Deriving the Heston Pricing PDE from Risk-Neutral Dynamics

# Derivation of Heston PDE


# Derivation of Heston PDE












I have some doubts about the derivation of the pricing PDE in stochastic volatility models like the Heston model. I read some books like The Volatility Surface by Jim Gatheral and there's a bit I'm confused about. Before I ask my question, I'll lay out the basic concepts as I understand them. I have a system of two SDE \begin{equation*} \begin{cases} dS_t = \mu S_t dt + \sqrt{V_t}S_t dW^1_t \\ dV_t = p_t dt + q_t \left( \varrho dW^1_t + \sqrt{1-\varrho^2} dW^2_t \right) \end{cases} \end{equation*} where $(W^1_t)_t$ and $(W^2_t)_t$ are independent Brownian motions. For example, in Heston's model we have \begin{equation*} p_t = \kappa (m-V_t), \qquad q_t = \eta\sqrt{V_t}. \end{equation*} The volatility matrix is given by $$ \sigma_t = \left[ \sqrt{V_t} \quad 0 \right], $$ and a price market of risk related to this model is a solution $(\theta_t)_t$ of the equation $$ \sigma_t \theta_t = \mu -r $$ where $r$ is the risk-free rate. For every process $(\lambda_t)_t$, $$ \begin{bmatrix} \frac{\mu - r}{\sqrt{V_t}}\\ \lambda_t \end{bmatrix} $$ is a market price of risk. There's no uniqueness of the market price of risk, hence the risk-neutral measure is not unique and the market model is incomplete. The Girsanov theorem states that the processes $(W^{\lambda,1}_t)_t$ and $(W^{\lambda,2}_t)_t$ given by \begin{equation*} \begin{cases} dW^{\lambda,1}_t = dW^1_t + \frac{\mu - r}{\sqrt{V_t}} dt \\ dW^{\lambda,2}_t = dW^2_t + \lambda_t dt \end{cases}. \end{equation*} are independent Brownian motions (*). In Heston model's literaure I have read the correlated version: if we define the correlated Brownian motions

\begin{equation*} \begin{cases} Z^{\lambda,1}_t = W^{\lambda,1}_t \\ Z^{\lambda,2}_t = \varrho W^{\lambda,1}_t + \sqrt{1 - \varrho^2} W^{\lambda,2}_t \end{cases}, \end{equation*} then after some calculations we can prove that \begin{equation*} \begin{cases} dZ^{\lambda,1}_t = dZ^1_t + \frac{\mu - r}{\sqrt{V_t}} dt \\ dZ^{\lambda,2}_t = dZ^2_t + \Lambda_t dt \end{cases}, \end{equation*} where \begin{equation*} \Lambda_t := \varrho \frac{\mu - r}{\sqrt{V_t}}+ \sqrt{1 - \varrho^2} \lambda_t \end{equation*} is the market price of volatility risk. We can easily prove that the risk-neutral dynamics become \begin{equation*} \begin{cases} dS_t=r S_t dt + \sqrt{V_t} S_t dZ^{\lambda,1}_t\\ dV_t = (p_t-\Lambda_t q_t) dt + q_t dZ^{\lambda,2}_t \end{cases}. \end{equation*}

For what concerns the derivation of the PDE, I know that the market is incomplete and hence the arbitrage pricing is not unique. The idea is to complete the market by adding an option $U_t=(t,S_t,V_t)$. Let $F(t,S_t,V_t)$ denote the arbitrage price in the market completion, with some calculation we obtain the equation \begin{align*} &\frac{\left(\partial_t F + \frac{1}{2}S^2 V \partial_{ss}F + \frac{1}{2} q^2 \partial_{vv}F + \varrho q S \sqrt{V}\partial_{sv}F \right) + rS \partial_s F - rF}{\partial_vF}=\\ &=\frac{\left(\partial_t U + \frac{1}{2} S^2 V \partial_{ss}U + \frac{1}{2} q^2 \partial_{vv}U + \varrho q S \sqrt{V} \partial_{sv}U \right) + rS \partial_s U - rU}{\partial_vU} . \end{align*} The left-hand side is a function of $F$ only and the right-hand side is a function of $U$ only. The only way that this can be is for both sides to be equal to some function $\varphi(t,S_t,V_t)$. Here comes the tricky part: in the books I've read, the function $\varphi$ is assumed to be $$\varphi = p \Lambda -q$$ without loss of generality. What does this actually mean? The function $\varphi$ is fixed, so I guess we are fixing a process $\Lambda$ (and hence a price market of risk). If so, why are we choosing $\Lambda$ such that $\varphi = p \Lambda -q$ rather than $\varphi = \Lambda$ or $\varphi = g(\Lambda)$ for some other function $g$? I know that $p-\Lambda q$ is the drift of the process $(V_t)_t$ under the risk neutral measure, but I don't understand this choice. Is it maybe related to the (unique) market price of risk of the market completion? I guess that this choice of $\Lambda$ makes the arbitrage price in the (not completed) model equal to the (unique) arbitrage price in the market completion. Could someone clarify this for me, from a mathematical point of view?

I also have 2 other questions:

- In order to apply the Girsanov theorem in (*), we must verify that the price market of risk $$ \begin{bmatrix} \frac{\mu - r}{\sqrt{V_t}}\\ \lambda_t \end{bmatrix} $$ is such that its exponential martingale is a true martingale (for instance, we can verify the Novikov condition). Is this proved somewhere?

- In the Heston model, the process $(\Lambda_t)_t$ is assumed to be $\Lambda_t = \xi \sqrt{V_t}$ for some constant $\xi$. Why is it a good assumption?

## Answer by James Cartwright (score 0)

https://quant.stackexchange.com/a/85486

In the Heston model you have two state variables: the stock price and its instantaneous variance. Under the risk-neutral measure, the stock’s expected drift is the risk-free rate minus dividends, while the variance mean-reverts with a volatility-of-variance term, and the Brownian shocks to stock and variance are correlated.

Now take any derivative whose value can be written as a function of time, stock price, and variance. Apply Itô’s lemma to that value function. This decomposes the change in the derivative value into:

a drift part (deterministic change over an instant), and

two random parts driven by the two Brownian motions (one from the stock shock, one from the variance shock).

Because stock and variance shocks are correlated, Itô’s lemma also produces a mixed second-order term: the cross-derivative with respect to stock and variance, multiplied by the correlation and the volatility-of-variance. The other second-order terms come from the stock’s instantaneous variance (giving the second derivative with respect to stock) and from the variance process (giving the second derivative with respect to variance).

Next use the no-arbitrage principle in risk-neutral pricing: the discounted derivative price must be a martingale. Equivalently, once you discount by the risk-free rate, the drift of that discounted process must be zero. When you write the discounted differential, discounting subtracts “risk-free rate times the derivative value” from the drift you got from Itô’s lemma.

Setting that total drift to zero gives the pricing partial differential equation: time derivative of the option value plus the stock-drift term times the first stock derivative plus the variance-drift term times the first variance derivative, plus one half times the stock variance times the second stock derivative, plus one half times the variance diffusion variance times the second variance derivative, plus the correlation term times the cross derivative, minus the risk-free rate times the option value, equals zero.

The terminal condition is simply that at maturity the option value equals its payoff as a function of the stock price. Boundary conditions depend on the payoff (for example, the large-stock and near-zero-stock behavior for calls/puts, and appropriate behavior as variance goes to zero or becomes large).