Stochastic volatility Levy models

Question

Hey I have some questions about stochastic volatility for Levy processes. If I understand correctly, if we change the time in Levy's process by CIR process, the newly received process is not Levy's process anymore? We model stock price as $$S(t)=S(0)\frac{\exp((r-q)t+Z(t))}{E[\exp(Z(t))]}$$ where $Z(t)$ is stochastic volatility ,,Levy'' (?) process. In this article is written that this process is not a martingale. So why we use it for pricing? Can anyone explain me this as simple as possible?

Answer 1

VGSV, NIGSV and CGMYSV

Let $X_t$ be a variance gamma process (or NIG or CGMY) and let $Y_t=\int_0^t y_s\mathrm{d}s$ where $y_s$ is a CIR (Heston) square-root process. We then set $$Z_t=X_{Y_t},$$ which means we first subordinate a Brownian motion with drift with a gamma process (to get the VG process $X_t$) and then again subordinate with an integrated CIR process, $Y_t$. The semimartingale $Z_t$ is called VGSV/NIGSV/CGMYSV (where SV stands for stochastic volatility) or, in general, stochastic volatility Lévy processes (SVLP). Note that whilst a Brownian motion and Gamma process have independent and stationary increments, neither $y_s$ nor $Y_t$ are Lévy processes themselves ($y_s$ mean-reverts). However, $Y_t$ is obviously non-negative and non-decreasing and thus a valid subordinator. The CIR process can be interpreted as the instantaneous rate of the time change. Carr et al. (2003) show that $$\varphi_{Z_t}(u)=\varphi_{Y_t}\left(-i\Psi_{X_t}(u)\right).$$ The characteristic exponent of a VG (or NIG or CGMY) process, $\Psi_{X_t}$, is well-known whereas the characteristic function of $Y_t$ can also be easily obtained in closed-form (it's stated in the paper below Equation (3.2)).

Importantly, as you say, the process $Z_t$ does not have independent increments anymore. Thus, it's not a Lévy process but it can model volatility clusters. Remember that standard Lévy processes (random clocks) are obtained by subordination (calendar time vs business time) and model a rich jump structure. However, they fail to incorporate stochastic volatility elements.

VGSA, NIGSA and CGMYSA

Carr et al. (2003) suggest two ways how to construct a stock price model based on $Z_t$. You refer to the first one which is ``superior in [its] ability to capture the information content of the option surface''. Set $$S_t=S_0\frac{e^{(r-q)t+Z_t}}{\mathbb{E}^\mathbb{Q}[e^{Z_t}]},$$ where $r$ and $q$ are the risk-free rate of return and dividend yield respectively. Then, \begin{align*} \varphi_{\ln(S_t)}^\mathbb{Q} &= \mathbb{E}^\mathbb{Q}[e^{iu\ln(S_t)}] \\ &= \mathbb{E}^\mathbb{Q}[e^{iu\ln(S_0e^{(r-q)t})}e^{iuZ_t}e^{-iu\ln(\mathbb{E}^\mathbb{Q}[e^{Z_t}])}] \\ &= e^{iu\ln(S_0e^{(r-q)t})}\frac{\varphi_{Y_t}\left(-i\Psi_{X_t}(u)\right)}{\varphi_{Y_t}\left(-i\Psi_{X_t}(-i)\right)^{iu}}. \end{align*} Remember that $\Psi_{X_t}$ and $\varphi_{Y_t}$ are well-known. The exponential stock price process $S_t$ is called VGSA/NIGSA/CGMYSA. In addition to $r$, $q$ and the three parameters of the VG process, the VGSA process furthermore depends on three CIR parameters. Section 7 of the paper shows how to additionally introduce leverage (negative correlation between returns and volatility) as yet another parameter. The leverage effect is an important empirical stylised fact.

Arbitrage and Martingales

Because $Z_t$ is not a Lévy process, it's not trivial how to ensure that $e^{-rt}S_t$ is a $\mathbb{Q}$-martingale. Thus, our classical martingale (risk-neutral) pricing approach doesn't directly apply in their setting. That's why they introduce the notion of martingale marginals. In particular, you'll want to read section 4.1 of their paper. They prove for example that (iff there is no static arbitrage) risk-neutral densities satisfy this martingale marginal property. In particular, the martingale marginal property is a more fundamental concept than pricing by an equivalent martingale measure (which requires the absence of both static and dynamic arbitrage strategies). They provide a simple two-step binomial tree example where no equivalent martingale measure exists but the martingale marginal property holds.

Carr et al. (2003) describe a second way of constructing stock prices which are ensured to be martingales after discounting (named VGSAM, NIGSAM and CGMYSAM) but perform worse in fitting observed option data. The more conservative processes we constructed above (VGSA, NIGSA and CGMYSA) don't allow for static arbitrage and satisfy the Lévy marginal (LM) property if the CIR processed used in their construction, $y_s$, starts at zero (Theorem 5.1). In Section 8, the authors illustrate how to apply Carr and Madan's (1999) fast Fourier transform (FFT) option pricing formula in their setting.