Approximate Maximum Likelihood Estimation in Sub-fractional Hybrid Stochastic Volatility Model

We consider the hybrid stochastic volatility model under the risk-neutral measure given by d S t = X t d t + V t − S t d W t + ρ λ t d L τ λ t , $$\displaystyle \begin{aligned} dS_t = X_t dt + \sqrt {V_{t-}} S_t dW_t + \rho _{\lambda t} dL_{\tau _{\lambda t}},\end{aligned} $$ d V t = − λ V t d t + υ λ t − d L τ λ t , $$\displaystyle \begin{aligned} dV_t = -\lambda V_t dt + \upsilon _{\lambda t-}dL_{\tau _{\lambda t}},\end{aligned} $$ d X t = α ( β − X t ) d t + σ X t γ t d W t H , $$\displaystyle \begin{aligned} dX_t = \alpha (\beta - X_t) dt + \sigma X_t^{\gamma _t} dW_t^H,\end{aligned} $$ d ρ t = ( ( 2 ζ − η ) − η ρ t ) d t + θ ( 1 + ρ t ) ( 1 − ρ t ) d Z t , $$\displaystyle \begin{aligned} d\rho _t = ((2 \zeta -\eta ) -\eta \rho _t) dt + \theta \sqrt {(1+\rho _t)(1-\rho _t)}dZ_t,\end{aligned} $$ d ξ t = κ ( μ − ξ t ) d t + ς ξ t d B t , $$\displaystyle \begin{aligned} d\xi _t= \kappa (\mu -\xi _t) dt + \varsigma \sqrt {\xi _t} dB_t,\end{aligned} $$ d γ t = ϖ ( ψ − δ ) ) d t + χ d M t , $$\displaystyle \begin{aligned}d\gamma _t = \varpi (\psi -\delta )) dt + \sqrt {\chi } dM_t,\end{aligned} $$ d τ t = ξ t − d t , $$\displaystyle \begin{aligned}d\tau _t = \xi _{t-} dt,\end{aligned} $$ where L t is a Levy process, W H is a sub-fractional Brownian motion, and B t, Z t, and M t are the standard Brownian motions. Here, S t is the asset price that is a geometric jump-diffusion, V t is the stochastic volatility that is a Levy O–U process, X t is the stochastic interest rate that is a sub-fractional CKLS process, ρ t is the stochastic leverage Jacobi (Beta) process, ξ t is a volatility modulation of the driving Levy subordinator that is a CIR process, τ t is the stochastic time change of the driving Levy subordinator that is a time integral of the CIR process, γ t is the stochastic elasticity model that is a CIR process, v is the volatility that is independent of L, and all the 14 parameters in the model are positive.