A Four-Factor Markovian Path-Dependent Volatility Model as a Pricing Engine

Gazzani and Guyon develop the operational pricing and calibration infrastructure for the four-factor Markovian path-dependent volatility (PDV) model — the low-dimensional approximation to Guyon–Lekeufack's non-Markovian PDV specification in which each power-law kernel is replaced by a sum of two exponentials, reducing the infinite-dimensional path history to a four-dimensional exponentially-weighted state vector — and demonstrate that the resulting model achieves competitive joint SPX/VIX smile calibration using neural-network-based approximation of the pricing functionals. The core construction is the approximation of each power-law kernel K_j(u) = c_j(1 + u/θ_j)^{−α_j} by a two-exponential sum Σ_{k=1}^2 w^j_k exp(−β^j_k u) with weights and decay rates calibrated to minimize the L² approximation error over the relevant timescale range, yielding four state variables Y^k_t = ∫_0^t e^{−β^j_k(t−s)} r_s^{p_j} ds (p_1 = 1, p_2 = 2) that together encode the trend and activity path features in a four-dimensional Markov process driven by the return increments. The Markovian state Y_t = (Y^1_t, Y^2_t, Y^3_t, Y^4_t) ∈ R^4 satisfies a linear SDE of the form dY^k_t = −β^k Y^k_t dt + c^k dX^k_t where X^k_t depends on r_t (for the trend factors, k=1,2) or r_t² (for the activity factors, k=3,4), making the full state process (S_t, Y_t) a five-dimensional Markov system amenable to PDE methods. The option price V(y, t; K, T) at current state y and calendar time t is approximated by a neural network V_θ(y, t; K, T) trained to satisfy the pricing PDE L V_θ = 0 via a physics-informed neural network (PINN) penalty, and the VIX is computed as the expectation of the variance swap rate over the 30-day forward window, itself approximated by a second neural network VIX_θ(y, T). The paper establishes that the four-factor Markovian approximation achieves implied volatility accuracy within 0.1–0.3 vol points relative to the true non-Markovian PDV model (as estimated by Monte Carlo of the full path-dependent model), and that the neural network pricing functions, once trained offline, provide inference at millisecond timescales per option price, enabling real-time joint SPX/VIX recalibration that would be infeasible with Monte Carlo simulation of the non-Markovian process.

The four Markovian factors are Y^k_t for k = 1, 2, 3, 4, satisfying dY^k_t = −β^k Y^k_t dt + dI^k_t where I^k_t = ∫_0^t e^{−β^k(t−s)} dz^k_s with z^1_s = z^2_s = r_s and z^3_s = z^4_s = r_s²; the decay rates β^k are calibrated to approximate the power-law kernels K_1 and K_2. The PDV instantaneous variance is σ_t² = f(Y_t) where f : R^4 → R_+ is a non-negative function calibrated from the historical PDV regression. The full state (S_t, Y_t) ∈ R^5 satisfies the Markov SDE dS_t = S_t σ(Y_t) dB_t with B a Brownian motion correlated with the return process via ρ. The generator of (S, Y) is L = Σ_k (−β^k y^k) ∂_{y^k} + (1/2)σ²(y) ∂²_{ss} + σ(y) Σ_k ρ_k ∂_{sy^k} + (1/2) Σ_{k,l} C_{kl}(y) ∂_{y^k y^l}. The neural network V_θ(y, t) is trained to minimize ℒ(θ) = ℒ_market + λ_PDE ||LV_θ||²_{L²(D)} where ℒ_market is the weighted smile fitting error and the PINN penalty enforces approximate PDE satisfaction on a training domain D ⊂ R^4 × [0,T_max].

