Functional Derivatives in the Bergomi Pricing Equation

Summary

The document asks for a rigorous interpretation of the functional derivative terms in Bergomi’s pricing equation for a European claim. The price is viewed as depending on spot and a family of stochastic processes indexed by future times, which raises questions about derivatives with respect to the volatility processes and mixed spot-process derivatives.

It presents the pricing equation and identifies the mathematical issue, but supplies no derivation or answer. The useful contribution is therefore framing a conceptual question about functional calculus in stochastic volatility models, rather than teaching a method or providing evidence. Readers will need additional material on functional derivatives and the model’s assumptions to resolve it; the document does not specify a function space, regularity conditions, or a precise construction of the derivatives.

Key ideas

• The Bergomi pricing equation includes second-order derivatives with respect to volatility processes.
• The price is described as depending on an indexed family of stochastic processes, not just terminal values.
• The document raises but does not resolve how these functional derivatives can be defined rigorously.
• It provides no assumptions or function-space framework for interpreting the notation.

Tags

# Derivation of Bergomi model


# Derivation of Bergomi model












In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) : $$ \frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^T du'\nu(t,u,u',\xi)\frac{d^2P}{\delta\xi^u\delta\xi^{u'}}+\int_t^T du \mu(t,u, \xi)S\frac{d^2P}{dS\delta\xi^u}=rP, $$ where $P$ is the time-t no-arbitrage price of a European contingent claim of maturity $T>t$ and $\xi$ is an infinite set of stochastic processes, each stochastic process $\xi^u$ being indexed by a time $u>0$. Intuitively, the derivation of the above makes sense. However, I am looking for a more rigorous understanding of this formula.

Specifically, $P$ is dependent on an entire set of stochastic processes $\{\xi^u-\text{stochastic process up to time }u|u\geq0\}$. It is as if $P$ depends on an infinite number of stochastic processes (not just their terminal values). In this context, I am trying to understand what derivatives such as:

- $\frac{d^2P}{\delta\xi^u\delta\xi^{u'}}$

- $\frac{d^2P}{dS\delta\xi^u}$

even mean. Is there a way to rigorously define mathematically such notations/constructs?

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