Testing Derivatives Pricing Models under Higher-Order Moment Swaps

This paper aims to test three parametric models in pricing and hedging higher-order moment swaps. Using vanilla option prices from the volatility surface of the Euro Stoxx 50 Index, the paper shows that the pricing accuracy of these models is very satisfactory under four different pricing error functions. The result is that taking a position in a third moment swap considerably improves the performance of the standard hedge of a variance swap based on a static position in the log-contract and a dynamic trading strategy. The position in the third moment swap is taken by running a Monte Carlo simulation.


Introduction
Moment swaps are derivative securities whose payoff depends on the realized higher moments of the underlying asset price or state variable. This payoff is linked to the powers of the log-returns and provides protection against various types of market conditions. Variance swaps which are liquidly traded today are obtained in the case of squared log-returns and offer protection against changes in the volatility process. Higher-order moment derivatives can be useful to protect against inaccurately estimated skewness or kurtosis. Rompolis and Tzavalis (2017) and Schoutens (2005) show that moment swaps can be used to significantly improve the standard hedge of the variance swap.
Some recent studies in the finance literature suggest that power-jump assets can be used to complete the market. For instance, an incomplete Levy market where power assets of any order can be traded will yield a complete market. Power assets and realized higher moments are highly linked and they are virtually the same in a discrete time framework (Corcuera et al., 2005).
There has been a proliferation of studies extending the Black-Scholes (BS) option pricing model. However, only a few of these studies addressed hedging contingent claims under more general assumptions about the state variable stochastic process. Well known examples of such studies are the stochastic volatility (SV) model of Heston (1993); the SV and random jumps model (SVJ) of Bates (1996); and the SV, stochastic interest rates and random jumps model of Doffou and Hilliard (2001). Using delta hedging strategies in an incomplete market cannot lead to a perfect hedge against jump risk and volatility risk linked to a position in contingent claims.
Locally risk-minimizing delta hedging strategies that attempt to hedge the option contract using only the state variable and minimizing the variance of the cost process of a non-self-financed hedging position are not adequate. In the presence of jumps, these strategies perform very poorly like the classic delta hedging strategies (Tankov et al., 2007). Other hedging strategies use option contracts to reduce or eliminate volatility risk (Bakshi et al., 1997) or protect against jump risk (Coleman et al., 2006;Cheang et al., 2015). Cross hedging strategies or delta-vega hedging can totally remove an option contract exposure to volatility risk but not its exposure to jump risk. Jump risk can be hedged either by using a risk-minimization strategy (Coleman et al., 2006;Tankov et al., 2007) or by discretizing jump sizes to compute the hedge ratios of the other options (Utzet et al., 2002). However, these two methodologies are limited. First, they do not characterize the maturity and moneyness of the option contracts to be used to efficiently and effectively set up the hedge. Further, these two approaches do not pick up the options' exposure to volatility risk. Finally, empirical evidences show that these two methods do not outperform the delta hedging strategy (Cheang et al., 2015). This paper tests empirically the Schoutens (2005) along with the Heston (1993) and the time-changed Levy models in their effectiveness in pricing and hedging moment swaps, particularly the variance swaps which are liquidly traded today. This paper is organized as follows. Section 2 defines moment swaps. Section 3 introduces the pricing and hedging model. The model parameters are estimated in Section 4. Section 5 tests the out-of-sample pricing performance of the model. Finally, Section 6 concludes the paper.

