Changing Measure in the Heston Model with Correlated Brownian Motions

# Derivation of Heston Stochastic Volatility (uncorrelated Brownian-Motions)


# Derivation of Heston Stochastic Volatility (uncorrelated Brownian-Motions)












Under the physical measure $ \mathbb{P} $, the dynamics of the Heston model are given by: $$ dS_t/S_t = \mu \, dt + \sqrt{v_t} \, dZ_{1,t}, $$ $$ dv_t = \kappa (\theta - v_t) \, dt + \sigma \sqrt{v_t} \, dZ_{2,t}, $$ with the correlation structure: $$ \langle dZ_{1,t}, dZ_{2,t} \rangle = \rho \, dt. $$

To facilitate the transformation to the risk-neutral measure, two uncorrelated Brownian motions are introduced with the following relationship: $$ \langle dB_{1,t}, dB_{2,t} \rangle = 0, \quad \text{and} \quad dZ_{2,t} = dB_{2,t}, \quad dZ_{1,t} = \rho \, dB_{2,t} + \sqrt{1 - \rho^2} \, dB_{1,t}. $$

The Girsanov theorem is then applied to these uncorrelated Brownian motions to account for the change of measure. Under the risk-neutral measure $ \mathbb{Q}_\lambda $, the Brownian motions transform as: $$ dB^{\mathbb{Q}_\lambda}_{2,t} = dB_{2,t} + \lambda \, dt, $$ $$ dB^{\mathbb{Q}_\lambda}_{1,t} = dB_{1,t} + \frac{\mu - r - \lambda \rho \sqrt{v_t}}{\sqrt{1 - \rho^2} \sqrt{v_t}} \, dt. $$

Finally, the dynamics of the processes under the risk-neutral measure, depending on the choice of the market price of risk parameter $ \lambda $, are rewritten as: $$ dS_t/S_t = r \, dt + \sqrt{1 - \rho^2} \sqrt{v_t} \, dB^{\mathbb{Q}_\lambda}_{1,t} + \rho \sqrt{v_t} \, dB^{\mathbb{Q}_\lambda}_{2,t} = r \, dt + \sqrt{v_t} \, dZ^{\mathbb{Q}_\lambda}_{1,t}, $$ $$ dv_t = \big( \kappa (\theta - v_t) - \lambda \sigma \sqrt{v_t} \big) \, dt + \sigma \sqrt{v_t} \, dB^{\mathbb{Q}_\lambda}_{2,t} = \big( \kappa (\theta - v_t) - \lambda \sigma \sqrt{v_t} \big) \, dt + \sigma \sqrt{v_t} \, dZ^{\mathbb{Q}_\lambda}_{2,t}. $$

These stochastic differential equations (SDEs) are subsequently connected to the two-dimensional Feynman-Kac partial differential equation (PDE).

A natural question arises: why is it necessary to transform from the initial correlated Brownian motions to uncorrelated ones?

## Answer by Andrea (score 2, accepted)

https://quant.stackexchange.com/a/81745

It is basically not necessary to introduce uncorrelated Brownian motions. Or, at least, not necessary as long as the rank of your covariance matrix is full.

Let's call $B_t$ the uncorrelated Brownian motions, and $\Sigma$ the desired covariance matrix.

$\Sigma = \sigma A A' \sigma'$ where $\sigma$ is meant as a diagonal matrix: (you can call $A$ the correlation decomposition).

So $A \, B_t$ are correlated Brownian motions.

If the target drift is $\mathbb{r}$, one needs to find $\mathbb{\lambda}$ such that

$\mathbb{r} = \mathbb{\mu} + \sigma A \mathbb{\lambda}$, or

$\sigma^{-1} ( \mathbb{r} - \mathbb{\mu} ) = A \mathbb{\lambda}$

Now, as long as this is possible (which is, if the correlation is not singular and $\sigma>0$), we don't really care about the exact value of $\mathbb{\lambda}=A^{-1} \sigma^{-1} ( \mathbb{r} - \mathbb{\mu} )$, but we can work with $A \mathbb{\lambda}$ which is the drift change for the correlated Brownian Motions.

So, $\lambda$ is the drift change for $B_t$, while $A \mathbb{\lambda}$ is the drift change for $A B_t$

This is what one normally does for a model with multiple stocks, where the drift change is applied to each of them individually, ignoring the correlation structure.

If the correlation is singular, your will have problems, and will have to decide how to resolve the model incompleteness. Better not to go there.