. Section 3 describes the new methodology used to infer starting values for v0 , η and κ. Section 4 compares the goodness of fit and the parameter stability of the Heston model calibrated by using the inverse calibration problem, either with market implied or random starting values for the parameters v0 , η and κ. Section 5 concludes. 2 2 The Heston Model In [8], Heston proposed a model which extends the Black-Scholes model by making the volatility parameter σ stochastic. The stock price process is modeled by the Black-Scholes stochastic differential equation: √ dSt = (r − q)dt + vt dWt , S0 ≥ 0, St where r is the risk-free interest rate and q the dividend yield, and where the squared volatility process follows the CIR (Cox-Ingersoll-Ross) stochastic differential equation: √ dvt = κ (η − vt ) dt + λ vt dW̃t , v0 = σ02 ≥ 0, n o where W = {Wt , t ≥ 0} and W̃ = W̃t , t ≥ 0 are two correlated standard Brownian motions   such that Cov dWt , dW̃t = ρ dt and where v0 is the initial variance, κ > 0 the mean reversion rate, η > 0 the long run variance, λ > 0 the volatility of variance and ρ the correlation. The variance process is always positive and cannot reach zero if 2κη > λ2 , which is known as the Feller condition. Moreover, the deterministic part of the CIR process is asymptotically stable if κ > 0 and tends towards the equilibrium point vt = η. The model parameters can be determined either by matching data or by calibration. In practice, calibrated parameters turn out to be unstable and often unreasonable (see [9]). This can be partially explained by the fact that adequate starting values play a crucial role in the value of the calibrated parameters. R t Under the Heston model, the total (or cumulated) variance Vt := 0 vs ds follows an integrated CIR process and has thus    2  2v0 iu exp κλ2ηt exp κ+γ coth(γt/2) φVt (u) := E [exp(iuVt )] =  2κη/λ2 , cosh(γt/2) + γ1 κ sinh(γt/2) where γ := p κ2 − 2λ2 iu as characteristic function (see [10]). The expected annualized variance for the time horizon T is thus given by 1 ∂φVT (u) 1 E [VT ] := T iT ∂u =η+ u=0 1 (v0 − η)(1 − exp(−κT )). κT (2.1) (2.1) indicates that the Heston expected variance is independent of the volatility of variance λ and of the correlation ρ: it only depends on v0 , η and κ. 3 3.1 The VS (Variance Swap) Starting Value Methodology A Market Implied Approximation of the Total Variance The market volatility index, or VIX for short, was introduced in 1993 by the CBOE to provide a benchmark for the short-term expected future market volatility. The VIX is currently implied by 3 S&P 500 index option prices. Indeed, on the 22nd of September 2003, the CBOE proposed a new methodology to extract the volatility index VIX from quoted index option prices, which is based on the concept of fair value of future variance developed by Demeterfi et al. in [11] and which is, consequently, model independent. The CBOE methodology can be applied to any index option market, provided that the underlying index option market has deep and active trading across a broad range of strike prices. In particular, it has already been applied to the NASDAQ 100, the DJIA, the AEX, the BEL20 and the FTSE 100 index option markets, among others (see [12]). This gives evidence of the wide-scope characteristic of the proposed VS procedure. By following a similar methodology, we can compute a model independent approximation for the expected future market variance for a time horizon T . Indeed, assuming a log-normal pure diffusion √ t vt dWt , the expected future variance can be expressed process for the future stock price, dF Ft = in terms of the forward price F0 and call and put option prices with strike K and maturity T , i.e. C(K, T ) and P (K, T ) (see [7]): "Z # Z F0 T 1 v(T ) := E [VT ] := E vt dt = 2 exp(rT ) P (K, T )dK 2 K 0 0 (3.1)  Z ∞ 1 + C(K, T )dK . 2 F0 K Since options are traded for a discrete range of strikes only, and not for a continuum of strikes, we can extract a model independent approximation of future market variance as follows: 2 v(T ) = T VIX (T ) ≈ 2 N X ∆Ki i=1 Ki2 exp(rT )Q(Ki ) −  F0 −1 K0 2 , (3.2) where ❼ F0 is the current forward price: F0 := F0 (T ) := S0 exp((r − q)T ). Note that for the numerical implementation, we will typically derive the forward price from the at the money option prices by making use of the put-call parity, where we approximate the at the money strike by the listed strike at which the difference between the quoted call and put prices is minimal. ❼ T is the option maturity in years ; ❼ r is the risk-free interest rate corresponding to the maturity T ; ❼ Ki is the strike price of the ith out of the money T -option1 ; ❼ ∆Ki is the interval between the strikes:   ∆K1 := K2 − K1 , i−1 ∆Ki := Ki+1 −K , ∀i 6= 1 or N 2  ∆KN := KN − KN −1 ; 1 An out of the money call option is characterized by Ki > F0 and an out of the money put option by Ki < F0 . 