Pricing Claims Under Path-Dependent Volatility with Functional Calculus

Summary

The document asks how to value a European payoff when a stock follows risk-neutral geometric Brownian motion whose volatility depends on its history. It contrasts this setting with the standard Markovian case, where volatility depends only on the current time and price and the conditional expectation solves the Black–Scholes PDE with a terminal payoff condition.

For the path-dependent case, it proposes functional Itô calculus and Dupire’s functional Feynman–Kac framework as possible tools. It notes that this framework can accommodate payoffs depending on the whole path, and that conditioning on the observed path differs from conditioning on the filtration. The document does not derive a pricing equation or establish conditions under which one applies. Its value is in identifying the Markov assumption behind the ordinary PDE and pointing toward a framework for extending pricing to path-dependent dynamics; further technical work is needed to resolve the question.

Key ideas

• When volatility depends only on the current price and time, the standard Feynman–Kac result gives a Black–Scholes PDE for the conditional claim value.
• A volatility functional of the past stock path generally makes the price process non-Markovian in the current price alone.
• Functional Itô calculus may provide a route to a path-dependent pricing equation.
• Path-dependent payoffs and conditioning on the observed path require care beyond the ordinary state-based formulation.

Tags

# Is there a Black Scholes PDE for a GBM with path-dependent volatility?


# Is there a Black Scholes PDE for a GBM with path-dependent volatility?












Question: Is there a known path-dependent Black-Scholes PDE?

To be a little more precise, let $S$ be a stock price under a risk-neutral measure such that $S$ satisfies the SDE with path-dependent volatility: $$dS_t = r S_t dt + \sigma(t, S_.) S_t dB_t.$$ Here $\sigma: [0,\infty)\times C([0, \infty), (0,\infty)) \to \mathbb{R}$ is a previsible path functional, i.e. for any $t\geq 0$, $\sigma(t, S_.) \in \mathscr{F}_t$. In other words $\sigma$ depends on the path of $S$ up to time $t$ and is known at time $t$.

When $\sigma(t, S_.)=\sigma(t, S_t)$ only depends on the current state $S_t$, the ordinary Feynman-Kac formula says that $$u(t, s) = \mathbb{E}(e^{-r(T-t)}h(S_T) | S_t=s),$$ if and only if $u$ solves the PDE $$u_t+r s u_s + \frac12 \sigma(t,s)^2 s^2 u_{ss}-ru=0$$ with terminal condition $u(T, s) = h(s)$.

What can we say about this path-dependent case? $$U_t = \mathbb{E}(e^{-r(T-t)} h(S_T) | \mathscr{F}_t)?$$

My only hunch is that Bruno Dupire's functional Ito calculus might be applied to this particular case but it is taking me some time to read through this paper and check everything. Also his Feynman Kac has a slightly different form, 1) it is more general as it lets $h$ also be a functional, in my notation $h(t, S_.)$, and 2) his condition in the conditional expectation is $S_t$ is a little different. It involves conditioning on the path rather than the filtration.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.