# Risk-neutral option pricing on a quasi-reverting underlying asset

Consider $$(S_t)_{t\geq0}$$, on the probability space $$(\Omega,\mathcal{F},\mathbb{Q})$$, which evolves according to $$\begin{equation} \frac{\mathop{dS_t}}{S_t}=\mu\mathop{dt}+\sigma_{t,S_t}\mathop{dW_t}, \end{equation}$$ such that $$\begin{equation} \sigma_{t,S_t}=\sqrt{v_0+\omega_t\left(\ln\Lambda_t-\ln S_t\right)^2}, \end{equation}$$ and $$\omega,\Lambda:t\rightarrow\mathbb{R}_{>0}$$. In the Black-Scholes model, the option payoff may be replicated by a self-financing portfolio of stock and riskless bonds and must at any time have the same value as the portfolio. So, under $$\mathbb{Q}$$ we require that $$\mu=r$$. In our case, should a similar requirement be placed on $$\Lambda_t$$ to guarantee no arbitrage? Referring to the Figure below, as $$S_t$$ diverges from $$\Lambda_t$$, volatility increases (which may revert $$S_t$$ to $$\Lambda_t$$, but not necessarily). Note that $$v_0=0.02$$, $$\mu=0$$, $$\omega_t=\omega=50$$, and $$\Delta t=0.01$$ in the Euler approximation. Also, the expression for $$\sigma_{t,S_t}$$ was just my simplest guess to produce this behaviour and if there is anything in the literature which describes a similar but more realistic path for $$S_t$$ I would be grateful to know.

• No, change of measure does not affect the volatility, even if it depends on the share price. Jun 5 at 5:01
• Thanks for your reply. So, we are still under $\mathbb{Q}$ for all $\Lambda_t$ and $\omega_t$. I was also interested in whether the distribution of $\mathop{dS_t}/S_t$ could be determined from the distribution of $\omega_t$ and $\Lambda_t$. This might be impossible analytically but expectation and variance would be enough. Jun 5 at 5:37