4 ❼ K0 is the first listed strike below the forward stock index level F0 : K0 := max{Ki : Ki ≤ F0 } ; ❼ Q(Ki ) is the midpoint of the bid-ask spread for the option with strike Ki and maturity T :  iff Ki < K0  Q(Ki ) := P (Ki , T ) C(K0 ,T )+P (K0 ,T ) iff Ki = K0 Q(Ki ) := 2  Q(Ki ) := C(Ki , T ) iff Ki > K0 , where the option prices C(K, T ) and P (K, T ) are taken equal to the mid-point of the bid and offer. ❼ Selection of liquid options After sorting the T -options in ascending order of strike, we select the calls with a strike price greater than K0 and a positive bid price. After encountering two consecutive call options with a zero bid price, we do not select any other calls with higher strike. We proceed similarly for the put options with a strike price lower than K0 . Moreover, we select both the call and the put with a strike equal to K0 and we replace these two options by an option with a price equal to (C(K0 , T ) + P (K0 , T ))/2. This methodology, which is used by the CBOE, allows to select the most liquid options. 3.2 Getting Rid of Potential Arbitrage Opportunities From a no-arbitrage argument, E hR T 0 i vt dt is clearly a non-decreasing function of the time-to- maturity T since vt ≥ 0. Nevertheless, the expected future total variance term structure v(T ) inferred from the above methodology (3.2) can exhibit some deviation from the arbitrage-free nondecreasing trend, which is due to the discrete nature of listed strikes. It might indeed turn out that, for less liquid maturities, a relatively low number of out of the money options is liquidly traded on the market. This will inevitably lead to a poor approximation of the two integrals over the continuum of strikes included in the expression of the expected future variance (3.1). It is thus not impossible to observe some “arbitrage” in the term structure of the non-annualized variance when using the methodology as it is2 . Hence, before fitting the market expected future variance term structure, we make sure that it is absent of any arbitrage opportunity by either (Figure 1) ❼ removing the maturities that exhibit some arbitrage (i.e. a change in the concavity of the expected total variance) ; ❼ replacing the expected total variance points corresponding to arbitrage opportunities by a linear interpolation of the closest no-arbitrage expected total variance points. Figure 1 illustrates the two adjustments we consider in order to get rid of potential arbitrage opportunities in the term structure of the expected future total variance for the 1st of April 2008. 2 Arbitrage opportunities can only be detected from the term structure of the cumulated variance v(T ) and not from the term structure of the annualized variance, i.e. VIX2 (T ) := T1 v(T ), due to the scaling by the time horizon T in the annualized variance. This explains why we have opted for fitting the total variance curve instead of the annualized variance one. 5 0.18 Market price Adjusted market price (reduced set) Adjusted market price (interpolation) 0.16 0.14 0.12 V(T) 0.1 0.08 0.06 0.04 0.02 0 0 0.5 1 1.5 T 2 2.5 3 Figure 1: Arbitrage-free market expected future total variance term structure (01/04/08) obtained by either removing the “arbitrage” maturities or by interpolating the no-arbitrage expected total variances For that particular quoting day, four maturities lead to arbitrage opportunities, namely T5 = 0.2466 (based on 9 liquid out of the money options), T8 = 0.4986 (based on 8 liquid out of the money options), T10 = 0.7507 (based on 8 liquid out of the money options) and T12 = 0.9973 (based on 12 liquid out of the money options). As expected, the number of out of the money options selected by the VS methodology for these four maturities is much lower than the average among maturities, which amounts to more than 32 option quotes by maturity, explaining the poor quality of these total variances. In the following, we will denote the arbitrage-free approximation of the expected future total variance for a time horizon T by va (T ). 3.3 Calibration of the Expected Total Variance Term Structure From (2.1) and (3.2), we can infer market implied starting values for the parameters v0 , η and κ by matching the expected total variance that we observe in the market with the one that we infer from the Heston model (i.e. from the integrated CIR process), and this for the whole set of maturities. Note that we use the model-free formula (3.2) to derive market VS prices instead of using directly market quotes for the sake of generality since there might not exist liquid quotes for 6 volatility derivatives on all relevant underliers. Hence, the VS starting value methodology only requires the existence of a liquid market for vanilla options, which constitute the set of calibration (or benchmark) instruments for the widely used inverse calibration problem. The starting value phase uses thus the same market quotes as the global calibration (i.e. on the whole option surface) exercise ; which makes the method especially suited as a preliminary step to calibrate the Heston model