A Rough-Path PDE Representation for Local Stochastic Volatility

Bank, Bayer, Friz, and Pelizzari develop a rough-PDE representation of option pricing functions in the class of local stochastic volatility (LSV) models, proving that the pricing functional V(t, s; 𝐖_{[0,t]}) — where 𝐖 is the geometric rough-path lift of the driving noise process — satisfies a linear PDE driven by the rough path 𝐖 in the sense of Lyons' rough-path integration theory, with the PDE coefficients being measurable functions of the current state (t, s) and the rough path 𝐖 up to time t. The Feynman–Kac correspondence is extended from the classical Itô setting — where the pricing PDE is a deterministic PDE driven by the current state — to the rough-path setting, where the pricing PDE is a random PDE driven by 𝐖 as a rough-path signal, with the solution given by the conditional expectation of the payoff under the rough-path-driven stochastic flow. The local stochastic volatility model combines a local volatility function σ_loc(t, S_t) with a stochastic volatility factor ξ_t driven by rough or diffusive noise, so that the instantaneous variance is σ_t² = σ_loc(t, S_t)² · ξ_t. The local volatility function σ_loc is calibrated to vanilla option prices via the Dupire formula, ensuring exact smile calibration at each maturity, while ξ_t introduces the stochastic dynamics required to explain the dynamics of the smile and the correlation between spot and volatility that a pure local volatility model misses. The rough-PDE framework handles the case where ξ_t is driven by rough noise (ξ_t = exp(ν W^H_t − ν²t^{2H}/2) in the rough Bergomi convention) by treating the rough path 𝐖 = (W^H, (W^H)^{⊗2}) — the rough path lift including the iterated integral — as a fixed input signal and writing the pricing equation as a PDE conditional on 𝐖. The extension of Feynman–Kac to the rough-path setting requires careful treatment of the composition of rough-path integration with the conditional expectation: the classical proof of Feynman–Kac uses Itô's formula to show that the conditional expectation solves the backward Kolmogorov PDE, and in the rough-path setting this step requires the rough-path chain rule (the Lyons–Gubinelli change of variables formula) applied to the composition of the pricing functional with the rough-path flow, producing the rough-PDE with drivers given by the controlled rough-path structure of the solution.

The rough path is 𝐖 = (W, W^{⊗2}) ∈ C^{0,p-var}([0,T]; R^d ⊕ R^{d⊗2}) for p ∈ (2, 3), satisfying the Chen's identity W^{⊗2}_{s,t} = W^{⊗2}_{s,u} + W^{⊗2}_{u,t} + W_{s,u} ⊗ W_{u,t} for all s ≤ u ≤ t. The LSV model is dS_t = S_t σ_loc(t, S_t) ξ^{1/2}_t dW^1_t where W^1 is the first component of W and ξ_t = f(𝐖_{[0,t]}) is a functional of the rough path up to time t (e.g. ξ_t = exp(ν W^H_t) in rough Bergomi). The pricing function V(t, s; 𝐖_{[0,t]}) = E^Q[Φ(S_T) | S_t = s, 𝐖_{[0,t]}] is the conditional expectation given the rough path up to t. The rough PDE satisfied by V is dV = L_{σ,ξ} V dt + Γ^k V d𝐖^k_t, interpreted in the rough-path sense, where L_{σ,ξ} = (1/2)σ_loc²(t,s)ξ_t s² ∂²_{ss} is the instantaneous second-order operator, 𝐖^k are the rough-path components, and Γ^k V encodes the interaction of the pricing functional with the k-th rough-path driver via the rough Feynman–Kac formula.

