1
$\begingroup$

I’m trying to follow Gatheral’s Volatility Surface Ch. 1, i.e. the text (pg. 5 and 6) linked to in this question, with further text discussed in this question. I can’t figure out how to arrive at the initial basic equation giving the change in the value of the portfolio, so if anyone can kindly please help wrap my head around it.

We have, quoting from the text, the following two SDE’s which we assume the stock price $S_t$ and variance $v_t$ take:

$$dS_t=\mu_tS_tdt+\sqrt{v_t}S_tdZ_1$$ $$dv_t=\alpha(S_t,v_t,t)d_t+\eta\beta(S_t,v_t,t)\sqrt{v_t}dZ_2$$

with:

$$d\langle Z_1,Z_2\rangle_t=\rho dt$$

where $\mu_t$ is the (deterministic) instantaneous drift of stock price returns, $\eta$ the volatility of volatility, and $\rho$ the correlation between random stock price returns and changes in $v_t$. $dZ_1$ and $dZ_2$ are Wiener processes.

We form a portfolio:

$$\Pi=V-\Delta S-\Delta_1V_1$$

of $V=V(S,v,t)$ being the (value of the) option being priced, a quantity $\Delta$ of the underlying stock $S$ and a quantity $\Delta_1$ of an asset $V_1$ whose value depends on volatility (which I assume follows the same valuation notation of $V_1=V_1(S,v,t)$).

The change in the value of this portfolio in a time $dt$ is:

$$d\Pi=\{\frac{\partial V}{\partial t}+\frac{1}{2}vS^2\frac{\partial^2V}{\partial S^2}+\rho\eta v\beta S \frac{\partial^2V}{\partial v \partial S}+\frac{1}{2}\eta^2 v\beta^2\frac{\partial^2V}{\partial v^2}\}dt -\Delta_1 \{\frac{\partial V_1}{\partial t}+\frac{1}{2}vS^2\frac{\partial^2V_1}{\partial S^2}+\rho\eta v\beta S \frac{\partial^2V_1}{\partial v \partial S}+\frac{1}{2}\eta^2 v\beta^2\frac{\partial^2V_1}{\partial v^2}\}dt \\ +\{\frac{\partial V}{\partial S}-\Delta_1\frac{\partial V_1}{\partial S}-\Delta\}dS \\ +\{\frac{\partial V}{\partial v}-\Delta_1\frac{\partial V_1}{\partial v}\}dv$$

where, for clarity, we have eliminated the explicit dependence on $t$ of the state variables $S_t$ and $v_t$, and the dependence of $\alpha$ and $\beta$ on the state variables.

Now, with my (limited and insufficient) understanding of calculus, taking the partial derivative of $V(S,v,t)$ with regard to $t$ yields only simply $\frac{\partial V}{\partial t}$ and conversely for $V_1$ like the equation above shows for $dS$ and $dv$ terms. I’m simply puzzled how we get the higher order terms with regard to $dt$. Intuitively, I feel we are looking at $V(dS_t, dv_t, t)$ but I’m not sure how to work through the math in that case. (Perhaps I should try applying the chain rule?)

Only a brief pointer should suffice, and I’ll try working through it. Also, could you please suggest some calculus reading as a precursor to this book?

Many thanks.

$\endgroup$
  • 1
    $\begingroup$ Having extra $dt$ terms is a consequence of the stochastic nature of the $S_t$ and $v_t$ processes. You should spend a little time studying the basics of stochastic calculus and in particular Ito's Lemma if you want to get the most out of more advanced books like Gatheral's. The relevant chapters in Hull would be a good place to start. $\endgroup$ – Ivan Mar 4 '18 at 22:03
  • $\begingroup$ Figured it out - it is an application of Itô's Lemma, final slide. Also ran into a very useful resource, I will study it in more detail. Thank you. $\endgroup$ – h.alex Mar 4 '18 at 23:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.