on the whole set of options (i.e. on the whole range of maturities and strikes). In order to fit the expected future total variance term structure, we minimize either the root mean square error (RMSE), the average absolute error as a percentage of the mean variance (APE) or the average relative percentage error (ARPE) functional: v uN N uX (va (Ti ) − v̂(Ti ))2 1 X |va (Ti ) − v̂(Ti )| , APE := , RMSE := t N v̄a i=1 N i=1 ARPE := N 1 X |va (Ti ) − v̂(Ti )| N i=1 va (Ti ) where N denotes the number of maturities, v̂(T ) the model expected future total variance for a time horizon T , and v̄a the average of the market adjusted total variance over the times to maturity v̄a := N 1 X va (Ti ). N i=1 The choice of the objective function impacts the weight which is assessed to the different future total variances. The RMSE and the ARPE give more weight to the long term and to the short term of the total variance curve, respectively ; whereas the APE functional assesses the same weight to each time horizon. Figure 2 shows the expected total variance term structure goodness of fit for a period ranging from the beginning of the second quarter of 2008 until the end of the first quarter of 2009. We focus on this one-year period in order to include both quoting days during the heart of the recent credit crunch as well as days before and after the crisis period. This will allow us to compare the influence of the choice of the initial guess for the model parameters for different levels of market fear. The results clearly indicate that the Heston model is able to fit pretty well the expected total variance term structure observed in the market. This observation should not be confused with the empirical evidence highlighted by Gatheral that the at the money implied volatility term structure can be, sometimes, fitted by a stochastic volatility model such as the Heston model, but that, in general, the term structure of implied volatility is quite intricate at the short end (see [13]). Indeed, our approach is different in the sense that we do not fit the implied volatility smirk for one particular moneyness, but we fit the expected total variance of the asset log-return, which integrates information about the whole option surface. Some exception to the overall high quality of the expected total variance curve fit can be noticed, at first sight, for a period ranging from October 2008 until November 2008. Indeed, during that period, the RMSE and/or the APE reach(es) some relatively high levels, although the ARPE remains at the same average level for the whole period of time under consideration. This 2-month time span corresponds roughly to the credit crisis period which was triggered by the bankruptcy of Lehman Brothers, which occurred on the 15th of September 2009. In order to have more insight into the quality of the fit for the different objective functions, we can have a look at the market expected total variance fit obtained by using the different possible combinations of no-arbitrage 7 RMSE 0.01 RMSE reduced RMSE interpolation 0.005 0 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 0.1 APE APE reduced APE interpolation 0.05 0 01/04/08 0.2 ARPE 0.15 ARPE reduced ARPE interpolation 0.1 0.05 0 01/04/08 Figure 2: Evolution of the expected total variance term structure goodness of fit measured by the RMSE (above), APE (center) and ARPE (below) functionals adjustment/objective function to fit the adjusted expected total variance term structure on two particular quoting days. The first day is characterized by a relatively low value of the RMSE and APE functionals (1st of August 2008), and the other is characterized by a relatively high value of the RMSE and APE objective functions (8th of October 2008) (see Figure 3). This picture clearly indicates that, although the RMSE and the APE turn out to be relatively high during the credit crisis, compared to quoting days characterized by a low level of market fear, the Heston model fit is still pretty good, whatever the choice of the objective function. Hence, none of the combinations no-arbitrage adjustment/objective function should be set aside at this stage of the empirical study. We will thus consider each starting value set separately to determine which combination leads to the most significant gain of precision and/or computation time for the global calibration exercise. 3.4 The VS Calibration in Two Steps The proposed calibration can be implemented in two successive steps: 1. determine market implied starting values for the parameters v0 , κ and η to replicate as best as possible the term structure of the market implied expected future total variance, i.e. minimize the distance between "Z # T
