Maths
Formulation
The original 1993 paper by Heston can be found here
Suppose under a real world probability measure $\mathbb{P}$, the price process $S_t$ of the underlying security is governed by the following diffusion process
$$dS_t = \mu S_t dt + \sigma_t S_t dW_{1t}$$
and the volatility process $\sigma_t$ follows the diffusion process:
$$d \sigma_t = -\alpha \sigma_t dt + \beta dW_{2t}$$
where $\mu$ is the expected rate of return, $W_{1t}$ and $W_{2t}$ are two standard Brownian motions so that:
$$Cov(dW_{1t}, dW_{2t}) = \rho dt$$
where $\rho$ is the constant correlation coefficient between $W_{1t}$ and $W_{2t}$
We also assume the continuously compounded rate of interest is a constant $r$
To make the algebra easier we let: $V_t = \sigma_t^2$
Analysis $\mathbb{P}$-World
Applying Ito's Lemma to $V_t=\sigma_t^2 = f(\sigma_t)$ where $f(x)=x^2$ gives:
$$dV_t = \kappa(\theta - V_t)dt + \gamma \sqrt{V_t} dW_{2t}$$
where:
- $\theta = \frac{\beta^2}{2 \alpha}$ is the long term mean of the variance
- $\kappa = 2 \alpha$ is the mean-reverting speed of the variance
- $\gamma = 2 \beta$ is the volatility of the variance
Proof
$df(t, X_t) = \left[\frac{\partial f}{\partial t} + \mu(t, X_t) \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2(t, X_t) \frac{\partial^2 f}{\partial x^2} \right]dt + \sigma(t, X_t) \frac{\partial f}{\partial x} dWt$
where
$dX_t = \mu dt + \sigma dW_t$
But now we have
$d \sigma_t = -\alpha \sigma_t dt + \beta dW_{2t}$
so we re-write Ito's Lemma for $\sigma$ rather than $X$
$df(t, \sigma_t) = \left[\frac{\partial f}{\partial t} + -\alpha \sigma_t \frac{\partial f}{\partial \sigma} + \frac{1}{2} \beta^2 \frac{\partial^2 f}{\partial \sigma^2} \right]dt + \beta \frac{\partial f}{\partial \sigma} dWt$
We want to get to $dV_t$ and $V_t = \sigma_t^2$ so we use $f(x)=x^2$
So $\frac{\partial f}{\partial t}=0$, $\frac{\partial f}{\partial x}=2x$ and $\frac{\partial^2 f}{\partial x^2}=2$
So Ito's Lemma becomes
$dV_t = d\sigma^2_t = df(\sigma_t) = \left[0 - \alpha \sigma_t . 2 \sigma_t + \frac{1}{2} \beta^2 . 2\right]dt + \beta.2\sigma_t.dWt$
$dV_t = \left[\beta^2 - 2 \alpha \sigma_t^2\right]dt + 2\beta\sigma_t.dWt$
$dV_t = 2\alpha\left[\frac{\beta^2}{2\alpha} - \sigma_t^2\right]dt + 2\beta\sigma_t.dWt$
$dV_t = 2\alpha\left[\frac{\beta^2}{2\alpha} - V_t\right]dt + 2\beta \sqrt{V_t}.dWt$
Defining $x_t = ln S_t$, we see that by Ito's Lemma
$$dx_t = \left(\mu - \frac{1}{2} V_t \right) dt + \sqrt{V_t} dW_{1t}$$
The Cholesky Decomposition gives us that:
$$W_{2t}=\rho W_{1t} + \sqrt{1-\rho ^2} W_{0t}$$
where $W_{0t}$ and $W_{1t}$ are two independent Brownian motions under $P$. This gives us the correlation of $\rho$ between $W_{1t}$ and $W_{2t}$
Check
$cov(W_{1t}, W_{2t}) = cov(W_{1t}, \rho W_{1t} + \sqrt{1-\rho ^2} W_{0t})$,
$cov(W_{1t}, W_{2t}) = cov(W_{1t}, \rho W_{1t}) + cov(W_{1t}, \sqrt{1-\rho ^2} W_{0t})$,
$cov(W_{1t}, W_{2t}) = \rho \times var(W_{1t}) + \sqrt{1-\rho ^2} \times cov(W_{1t}, W_{0t})$,
$cov(W_{1t}, W_{2t}) = \rho.dt$
and $corr(W_{1t}, W_{2t}) = \frac{cov(W_{1t}, W_{2t})}{\sqrt{var(W_{1t}).var(W_{2t})}} = \frac{\rho.dt}{dt} = \rho$
We can now rewrite the HSV as follows:
$$dx_t = \left(\mu - \frac{1}{2} V_t \right) dt + \sqrt{V_t} dW_{1t}$$
$$dV_t = \kappa (\theta - V_t) dt + \rho \gamma \sqrt{V_t} dW_{1t} + \sqrt{1-\rho^2} \gamma \sqrt{V_t} dW_{0t}$$
Analysis $\mathbb{Q}$-World
So far this is all well and good. Now we need to map our stochastic volatility process into $\mathbb{Q}$ world
Let us start with a Cameron Martin Girsanov review
If $W_t$ is a S.B.M. under $P$ and $\gamma_t$ is a previsible process then:
$\overset{\sim}{W_t} = W_t + \displaystyle\int_{0}^{t} \gamma_s ds$ is a S.B.M under $\mathbb{Q}$ where $\mathbb{Q}$ is defined by:
$\frac{d\mathbb{Q}}{d\mathbb{P}}=exp\left(-\displaystyle\int_{0}^{T} \gamma_t dW_t - \frac{1}{2} \displaystyle\int_{0}^{T} \gamma_t^2 dt \right)$
