# Gamma vs. Volatility Risk

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

- 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 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.

• 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 '14 at 16:02

This is a not a theoritcal/academic answer relating the two by an equation. But from a practicioners stand point. The relationship between vol and gamma depends on the strategy your putting on.

For example. In a Short Straddle/Strangle/Butterfly/Iron Condor. Your short theta and the risks your taking are gamma risk, even though your delta neutral, and implied vol risk. If Implied vol rises, your contracts go up in value and since your short the contracts, your position takes a negative hit. Not to mention if realized volatility does rise, and the stock moves toward your break-even zones, either by trending up or trending down, you also lose money.

Which raises another question is this always the case, can there be a position constructed such that one is long theta (same thing as saying short gamma) but long volatility (long vega)?? The answer is yes. In a long calendar spread your technically short theta, the theta on the further out option that your selling is greater than theta on the shorter maturity contract which is being bought, your net short premium from the spread, your long theta, short gamma, but yet your long vega/volatility.

So the answer to your question - it depends. Just my 2 cents

@Paul, I think you are correct. Your expression relates Gamma and Volatility Risk, as volatility risk is the risk of mis-estimating the future realised volatility.

My only comment relates to your last bullet point: I have always viewed this formula outside of hedging strategies that target Gamma, as we only have the replication strategy. If you are replicating a derivative with near zero Gamma, then you are dealing with a boring near-linear instrument in the first place.

I prefer to view the formula as 'volatility P&L generated for a given path'. If the path brings you far-away-from-the-money, then Gamma is zero and there is no P&L due to volatility.