v0 − η 1 − e−κT E [VT ] := E vt dt = ηT + κ 0 8 0.16 0.16 Market price Adjusted market price (reduced set) Model fit (RMSE reduced) Model fit (APE reduced) Model fit (ARPE reduced) 0.12 0.12 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0.5 1 1.5 Market price Adjusted market price (interpolation) Model fit (RMSE interpolation) Model fit (APE interpolation) Model fit (ARPE interpolation) 0.14 V(T) V(T) 0.14 2 0 2.5 0 0.5 1 T 1.5 2 2.5 2 2.5 T 0.25 0.25 Market price Adjusted market price (reduced set) Model fit (RMSE reduced) Model fit (APE reduced) Model fit (ARPE reduced) 0.2 Market price Adjusted market price (interpolation) Model fit (RMSE interpolation) Model fit (APE interpolation) Model fit (ARPE interpolation) 0.2 V(T) 0.15 V(T) 0.15 0.1 0.1 0.05 0.05 0 0 0.5 1 1.5 2 2.5 0 0 0.5 T 1 1.5 T Figure 3: Expected total variance term structure goodness of fit on the 01/08/08 (above) and on the 08/10/08 (below). On the left, the arbitrage-free adjustment consists in removing the “arbitrage” maturities, whereas on the right it consists in interpolating the no-arbitrage total variances 9 and va (T ). Hence, the optimal starting values are such that {v0⋆ , η ⋆ , κ⋆ } : f ({E[VT ]}, {va (T )}, v0⋆ , η ⋆ , κ⋆ ) ≤ f ({E[VT ]}, {va (T )}, v0 , η, κ), where f is the distance to be minimized and is chosen to be the RMSE, APE or ARPE functional. 2. Calibrate the whole parameter set of the Heston model on liquid out of the money European vanilla options, starting from {v0⋆ , η ⋆ , κ⋆ , λ(0) , ρ(0) }, where λ(0) and ρ(0) are some plausible arbitrary values. In the numerical study, we will compare the calibration results with those obtained by taking (0) {v0 , η (0) , κ(0) , λ(0) , ρ(0) } as initial guess. Note that in the case of the VS calibration methodology, we might have to adjust λ(0) to make sure that the Feller condition is satisfied by the initial parameter set. 4 Numerical Study For the numerical study, we calibrate the Heston model on the whole set of liquid out of the money options for a one-year time period ranging from the beginning of the second quarter of 2008 until the end of the first quarter of 2009. We consider as starting values for the least-squares search algorithm (0) ❼ either “random” starting values, namely {v0 , η (0) , κ(0) , λ(0) , ρ(0) } = {0.05, 0.05, 2, 0.2, −0.8} ; ❼ or starting values fitting the expected total variance term structure, namely {v0⋆ , η ⋆ , κ⋆ , λ(0) , ρ(0) } = {v0⋆ , η ⋆ , κ⋆ , 0.2, −0.8}, where v0⋆ , η ⋆ and κ⋆ are determined by one of the possible combinations no-arbitrage adjustment/objective function. Note that for a few quoting days under investigation, the value η ⋆ or κ⋆ minimizing the distance between the model and market expected future total variance curves, was set at some exceptionally low level (η ⋆ or κ⋆ ≈ 0) or high level (κ⋆ > 50). Such values indicate that the Heston stochastic volatility model is not a good candidate to fit the market situation of that day. Hence, we have removed these quoting days from our sample since alternative models should be then considered to fit the market data. Table 1 shows the average (relative) improvement of precision and computation time that arises by considering VS starting values instead of random ones in the global search algorithm. A positive percentage corresponds to a gain and a negative one to a loss. We clearly see that starting from v0⋆ , η ⋆ and κ⋆ leads to a significant improvement of the option surface goodness of fit, whatever the adjustment we choose to eliminate arbitrage, and whatever the distance between the market and model variance curves we minimize. Indeed, the RMSE functional decreases, on average, of more than one fourth for each combination no-arbitrage adjustment/objective function ; the improvement being slightly better when we eliminate the maturities corresponding to arbitrage opportunities to obtain an arbitrage-free expected total variance curve. Moreover, opting for the RMSE to infer the starting values v0⋆ , η ⋆ and κ⋆ that fit as best as possible the term structure of the expected total variance, leads, on average, to a better fit of the option surface than opting for the APE and, to a larger extent, for the ARPE functional to determine market implied starting values for 10 Table 1: Average relative precision and computation time improvement arising from the VS starting values. A positive number corresponds to a gain and a negative one to a loss. Reduced set Interpolation RMSE cpu RMSE APE ARPE RMSE APE ARPE 31.6752 % 6.4233 % 30.3578 % -5.4416 % 28.9367 % -3.4043 % 30.6661 % 3.7290 % 30.3250 % 2.3255 % 28.0744 % 1.2240 % v0 , η and κ. We also note that the combination reduced set/RMSE leads to the most significant decrease of computation time, although the gain in cpu is far from being comparable to the gain in precision. Hence, to study the parameter stability, we will focus on this particular combination to infer adequate starting values for the parameters v0 , η and κ. Figure 4 to Figure 8 show the evolution of the Heston parameters through time obtained by considering either “random” starting values or market implied starting values for v0 , η and κ. The RMSE functional measuring the distance between market and model option prices is shown on