Implied Volatility as a Dollar-Gamma-Weighted Average

Summary

This note gives a mathematical interpretation of an option’s implied volatility under a stochastic volatility model. It describes the result for a European call as an average of the underlying instantaneous variance over time and across risk-neutral paths. The weights come from discounted dollar gamma, evaluated using the same constant implied volatility being interpreted. In this sense, implied variance reflects which times and price regions contribute most to the option’s sensitivity to variance.

The cited discussion is presented as a detailed treatment of why at-the-money volatility can be a useful reference, and the note says the source proceeds to additional approximations. However, the expression is specific to the stated option-pricing setup and model assumptions; it is not a general recipe for every derivative or market. The note does not provide the derivation or the later approximations, so applying the relationship requires consulting the underlying treatment.

Key ideas

• The note interprets call implied variance as a weighted average of instantaneous variance.
• The weights are discounted dollar gamma computed at the option’s implied volatility.
• The averaging accounts for risk-neutral paths generated by the stochastic volatility model.
• The relationship is presented for a European call in a specified pricing setup, with further approximations left out.

Tags

# Explaining mathematically why to use the ATM vol


# Explaining mathematically why to use the ATM vol












In this question, I got an answer that is much explaining in words what could be explained mathematically. The user who answered referenced the book "The Volatility Surface, by Jim Gatheral's". But this book is too summarized. Could you please indicate another more detailed reference to help me understand the argument given in the answer of this question?

## Answer by Quantuple (score 1)

https://quant.stackexchange.com/a/35068

This is detailed in the chapter 2 of Lorenzo Bergomi's book "Stochastic volatility modeling", section 2.4.1 - Implied volatilities as weighted averages of instantaneous volatilities. Samples of the book, notably chapter 2, are available for download here.

The author shows that

$$ \sigma_{KT}^2 = \frac{\Bbb{E}^\Bbb{Q} \left[ \int_0^T e^{-rt} S_t^2 \frac{dP_{\sigma_{KT}}}{dS^2} \sigma_{t}^2 dt \right]}{\Bbb{E}^\Bbb{Q} \left[ \int_0^T e^{-rt} S_t^2 \frac{dP_{\sigma_{KT}}}{dS^2} dt \right]} $$

where $\sigma_{KT}$ is the implied volatility of a European call option of strike $K$ and maturity $T$ of $P_{\sigma_{KT}}$ priced under the stochastic volatility model $$ dS_t = (r-q) S_t dt + \sigma_t S_t dW_t^\Bbb{Q} \tag{1}$$

$\sigma_{KT}^2$ is thus the average value of $\sigma^2_t$, weighted by the dollar gamma computed with the constant volatility $\sigma_{KT}$ itself, over paths generated by the stochastic volatility model $(1)$.

Bergomi then discusses further approximations.

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.