A jump diffusion model for VIX volatility options and futures

Abstract

Volatility indices are becoming increasingly popular as a measure of market uncertainty and as a new asset class for developing derivative instruments. Although jumps are widely considered as a salient feature of volatility, their implications for pricing volatility options and futures are not yet fully understood. This paper provides evidence indicating that the time series behaviour of the VIX index is well approximated by a mean reverting logarithmic diffusion with jumps. This process is capable of capturing stylized facts of VIX dynamics such as fast mean-reversion at higher levels, level effects of volatility and large upward movements during times of market stress. Based on the empirical results, we provide closed-form valuation models for European options written on the spot and forward VIX, respectively.

Appendices

Appendix 1: Derivation of the characteristic functions for the jump-diffusion processes

Duffie et al. (2000) prove that, under technical regularity conditions, the characteristic function for affine diffusion/jump diffusion processes, such as the SRJ and SRPJ, has the following exponential affine form¹³:

$$ \phi (V_{t} ,T - t;s) = \exp \left( {A(T - t;s) + B(T - t;s)V_{t} } \right) $$

(26)

Thus, for the case of the SRJ, $A(T - t;s)$ and $B(T - t;s)$ are given by¹⁴:

$$ A(T - t;s) = a\left( {T - t;s} \right) + z\left( {T - t;s} \right) $$

(27)

$$ a(T - t;s) = - {\frac{2k\theta }{{\sigma^{2} }}} \times ln\left( {{{\left( {k - {\frac{1}{2}}\,i\sigma^{2} s\left( {1 - e^{{ - k\left( {T - t} \right)}} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {k - {\frac{1}{2}}i\sigma^{2} s\left( {1 - e^{{ - k\left( {T - t} \right)}} } \right)} \right)} k}} \right. \kern-\nulldelimiterspace} k}} \right) $$

(28)

$$ z(T - t;s) = {\frac{2\lambda \rho }{{2k - \eta \sigma^{2} }}} \times ln\left( {{{\left( {k - {\frac{1}{2}}\,i\sigma^{2} s + is\left( {{\frac{{\sigma^{2} }}{2}} - {\frac{k}{\eta }}} \right)e^{{ - k\left( {T - t} \right)}} } \right)} \mathord{\left/ {\vphantom {{\left( {k - {\frac{1}{2}}i\sigma^{2} s + is\left( {{\frac{{\sigma^{2} }}{2}} - {\frac{k}{\eta }}} \right)e^{{ - k\left( {T - t} \right)}} } \right)} {\left( {k - {\frac{isk}{\eta }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {k - {\frac{isk}{\eta }}} \right)}}} \right) $$

(29)

and,

$$ B(T - t;s) = {\frac{{ksie^{{ - k\left( {T - t} \right)}} }}{{k - {\frac{1}{2}}i\sigma^{2} s\left( {1 - e^{{ - k\left( {T - t} \right)}} } \right)}}} $$

(30)

The characteristic function of the LRJ expressed in logarithms is given by:

$$ \phi (lnV_{t} ,T - t;s) = \exp \left( {A(T - t;s) + B(T - t;s)\left( {lnV_{t} } \right)} \right) $$

(31)

where

$$ A(T - t;s) = is \theta (1 - e^{{ - k\left( {T - t} \right)}} ) - s^{2} \sigma^{2} \left( {{\frac{{1 - e^{{ - 2k\left( {T - t} \right)}} }}{4\kappa }}} \right) + {\frac{\lambda }{k}} \times ln\left( {{\frac{{\eta - ise^{{ - k\left( {T - t} \right)}} }}{\eta - is}}} \right) $$

(32)

$$ B(T - t;s) = ise^{{ - k\left( {T - t} \right)}} $$

(33)

Finally, in the case of the SRPJ the coefficients $A(T - t;s)$ and $B(T - t;s)$ cannot be solved in closed form and are estimated numerically. So, the conditional characteristic function $\phi (V_{t} ,T - t;s) = E(e^{{isV_{T} }} \left| {V_{t} } \right.)$ of the SRPJ must satisfy the following Kolmogorov backward differential equation:

$$ {\frac{\partial \phi }{{\partial V_{t} }}} + k(\theta - V_{t} ) + {\frac{1}{2}}{\frac{{\partial^{2} \phi }}{{\partial V_{t}^{2} }}}V_{t} \sigma^{2} - {\frac{\partial \phi }{\partial \tau }} + \lambda V_{t} {\rm E}\left[ {F(V_{t} + y) - F(V_{t} )} \right] = 0 $$