The moment swaps pricing model
Consider an asset (stock or stock index) with a continuous dividend yield y ! 0. The asset price process is modeled by an Ito semi martingale P = {P t , t ! 0} such that P > 0 and P_ > 0. In addition to this asset, a bond or money market account with a constant compound interest rate a is available with a price process H = {H t = e at , t ! 0}. Consider n equally spaced time intervals of length Dt such that t j = jDt, with j = 0, 1, 2, . . .., n and W = nDt is the expiration date of the derivative contract written on the state variable price P t . The price of the state variable at each interval j is denoted P j for simplicity. In practice, the t j are the daily closing times and P j is the closing price at day j. It follows that the daily log-returns are given by ln(P j )ln(P j-1 ), j = 1, 2, . . .. n.
Assume futures contracts written on the state variable price P exist with expiration date W. In the risk-neutral world, the price of the futures contract is given by F t = P t exp(ay) (Wt). For simplification, the futures price at the discrete time t j is denoted F j . The mthmoment swap on the stock is a contract in which the two counterparties agree to exchange at maturity a nominal amount multiplied by the difference between a fixed level contract price and the realized level of the mth-order non-central sample moment of the log-return over the life of the contract. The payoff function is defined by: where NA is the nominal or notional amount and n is the number of segments of length Dt within the time interval [0, W]. For m = 2, equation (1) gives the expression of the 2nd-moment swap or variance swap. Variance swaps are basically forward contracts in which the counterparties agree to exchange a notional amount multiplied by the difference between a fixed variance and the realized variance. The fixed variance is the variance swap rate or the variance forward price. Variance swaps offer protection against volatility shocks. The 3rd-moment swap is linked to the realized skewness and offers protection against changes in the symmetry of the underlying distribution. Changes in the tail behavior of the underlying distribution created by the occurrences of unexpected large jumps are shielded by the 4th-moment swap related to the realized kurtosis. If the futures price is the state variable driving the moment derivatives, then the payoff function of the mth-moment swap on the futures is: Using the link between the stock and futures prices, F j = P j exp(ay)(Wt j ), and setting the notional amount NA equal to one, the relationship between the futures and the stock moment swaps is derived as follows: The term X P is the log-contract on the stock and plays a critical role in the hedging of moment swaps. If Dt raised to the higher order powers is very small and therefore negligible, the above listed expression can be approximated by equation (3) below:

Hedging moment swaps driven by futures contracts
The focus is on hedging the moment swaps driven by the futures price as the state variable or underlying asset. Following Carr and Lewis (2004), as well as Schoutens (2005), a new expression for ln can be derived by combining the Taylor series expansion of the mth power of the logarithm function and the power series representation of the exponential function. Most specifically, we have: Equation (4) is obtained from equation (5) by replacing b by ln(l ), rearranging terms and considering the fact that j (ln l mþ1 ) = j ((l -1) mþ1 ). Replacing l by F j F jÀ1 in equation (4) leads to: where DF j = F j -F j-1 . Hence, with a notional amount of one, the moment swap given in equation (2) takes the following new expression: Equation (6) shows that the payoff function of the mth-moment swap on the futures can be decomposed into: the payoff from -m! number of log-contracts on the futures; the payoff from a dynamic trading strategy in futures given by m! X n j¼1 DF j F jÀ1 ; and the payoff from a series of moment swaps of order strictly lower than m.
Such decomposition is an approximation because it is derived from the Taylor series expansion of the mth power of the logarithm function and the power series representation of the exponential function which are both approximations.
The log-contract on the futures, X F , is given by X F = ln F Wln F 0 = X P -(ay)W. Carr and Lewis (2004) shows that this log-contract can be hedged by taking a static position in bonds and vanilla options. Equation (6) leads to the classical hedge of the variance swap in terms of the logcontracts on the futures when m = 2, namely: To hedge the variance swap requires taking a short position in two log-contracts and a dynamic strategy in futures. The dynamic strategy can be executed at a zero cost. For m = 3, equation (6) simplifies to: The variance swap on futures can now be derived as given in equation (8) below: Comparing equations (7) and (8) leads to the conclusion that taking an additional short position in 1/3 third moment swaps leads to an improvement of the hedging strategy of the variance swap.

The dynamics of some advanced models
To truly evaluate the effectiveness of the hedging strategies, the performance of the moment swap hedges must be compared to that of some advanced models' hedges. The Heston model and the Levy models with stochastic time and SV will be examined. These two classes of advanced models offer very acceptable global calibration fits and are well known and used in the world of derivatives practitioners. The dynamics of each advanced model is explicitly given by its characteristic function w (v , t) of the logarithm of the price process ln P t , namely, by w (v , t) = E [exp(iv ln(P t ))].

