Pricing VIX Futures

Question

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?

Answer 1

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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Let the Heston dynamics under the $\Bbb{P}$ measure be given by \begin{align} \frac{dS_t}{S_t} &= \mu_t dt + \sqrt{v_t} dW_S^{\Bbb{P}}(t) \\ dv_t &= \kappa(\theta-v_t)dt + \xi \sqrt{v_t} dW_v^{\Bbb{P}}(t) \\ d\langle W_S^{\Bbb{P}},W_v^{\Bbb{P}}\rangle_t &= \rho dt \end{align}

Define the Radon-Nikodym derivative of $\Bbb{Q}$ with respect to $\Bbb{P}$ as \begin{align} \frac{d\Bbb{Q}}{d\Bbb{P}} &= \mathcal{E}\left( -\lambda_S W_S^{\Bbb{P}}(t) - \lambda_{S,\bot} W_{S,\bot}^{\Bbb{P}}(t) \right) \\ &:= \mathcal{E}(X_t) \end{align} where $\lambda_S$ is the market price of equity risk $$ \lambda_S = \frac{\mu_t-r_t}{\sqrt{v_t}} $$ $\lambda_{S,\bot}$ another risk-premium (yet to be defined) and $\mathcal{E}[X_t]$ the Doélans-Dade exponential of process $X_t$. Assuming that $\lambda_S$ and $\lambda_{S,\bot}$ verify the Novikov condition , Girsanov theorem then stipulates that \begin{align} W_S^{\Bbb{Q}}(t) &= W_S^{\Bbb{P}}(t) - \left\langle W_S^{\Bbb{P}}, X \right\rangle_t \\ W_v^{\Bbb{Q}}(t) &= W_v^{\Bbb{P}}(t) - \left\langle W_v^{\Bbb{P}}, X \right\rangle_t \end{align} are 2 standard $\Bbb{Q}$-Brownian motions verifying $d\langle W_S^{\Bbb{Q}},W_v^{\Bbb{Q}} \rangle_t = \rho dt$.

From the bilinearity property of quadratic variation, the first of the above equation yields \begin{align} W_S^{\Bbb{Q}}(t) = W_S^{\Bbb{P}}(t) + \lambda_S t \end{align}

Using a Cholesky decomposition to re-express $W_v^{\Bbb{P}}(t)$ as $$ W_v^{\Bbb{P}}(t) = \rho W_S^{\Bbb{P}}(t) + \sqrt{1-\rho^2} W_{S,\bot}^{\Bbb{P}}(t) $$ the second equation in turn gives $$ W_v^{\Bbb{Q}}(t) = W_v^{\Bbb{P}}(t) + \lambda_S \rho t + \lambda_{S,\bot} \sqrt{1-\rho^2} t $$ which allows us to rewrite the dynamics under $\Bbb{Q}$ as \begin{align} \frac{dS_t}{S_t} &= r_t dt + \sqrt{v_t} dW_S^{\Bbb{Q}}(t) \\ dv_t &= \kappa(\theta-v_t)dt - \xi \sqrt{v_t} (\lambda_S \rho + \lambda_{S,\bot} \sqrt{1-\rho^2} ) dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}}(t) \\ d\langle W_S^{\Bbb{Q}},W_v^{\Bbb{Q}} \rangle_t &= \rho dt \end{align}

Observe that the change of measure we introduced makes discounted asset prices emerge as $\Bbb{Q}$-martingales. Also observe how, the dynamics under $\Bbb{Q}$ is yet merely defined up to a constant $\lambda_{S,\bot}$. In other words, it is not unequivocally determined until we make an assumption regarding the market price of volatility risk. Let's define $$ \lambda(t,S_t,v_t) = \xi \sqrt{v_t} (\lambda_S \rho + \lambda_{S,\bot} \sqrt{1-\rho^2} ) $$ so that \begin{align} \frac{dS_t}{S_t} &= r_t dt + \sqrt{v_t} dW_S^{\Bbb{Q}}(t) \\ dv_t &= \kappa(\theta-v_t)dt - \lambda(t,S_t,v_t) dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}}(t) \\ d\langle W_S^{\Bbb{Q}},W_v^{\Bbb{Q}} \rangle_t &= \rho dt \end{align}

Some common choices:

Choice 1 Assume $$ \lambda(t,S_t,v_t) = \alpha v_t $$ Then we get \begin{align} dv_t &= \kappa(\theta-v_t)dt - \alpha v_t dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}} \\ &= (\kappa \theta - (\kappa+\alpha)v_t) dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}} \\ &= (\kappa+\alpha) (\frac{\kappa \theta}{\kappa+\alpha} - v_t) dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}} \end{align} so that \begin{align} \kappa^* &= \kappa + \alpha \\ \theta^* &= \frac{\kappa \theta}{\kappa+\alpha} \end{align} This is the choice formulated in my answer above, and also the one in Heston's original paper [http://web.math.ku.dk/~rolf/Heston93.pdf] (end of p.329 and p.335).

Choice 2 Assume $$ \lambda(t,S_t,v_t) = \lambda_S \sqrt{v_t} \rho \xi = (\mu_t - r_t)\rho\xi := \alpha $$ \begin{align} dv_t &= \kappa(\theta-v_t)dt - \alpha dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}} \\ &= \kappa((\theta-\alpha/\kappa)-v_t) dt + \xi \sqrt{v_t} dW_v^{\Bbb{Q}} \end{align} so that \begin{align} \theta^* &= \theta-\alpha/\kappa \\ \end{align} This corresponds to a replication minimising the Delta-hedging standard error (minimum variance $\Delta$ à la Bergomi).