# The derivation of vega/gamma relationship

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.

• 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? Commented Mar 14, 2023 at 6:55
• @Frido: I added the screenshot of the cited derivation.
– Hans
Commented Mar 14, 2023 at 7:11
• 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? Commented Mar 14, 2023 at 7:16
• In other words, you're right, for just showing vega-gamma relationship in BS world you don't need Feynman-Kac Commented Mar 14, 2023 at 7:26
• @Frido: See my answer below.
– Hans
Commented May 26, 2023 at 21:29

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.

• 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.
– Hans
Commented May 28, 2023 at 6:14
• @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 Commented May 28, 2023 at 7:34
• OK, you echoed my point about operator semigroup and fair enough. +1 and accepted.
– Hans
Commented May 28, 2023 at 7:44

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.