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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?

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closed as unclear what you're asking by Quantuple, LocalVolatility, amdopt, JejeBelfort, David Addison Jul 11 '17 at 18:32

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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.

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  • $\begingroup$ But why do we use the ATM vol to price quanto options, even if the option is deep out of the money? I got the compect that the implied vol is an avergage of instantaneous vols and that local voltility is an average of stochastic volatilities, but I do not get why this results in always taking ATM vol to price a quanto option. Could you please help clarifying this point? $\endgroup$ – John Jul 10 '17 at 22:25
  • $\begingroup$ You cannot connect the dots because this is not "the" only good way to do it. Have you read the paper given in the accepted answer to your question. In the conclusion it clear states that: "We have also found that taking the equity volatility at-the-money is often not the best choice to price a Quanto or Compo option using the lognormal formula." $\endgroup$ – Quantuple Jul 11 '17 at 6:56
  • $\begingroup$ Also this is not what you are asking here so please keep to one question per post. Referring to other questions in a Russian dolls fashion (I can count at least 3 here) is not suitable given the site's format. Especially since you already got a reason for not using strike vol in your related question: it produces arbitrage opportunities. $\endgroup$ – Quantuple Jul 11 '17 at 7:07
  • $\begingroup$ My doubt is why the ATM vol is "a" good way of doing this, and not why it is is "the" good way. That is not clear in any of the answers. $\endgroup$ – John Jul 11 '17 at 10:40
  • $\begingroup$ This is explained in the paper which was referred to you. Consider the degenerate situation of no equity-fx correlation. Then you are basically pricing a vanilla option. But if you use the ATM vol it won't allow you to reach the market price of deep OTM options which is basically what you are saying as well. So ATM good because no arbitrage, but not good because in degenerate case you don't fall back on vanilla prices. The opposite for strike vol. The "good" ways to do it would require something more than the lognormal model. $\endgroup$ – Quantuple Jul 11 '17 at 11:24

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