(34)

subject to the boundary condition

$$ F(V_{t} ,T - t = 0;s) = e^{{isV_{t} }} $$

(35)

where $i = \sqrt { - 1}$. Differentiating the characteristic function given by Eq. (26) yields:

$$ \begin{gathered} \phi_{V} = BF \hfill \\ \phi_{VV} = B^{2} F \hfill \\ \phi_{T - t} = F\left( {A_{T - t} + VB_{T - t} } \right) \hfill \\ \end{gathered} $$

(36)

where the subscripts denote the corresponding partial derivatives. Substituting Eq. (36) into Eq. (34) and rearranging yields:

$$ V_{t} \left( { - kB - B_{T - t} + {\frac{1}{2}}\,\sigma^{2} B^{2} + \lambda {\rm E}\left[ {e^{yB} - 1} \right]} \right) + \left( {k\theta B - A_{T - t} } \right) = 0 $$

(37)

Also, ${\rm E}\left[ {e^{yB} - 1} \right] = \int_{0}^{ + \infty } {\eta e^{ - \eta y} e^{yB} } dy - 1\,\, = {\frac{\eta }{{\eta - {\rm B}}}} - 1$, and since $V_{t} \ne 0$, the expressions in the parentheses in Eq. (37) must equal zero. Therefore, we obtain the following ordinary differential equations (ODEs)

$$ - kB - B_{T - t} + {\frac{1}{2}}\,\sigma^{2} B^{2} + \lambda \left( {{\frac{\eta }{{\eta - {\rm B}}}} - 1} \right) = 0 $$

(38)

$$ k\theta B - A_{T - t} = 0 $$

(39)

Although the ODEs cannot be solved in a closed form, numerical solutions are possible subject to the boundary conditions $A(T - t = 0;s) = 0$, and $B(T - t = 0;s) = is$.

Appendix 2: Maximum-likelihood estimation

Maximum Likelihood estimation requires the conditional (transition) density function $f[V(t + \tau )\left| {V(t)} \right.,\Uptheta ]$ (τ > 0) of the process V _t , where τ denotes the sampling frequency of observations and Θ is the set of parameters to be estimated. For a sample $\left\{ {V(t)} \right\}_{t = 1}^{T}$, the log-likelihood function that is maximized is given by: $LL = \mathop {\max }\limits_{\Uptheta } \sum\nolimits_{t = 1}^{T - \tau } {\log \left( {f\left( {V(t + \tau )} \right)\left| {V(t),\Uptheta } \right.} \right)}$.

In the case of the SR and LR processes, the conditional density is known in closed form (see Dotsis et al. 2007 and Detemple and Osakwe 2000, respectively). The conditional density of the jump diffusion processes is derived from the characteristic function as described below (see also Singleton 2001). Assume that we stand at time t, and τ denotes the sampling frequency of observations. Then, the Fourier inversion of the characteristic function $\phi (V(t),T - t;s)$ provides the required conditional density function $f[V(T)\left| {V(t)} \right.]$:

$$ f[V(T)\left| {V(t)} \right.] = {\frac{1}{\pi }}\int_{0}^{\infty } {\text{Re} [e^{ - isV(T)} \phi (V(t),T - t;s)]ds} $$

(40)

where $\text{Re}$ denotes the real part of complex numbers. For a sample $\left\{ {V(t)} \right\}_{t = 1}^{T}$, the conditional log-likelihood function to be maximized is given by:

$$ LL = \mathop {\max }\limits_{{\left\{ \Uptheta \right\}}} \sum\limits_{t = 1}^{T} {\log \left( {{\frac{1}{\pi }}\int_{0}^{\infty } {\text{Re} [e^{ - isV(t + \tau )} \phi (V(t),T - t;s)]ds} } \right)} $$

(41)

where Θ = {κ, θ, σ, λ, η} is the set of parameters to be estimated.¹⁵ The standard errors of the ML estimators are retrieved from the inverse Hessian evaluated at the obtained estimates.