The Heston stochastic volatility model
The Heston SV model has a state variable (the stock price) process that follows the Black and Scholes (1973) stochastic differential equation, that is: with the squared volatility driven by the Cox et al. (1985) process (Cox, Ingersoll and Ross [CIR] process): where dW t and dZ t , t ! 0, are increments to two correlated standard Brownian motions with Cov(dW t , dZ t ) = r dt.
The characteristic function w (v , t) for this advanced model is provided in Heston (1993) and in Bakshi et al. (1997) as follows: 4.2 Levy models with stochastic time and stochastic volatility The BS model has well known limitations because its underlying source of randomness is a Brownian motion. One practical way to improve this model is to replace the Brownian motion by a Levy process. Some of these Levy models can also incorporate SV. Two independent stochastic processes are used to build the Levy models with stochastic time considered here. This approach was initiated by Mandelbrot and Taylor (1967). The asset price dynamics is modeled by the exponential of the Levy process, Y = {Y t , t ! 0} preferably time-changed and by a second process, a stochastic clock, which incorporates SV allowing time to be stochastic.
In t calendar time units, the economic time that has passed is given by the integral C t of a positive process g = {g t , t ! 0} such that C t ¼ Ð t 0 g s ds and C = {C t , t ! 0}. If the characteristic function of C t , given g 0 , is f (v ; t, g 0 ), then the price process in the risk neutral world P = {P t , t ! 0} can be modeled as: is a Levy process. The stock price process is modeled as the exponential of a time-changed Levy process which incorporates jumps and SV. The Levy process Y t picks up the jumps while the SV is accounted for by the time change C t . The characteristic function w (v , t) for the logarithm of the stock price is: is named the characteristic exponent of the Levy process. The subparagraphs that follow will examine: the Cox et al. (1985) process (CIR process) for g ; and two Levy processes which are the normal inverse Gaussian (NIG) process and the variance gamma (VG) process.
This characteristic function is infinitely divisible and therefore can help define the NIG ¼ 0, because the process has stationary and independent NIG distributed increments. Each increment satisfies the NIG(h , m , #t) law over the time slot [k, k þ t].
4.2.2 The VG levy process. Madan et al. (1998) defines the characteristic function of the VG(B, E, N) distribution with parameters B > 0, E > 0 and N > 0 as follows: Because this distribution is infinitely divisible, the VG-process can be defined as the one which begins at time zero with independent and stationary increments. The VG-process 4.2.3 The Cox, Ingersoll and Ross stochastic clock. The CIR process that solves the stochastic differential equation listed below is used as the rate of time change by Carr et al. (2003): where dZ t is an increment to a standard Brownian motion, the parameter d is the long run rate of time change, K is the rate of mean reversion and e dictates the volatility of the time change. The process g t is the instantaneous rate of time change and the new clock is given by its integral C t ¼ Ð t 0 g s ds. The characteristic function of C t , given g 0 , is explicitly provided in Cox et al. (1985) as follows: For the new clock to be increasing, the instantaneous rate of time change must be positive. Moreover, for the random time changes to persist, the rate of time change must be mean reverting. The square root process of CIR considered here is a classic example of a mean-reverting positive process.

The log-contract
The log-contract on the stock and the log-contract on the futures were already defined in Sections 2 and 3, respectively. Further, it was established that the log-contract plays a critical role in the hedging of moment swaps. Closed form solutions for the price of the logcontract are achieved under many models. For example, under the BS model, the logcontract price on the stock is given by: where s is the volatility of the stock price P. The log-contract price for more elaborate models can be directly obtained from the riskneutral characteristic function of ln(P W ), that is:

Higher-order moment swaps
Taking the partial derivative of w (v , t) with respect to v leads to: Therefore, the general expression for the price of the log-contract is given by: In the case of the Heston model, the log-contract price on the stock takes the following expression: while in the case of the time-changed Levy models with a characteristic function w (v , t) shown in equation (12), the log-contract price on the stock is given by:

