The Stochastic Volatility and Jump Diffusion Processes

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affine jump-diffusion models in capturing the joint dynamics of stock and option prices with daily spot and option prices of the S&P500 index from January, 1993 to December, 1993. But they do not compare the performance of L´evy jump models with that of the double-exponential jump in returns and volatility model in both the spot and option markets.

Thus, the purpose of our paper is to fill this gap and conduct a comprehensive em-pirical study on the relative merits of option pricing models. To this goal, we follow the closed form option pricing model that admits the stochastic volatility model (SV) seen in Heston(1993), the stochastic volatility model with normal jumps (SV-MJ) seen in Bakshi (1997), double-exponential jumps (SV-DEJ) seen in Kou (2002), and correlated jump in volatility and normal jumps (SV-MJ-VCJ) seen in Eraker (2003). Moreover, we develop the closed form option pricing model, such as the stochastic volatility model with Variance-Gamma jumps (SV-VG), Normal Inverse Guassian jumps (SV-NIG), and correlated jump in volatility and double-exponential jumps (SV-DEJ-VCJ) in Kou, Yu, and Zhong (2016) by Esscher Transforms (Gerber, Shiu, and Elias 1994).

2.2 The Stochastic Volatility and Jump Diffusion Pro-cesses

Volatility is an important role in the option pricing and also in hedging strategy. There are many hedging tools which use the volatility index as underlying asset, such as VIX Futures, VIX Option and Variance futures. These derivatives indicate that option price has been largely affected by the volatility. However, Black & Sholes (1973) model assume that the volatility is an constant that is too restrictive and therefore it cannot match the market requirements, such as the volatility smile and the volatility clustering. In order to find the right distribution assumption to close the market price, the researchers developed the series of the related volatility researches and applications.

After Hull and White (1987) firstly developed the fundamental concern of the stochas-tic volatility whose variation dynamic follows the geometric Brownian motion, the series of the related stochastic volatility researches are generated by many financial researchers.

Among these researches, Heston (1993) submitted the most popular model that the vari-ation of index return follows the square-root process (used in Cox, Ingersoll, and Ross

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(1985)). This model have several advantages: (i) It can avoid the negative variations.

(ii) The variation of index return have the mean-reverting property. (iii) The correlated between volatility shocks and underlying stock returns serves to control to the level skew-ness and the volatility variation coefficient serves to the level of kurtosis. These benefits can make it fit index returns well. Moreover, Heston (1993) use the Fourier transform to obtain the closed form of European call option price by the CIR model.

However, since the stochastic volatility model belong to continuous-time model, it cannot capture the heavy-tailed feature, i.e. large return jumps, in short-term period and thus prices short-term options properly is limited. Therefore, Merton (1976) firstly added the compound Poisson normal jump diffusion model to the Black & Sholes (1973) model. Bates (1996) based on the the stochastic volatility model and Merton (1973) jump to develop the stochastic volatility with normal jump model (SV-MJ). Bates (1996) applied this model to American options and used the deutsche mark foreign currency options to examine the fitting performance. Bakshi et al. (1997) compare the pricing error and hedging error of the Black-Sholes, stochastic volatility, stochastic volatility stochastic interest rate and stochastic volatility with normal jump models. They find that stochastic volatility with normal jump model outperform in the option pricing error.

However, for hedging purposes, the SV model achieve the best hedging results among all the models. There are several advantages of the compound Poisson jump diffusion model. First, it can capture the property of implied volatility smile. Second, it can describe the heavy-tail feature of index return. That is, when the jump amplitude be-come more larger, the return distribution which is constructed by the model will also become more heavier (Matsuda (2004), Gatheral (2006), Hull (2014)). Third, the com-pound Poisson jump diffusion model have analytical property, and we can solve the partial integro-differential equation (PIDE) to get the closed form formula which can make pricing and hedging more effective than monte carlo method. (such as Hilliard and Reis (1999), Koekebakker and Lien (2004), Cartea and Figueroa (2005), Chevallier (2013), Wilmot and Mason (2013), Schmitz et al. (2014), Mayer et al. (2015), Xiao et al. (2015)).

On the other hand, Kou (2002) applied double-exponential distribution and equilib-rium theory to acquire the option pricing model and used the Japanese LIBOR Caplets in May 1998 to illustrate that the model cam produce implied volatility smile. Since the

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sity of double-exponential distribution for jump sizes is monotonically which is decreasing for both negative jumps and positive jumps, it better fit for small jumps than Merton jumps (seen in Kou et al. (2016)). Further, asymmetric heavy-tail (e.g. leptokurtosis) feature of double-exponential distribution also helps fit both positive and negative large jumps. Therefore, many literatures start to use double-exponential distribution to model the return distribution. Ramezani and Zeng (2002) indicated that double-exponential jump diffusion model can fit better than normal jump model when the index return have the jump events around the time peiod. Moreover, Kou and Wang (2004) demonstrate that a double-exponential jump diffusion model can lead to an analytic approximation for finite-horizon American options and analytic solutions for popular path-dependent options.