Figure 9 for the two sets of starting values. Figure 9 indicates that starting from a random initial guess leads, sometimes, to a pretty high value of the RMSE. This typically occurs when the search algorithm gets stuck in one of the “bad” local minima of the RMSE functional. In our sample period, this happened from November 2008 onwards and can thus occur for quoting days characterized by a high degree of stress in the market (November 2008) but also for days characterized by an average stress level (end of March 2009). Indeed, to prevent additional failures of financial institutions, on the third of October 2008, the Troubled Asset Relief Program was passed and became immediately effective by injecting capital, by guaranteeing or buying assets, and by providing liquidity to leading financial institutions (see [14]). These rescues led to a slow increase of the confidence in the whole market, and, in particular, to a slow decrease of the overall level of market volatility from the late November 2008 on. This observation gives evidence of the necessity of considering starting values that are inferred from current market data. Indeed, we see that the value of the RMSE resulting from the market implied starting values v0⋆ , η ⋆ and κ⋆ either almost coincides with, or is significantly (0) lower than the RMSE resulting from “random” initial guesses v0 , η (0) and κ(0) . The evolution of the calibrated parameters through time gives evidence of the fact that considering market implied starting values instead of random ones for the parameters v0 , η and κ allows to stabilize the value of the calibrated parameters over time and avoids the problem of getting stuck in one “bad” (and unrealistic) local minimum. We also note that the calibrated correlation is sometimes touching its boundary -1, which gives us some insight into the inability of the Heston model to capture the strong volatility skew that is observed in the equity market during distressed periods. 5 Concluding Remarks This paper provides a market implied way of inferring starting values for the parameters v0 , η and κ of the Heston stochastic volatility model based on the fitting of the term structure of the expected future total variance. The main advantage of this method compared to existing methods such as the ones derived from asymptotic theory resides in the fact that the market expected total variance 11 0.7 0.6 random RMSE reduced 0.5 v0 0.4 0.3 0.2 0.1 0 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 Figure 4: Evolution of the spot variance v0 obtained by the least-squares search algorithm (i.e. the global calibration exercise) 0.2 random RMSE reduced 0.15 η 0.1 0.05 0 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 Figure 5: Evolution of the long run variance η obtained by the least-squares search algorithm (i.e. the global calibration exercise) 30 25 random RMSE reduced κ 20 15 10 5 0 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 Figure 6: Evolution of the mean reversion rate κ obtained by the least-squares search algorithm (i.e. the global calibration exercise) 12 3 2.5 random RMSE reduced 2 λ 1.5 1 0.5 0 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 Figure 7: Evolution of the volatility of variance λ obtained by the least-squares search algorithm (i.e. the global calibration exercise) 0 −0.2 random RMSE reduced ρ −0.4 −0.6 −0.8 −1 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 Figure 8: Evolution of the correlation parameter ρ obtained by the least-squares search algorithm (i.e. the global calibration exercise) 20 random RMSE reduced RMSE 15 10 5 0 01/04/08 01/07/08 01/10/08 Trading day 02/01/09 31/03/09 Figure 9: Evolution of the option surface goodness of fit obtained by the least-squares search algorithm (i.e. the global calibration exercise) 13 term structure summarizes the information of the whole option surface instead of considering a reduced portion of the volatility surface only. As possible future research line, we could apply the same Variance Swap calibration methodology to other stochastic volatility models, such as the Schöbel and Zhu model [15] or the BarndorffNielsen and Shephard model [16]. Another research perspective could be to consider, additionally to the term structure of the expected total variance, the term structure of some of the moments of the underlying asset in the calibration exercise of the Heston model. Indeed, starting from the spanning option payoff formula of Breeden and Litzenberger [6], we can derive a model independent risk-neutral formula for any moment of the asset log-return as shown in [17]. In particular, a calibration based on a combination of the matching of the second and third moments of the asset log-return seems promising to infer initial starting values for the whole set of parameters. References [1] Guillaume, F. and Schoutens, W.: Illustrating the impact of calibration risk under the Heston model. 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