Sample path generation of the stochastic volatility CGMY process and its application to path-dependent option pricing

This paper proposes the sample path generation method for the stochastic volatility version of CGMY process. We present the Monte-Carlo method for European and American option pricing with the sample path generation and calibrate model parameters to the American style S\&P 100 index options market, using the least square regression method. Moreover, we discuss path-dependent options such as Asian and Barrier options.


Introduction
Since Heston (1993) applied the CIR model by Cox et al. (1985) to option pricing, the model has been the standard framework for option pricing because it allows for stochastic volatility and volatility smile effect that observed for the Black Scholes model (Black and Scholes (1973)). However, the Lévy process models with time-varying volatility have been used in option pricing in descrete time model, since empirical studies based on the stochastic volatility model show that the Brownian Motion is often rejected (See Rachev and Mittnik (2000), Kim et al. (2010), Kim et al. (2011)). Carr et al. (2003) defined the class of continuous time stochastic volatility model on Lévy processes (SVLP) using the time changed Lévy process. The SVLP has been successfully applied in European option pricing, but the absence of an efficient sample path generation method makes the SVLP model hard to be applied to path-dependent options such as American, Barrier or Asian option. * College of Business, Stony Brook University, New York, USA (aaron.kim@stonybrook.edu) This paper proposes the sample path generation method for the stochastic volatility version of CGMY (CGMYSV) process, which is a subclass of the SVLP model. The method is constructed by an approximation of the series representation of CGMY process with the time-varying scale parameter. The series representations for the tempered stable and tempered infinitely divisible processes are discussed in Rosiński (2007) and Bianchi et al. (2010), and it is applied to Monte-Carlo simulation for CGMY market model with a GARCH volatility in Kim et al. (2010). Yet, the CGMYSV model is a continuous-time model different from the GARCH model with CGMY innovations. We develop an algorithm of the CGMYSV sample path generation, and it will be applied Monte-Carlo simulation (MCS). The algorithm will be used to price European and American option and to calibrate risk-neutral parameters to the S&P 500 index option (European style) and S&P 100 option (American style) data. We will use the least square regression method by Longstaff and Schwartz (2001) for American option pricing with MCS. We verify that the new sample path generation method performs well in American option pricing by that empirical study. We also apply the algorithm to Asian and Barrier option pricing with MCS.
The remainder of this paper is organized as follows. The CIR process and the CGMY process with the series representation are presented in Section 2. The sample path generation method based on the series representation is constructed in Section 3. In Section 4, we perform the CGMYSV model calibration for S&P 500 index option and S&P 100 index option. Also, the Asian and Barrier option prices are discussed.

Preliminary
In this section, we briefly discuss CIR model and CGMY process.

CIR model
The CIR model is given by for κ, η, ζ > 0 and Brownian motion {W t } t≥0 . Let F v t be a smallest σ-algebra generated by process (1−e −κ∆t )ζ 2 and the random variable ξ is non-central χ 2distributed with degrees of freedom 4κη/ζ 2 and noncentrality parameter 2cv t e −κ∆t .
given as following (Proposition 6.2.5 in Lamberton and Lapeyre (1996)

Stochastic volatility version of the CGMY process
Suppose {Z t } t≥0 is the standard CGMY process with parameters (α, λ + , λ − ) and {v t } is the stochastic volatility process given by CIR model in(1). We define a process {L t } t≥0 by where V t = t 0 v s ds, and {Z t } t≥0 is independent of the process {v t } t≥0 . The process {L t } t≥0 is referred to as the stochastic volatility version of the CGMY process or simply CGMYSV process 2 with parameters (α, λ + , λ − , κ, η, ζ, ρ, v 0 ). By (2), we obtain the characteristic function of L t as where φ stdCGMY (u; α, λ + , λ − ) is the characteristic function of Z 1 defined in (3).

