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). Figure

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.

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    $\begingroup$ No, change of measure does not affect the volatility, even if it depends on the share price. $\endgroup$ Jun 5 at 5:01
  • $\begingroup$ 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. $\endgroup$
    – UNOwen
    Jun 5 at 5:37


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