The rough-PDE framework for LSV generalizes two classical results simultaneously: the Feynman–Kac theorem (which connects PDE solutions to stochastic expectations) and the rough-path chain rule (which extends Itô's formula to non-semimartingale drivers). The combination requires establishing that the composition of a smooth function with a rough-path flow satisfies a controlled rough-path equation, a result that is a consequence of the Lyons–Gubinelli flow theorem for rough differential equations and that provides the mathematical foundation for interpreting the pricing equation as a well-posed rough PDE rather than merely as a stochastic conditional expectation. The practical significance of the rough-PDE representation for LSV is twofold. First, it provides a rigorous framework for analyzing the calibration of LSV models with rough volatility: the local volatility function σ_loc must be chosen so that the rough-PDE solution reproduces the market smile, a condition analogous to the Dupire formula in classical local volatility but now involving the rough-path driver. The rough Dupire formula — the analogue of the classical ∂_T C = (1/2)σ_loc² ∂²_K C for the rough case — can be derived from the rough-PDE and involves the rough-path covariance structure of 𝐖. Second, the rough-PDE provides a PDE-based alternative to Monte Carlo simulation for computing option prices and Greeks in the rough LSV model, where the rough-path driver 𝐖 is treated as a fixed realization and the PDE is solved conditional on 𝐖 using classical numerical methods. The comparison with the classical LSV literature — where the PDE is a deterministic Fokker–Planck equation for the conditional density of S given the current stochastic volatility state — reveals that the rough-PDE framework and the classical PDE framework are complementary rather than competing: the classical framework works conditional on the Markovian vol state, while the rough framework works conditional on the rough-path realization, and the two are connected by integrating the rough-PDE solution against the distribution of the rough-path noise to recover the unconditional pricing function.

For the desk's skew-sensitive structured products — which include barriers, cliquets, and range-accrual structures where the smile shape at each observation date matters — the rough LSV model with the rough-PDE pricing framework provides a rigorous theoretical basis for the two-model approach (local vol for smile calibration + stochastic vol for dynamics) that is standard practice in structured products desks. The rough-PDE framework validates the Dupire local volatility extraction from the current smile as an exact calibration of the rough-PDE solution, while the rough volatility driver ξ_t provides the smile dynamics consistent with the empirically observed rough vol scaling. Computing the rough-PDE Green's function — the transition density of S under the rough LSV dynamics conditional on a fixed rough-path realization 𝐖 — via the Fokker–Planck equation associated with the rough PDE provides a deterministic method for computing the conditional distribution of S at expiry, which can be integrated against the payoff function to produce option prices without Monte Carlo simulation. The computational cost of this approach is one classical PDE solve per rough-path scenario, and the total cost is n_paths × cost-per-PDE-solve, where n_paths is the number of rough-path scenarios used for averaging; for n_paths = 1000 and a 2D state space (S, ξ), the total cost is comparable to 1000 finite-difference PDE solves, which is feasible for daily calibration but too slow for real-time recalibration at tick frequency.

The rough-path framework requires the driving noise 𝐖 to be a geometric rough path — satisfying the algebraic Chen identity and having finite p-variation for some p < 3 — which is established for fractional Brownian motion with H > 1/4 but fails for H ≤ 1/4. Since the empirical Hurst index estimate H ≈ 0.1 is below this threshold, the theoretical rough-PDE framework does not apply directly to the RFSV model with H = 0.1, and extension to H ≤ 1/4 requires the theory of branched rough paths or regularity structures, which are more complex and less developed numerically. In practice, the desk uses H = 0.1 in the RFSV model as an effective roughness parameter without claiming mathematical rigour of the rough-path framework at that value of H. The regularity conditions on the coefficients σ_loc and f (the function mapping the rough path to the stochastic volatility factor ξ) required for the rough-PDE existence and uniqueness result are more stringent than the conditions for classical stochastic volatility models: in particular, f must be a C^{3,α} function of the rough path in the controlled rough-path sense, a condition that excludes the standard log-normal specification ξ_t = exp(ν W^H_t) unless the exponential is approximated by a polynomial or the rough path is regularized by a mollification. The mismatch between the mathematical framework (requiring smooth f) and the practical specification (log-normal f) is a standing tension in the rough LSV literature that has not been fully resolved.