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Original Question: What is the link between Gamma and the Volatility Risk?

It leads me to ask:

- What is the Volatility Risk definition and what are the good practices to measure it?

Thinking about that question, all I could figure out liking with it is this:

Consider a market $(S^0, S)$ composed of one non-risky asset $S^0$ and one risky asset $S$. The interest rate in this market is $r$ (supposed constant just to simplify. Also, consider an adapted square integrable process $\sigma$ and suppose that $S$ follows $$S_t= S_0 + \int_0^t r S_u ~du +\int_0^t \sigma_u S_u dW_u \quad , t \geq 0$$ under the risk-neutral probability. Let's suppose a trader evaluate the price $v$ of an option of maturity $T$ by fixing $\sigma_t=\bar{\sigma}(t,S_t)$, a function of local volatility. Then, I know that the coverage error is given by (one can show it by a simple aaplication of Feyman-Kac theorem and Itô's Lemma) \begin{align} \text{Err} = V_T- v(T, S_T) &= \int_0 ^T e^{r(T-t)} (\bar{\sigma}(t,S_t) - \sigma_t) \partial^2_x v(t,S_t)dt \\&=\int_0 ^T e^{r(T-t)} (\bar{\sigma}(t,S_t) - \sigma_t) \Gamma(t,S_t)dt,\end{align} where $V = (V_t)_{0 \leq t \leq T}$ is the replicating portfolio and $v(t,x)$ is the intrinsic value of the option at time $t \in [0,T]$ and spot $S_t=x$.

So, we conclude that if:

  • $\Gamma \geq 0$ (i.e. a convex price): an over-estimation of $\bar{\sigma}(t,S_t)$ secures a gain, and an under-estimation of $\bar{\sigma}(t,S_t)$ secures a loss.
  • $\Gamma\leq 0$ (i.e. a concave price): an under-estimation of $\bar{\sigma}(t,S_t)$ secures a gain, and an over-estimation of $\bar{\sigma}(t,S_t)$ secures a loss.
  • $\Gamma \approx 0$ (i.e. neutral Gamma hedging): the hedging is weakly sensitive to realized volatility.

Am I going in the right direction to answer that?

I would appreciate any advice. Thanks in advance.

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1 Answer

I think what you are missing is simply the Vega-Gamma relation in the Black-Scholes model. Namely: $$ Vega = \frac{\partial v}{\partial \sigma} = \sigma(T-t)S^2 \frac{\partial^2 v}{\partial S^2} = \sigma \tau S^2 \Gamma $$ Plugging this into your coverage error, you get its expression in terms of the Vega which is the most natural measurement of your exposure to implied volatility.

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That's perfect for B&S model. But how to relate Vega and gamma a in more general context like local and stochastic vol. models ? –  Paul Mar 30 at 16:02
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