Heston Model Expectations Involving the Variance Process

Summary

The document sets up a Heston stochastic volatility model for log price and variance, then states an exponential-affine conditional moment-generating function in the terminal log price and variance. It also presents a known expression for an unconditional expectation involving the exponential transform multiplied by variance. The practical target is a related expectation used in a Fourier-style option pricing calculation, with a log-moneyness term, strike, integration bounds, and an integer frequency index.

The author seeks to simplify the expression by substituting the exponential price process into the log-price term, but the document contains no answer or derivation. It therefore records a mathematical problem rather than a completed method. Computing the requested quantity would require careful handling of the complex Fourier argument, conditioning, and differentiation of the affine transform with respect to the variance argument; the displayed unconditional formula alone does not resolve those steps.

Key ideas

• The Heston setup models log price and variance with correlated stochastic drivers implied by the model specification.
• The conditional exponential transform is written in affine form using functions of time and transform arguments.
• The target expectation combines the transform with the current variance and a Fourier frequency term.
• The document poses the computation but provides no solution or evidence validating a derivation.

Tags

# Computing expectation of conditional characteristic function of the Heston model and variance process $V_t$


# Computing expectation of conditional characteristic function of the Heston model and variance process $V_t$












I'm using the following Heston model:

\begin{align} \text{d}X_t &= -\dfrac{1}{2} V_t \text{d}t +\sqrt{V_t} \text{d}B_t, \\ \text{d}V_t &= -\lambda(V_t-\kappa) \text{d}t + \sigma \sqrt{V_t} \text{d}W_t. \end{align}

The conditional moment generating function of this model is

\begin{align} \varphi_t(u,w) =& \mathbb{E}[\exp(uX_T+wV_T)\vert \mathcal{F}_t] \notag \\ =& \exp(\phi_{T-t}(u,w)+V_t\psi_{T-t}(u,w)+uX_t), \text{ } (u,w) \in \mathbb{R}^2, \end{align}

I already know the unconditonal expectation

\begin{equation*} \mathbb{E}[\exp(uX_t+wV_t)V_t] = \left[\partial_w \phi_t(u,w) + V_0 \partial_w \psi_t(u,w)\right] e^{X_0u} \phi_t(u,w) \exp(\phi_t(u,w) + V_0\psi_t(u,w)+iuX_0). \end{equation*}

For my applications I want to compute for the price process $S_t = \exp(X_t)$ and strike $K$ the expectation

\begin{equation*} \mathbb{E}\left[\exp\left(\phi_{T-t}(\frac{k\pi}{b-a},0)+V_t\psi_{T-t}(\frac{k\pi}{b-a},0) + \frac{ik\pi}{b-a} (\log(S_t /K)-a) \right) V_t\right] \end{equation*}

for $a<0$ and $b>0$ and $k$ as integer. My approach was to insert $\exp(X_t)$ for $S_t$ and use the logarithm rules, but than I was stuck. I would be thankful, if you can help me to compute this expectation.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.