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1) An option $V$ under the Black-Scholes model is perfectly hedged when it is delta hedged continuously with the underlying $S$. When the hedging time is discrete, the delta $\Delta$ needs to take into account the gamma $\Gamma$. One way to do this is to Taylor expand the hedge portfolio $V(t,S)-\Delta S$ over a finite time interval $\delta$ and minimize the variance var$[\delta(V(t,S)-\Delta S)]$. We can obtain, $$\Delta = V_S+cSV_{SS}\delta t+O((\delta t)^2)$$ for some constant $c$.

For the discrete time hedging, is it better to introduce another option $H$ with weight $w$ on $S$ to form the portfolio $\Pi:=V-wH-\Delta S$ to hedge $V$, as $H$ has its gamma to hedge that of $V$? I Taylor-expand the $\Pi$ over $\delta t$ and have obtained a result with a complicated expression for $(\Delta,w)$ which I am not entirely sure of. Are there any references on this topic of "gamma hedging"?


2) Thanks to Quantuple's pointer to Chapter 1 of Lorenzo Bergomi's book Stochastic Volatility Modeling, it is clear now that the gamma comes into the hedge for a purpose that is very distinct from what I described above. It is first and foremost a hedge against the stochasticity of the difference between the realized variance and the implied variance, rather then coming into the delta hedging as a first-order correction for the finiteness of the hedging time interval. Here is the follow-up question.

Suppose the option $V$ has $2$ underlyings $S_1$ and $S_2$ which are correlated with each other with correlation $\rho$ and we have two hedging options $H_1$ with only underlying $S_1$ and $H_2$ with only underlying $S_2$. The volatility of $S_1$ is $\sigma_1$ and that of $S_2$ is $\sigma_1$. I would like to determine the impact of $\rho$ on hedging coefficients $w_1$ of $H_1$ and $w_2$ of $H_2$. Is the following approach correct?

We delta hedge all the options. We assume the volatilities are stochastic. Let $\langle\cdot\rangle:=\mathbf E[\cdot]$. \begin{align} \text{p&l} = &\frac12S_1^2\frac{\partial^2(V-w_1H_1)}{\partial S_1^2}(\sigma_1^2-\sigma_{1,\text{imp}}^2)+\frac12S_1^2\frac{\partial^2(V-w_2H_2)}{\partial S_2^2}(\sigma_2^2-\sigma_{2,\text{imp}}^2) \nonumber\\ &+S_1S_2\frac{\partial^2V}{\partial S_1\partial S_2}(\rho\sigma_1\sigma_2-\langle\rho\sigma_1\sigma_2\rangle) \nonumber\\ &+\text{functional of }(V-w_1H_1-w_2H_2)\text{ the volatilities, realized and implied}, \end{align} where "imp" on the subscript denotes that the variable is implied from the option price. Ignore the last term for now. Denote $\alpha_1:=S_1^2\frac{\partial^2}{\partial S_1^2}(V-w_1H_1)$ and similarly for $\alpha_2,\, v_{1,2}:=\rho\sigma_1\sigma_2-\langle\rho\sigma_1\sigma_2\rangle,\, v_1:=\sigma_1^2-\sigma_{1,\text{imp}}^2$ and similarly for $v_2$. It is reasonable to assume $\langle v_1\rangle=\langle v_2\rangle=\langle v_{12}\rangle=0$. We have \begin{equation} \text{var[p&l]} = \frac14\alpha_1^2 \langle v_1^2\rangle+\alpha_1\alpha_{1,2}\langle v_1v_{1,2}\rangle+\frac14\alpha_2^2 \langle v_2^2\rangle+\alpha_2\alpha_{1,2}\langle v_2v_{1,2}\rangle+ \alpha_{1,2}^2\langle v_{1,2}^2\rangle. \end{equation} Minimizing the variance above can be viewed geometrically as the vector $\alpha_{1,2}v_{1,2}$ projecting on the hyperplane spanned by the vectors $\{\frac12\alpha_1v_1,\,\frac12\alpha_2v_2\}$. So $\alpha_{1,\min} = -2\frac{\langle v_1v_{1,2}\rangle}{\langle v_1^2\rangle}\alpha_{1,2}$ and $\alpha_{2,\min} = -2\frac{\langle v_2v_{1,2}\rangle}{\langle v_2^2\rangle}\alpha_{1,2}$ minimizes var[p&l]. \begin{equation} w_{1,\min} = \frac{\Gamma_1+2\frac{\langle v_1v_{1,2}\rangle}{\langle v_1^2\rangle}\frac{S_2}{S_1}\Gamma_{1,2}}{\Gamma_{H_1}} \end{equation} and exchanging $1$ and $2$ on the subscript gives the expression for $w_{2,\min}$.

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    $\begingroup$ Hi Hans. I would recommend reading the first chapter of Lorenzo Bergomi's book "Stochastic Volatility Modeling" if you haven't already. Indeed Gamma needs to be hedged with another option. You are then exposed to implied vol realisations rather than realised vol itself. $\endgroup$ – Quantuple Nov 4 '18 at 12:01
  • $\begingroup$ Wouldn't it be easier to open a chat room here? $\endgroup$ – Quantuple Nov 5 '18 at 7:52
  • $\begingroup$ @Quantuple: I have added a followup question together with my solution. I would like to hear your opinion on it. Thank you. $\endgroup$ – Hans Nov 7 '18 at 9:36
  • $\begingroup$ Hi @Hans: I agree with your P\&L equation as long as $\sigma_i := d\langle S_i \rangle_t/S_i(t)^2$ (realised volatility), $\sigma_{i,imp}^2$ is the vol for the asset $i$ you are using in your model to price both $V$ and $H_i$ and you are looking at a portfolio formed of delta-hedged $V$ plus $w_i$ times delta-hedged $H_i$. For the remaining part, I guess it's a matter of whether it is indeed reasonable to assume that realised vol and implied vol are the same on average, which does not seem reasonable at all to me. $\endgroup$ – Quantuple Nov 12 '18 at 15:48
  • $\begingroup$ Since this is a hard quantity to model, I'd say it's more frequent for practitioners to just pick $w_i$ such that the Gamma terms vanish. In which case you are left with a Vega, Vanna, Volga P\&L (+ the cross terms) as explained in Bergomi's book. $\endgroup$ – Quantuple Nov 12 '18 at 15:49

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