Pricing VIX derivatives with free stochastic volatility model

Abstract

This paper aims to develop a new free stochastic volatility model, joint with jumps. By freeing the power parameter of instantaneous variance, this paper takes Heston model and 3/2 model for special examples, and extends the generalizability. This model is named after free stochastic volatility model, and it owns two distinctive features. First of all, the power parameter is not constrained, so as to enable the data to voice its authentic direction. The Generalized Methods of Moments suggest that the purpose of this newly-added parameter is to create various volatility fluctuations observed in financial market. Secondly, even upward and downward jumps are separately modeled to accommodate the market data, this paper still provides the quasi-closed-form solutions for futures and option prices. Consequently, the model is novel and highly tractable. Here, it should be noted that the data on VIX futures and corresponding option contracts is employed to evaluate the model, in terms of its pricing and implied volatility features capturing performance. To sum up, the free stochastic volatility model with asymmetric jumps is capable of adequately capturing the implied volatility dynamics. Thus, it can be regarded as a model advantageous in pricing VIX derivatives with fixed power volatility models.

Appendix: Derivation of 1st and 2nd order moment of \(\varepsilon _{t+1}\)

To derive $\mathbb {E}^{\mathbb {Q}}[\varepsilon _{t+1}]$ and $\mathbb {E}^{\mathbb {Q}}[\varepsilon _{t+1}^{2}]$, we split the Brownian motion $W^{\mathbb {P}}$ into $Z^{\mathbb {P}}$ and its orthogonal part $Z^{\mathbb {P},\bot }$ and obtain

$$\begin{aligned} \varepsilon _{t+1}&=\delta _s^{*}V_{t+1}\Delta t+\gamma ^{*}V_{t+1}^{\alpha }W^{\mathbb {P}}(\Delta t)\\&=\delta _s^{*}V_{t+1}\Delta t +\gamma ^{*}V_{t+1}^{\alpha } \left[ \rho Z^{\mathbb {P}}(\Delta t)+\sqrt{1-\rho ^2}Z^{{\mathbb {P}},\bot }(\Delta t)\right] . \end{aligned}$$

Taking expectation to both sides, it follows that

$$\begin{aligned} \mathbb {E}_t[\varepsilon _{t+1}]&=\delta _s^{*}\mathbb {E}_t[V_{t+1}]\Delta t+\rho \gamma ^{*}\mathbb {E}_t\left[ V_{t+1}^{\alpha }Z^{\mathbb {P}}(\Delta t)\right] \\&\quad +\gamma ^{*}\sqrt{1-\rho ^2}\mathbb {E}\left[ V_{t+1}^{\alpha }Z^{\mathbb {P},\bot }(\Delta t)\right] \\&=\delta _s^{*}\mathbb {E}_t[V_{t+1}]\Delta t+\rho \gamma ^{*}\mathbb {E}_t\left[ V_{t+1}^{\alpha }\right] *\mathbb {E}_t\left[ Z^{\mathbb {P}}(\Delta t)\right] \\&\quad +\gamma ^{*}\sqrt{1-\rho ^2}\mathbb {E}\left[ V_{t+1}^{\alpha }\right] *\mathbb {E}\left[ Z^{\mathbb {P},\bot }(\Delta t)\right] \\&=\delta _s^{*}\Delta tG(\kappa ^{*}, \theta ^{*}, \sigma , -1; \Delta t, V_t). \end{aligned}$$

The second and third terms in the last equation equal 0 due to the independence and independent increments of Brownian motion. On the other hand, the second moment of $\varepsilon _{t+1}$ is explicitly known as

$$\begin{aligned} \mathbb {E}[\varepsilon ^{2}_{t+1}]&=\mathbb {E}_t\left[ \delta _s^{*2}V_{t+1}^2\Delta t^2 +\gamma ^{*2}V_{t+1}^{2\alpha }(W^{\mathbb {P}}(\Delta t))^2+2\delta _s^{*}\Delta t\gamma ^{*}V_{t+1}^{\alpha +1}W^{\mathbb {P}}(\Delta t)\right] \\&=\delta _s^{*2}\Delta t^2\phi (\Delta t, V_t; {-}2, 0, 0, 0) {+}\mathbb {E}_t\left[ \gamma ^{*2}V_{t+1}^{2\alpha }(\rho Z^{\mathbb {P}}(\Delta t){+}\sqrt{1-\rho ^2}Z^{\mathbb {P},\bot }(\Delta t))^2\right] \\&=\delta _s^{*2}G(\kappa ^{*}, \theta ^{*}, \sigma , -2; \Delta t, V_t)\Delta t^2+\gamma ^{*2}G(\kappa ^{*}, \theta ^{*}, \sigma , -2\alpha ; \Delta t, V_t)\Delta t \end{aligned}$$

where we use Remark 2.1 by setting $\eta$ to be different values and the fact that $V_t$ is a Markov process.