Series representation of the CGMYSV Process
Suppose that we have a CIR process {v t } t≥0 with parameters κ, η, and ζ as defined in (1) Algorithm 1: CGMYSV sample path generation Result: SVMYSV sample path Let T be the time horizon ; Let M , J, and N be large positive integer ; and let ||P || = max{∆t m | m = 1, 2, · · · , M }. Suppose that {v tm } tm∈P and {L tm } tm∈P are discrete sub-sequences of the CIR process and the CGMYSV process, respectively. Let ∆L tm = L tm − L t m−1 , By the series representation, we have and {Γ j } j=1,2,··· are given in Section 2.3. The same argument as the relation between series representation of the CGMY process presented in Section 2.3, we can define a series representation of Y tm as follows: Combining equations (6) and (7), we can generate sample path of the CGMYSV process as Algorithm 1.

Simulation of the CGMYSV Process
In order to verify the performance of Algorithm 1, we generate a set of example sample paths of the The second plate of the figure is for 20 sample path of CIR process. For goodness of fit test for the generated path, we perform Kolmogorov-Smirnov test. We compare the distribution of 10-days simulated random numbers {L n,10 |n = 1, 2, · · · , N } with the distribution of L 10∆t . The cumulative distribution function of L 10∆t can be obtained by the Ch.F of the CGMYSV using the inverse Fourier-Transform methos (See Carr et al. (2002) and Rachev et al. (2011) more details). Table 1 presents the result of the KS test, and it has 70.29% p-value and it is not rejected at the 5% significant level. Using the same arguments, we perform KS test for 25-days, 50-days, and 100-days simulated random numbers. They are not rejected at the 5% significant level, either. We graphically compare the empirical probability density function (pdf) of the simulated sample path and the CGMYSV pdfs for those four cases. We draw empirical pdfs using gray bar charts and draw solid lines for CGMYSV pdfs in four plates in Figure 2.

The CGMYSV Option Pricing Model
In this section we discuss the option pricing model on the CGMYSV model. We define the model and calibrate parameters using European style S&P 500 index option (SPX option) and American style S&P 100 index option (OEX option).
Let r and q be the risk free rate of return and the continuous dividend rate of a given underlying asset, respectively. The risk-neutral price process {S t } t≥0 of a given underlying asset is assumed as where {L t } t≥0 is the CGMYSV process with parameters (α, λ + , λ − , κ, η, ζ, ρ, v 0 ). By (5), we also have

Calibration to European Options
On the risk neutral price process {S t } t≥0 defined by (8) Carr and Madan (1999) and Boyarchenko and Levendorskiȋ (2000), we can calculate European call/put prices numerically.
We calibrate the CGMYSV parameters (α, λ + , λ − , κ, η, ζ, ρ, v 0 ) using the SPX option prices on September 11, 2017. We observed 247 call prices and 289 put prices on the day. The S&P 500 price S 0 , risk-free rate of return, and continuous dividend rate at the day were S 0 = 2488.11, r = 1.213%, and q = 1.884% respectively. The calibration results for SPX calls and puts are provided in Table 2. Figure 3 shows observed SPX call and put prices ( drawn by '•'), and calibrated CGMYSV prices using FFT (drawn by '+').
We recalculate the European call and put prices using Monte-Carlo Simulation (MCS) method with the calibrated parameters in Table 2. The sample paths of the MCS method are generated by Algorithm 1. The number of sample paths is 10,000 in this investigation.
To compare the MCS method with the FFT method, we use the four error estimators:the average absolute error (AAE), the average absolute error as a percentage of the mean price (APE), the average relative percentage error (ARPE), and the root mean square error (RMSE) (see Schoutens (2003) In this option pricing with MCS method, we also obtain standard error for each 247 call and 289 put options, but we do not provide them all because of the space limitation. Instead, we show MCS prices with the 95% confidence interval in Figure 4 only for the case 2, 400 < K < 2, 600 and time to maturity 48 days.
Finally, we perform the bootstrapping. We select an at-the-money call and an at-the-money put of K = 2, 500 and T = 28 days as an example, and calculate call and put prices with MCS parameters in Table 2, respectively. Table 4 shows that the MCS prices and their standard errors for 100, 1,000, 5,000, and 10,000 number of sample paths. We repeat this process 100 times and draw boxplots. Boxplots for call and put for each number of sample paths are the up plate and the bottom plate of Figure 5. Stars in those boxplots are the call/put prices using FFT method. We can observe that the number of sample paths increases, then the MCS prices close to the FFT price and dispersions are reduced.

