Heston Option Pricing and the Characteristic Function PDE
Summary
The document asks why the characteristic function used in Heston stochastic-volatility option pricing satisfies the same partial differential equation as the probability terms in a call-price representation. It presents the governing PDE for those terms, including the mixed derivative induced by correlation between the asset and variance processes, and defines the associated model parameters and two probability measures.
The supplied answer does not explain the Feynman–Kac connection in detail. Instead, it identifies a typo in a cited derivation: the variance terms in an equation for the characteristic-function coefficient have misplaced or missing volatility-scale factors. Thus, the material is useful as a narrowly scoped derivation correction, but it offers little evidence beyond the stated correction and linked answer. It does not provide a full derivation or establish broader conditions for applying the PDE.
Key ideas
- The Heston call-price representation expresses value using two probabilities under different measures.
- The probability terms satisfy a PDE that includes asset–variance correlation and variance dynamics.
- The question concerns why the characteristic function obeys the same PDE, but the included answer does not give that explanation.
- The answer flags a derivation typo involving volatility-scale factors in the coefficient equation.
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Full text
# Heston model characteristic function
# Heston model characteristic function
The characteristic function of $x=ln(S_T)$ in the framework of Heston model is guessed to be: $$f_j(\phi,x,v)=e^{C_j(\tau,\phi)+D_j(\tau,\phi)+i\phi x}$$
The call price is guessed to have the form: $$C_T(K)=e^{x_t}P_1(x,v,\tau)-e^{r \tau}KP_2(x,v,\tau)$$
where $P_1$ and $P_2$ are probabilities that an option expires in-the-money w.r.t proper measures.
Now, I can follow the derivation of the PDE for $P_1$ and $P_2$ $$\frac{\partial P_j}{\partial \tau}+\rho \sigma v\frac{\partial^2 P_j}{\partial x \partial v}+\frac{1}{2}v\frac{\partial^2 P_j}{\partial x^2}+\frac{1}{2}v\sigma \frac{\partial^2 P_j}{\partial v^2}+(r+u_jv)\frac{\partial P_j}{\partial x}+(a-b_jv)\frac{\partial P_j}{\partial v}=0$$ where $j=1,2$, $u_1=\frac{1}{2}$, $u_2=-\frac{1}{2}$, $a=\kappa \theta$, $b_1=\kappa + \lambda -\rho \sigma$, $b_2 = \kappa + \lambda$ and $\tau = T - t$.
However, I cannot understand and couldn't find any resources online which would justify why characteristic functions $f_j(\phi,x,v)$ also have to satisfy the very same PDE. Most of them just say it's the result of Feynman-Kac formula, but I cannot understand the link. Could someone explain it or point me to an appropriate source (article, textbook, whatever), please?
## Answer by javier quintanilla (score 1)
https://quant.stackexchange.com/a/79644
To answer your comment about (A7), it is a typo. It should read as: $$\boldsymbol{-\frac{1}{2}\phi^2} + \rho\sigma\phi i D + \boldsymbol{\frac{1}{2}\sigma^2D^2}+u_j\phi i-b_jD+\frac{\partial D}{\partial t}=0 $$ In the original paper the first term has a $\sigma^2$ that shouldn't be there, and the third term is missing a $\sigma^2$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.