Module slv 

Stochastic Local Volatility (SLV): Heston-style stochastic variance multiplied by a leverage function calibrated so the model reprices the market’s vanilla surface exactly (in the limit):

dS/S = (r - q) dt + L(S, t) sqrt(v) dW1
dv   = kappa (theta - v) dt + xi sqrt(v) dW2,   d<W1, W2> = rho dt

By Gyongy’s theorem the model matches the market when L^2(S, t) = sigma_LV^2(S, t) / E[v_t | S_t = S], with sigma_LV the Dupire local vol. The conditional expectation is estimated by the standard particle / binning method: simulate forward, bin paths by spot at each step, average the variance per bin, and use the resulting leverage for the next step. Both the calibration and the pricing simulations step the variance with the Andersen QE transition and the spot through the Broadie-Kaya decomposition (see [SlvStepper]), so the particle distribution carries no variance-truncation bias.

SLV interpolates between the two pure models: xi -> 0 recovers pure local vol (E[v|S] -> v0, L -> sigma_LV / sqrt(v0)), while a flat market surface makes L collapse the stochastic vol back to flat vanilla prices — but forward smiles and path-dependent payoffs keep genuine stochastic-vol dynamics. Vanilla repricing is the calibration test, forward-smile richness the reason to use it.

Structs

ConditionalVariance

The binned conditional-variance curves E[v_t | S_t], one per time step — together with the Dupire surface they define the leverage.

Slv

A calibrated SLV model (borrows the Dupire local vol it was built on).

SlvConfig

Simulation / calibration controls.

Functions

calibrate

Calibrate the leverage function to local_vol out to horizon years: forward simulation with per-step binning of E[v | S]. Deterministic for a given config.