Path-Dependent PDEs for Volterra Models and the Weak Error of Discretization

Bonesini, Jacquier, and Pannier establish two connected results for stochastic Volterra models: first, that conditional expectations of smooth functionals of the solution path — the option pricing functions in a rough volatility model — are the unique classical solutions of associated path-dependent partial differential equations (PPDEs) in the sense of Dupire's functional Itô calculus; and second, that the weak error of Euler-type discretization schemes for smooth functionals of the Riemann–Liouville fractional Brownian motion W^H with H ∈ (0, 1/2) converges to zero at a rate of order Δt^H, where Δt is the time step, providing a rigorous quantitative bound on the discretization error introduced by any numerical scheme for rough volatility option pricing. The path-dependent PDE is the central theoretical object: in a classical Markovian setting, the conditional expectation V(t, x) = E_t[Φ(S_T) | S_t = x] satisfies the standard backward Kolmogorov PDE ∂_t V + (1/2)σ²(x)∂²_x V = 0, an equation in the finite-dimensional state (t, x). In a non-Markovian rough volatility setting, the state at time t is the entire path (X_s : s ≤ t) of the underlying process, and the conditional expectation V(t, X_{·∧t}) is a function of this infinite-dimensional path state, satisfying the PPDE ∂_t V + ∂^h_X V · σ(X_t) + (1/2)∂²_{XX} V · σ²(X_t) = 0, where ∂^h_X V is the horizontal derivative (rate of change under horizontal path extension) and ∂_{XX} V is the vertical derivative (rate of change under path perturbation at the endpoint) in the sense of Dupire's functional Itô calculus. The existence and uniqueness of classical solutions to this PPDE under the rough Volterra dynamics is the first main result, providing a rigorous PDE description of rough volatility option pricing. The weak error bound is the paper's second main contribution and its most practically significant result for numerical implementation: the H-dependent rate O(Δt^H) characterizes precisely how much of the option pricing error in any Euler scheme for rough vol is attributable to the coarse time-stepping, as opposed to the variance-reduction methods or payoff regularity, and it shows that the discretization error worsens as H decreases (rougher paths), with the extreme case H → 0 producing an error rate approaching O(1) — no convergence at all. For H = 0.1, the practical implication is that the weak convergence rate is Δt^{0.1}, meaning that reducing the time step by a factor of 2 reduces the weak error by only 2^{0.1} ≈ 1.07 — a dramatic slowdown relative to the Brownian case (H = 1/2) where the same step reduction cuts error by 2^{0.5} ≈ 1.41.

The Volterra price process satisfies X_t = X_0 + ∫_0^t b(X_s) ds + ∫_0^t K(t,s) σ(X_s) dW_s where K(t,s) = (t−s)^{H−1/2} / Γ(H+1/2) is the Riemann–Liouville kernel with H ∈ (0,1/2), W a standard Brownian motion, and b, σ are smooth Lipschitz functions. The PPDE satisfied by V(t, x_{·}) = E[Φ(X_T) | (X_s)_{s≤t} = (x_s)_{s≤t}] is (D_t V)(t, x_{·}) + (1/2)(D²_x V)(t, x_{·}) σ²(x_t) = 0 for t < T with terminal condition V(T, x_{·}) = Φ(x_{·}), where D_t is the horizontal derivative and D_x, D²_x are the first and second vertical derivatives in Dupire's functional calculus. The horizontal derivative D_t V(t, x_{·}) = lim_{h→0} [V(t+h, x^{(h)}_{·}) − V(t, x_{·})] / h where x^{(h)} is the horizontal extension of x — the path that equals x on [0,t] and is constant at x_t on [t, t+h] — captures the rate of change of V along flat extensions, while the vertical derivative D_x V encodes sensitivity to perturbations of x at the terminal point t. The Euler discretization of X on a grid with n steps gives X^n_t = X_0 + Σ_{k: t_k < t} K(t, t_k)(b(X^n_{t_k}) Δt + σ(X^n_{t_k}) ΔW_{t_k}), and the weak error is |E[Φ(X_T)] − E[Φ(X^n_T)]| = O(n^{−H}).

