Using Itô’s Lemma to Derive a Delta- and Vega-Hedged Portfolio Change
Summary
The document sets up a portfolio containing an option, the underlying stock, and a second asset whose value depends on both the stock and variance. The stock and variance follow correlated stochastic differential equations. The question is why the portfolio’s change includes second-order derivatives with respect to the stock and variance, as well as a mixed derivative, even though ordinary partial differentiation with respect to time produces only the direct time derivative.
The relevant method is Itô’s lemma: for stochastic state variables, their squared increments contribute terms of order dt, and correlated increments create a cross term. Applying the lemma to each derivative security produces the displayed drift and curvature components; holdings in the stock and volatility-sensitive asset are then used to hedge selected exposures. The note is a request for a calculus explanation, not a worked derivation or a discussion of calibration, hedging performance, or model limitations.
Key ideas
- Itô’s lemma adds second-order terms when differentiating a function of stochastic state variables.
- The stock and variance processes generate curvature terms in the portfolio’s change.
- Correlation between the two processes creates a mixed stock-variance derivative term.
- The portfolio combines the option, stock, and volatility-sensitive asset to hedge risk exposures.
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# How to derive the change in portfolio value as given by Gatheral in The Volatility Surface?
# How to derive the change in portfolio value as given by Gatheral in The Volatility Surface?
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.