Parameters estimation and results
The parameters to be estimated are those related to the Heston model, the NIG Levy process with CIR stochastic clock (NIG-L-CIR) and the VG Levy process with CIR stochastic clock (VG-L-CIR). For the Heston model, the parameters are s 2 0 , K, d , b and r . The parameters are h , m , #, K, d , e and g 0 for the NIG-L-CIR model. Finally, the parameters for the VG-L-CIR model are B, E, N, K, d , e , and g 0 . The calibration of the models follows the methods developed in Carr and Madan (1998) for pricing classical vanilla options. In general, these methods can be used when the characteristic function of the risk-neutral stock price process is well specified. Carr and Madan (1998) shows that the price C E (E, W) of a European call option with exercise price E and maturity date W is given by: where « is a positive constant such that the «th moment of the stock price exists and the function g(x ) is defined by: computation methodology is applied in the calibration procedure to estimate the model parameters by minimizing the price differential (market observed price minus model price) in a least-squares configuration. The sample data used consists of 432 plain vanilla call option prices with maturities from less than one month to 5.165 years. These prices are taken from the volatility surface of the euro Stoxx 50 Index. The models were tested on August 25, 2017, when the value of the index was e3,438.55. The risk-free interest rate is set at 3 per cent and the dividend yield at 0 per cent for the sake of simplicity and to focus on the essence of the stochastic behavior of the asset. The calibration procedure is like the one in Schoutens et al. (2004). The risk-neutral parameters for all three models are reported below: The out-of-sample performance of an option pricing model is measured by the stability of the parameters and accuracy of the parameter estimates. Testing the out-of-sample performance of these three models is in effect a test of how slowly the parameters change over time. The pricing biases of these models are computed and contrasted. These pricing biases are captured through four global measures of fit. These global measures of fit are: (1) the root mean square pricing error (RMSPE); (2) the mean absolute percentage pricing error (MAPPE); (3) the mean absolute pricing error (MAPE); and (4) the mean relative percentage pricing error (MRPPE).
These global measures of fit are defined as follows: where n is the number of options, MP i the market price of option i, d MP i the model price of option i and MP the mean option price. The computations of these pricing errors are listed in Table I.
The prices of the log-contracts are computed from equations (13) and (14) using the parameters given above and are listed in Table II.
The pricing biases of the Heston model are compared to those of the NIG-L-CIR model and those of the VG-L-CIR model using the RMSPE, the MAPPE, the MAPE and the MRPPE statistics. These statistics are computed in Table I. Under the RMSPE measure, the NIG-L-CIR model performs better than the VG-L-CIR model which in turn performs better than the Heston model. Using the MAPPE measure, the NIG-L-CIR and the VG-L-CIR models show the same performance which is superior to that of the Heston model. Under the MAPE statistic, the NIG-L-CIR model outperforms the VG-L-CIR model which in turn outperforms the Heston model. Finally, the pricing errors obtained under the MRPPE statistic confirm the superior performance of the NIG-L-CIR model over that of the VG-L-CIR model and over that of the Heston model.
As indicated earlier, the prices listed in Table II are taken from the volatility surface of the euro Stoxx 50 Index. All models appear to fit this surface very well leading to almost similar vanilla prices under the different models. However, the fact that model risk does exist and is different from model to model explains why exotic prices under the various models can be significantly different. This difference is illustrated in Table II where the logcontract prices differ by 48.82 per cent between the Heston and NIG-L-CIR models and by 48.73 per cent between the Heston and VG-L-CIR models.
Moneyness, maturity and volatility effects on pricing bias can be further investigated using a regression analysis. The dependent variable is the percentage pricing error of a given call in the sample at a given date. The independent variables are the moneyness, the time to maturity and the volatility of the euro Stoxx index return. The regression equation is given by: where PPE i (t) is the percentage pricing error of call option i on date t; P/E i and TM i represent the moneyness and time to maturity, respectively, of the call option contract; VOL(t -1) stands for the previous day's annualized standard deviation of the euro Stoxx 50 index return; and O i t ð Þ is the error term. This is a cross-sectional regression; therefore, the standard errors (in parentheses) are computed using the White (1980) heteroskedasticity consistent estimator. The regression is run for each of the three models considered and the results are summarized in Table III. Each independent variable has statistically significant explanatory power of the remaining pricing errors regardless of the model examined. Hence, each model pricing errors have some maturity, intra-daily volatility and moneyness related biases with different magnitudes. The pricing errors have the same sign and therefore are biased in the same direction. The pricing errors relative to the euro Stoxx index's volatility on the previous day are almost stationary, confirming the importance of modeling SV. The mispricing of the NIG-L-CIR and VG-L-CIR models is lower than that of the Heston model, confirming that modeling both SV and jumps is important. The adjusted R 2 of 8.7 per cent for the Heston model, 4.6 per cent for the NIG-L-CIR model and 4.8 per cent for the VG-L-CIR model indicate that the collective explanatory power of these independent variables is quite low. The regression results confirm the results in Table I, that the NIG-L-CIR model is superior to the VG-L-CIR model which in turn is superior to the Heston model in terms of pricing accuracy.