Duffie, Pan, and Singleton (2000), Eraker (2004), Eraker et al. (2003), and Kou et al. (2016) consider the stochastic volatility model incorporating jumps in returns and in volatility. The empirical result of Eraker (2003) show that both of these jump components are important for both S&P500 and Nasdaq100 index from January 2, 1980, to December 31, 1999 and September 24, 1985, to December 31, 1999. Models with only diffusive stochastic volatility and jumps in returns are misspecified, because they do not have a component driving the conditional volatility of returns, which is rapidly moving. However, Kou et al. (2016) show that a simple affine jump-diffusion model with the stochastic volatility and double-exponential jumps fits both the S&P500 and the NASDAQ-100 daily returns from 1980 to 2013 well. That is, the SV-DEJ-VCJ model may not outperform the SV-DEJ model, since by construction, jumps in volatility occur at the same time as jumps in returns, a constraint that impedes the double-exponential distribution in picking up small jumps.

Madan and Seneta (1990), Madan, Carr, and Chang (1998) and Huang and Wu (2004) submitted the Variance Gamma (VG) process which is obtained by evaluated Brownian motion (with constant drift and volatility) at a random time change given by a gamma process. That is, VG process allow non-normal increment as compared to normal increments of Brownian motion, and thus the jump component of variance gamma process is much more flexible than a compound Poisson process. Moreover, Li, Wells, and Yu (2008, 2011) develop efficient MCMC methods for estimating parameters and latent volatility/jump variables of the L´evy jump model using stock and option prices.

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Li, Wells, and Yu (2011) show that the VG model of Madan, Carr, and Chang (1998) with stochastic volatility has the best performance among of all the models they consider in capturing both the physical and risk-neutral dynamics of the S&P500 index.

Barndoeff-Nielsen (1997, 1998) submitted normal inverse Gaussian (NIG) process which is obtained by evaluated Brownian motion (with constant drift and volatility) at a random time change given by a inverse Gaussian process. That is, NIG process allow non-normal increment as compared to normal increments of Brownian motion, and thus the jump component of NIG process is much more flexible than a compound Poisson process.

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Chapter 3 The Models

3.1 Stochastic Volatility Model

This model is shown in Heston (1993) who assume that the variation of index return follows the square-root process (used by Cox, Ingersoll, and Ross (1985)). This model have several characteristics for the index return: (i) It can avoid the negative variations.

(ii) The variation of index return have the mean-reverting property. (iii) The correlated between volatility and underlying spot returns serves to control to the skewness. (v) The volatility variation coefficient serves to the kurtosis. This model capture an important features of equity index return dynamics, namely stochastic volatility, and also provide analytical tractability for option pricing in Section 4.1.1 and model estimation in Section 6.

Consider a probability space Ω, F, P, {Ft}^T_t=0 , where Ω is the sample set of all outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {F_t}^T_t=0 be the filtration generated by the pair of the Wiener processes dW_y,t^P, dW_v,t^P at time t, 0 ≤ t ≤ T . According to above definition, the index returns, y_t, follows the stochastic volatility (SV) model, and the stochastic volatility v_t follows the mean-reverting square-root process. Then its dynamic process under physical probability measure P is:

dy_t=

 µ − 1

2v_t



dt +√

v_tdW_y,t^P, dv_t= κ (θ − v_t) dt + σ_v√

v_tdW_v,t^P,

(3.1)

where µ is the drift term which is a constant. The stochastic volatility v_tfollows a

mean-‧

reverting square root process with long term level θ > 0, the speed of adjustment is κ > 0, and the variation coefficient of stochastic volatility is σ_v > 0. Moreover, dW_y,t^P and dW_v,t^P are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^P ∼ ¹N (0, dt), dW_v,t^P ∼ N (0, dt), and Corr(dW_y,t^P, dW_v,t^P) = ρdt. Therefore, by the Cholesky decomposition, the correlated Wiener processes can be decomposed into two independent Wiener processes, i.e., dW_y,t^P = dB_1,t^P and dW_v,t^P = ρdB_1,t^P +p1 − ρ²dB_2,t^P. Thus, the equation (3.1) can be rewritten as following

dy_t=

We transform (3.1) into discrete-time version:

y_t= y_t−1+

where ∆ is the time interval, µ is the drift term. The stochastic volatility v_t follows a mean-reverting square root process with long term level θ > 0, the speed of adjustment κ > 0, and the variation coefficient of stochastic volatility σ_v > 0. Moreover, ε^P_y,t and ε^P_v,t are diffusion noises with standard normal distribution with correlation coefficient ρ, i.e., ε^P_y,t ∼ N (0, 1), ε^P_v,t ∼ N (0, 1), and Corr(ε^P_y,t, ε^P_v,t) = ρ. Therefore, by the Cholesky decomposition, the correlated Wiener processes can be decomposed into two independent Wiener processes, i.e., ε^P_y,t= ε^P_1,t and ε^P_v,t = ρε^P_1,t+p1 − ρ²ε^P_2,t. Thus, under P measure, the equation (3.3) can be rewritten as following

y_t= y_t−1+

For this dynamics process, we have observations {y_t}^T_t=0, latent volatility variables {v_t}^T_t=0 and model parameters space Θ = {µ, κ, θ, σ_v, ρ}. There is a drawback under the SV model. Since the SV model belong to continuous-time model, it cannot capture the heavy-tailed feature, i.e. large return jumps, in short-term and thus it cannot explain the smirkiness exhibited in the cross-sectional option data (Bakshi et al., (1997), Bates

1

The notation ∼ N (µ, σ

²

) means that the distribution is normal with parameter µ, σ

²

and the density

1/ √

2πσ

²

e

⁻

(

^{(x−µ)2/2σ2}

), where x ∈ R

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(2000)). For modifying this drawback, we introduce the stochastic volatility with L´evy jumps model in the next section.

3.2 The Characteristic Exponent of L´ evy Jump

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