Deriving the Risk-Neutral Heston Model with Girsanov's Theorem

# Change of Measure in the Heston Model Using Girsanov's Theorem


# Change of Measure in the Heston Model Using Girsanov's Theorem












The Heston model assumed the price $S(t)$ of an asset and its variance $v(t)$ follow $$ dS(t)=\mu S(t)dt+\sqrt{v(t)}S(t)dW_{1}(t), \\ dv(t)=\kappa[\theta-v(t)]dt+\sigma\sqrt{v(t)}dW_{2}(t), $$ where $W_{1}(t)$ and $W_{2}(t)$ are two correlated Wiener processes of rate $\rho$ such that $$ \mathbb{E}[dW_{1}(t)dW_{2}(t)]=\rho dt. $$ This model is under the historical measure $\mathbb{P}$. However, the model must be under the risk-neutral measure $\mathbb{Q}$ for options (or any other derivatives) pricing. From the book The Heston Model and Its Extensions in Matlab and C#, the change of measure is done by admitting Girsanov's theorem for each process above. The book, however, did not provide further explanation about it. Instead, it directly provides the result: $$ dS(t)=rS(t)d{t}+\sqrt{v(t)}S(t)d\tilde{W}_{1}(t), \\ dv(t)=\tilde{\kappa}\left[\tilde{\theta}-v(t)\right]dt+\sigma\sqrt{v(t)}d\tilde{W}_{2}(t), $$ This model is risk-neutral:

- The asset price $S(t)$ growth rate is only based on the risk-free interest rate $r$ under $\mathbb{Q}$ (the asset price growth is based on the drift $\mu$ under $\mathbb{P}$).

- The volatility risk premium is $\lambda(t,S(t),v(t))=\lambda v(t)$ (Heston was motivated by Breeden's CCAPM model when choosing this).

- $\lambda$ is the variance risk premium.

- $\tilde{\kappa}=\kappa+\lambda$ and $\tilde{\theta}=\kappa\theta/(\kappa+\lambda)$ are the risk-neutral parameters.

- $\tilde{W}_{1}(t)=W_{1}(t)+\frac{\mu-r}{\sqrt{v(t)}}t$ is the Wiener process of $S(t)$ under $\mathbb{Q}$ and $\tilde{W}_{2}(t)=W_{2}(t)+\frac{\lambda\sqrt{v(t)}}{\sigma}t$ is the the Wiener process of $\sqrt{v(t)}$ under $\mathbb{Q}$.

I am trying to derive this risk-neutral model using Girsanov's theorem, but I am too far off from knowing where I have to begin with. The rough step-by-step would be:

- Define the market prices of risk $\gamma_{1}$ and $\gamma_{2}$.

- Given $\gamma_{1}$ and $\gamma_{2}$, revisit the Radon-Nikodym derivative: \begin{align*} \frac{d\mathbb{Q}}{d\mathbb{P}}\Bigg\vert_{\mathcal{F}_{t}}= \exp\bigg[-\bigg(\int_{0}^{t}\gamma_{1}(s)d\tilde{W}_{1}(s)+\int_{0}^{t}\gamma_{2}(s)d\tilde{W}_{2}(s)\bigg)-\frac{1}{2}\bigg(\int_{0}^{t}\gamma_{1}^{2}(u)du+\int_{0}^{t}\gamma_{2}^{2}(u)du\bigg)\bigg]. \end{align*}

- Girsanov's theorem states that $\tilde{W}_{1}(t)$ and $\tilde{W}_{2}(t)$ follow standard Brownian motion (or equivalently are Wiener processes) under $\mathbb{Q}$.

- The math worked wonders then.

I will number my questions for convenience and readability:

- Is this procedure correct? Any answers would help me significantly.

- How can I start with step 1 with just the Heston model?

## Answer by Jiaji Qu (score 3, accepted)

https://quant.stackexchange.com/a/81568

Your procedure is correct.

The first step is that if we define market prices of risk $\gamma_1(t)$ and $\gamma_2(t)$ (also called shifts, like of the Brownian motion) i.e. we set $\widetilde{W}_i(t) \;=\; W_i(t) \;+\; \int_{0}^{t}\gamma_{i}(s)\,ds, \quad i=1,2.$

then $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ will be Brownian motions under a new measure $\mathbf{Q}$ given the appropriate Radon-Nikodym derivative (which you so generously provided) by Girsanov's theorem:

$\frac{d\mathbf{Q}}{d\mathbf{P}}\Bigg|_{\mathcal{F}_t} \;=\; \exp\!\Biggl[ \;-\;\int_{0}^{t}\gamma_{1}(s)\,dW_1(s) \;-\;\int_{0}^{t}\gamma_{2}(s)\,dW_2(s) \;-\;\tfrac12 \int_{0}^{t} \bigl(\gamma_{1}^2(u) \;+\;\gamma_{2}^2(u)\bigr)\,du \Biggr]$

Under $\mathbf{Q}$, our BM looks something like:

$d\widetilde{W}_i(t) \;=\; dW_i(t) \;+\; \gamma_i(t)\,dt, \quad i=1,2.$

(with same correlation $\rho$). Now we play to match drifts, which is just a bit of calculation:

Under physical $\mathbf{P}$ our asset takes the form: $ dS(t) \;=\; \mu\,S(t)\,dt \;+\; \sqrt{v(t)}\,S(t)\,dW_1(t). $.

