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In a 2006 paper Zhang and Zhu propose a model for VIX and VIX Futures based on Heston.

I am struggling in understanding how they get equation 6 and 8 (where they define the parameters).

Can anyone of you help me?enter image description here

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Heston - Change of measure

Consider the following Heston dynamics written under the real world measure $\Bbb{P}$ \begin{gather} \frac{dS_t}{S_t} = \mu_t dt + \sqrt{v_t} dW_S^{\Bbb{P}}(t),\ S(0) = S_0 \\ dv_t = \kappa(\theta-v_t)dt + \xi \sqrt{v_t} dW_v^{\Bbb{P}}(t),\ v(0) = v_0 \\ d\langle W_S^\Bbb{P}, W_v^\Bbb{P} \rangle_t = \rho dt \end{gather} In order to be able to use that model to price financial instruments, arbitrage-free pricing theory (APT) tells us that we need to move to an equivalent measure $\Bbb{Q}$ under which discounted asset prices are martingales (or more generally: the value of any self-financing portfolio, when expressed in the risk-free money market account numéraire, should emerge as a $\Bbb{Q}$ martingale).

Because the Heston model is incomplete, there exists infinitely many such measures. Mathematically, these will differ by the drift attributed to the instantaneous variance process i.e. $$ dv_t = \kappa(\theta-v_t)dt - \lambda(t,S_t,v_t) dt + \xi \sqrt{v_t} dW_v^\Bbb{Q}(t) $$ where the term $\lambda(t,S_t,v_t)$ is often referred to as the market price of volatility risk.

In his original 93 paper, Heston makes a particular assumption regarding the market price of volatility risk which he considers being proportional to $v_t$ relying on some economic arguments $$ \lambda(t,S_t,v_t) = \lambda v_t$$ In that particular case, the dynamics under (Heston's) $\Bbb{Q}$ may be rewritten \begin{gather} \frac{dS_t}{S_t} = (r_t - q_t) dt + \sqrt{v_t} dW_S^{\Bbb{Q}}(t),\ S(0) = S_0 > 0 \\ dv_t = \kappa^*(\theta^*-v_t)dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}}(t),\ v(0) = v_0 \\ d\langle W_S^\Bbb{Q}, W_v^\Bbb{Q} \rangle_t = \rho dt \end{gather} with \begin{align} \kappa^* = \kappa + \lambda \\ \theta^* = \theta \frac{\kappa}{\kappa + \lambda} \tag{1} \end{align} and $r_t$ (resp. $q_t$) figures the risk-free rate (resp. equity dividend yield).

Heston - Variance swaps

For a pure diffusion model, the fair variance strike $\hat{\sigma}^2_T(0)$ of a fresh-start variance swap of maturity $T$ calculated at $t=0$ is defined as $$ \hat{\sigma}_T(0)^2 = \frac{1}{T} \Bbb{E}_0^\Bbb{Q} \left[ \int_0^T d\langle \ln S \rangle_t \right] $$ In the particular case of the Heston model we can further write \begin{align} \hat{\sigma}_T(0)^2 &= \frac{1}{T} \Bbb{E}_0^\Bbb{Q} \left[ \int_0^T v_t dt \right] \tag{2} \\ &= \theta^* + (v_0 - \theta^*) \frac{1-e^{-\kappa^* T}}{\kappa^* T} \end{align} where the second equality can be obtained either by:

  • Permuting integral and expectation operators in $(2)$ (Fubini), noting that $v_t$ is CIR so that its conditional expectation is known in closed form for any time $t$, integrating the result;
  • Integrating the SDE verified by $v_t$, taking the expectation, solving the resulting ODE for $\Bbb{E}_0[v_t]$ and again integrating the result.

VIX

Because the VIX squared is by definition the fair variance strike of an (idealised) variance swap of maturity $T=\tau_0$ equal 30 days we then have, under Heston \begin{align} VIX^2(0) &= \theta^* + (v_0 - \theta^*) \frac{ 1-e^{-\kappa^* \tau_0}}{\kappa^* \tau_0} \\ &= \underbrace{\theta^* \left( 1 - \frac{1-e^{-\kappa^* \tau_0}}{\kappa^* \tau_0} \right)}_{A} + v_0 \underbrace{\frac{1-e^{-\kappa^* \tau_0}}{\kappa^* \tau_0}}_{B} \tag{3} \end{align} which is exactly the equation you mention, with the risk-neutral parameters of the Heston dynamics $(\kappa^*, \theta^*)$ related to the parameters under the real world measure $(\kappa, \theta)$ through $(1)$

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  • $\begingroup$ @Davide don't hestitate to mark this answer as accepted if it helped you. $\endgroup$ – Quantuple Mar 19 '18 at 16:02

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