Effectiveness of the variance swap hedge using moment swaps
To assess the effectiveness of the variance swap hedge using moment swaps, this paper will contrast the strategy that includes a third moment swap outlined in equation (8) with the one which excludes moment swaps as described in equation (7). This comparison is made for the different models discussed in Section 4. The parameters used are still the same listed earlier.
Exactly 20,000 variance swaps contracts are hedged. The profit and loss distributions of these hedges under the various models examined are obtained by running a Monte Carlo simulation with one million scenarios. To simulate a NIG process, a NIG(h , m , #) random numbers are first simulated. The NIG random numbers are obtained by combining inverse Gaussian (IG) random numbers and standard normal numbers. An IG(c, d) random variable Y has the following characteristic function: First, the IG 1; # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h 2 À m 2 p random numbers i m are simulated using for instance the IG generator of Michael, Schucany and Haas [Devroye (1986)]. Next, a series of standard normal random variables v m are sampled. The NIG random numbers n m are then generated using the expression below: Finally, the sample paths of a NIG(h , m , #) process Y = {Y t , t ! 0} in the time slots t n = nDt, n = 0, 1, 2, . . . are generated using the independent NIG(h , m , #Dt) random numbers n m given by: The simulation of a VG process is straightforward because a VG process can be seen as the difference of two independent Gamma processes. Gamma numbers are generated using the Johnk's gamma generator or the Berman's gamma generator [Devroye (1986)]. The simulation of a CIR process g = {g t , t ! 0} is also straightforward by discretizing the stochastic differential equation: The sample path of the CIR process g = {g t , t ! 0} in the time slots t = nDt, n = 0, 1, 2, . . . is given by: where {x n , n = 1, 2, . . .} is a series of independent standard normally distributed random numbers.
The means and standard deviations of the profit and loss distributions for each of the three models are computed both under a hedging strategy without moment derivatives and under a hedging strategy with moment derivatives. The mean and standard deviation of the hedging without moment derivatives are respectively m 1 and s 1 while m 2 and s 2 are the same descriptive statistics of the hedging that uses a third moment swap. All these descriptive statistics are listed in Table IV.
Table IV clearly shows that the mean of the profit and loss distribution is closer to zero and its standard deviation is much smaller under the hedge with a third moment swap. Hence, consistent with the theory, the empirical results illustrated in Table IV confirm that the use of a third moment swap in hedging a variance swap dramatically improves the performance of the hedge. The tails of the basic hedge without the third moment swap are much fatter than those under the enhanced hedge, which incorporates the third moment swap. Under the basic hedge without the third moment swap, the standard deviation of the profit and loss distribution for the Heston model is already very low. This is justified by the fact that the Heston model exhibits continuous paths. The Taylor series expansion of the mth power of the logarithm function given by equation (4) is very accurate in a no-jump economy even if limited to the quadratic term. This is not the case in real markets where asset prices exhibit jumps of various magnitudes due of supply and demand shocks. These jumps are the main sources of large hedging errors.

Conclusions
This paper tests empirically the pricing accuracy of the Schoutens (2005) along with the Heston (1993) and the time-changed Levy models and their effectiveness in hedging moment swaps. Variance swaps which are now liquidly traded can be used to protect against changes in the volatility regime. Moments of higher order such as skewness and kurtosis are important, for example, in a jump economy. To protect against inaccurate estimates of skewness and kurtosis, moment derivatives are needed. The results obtained here show that using moment swaps leads to a significant improvement of the standard hedge of the variance swap using only log-contracts. Each hedge is set up as a position in a log-contract combined with a dynamic trading strategy and a static position in moment swaps of lower order. Most specifically, SV models such as the Heston and time-changed Levy models are introduced and closed-form solutions for the price of the log-contract under these models are derived from the risk-neutral characteristic function. Then, the effectiveness of the variance swap hedge using a third-order moment swap is evaluated. The result shows that taking a position in third moment swaps can significantly improve the performance of the standard hedge of a variance swap based on a static position in the log-contract and a dynamic trading strategy. The position in third moment swaps is taken by running a Monte Carlo simulation under the various SV models considered. Overall, the pricing accuracy of the three SV models considered, measured by RMSPE, MAPPE, MAPE and MRPPE shown in Table I, is very satisfactory. A possible extension of this research would be to examine how effective these models are in pricing and hedging moment swaps of order n with n ! 3, using moment derivatives of any order.