We want the drift under $\mathbf{Q}$ to be $rS(t)$. Using a formula for change of drift $dW_1(t) \;=\; d\widetilde{W}_1(t) \;-\; \gamma_1(t)\,dt,$. Our physical asset dynamics can thus be converted to the form:

$\begin{aligned} dS(t) &=\; \mu\,S(t)\,dt \;+\;\sqrt{v(t)}\,S(t)\,\bigl[d\widetilde{W}_1(t) - \gamma_1(t)\,dt\bigr]\\[3pt] &=\; \Bigl[\mu - \sqrt{v(t)}\,\gamma_1(t)\Bigr]\,S(t)\,dt \;+\; \sqrt{v(t)}\,S(t)\,d\widetilde{W}_1(t). \end{aligned} $

Equating the drift to $rS(t)$ yields $\mu \;-\; \sqrt{v(t)}\,\gamma_1(t) \;=\; r \quad\Longrightarrow\quad \gamma_1(t) \;=\; \frac{\mu - r}{\sqrt{v(t)}}.$

Now using the variance parameter, we can calculate the other market price of risk:

Under physical $\mathbf{P}$ the instanteous variance looks like: $dv(t) \;=\; \kappa\,[\theta - v(t)]\,dt \;+\; \sigma\,\sqrt{v(t)}\,dW_2(t).$

Under risk-neutral $\mathbf{Q}$, we use $dW_2(t) = d\widetilde{W}_2(t) - \gamma_2(t)\,dt,$

which gives $ \begin{aligned} dv(t) &=\; \kappa\,[\theta - v(t)]\,dt \;+\; \sigma\,\sqrt{v(t)}\,\bigl[d\widetilde{W}_2(t) - \gamma_2(t)\,dt\bigr]\\[3pt] &=\; \Bigl[\kappa\,[\theta - v(t)] - \sigma\,\sqrt{v(t)}\,\gamma_2(t)\Bigr]\,dt \;+\; \sigma\,\sqrt{v(t)}\,d\widetilde{W}_2(t). \end{aligned} $.

We want this drift to match $\widetilde{\kappa}\,[\widetilde{\theta} - v(t)]$, so let's use what you had for the risk-neutral parameters and set $\widetilde{\kappa} = \kappa + \lambda, \quad \widetilde{\theta} = \frac{\kappa\,\theta}{\kappa + \lambda}, $ where $\lambda$ is the variance risk premium.

Using the risk-neutral market price of risk you gave earlier, I have $\gamma_2(t) = \frac{\lambda\,\sqrt{v(t)}}{\sigma},$ which implies $-\,\sigma\,\sqrt{v(t)}\,\gamma_2(t) = -\,\lambda\,v(t).$

Then $\kappa\,[\theta - v(t)] - \lambda\,v(t) \;=\ (\kappa+\lambda)\,\Bigl[\tfrac{\kappa\,\theta}{\kappa+\lambda} - v(t)\Bigr],$

which matches the desired risk-neutral drift $\widetilde{\kappa} [\widetilde{\theta} - v(t)]$.

This shows that the market prices of risk must be:

$\gamma_1(t) \;=\; \frac{\mu - r}{\sqrt{v(t)}}, \quad \gamma_2(t) \;=\; \frac{\lambda\,\sqrt{v(t)}}{\sigma}.$.

To conclude, under $\mathbf{Q}$ we have the Brownian motions:

$\widetilde{W}_1(t) \;=\; W_1(t) \;+\; \int_{0}^{t}\gamma_1(s)\,ds, \qquad \widetilde{W}_2(t) \;=\; W_2(t) \;+\; \int_{0}^{t}\gamma_2(s)\,ds,%$

$ \begin{cases} \displaystyle dS(t) = r\,S(t)\,dt \;+\; \sqrt{v(t)}\,S(t)\,d\widetilde{W}_1(t), \\[6pt] \displaystyle dv(t) = (\kappa+\lambda)\,\Bigl[\tfrac{\kappa\,\theta}{\kappa+\lambda} - v(t)\Bigr]\,dt \;+\; \sigma\,\sqrt{v(t)}\,d\widetilde{W}_2(t), \end{cases} \quad d\widetilde{W}_1(t)\,d\widetilde{W}_2(t) \;=\; \rho\,dt. $

This is the so-called canonical risk neutral Heston model.