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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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    $\begingroup$ The $\xi$'s are forward start instantaneous variance swaps with maturity $T$ and starting times $u \in [t,T]$. Hence the terms are covariances of all these forward starting variance swaps among themselves and between the forward start varswaps and the spot process. Clearly simplification / approximations are needed as such a model is intractable. $\endgroup$
    – user34971
    Jul 26, 2021 at 8:16
  • $\begingroup$ @FridoRolloos Yes, I had understood that. However, the mathematical definition of the derivatives still eludes me. How can we take the partial derivative of P w.r.t. a stochastic process? And why do we use $\delta$ instead of $d$ or $\partial$ ? $\endgroup$
    – fwd_T
    Jul 26, 2021 at 12:35
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    $\begingroup$ It's not the derivative wrt a stochastic process. I believe it's the derivative wrt to the expectation of future instantaneous volatility: $\xi^u = E_t [\sigma_u^2]$. In other words Bergomi assumes that the option price depends on the expectation of all future instantaneous volatilities. As for the $\delta$, no clue why he uses that notation. One small thing that bothered me a bit in the book is the use of $d$ instead of $\partial$ even though from a LaTex perspective I understand it's faster to write $d$. $\endgroup$
    – user34971
    Jul 26, 2021 at 12:49
  • $\begingroup$ @FridoRolloos Yes, I think you are correct. This is the correct way to interpret it. $\endgroup$
    – fwd_T
    Jul 28, 2021 at 8:48

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