The PPDE characterization of rough volatility option prices is significant for several reasons beyond its immediate computational applications. Dupire's functional Itô calculus — developed to extend classical Itô calculus from functions of the current state to functionals of the entire path — provides the natural language for describing the sensitivities of path-dependent claims, and the PPDE result shows that even in the apparently non-classical setting of rough volatility, option prices admit a characterization in terms of this calculus. The vertical derivative D_x V is precisely the functional delta of the option with respect to the current value of the underlying — the analogue of the Black–Scholes delta for path-dependent models — and the horizontal derivative D_t V encodes the theta, while the second vertical derivative D²_x V is the functional gamma. The PPDE therefore provides a complete and rigorous set of Greeks for rough volatility options without requiring Monte Carlo differentiation. The weak convergence rate O(Δt^H) is derived by representing the discretization error as an integral of the PPDE solution against the error in the discrete increments, which involves the correlation structure of the discretized fractional increments and their deviation from the continuous-time covariance. The H-dependent rate arises because the fractional increments ΔW^H_{t_{k+1}} = W^H_{t_{k+1}} − W^H_{t_k} are not independent — unlike Brownian increments — and their inter-step dependence introduces error terms of order Δt^H in the first variation of the discrete sum. Comparing this to the classical Brownian case H = 1/2, where the Euler error for smooth payoffs is O(Δt) (twice the convergence rate), shows that the roughness parameter H is the direct determinant of how much computation must be invested to achieve a given pricing accuracy. Higher-order schemes can improve on the basic Euler rate: the Milstein scheme, which includes the first-order correction from the Itô–Taylor expansion, improves the strong convergence rate from O(Δt^{H+1/2}) to O(Δt^{min(2H+1/2, 1)}) but does not improve the weak rate beyond what the PPDE representation already implies. Predictor-corrector schemes and multi-level Monte Carlo (MLMC) with variance reduction can achieve near-optimal computation-to-error trade-offs: MLMC under rough Heston with H = 0.1 achieves a root-mean-square error of ε with computational cost O(ε^{−2/H}) = O(ε^{−20}) in the worst case, significantly more expensive than the classical O(ε^{−2}) for Brownian diffusions and reflecting the fundamental computational challenge of rough volatility.

The H-dependent weak error bound has a direct operational consequence for the desk's simulation-based pricing of rough volatility models: for H = 0.1 and a target pricing accuracy of 0.5 vol points (approximately 0.1% relative error in a 50% vol option), the required number of time steps per year of simulation is n = (ε^{-1/H}) where ε is the target weak error, which at H = 0.1 and ε = 0.001 requires n = (1000)^{10} = 10^{30} steps — manifestly infeasible with standard Euler discretization. In practice, the desk uses multi-level Monte Carlo with antithetic variates and the Hybrid scheme of Bennedsen–Lunde–Pakkanen for rough fractional Brownian motion, which achieves the optimal rate of the PPDE bound while requiring only a logarithmic factor more work per unit accuracy compared to the Brownian case; the practical simulation cost at H = 0.1 and one-basis-point accuracy is approximately 100 to 1000 times higher than for classical stochastic volatility. The PPDE structure provides a rigorous basis for functional sensitivity computation: the functional delta D_x V and gamma D²_x V can be computed by differentiation of the PPDE solution rather than by Monte Carlo bump-and-reprice, which is subject to Monte Carlo error and requires careful bump-size calibration to balance numerical differentiation error against variance. For rough volatility models where the PPDE can be solved numerically (using, e.g., the signature kernel PPDE solver of Pannier–Salvi), the functional sensitivities are available analytically as output of the PDE solve, providing a deterministic and variance-free greeks computation pipeline.

The weak convergence rate O(Δt^H) is established for smooth payoffs Φ satisfying a polynomial growth condition on the path space, which excludes digital and barrier options whose payoffs are indicator functions of path events and therefore discontinuous in the path topology. For discontinuous payoffs the weak convergence theory requires additional smoothing arguments or alternative discretization schemes (such as the Kushner–Dupuis Markov chain approximation), and the H-dependent rate may not hold without modification. The desk uses a Gaussian kernel smoothing approximation to discontinuous payoffs when applying the PPDE-based greeks framework to barrier options, accepting a bias of order σ_kernel (the kernel bandwidth) in exchange for differentiability. The PPDE existence-and-uniqueness result requires regularity conditions on the coefficients b and σ of the Volterra SDE — specifically global Lipschitz continuity and linear growth — that are not satisfied by the log-normal (Black-Scholes) specification σ(x) = σ₀ x (multiplicative volatility) or by the CIR (Heston) specification σ(v) = ν √v where v → 0 is a boundary singularity. Extensions to these practically important cases require either a regularization argument that perturbs the coefficients to be globally Lipschitz and then takes a limit, or a viscosity solution framework for the PPDE that accommodates non-classical solution concepts at the boundary; both approaches are active research areas not covered in the current paper.