This is far more powerful than the special case we actually need. In practice what we do is to chose the $\gamma_t$ to take out the market price of risk and so $\mathbb{Q}$ becomes the risk-neutral probability measure
We start with the following inspired definition:
$\eta _t = \left(\frac{\gamma \rho(\mu - r) + \lambda(t, S_t, V_t)}{\gamma \sqrt{1-\rho^2} \sqrt{V_t}}, \frac{\mu-r}{\sqrt{V_t}} \right)' \in R^2$
where $\lambda(t,S_t, V_t)$ is the market price of volatility risk
we now write $\textbf{W}_t = (W_{0t}, W_{1t})' \in R^2$ for convenience and define $\frac{d\mathbb{Q}}{d\mathbb{P}}$ by
$\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp \left(\displaystyle\int_0^t \eta^{'}_{u} d\textbf{W}_u - \frac{1}{2} \displaystyle\int_0^t ||\eta_u||^2 du \right)$
So by Girsanov's Theorem, the process $\textbf{W}^*_t$ defined by:
$$\textbf{W}^*_t=\textbf{W}_t - \displaystyle\int_0^t \eta _u du$$
is a standard Brownian motion under $\mathbb{Q}$
It may be helpful to write out $\textbf{W}^*_t$ longhand at this point to have a more detailed look at this vector
$W^*_{0t} = W_{0t} + \displaystyle \int_0^t \frac{\lambda(u, S_u, V_u)}{\gamma\sqrt{1-\rho^2}\sqrt{V_u}}du + \displaystyle \int_0^t \frac{\rho(\mu - r)}{\sqrt{1-\rho^2}\sqrt{V_u}}du$
$W^*_{1t} = W_{1t} + \displaystyle \int_0^t \frac{\mu-r}{\sqrt{V_u}}du$
So $W_{0t}^*$ and $W_{1t}^*$ are two independent standard Brownian motions under $\mathbb{Q}$
This brings us to the question of what is $\lambda$. Clearly $\lambda$ is a parameter which allows us to adjust for the market's attitude to the uncertainty around the volatility of the volatility
Unlike the first order market price of risk there is no arbitrage free construction of what $\lambda$ must be and so we have to observe this value from the market
Work by Breeden (1979) suggests that $\lambda(t, S_t, V_t)$ is proportional to $V_t$ and hence equal to $\lambda V_t$ where $\lambda$ is a constant so we will proceed on that basis from here
Ultimately this is a matter of fitting a parameter to the data, This analysis by PIMCO suggests the parameter fitting will be fairly stable
Then under $\mathbb{Q}$, the risk neutral dynamics of the logarithmic price process and the variance process are:
$dx_t=\left(r-\frac{1}{2} V_t\right) dt + \sqrt{V_t} dW_{1t}^*$
Check
$dx_t = (\mu - \frac{1}{2} V_t) dt + \sqrt{V_t} dW_{1t}$
$W_{1t}=W_{1t}^{*} - \displaystyle\int_0^t \frac{\mu-r}{\sqrt{V_u}} du$
$dW_{1t}=dW_{1t}^{*} - \frac{\mu-r}{\sqrt{V_t}}$
$dx_t = (\mu - \frac{1}{2} V_t) dt + \sqrt{V_t} \left(dW_{1t}^{*} - \frac{\mu-r}{\sqrt{V_t}} \right)$
$dx_t = (r - \frac{1}{2} V_t) dt + \sqrt{V_t} dW_{1t}^{*} $
$dV_t = \kappa^* (\theta^* - V_t) dt + \rho \gamma \sqrt{V_t}dW^*_{1t} + \sqrt{1-\rho^2} \gamma \sqrt{V_t} dW^*_{0t}$
where
$\kappa^*=\kappa + \lambda$ and
$\theta^*=\frac{\kappa \theta}{\kappa + \lambda}$
so $\kappa^*$ and $\theta^*$ are risk neutral model parameters
The check for this line would work the same way as the $dx_t$ check but is simply a little more complicated
We now define the process $W_{2t}^{*}$ by putting:
$W_{2t}^{*}=\rho W_{1t}^{*}+\sqrt{1-\rho ^2} W_{0t}^{*}$
It can be shown that $W_{2t}^{*}$ is a Brownian motion under $\mathbb{Q}$ (fairly trivial as $W_{0t}^{*}$ and $W_{1t}^{*}$ are) and that: $Cov(dW_{1t}^{*},dW_{2t}^{*})=\rho dt$ under $\mathbb{Q}$
So we can re-write the price and variance dynamics as:
$$dS_t=rS_t dt + \sqrt{V_t} S_t dW_{1t}^{*}$$
$$dV_t=\kappa ^{*}(\theta ^{*} - V_t) dt + \gamma \sqrt{V_t} dW_{2t}^{*}$$
In summary, so far we have managed to define our simultaneous stochastic differential equations under $\mathbb{P}$ and then we have recalculated them under $\mathbb{Q}$. You will notice the $\mu$ in $\mathbb{P}$ world has become a $r$ in $\mathbb{Q}$ world in particular
we now have two options for valuing an option:
- Monte Carlo
- Finding a closed form solution
We will look at the Monte Carlo option in this course
What fundamental fact about the modeling of share prices with a lognormal Monte Carlo process can we use to simply introduce price jumps into our analysis?