Calibration to American Options
We see that the sample path generation method using the series representation works for MCS of European option pricing in the previous section. In this section, we discuss the American option pricing with the same sample path generation method. We use Least Square Regression Method (LSM) by Longstaff and Schwartz (2001) for American option pricing with MCS. When we do the regression for the expected value of option, we use S t , S 2 t , σ t , σ 2 t and σ t S t as independent variables, following the idea in Chapter 15 of Rachev et al. (2011).
For empirical illustration, we use market prices of the OEX option, which is American style. We calibrate parameters of the CGMYSV model with fixed seed numbers for each random number generation. That 3 The measures are computes as follows: where N is the number of observations, and Pj and Pj denote the model price and the observed market call/put prices, respectively. is, we fix a seed number of χ 2 random number generator in the CIR process, and we generate uniform and exponential random numbers U j , U ′ j , E j , E ′ j and τ j with predefined seed numbers, and fix them. Then we set the model parameters, generate sample paths using Algorithm 1 with the fixed seed number and the fixed random number sets, and then calculate American option price using LSM. Repeat that process and find the optimal parameters to minimize RMSE. As a benchmark, we calibrate the parameters of the CGMY option pricing model (See Carr et al. (2002)) to the OEX option prices using LSM with sample path generated by the series representation explained in Section 2.3.
The calibration results are presented in Therefore, we can conclude that the CGMYSV option pricing model performs typically better than the CGMY option pricing model, except in a few cases in this investigation. Hence, LSM with Algorithm 1 works well in the American option calibration.
Finally, we perform the bootstrapping. We selected the at-the-money put for the strike price K = 910 and the days to maturity T = 31 days on April 6, 2016. Put prices are obtained by LSM using parameters calibrated to the day provided in Table 5. On the day, the underlying S&P 100 index price was 918.21, and the market put price was 13.95 for the strike price 910 and 31 days to maturity. Table 7 shows that the LSM prices and their standard errors for 100, 1,000, 5,000, and 10,000 number of sample paths. The LSM prices approach to the market price and the standard error decreases as the number of sample paths increases. We repeat this process 100 times and present boxplots for those 100 prices, as Figure 6. Stars in those boxplots are the market put prices. We can observe that the LSM prices close to the market price and dispersions are reduced as the number of sample paths increases. Additionally, Figure 7 provides a graphical illustration of the calibration for April 6, 2016. Calibrated CGMYSV prices are drawn by '×', the market observed prices are drawn by '•', and the 95% confidence intervals are marked by 'I' shape. The day to maturities T are written on the plate.

Asian and Barrier options
The For the Asian option, we consider the arithmetic average call and put where the strike price K = 2, 500 and the time to maturity T = 25days. Table 8 shows that the MCS prices for Asian call & put and their standard errors for 100, 1,000, 5,000, and 10,000 number of sample paths. The the standard error of the MCS prices decreases as the number of sample paths increases. We repeat this process 100 times and present boxplots for those 100 prices, as Figure 8. We can observe that the MCS prices converge, and dispersions are reduced as the sample paths increase.
With the same argument, we find MCS price for the Barrier options. We consider the down-and-out call and the up-and-out put Barrier options with the strike price K = 2, 500 and the time to maturity T = 25days. Barrier of the down-and-out call and the up-and-out put are 2, 400 and 2, 750, respectively. Table 9 shows that the MCS prices and their standard errors for 100, 1,000, 5,000, and 10,000 number of sample paths. The standard error of the MCS prices decreases as the number of sample paths increases. We repeat this process 100 times and present boxplots for those 100 prices, as Figure 9. We can also observe that the MCS prices converge, and dispersions are reduced as the sample paths increase.

Conclusion
In this paper, we develop the CGMYSV sample path generation algorithm using the series representation. The series representation method's performance is tested by comparing the simulated distribution to the pdf calculated by the inverse Fourier transform method. We apply the sample path generation method to European and American option pricing with MCS and LSM. We compare the MCS method to the FFT method in European option pricing with SPX option market data. Also, we calibrate the parameters of the CGMYSV model to the American style OEX option using LSM. We measure the performance of the calibration using four error estimators and the boot-strapping method. We conclude that the sample path        Table 9: MCS prices and standard errors for the down-and-out call and the up-and-out put.