0.1 Structure
We introduce the general framework of the stochastic volatility model where \(\sigma_{t}=f(Y_{t})\) is the volatility process and \(\{Y_{t}\}\) has dynamics given by \[ dY_{t}=\alpha(Y_{t})dt+\beta(Y_{t})dW_{t}^1, \] the stock price \(\{S_{t}\}\) has dynamics given by \[ dS_{t}=\mu(Y_{t})S_{t}dt+\sigma_{t}S_{t}dW_{t}^0, \] and Brownian Motions \(W^0, W^1\) have a given correlation coefficient \(\rho\). Equivalently \[ W^1=\rho W^0+\sqrt{ 1-\rho^2 }W^\perp, \] for independent BM \(W^0, W^\perp\).
Under risk-neutral pricing, we wish to find \(\mathbb{Q}\) under which \(\{e^{-rt}S_{t}\}\) is a martingale. A similar application of Girsanov's Theorem tells us \[ \begin{align} dS_{t} & =rS_{t}dt+(\mu(Y_{t})-r)S_{t}dt+f(Y_{t})S_{t}dW_{t}^0 \\ & = r S_{t}dt+f(Y_{t})S_{t}dW_{t}^{0,\mathbb{Q}}, \end{align} \] so \(\mathbb{Q}\) is chosen such that \[ W_{t}^{0,\mathbb{Q}}=W_{t}^0+\int _{0}^t \frac{{\mu(Y_{s})-r}}{f(Y_{s})} \, ds \] is a BM under \(\mathbb{Q}\). Different from BS model, in stochastic volatility model we also have the dynamics for \(\{Y_{t}\}\) which we shall care about under the change of measure. Luckily \[ W_{t}^{\perp,\mathbb{Q}}=W_{t}^\perp+ \int _{0}^t\gamma_{s} \, ds \] as a BM under \(\mathbb{Q}\) does not change the drift term in the stock price dynamics. As a result, different \(\{\gamma_{t}\}\) can be chosen to do the change of measure and they result in \(W^{0,\mathbb{Q}}\) and \(W^{\perp,\mathbb{Q}}\) being independent BM under \(\mathbb{Q}\). Thus it is necessary to denote the risk neutral measure as \(\mathbb{Q}^{(\gamma)}\) as it is induced by the choice of \(\{\gamma_{t}\}\).
The Radon-Nikodym Derivative is then \[ \frac{{d\mathbb{Q}^{(\gamma)}}}{d\mathbb{P}}=\exp\left( -\int _{0}^T \frac{{\mu(Y_{s})-r}}{f(Y_{s})} \, dW_{s}^0 - \frac{1}{2}\int _{0}^T \left(\frac{{\mu(Y_{s})-r}}{f(Y_{s})}\right)^2 \, ds - \int _{0}^T\gamma_{s} \, dW_{s}^\perp - \frac{1}{2 }\int _{0}^T \gamma_{s}^2 \, ds \right), \] from Girsanov’s theorem and it remains as an issue whether the condition of Girsanov holds. Here we assume that its always legitimate by force e.g. the stochastic Sharpe ratio \(\frac{{\mu(Y_{t})-r}}{f(Y_{t})}\) is always almost surely bounded bounded (which is not necessarily true in practice).
0.2 Incompleteness
In the BS model the risk-neutral measure \(\mathbb{Q}\) always exists and is unique, implying that the market is complete, however in the Stochastic Volatility Model, the risk-neutral measure is not unique (since any choice of \(\{\gamma_{t}\}\) results in a risk-neutral measure) and so the market is incomplete. Intuitively, this is due to the fact that there is only one stock in the market but there are two independent BM (two sources of randomness) leading to the failure of exact replication. This is equivalent to the second fundamental theorem of asset pricing which states that the market is complete iff the number of independent stocks equal the number of independent sources of randomness.
0.4 Option Pricing
A fixed \(\{\gamma_{t}\}\) induces the risk-neutral measure \(\mathbb{Q}^{(\gamma)}\) and if the payoff function is \(h(S_{[0,T]})\) then the arbitrage-free value of the option at time \(t\) is given by \[ e^{r(T-t)}\mathbb{E}_{{\mathbb{Q}^{(\gamma)}}}[h(S_{[0,T]})|\mathcal{F}_t] \] under the logic of risk-neutral pricing. Obviously the selection of \(\{\gamma_{t}\}\) is not unique so this leads to a family of arbitrage-free prices of the option, not a single value.
0.4.1 Exponential Ornstein-Uhlenbeck Model
Differing out choice of \(f,\alpha, \beta,\Lambda\) leads to different stochastic volatility models. Empirically, volatility is observed to have mean-reverting pattern so it is natural to set up [[ornstein-uhlenbeck-process|Ornstein-Uhlenbeck]] dynamics for \(\{Y_{t}\}\) together with \(\Lambda=0\), \(f(y)=e^y\). This is the Exponential Ornstein-Uhlenbeck Model and under this specific risk-neutral measure \(\mathbb{Q}\) induced by \(\Lambda\) we have \[ \begin{align} dS_{t} & =rS_{t}dt+e^{Y_{t}}S_{t}dW_{t}^{0,\mathbb{Q}} \\ dY_{t} & = \alpha(m-Y_{t})dt+\beta W_{t}^{1,\mathbb{Q}} \end{align} \] where \(W^{0,\mathbb{Q}}\), \(W^{1,\mathbb{Q}}\) have correlation coefficient \(\rho\).
0.4.2 Heston Model
Alternatively, we could use Cox-Ingersoll-Ross (CIR) Model as the dynamics of \(\{Y_{t}\}\) together with \(\Lambda=0\), \(f(y)=\sqrt{ {y} }\). This is the Heston Model and under this specific risk-neutral measure \(\mathbb{Q}\) we have \[ \begin{align} dS_{t} & = rS_{t} dt+\sqrt{ Y_{t} }S_{t}dW_{t}^{0,\mathbb{Q}} \\ dY_{t} & = \alpha(m-Y_{t})dt+\beta \sqrt{ Y_{t} }W_{t}^{1,\mathbb{Q}}, \end{align} \] where \(W^{0,\mathbb{Q}}\), \(W^{1,\mathbb{Q}}\) have correlation coefficient \(\rho\).
The Heston model is popular since under the PDE approach, after log transform, one gets a PDE with constant coefficient, solvable through [[fourier-transform]].