The multi-exponential approximation of power-law kernels — approximating K_j(u) ∝ u^{−α_j} by Σ_k w^j_k e^{−β^j_k u} — has a long history in numerical analysis as the problem of rational approximation of Stieltjes functions via partial fractions, and the optimal two-exponential fit in the L² sense over a logarithmic timescale grid can be computed by the Remez algorithm or by nonlinear least squares with warm starting. The approximation accuracy depends sensitively on the range of timescales over which the fit is required: over the range of one day to one year, a two-exponential sum approximates the power law to within 2–5% relative error, while over a wider range including shorter timescales (one minute to one day) the error grows substantially as the power-law singularity at short lags is incompatible with any finite exponential sum. The desk uses a three-exponential approximation for the activity kernel K_2 to achieve adequate accuracy at sub-daily timescales relevant for intraday signal generation. The physics-informed neural network pricing approach avoids discretizing the four-dimensional PDE on a grid — which would require grid sizes of order M^4 = 10^4 to 10^6 elements for M = 10 to 30 points per dimension — and instead trains a neural function V_θ(y, t) to satisfy the PDE at a set of collocation points drawn from the training domain D. The PDE residual ||LV_θ||² at each collocation point is computed by automatic differentiation of V_θ with respect to its inputs (y, t), so the PINN training requires no discretization of L and no forward simulation beyond the collocation point sampling. The tension between PDE accuracy (requiring more collocation points in D) and smile accuracy (requiring the network to match option prices at market strikes and maturities) is controlled by the hyperparameter λ_PDE and is resolved by training on a weighted objective that favours smile accuracy over PDE accuracy in the region near the initial state y_0 and relaxes PDE accuracy in distant regions of D visited rarely in calibration. The comparison with direct Monte Carlo simulation of the non-Markovian PDV process reveals that the four-factor approximation introduces a systematic vol point error of approximately 0.05–0.15 at one-month maturities and 0.01–0.05 at maturities beyond three months, reflecting the larger role of long-horizon path memory at short maturities where the non-Markovian path integral is most influential. This error profile implies that the four-factor model is most accurate precisely where the option market is deepest and the SPX and VIX smiles most liquid, a practical alignment that makes the approximation error less consequential for real-world calibration than the raw approximation metric suggests.

For the desk's joint SPX/VIX risk management, the four-factor PDV model provides a computationally practical state representation that is directly interpretable in terms of the observable return path: the four factors Y^k_t are weighted moving averages of r_t and r_t² at two timescales each, and their current values can be computed from intraday high-frequency return data within seconds of a new price quote, providing a live model state update that feeds directly into the neural pricing functions for intraday recalibration and real-time greeks computation. The Markovian structure enables a standard Kalman-type filtering framework for updating the model state as new market observations arrive: the state (S_t, Y_t) evolves according to the SDE system, and each new option price observation provides a likelihood function for the parameters (f, β^k, w^k) that can be updated recursively, potentially enabling online calibration that tracks slow parameter drift driven by changes in the PDV relationship between path history and volatility without requiring a complete recalibration from scratch at each update.

The exponential approximation error in the activity kernel K_2 at short timescales (u < 1 day) propagates into systematic misfits of the short-maturity ATM implied volatility: the four-factor model underestimates the high-frequency contribution to near-term variance, producing a model short-maturity vol that is systematically below market at one-week to one-month maturities during high-frequency volatile regimes. This systematic bias is absorbed into the neural network calibration as a constant offset in V_θ for short-maturity options, which degrades the network's generalization performance across different initial state values y_0 and makes the calibrated model unreliable for pricing short-dated options in regimes far from the calibration window. The training distribution for the PINN is a fundamental hyperparameter whose specification requires expert judgment: the collocation domain D must cover all realistic values of (y, t) that will be encountered in live trading, and the weighting of collocation points determines how the PDE accuracy is distributed across D; in practice, the domain is constructed from Monte Carlo simulation of the SDE system under the calibrated PDV dynamics, and rare but risk-relevant states — large negative Y^1_t (negative trend) combined with large Y^3_t and Y^4_t (high activity) — are underrepresented in the simulation-based collocation and may produce inaccurate neural network pricing precisely in the stressed regimes where accurate pricing is most critical for risk management.