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In Lorenzo Bergomi, Stochastic Volatility Modeling, Chapter 5 Appendix A.1, Equation (5.64), as shown below, seems to assume $\hat\sigma$ to be constant. If that is the case, why do we bother to invoke the Feynman-Kac formula? We can simply use the Black-Scholes formula.

enter image description here

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  • $\begingroup$ Hi Hans, would like to try to help you but don't have Bergomi's book with me (and don't have an electronic copy). Can you give more background / post a screenshot perhaps? $\endgroup$
    – Frido
    Mar 14, 2023 at 6:55
  • $\begingroup$ @Frido: I added the screenshot of the cited derivation. $\endgroup$
    – Hans
    Mar 14, 2023 at 7:11
  • $\begingroup$ I think Bergomi uses Feynman Kac to show that dollar gamma and dollar delta are martingales. But as you say, he is only showing this under Black-Scholes so could have just used BS PDE. Before I quit QFSE (and joined again) I had answered ((under user34971) a similar questions but for more general processes. In that case using expectations = Feynman Kac might have more added value: quant.stackexchange.com/a/45474/65759 Does this answer your question? $\endgroup$
    – Frido
    Mar 14, 2023 at 7:16
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    $\begingroup$ In other words, you're right, for just showing vega-gamma relationship in BS world you don't need Feynman-Kac $\endgroup$
    – Frido
    Mar 14, 2023 at 7:26
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    $\begingroup$ @Frido: See my answer below. $\endgroup$
    – Hans
    May 26, 2023 at 21:29

2 Answers 2

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Just want to add the observation that the pricing PDE solution can be formally written as $$ C(\tau) = e^{\tau \mathcal H} C(0) \quad (*) $$ where $\tau$ is time to maturity and $\mathcal H$ is a differential operator. For example, in the BS world with zero interest rate it is $$ \mathcal H = \tfrac12 \sigma^2 S^2 \frac{\partial^2}{\partial S^2} $$ Thus $U(\tau) = e^{\tau \mathcal H}$ is an 'evolution operator'.

In the BS case $\sigma$ is not a variable but a parameter. So you can differentiate both sides of equation (*) to very quickly obtain the vega gamma relation by noting that the operator $U(\tau)$ depends on the parameter $\sigma$. You could reinsert dividends and rates to also obtain sensitivities to $r$ and $q$ in the same manner.

In stochastic volatility models you can similarly show that the sensitivity of the option price to the correlation parameter is the stochastic volatility model vanna.

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  • $\begingroup$ Very nice. The operator expression which is not just formal but can be well defined via spectral operator integral i.e. Dunford integral. It relies on the $C(\tau)$ being a one-parameter analytic semigroup and the spectral theorem of a parabolic equation though, which are involved. It would be great to give more technical details and references so as to make the answer more substantial. $\endgroup$
    – Hans
    May 28, 2023 at 6:14
  • $\begingroup$ @Hans Agree that it could be more technical. If I could I would, but this is the theory of semigroups of linear operators which is somewhat beyond my knowledge at the moment. I 'just' apply it. However here is the canonical reference: link.springer.com/book/10.1007/978-1-4612-5561-1 $\endgroup$
    – Frido
    May 28, 2023 at 7:34
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    $\begingroup$ OK, you echoed my point about operator semigroup and fair enough. +1 and accepted. $\endgroup$
    – Hans
    May 28, 2023 at 7:44
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Even though it is true that the volatility is constant in this setting, the relationship is valid for all terminal condition or pay-off function -- beyond the typical $(\pm(S-K))_+$ -- so long as the pay-off function is independent of the volatility. We can certainly write out the integral expressions of vega and gamma (of arbitrary pay-off functions) and find their relationship. But it seems simpler dealing with the PDE directly. Moreover, this methodology can be used to find other high order partial derivatives.

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