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Deriving the Vega–Gamma Relation with Pricing PDEs

Article Quant Q&A · Author: Hans
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Summary

The document explains why a pricing partial differential equation can be useful for deriving the relationship between option vega and gamma even when volatility is constant. In the Black–Scholes setting, volatility is a parameter of the differential operator, and the option price can be represented as an evolution operator applied to the payoff. Differentiating with respect to that parameter connects sensitivities without requiring a fresh direct calculation from the Black–Scholes formula.

The discussion notes that the result applies to payoffs independent of volatility, not only standard calls and puts, and that the PDE approach can extend to other higher-order derivatives. It also describes an analogous stochastic-volatility result connecting sensitivity to correlation with vanna. These are conceptual derivations, not empirical findings; the stated extensions depend on the model setup and the payoff assumptions.

Key ideas

  • In Black–Scholes, volatility is a parameter in the pricing operator, so differentiating the pricing solution can relate vega and gamma.
  • The PDE method can cover volatility-independent payoffs beyond standard vanilla options.
  • The same operator approach can derive other higher-order sensitivities.
  • In stochastic-volatility models, correlation sensitivity is related to vanna.

Tags

OptionsDerivatives pricingVolatility
Full text
# The derivation of vega/gamma relationship


# 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.

## Answer by Frido (score 5, accepted)

https://quant.stackexchange.com/a/75682

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.

## Answer by Hans (score 2)

https://quant.stackexchange.com/a/75676

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.

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.