Stochastic models and simulation

Simulation provides the future rates, spot prices and discount factors needed for stochastic payoffs and exposure calculations. A model specifies their dynamics, while the date grid, path count and random seed control the numerical experiment.

QuantSupport supports stored path sets generated from a SimulationConfiguration and model-backed observations supplied by a MarketModel to scripting and exposure services. Stored paths are useful for inspection and numerical payoff valuation. The model-backed services also retain the path-construction dependencies used for automatic differentiation.

Materialized numerical paths

SimulationBuilder constructs a date grid strictly after the market reference date through the configured horizon. Model inputs include index, path count, seed, frequency and day counter. It converts required curves to f64 and generates Hull–White, LGM or lognormal Brownian paths using standard-normal draws from seeded StdRng. This route does not inherit the Sobol/antithetic construction of LgmMarketModel.

GeneratedMonteCarloSimulation stores the resulting values in DualFwd containers with generation inputs held fixed. It exposes no simulation pillars, so subsequent differentiation covers only the operations performed on the stored values. The MC equity pricer can differentiate its subsequent discounting without providing path-generation delta or vega.

For lognormal Brownian motion, the interval step has the exponential form

\[ S_{t+\Delta}=S_t\exp\left[ (r-q-\tfrac12\sigma^2)\Delta+\sigma\sqrt{\Delta}Z\right]. \]

Here \(\sigma\) is sampled at the interval endpoint. The step does not integrate time-dependent variance across sigma boundaries; aligning or refining the grid matters when the volatility schedule changes. Calibrated term-volatility stripping matches accumulated variance at its selected expiries, but does not change that path-stepping rule. Spot comes from the reference-date fixing. The builder’s rate is obtained from the terminal discount factor; a general term structure does not thereby become a fully time-dependent equity carry model.

Hull–White representation

The one-factor Gaussian short-rate model is conventionally written

\[ dr_t=(\theta(t)-a r_t)\,dt+\sigma(t)\,dW_t. \]

Its affine bond representation uses \(P(t,T\mid r)=A(t,T)e^{-B(t,T)r}\), with \(B(t,T)=(1-e^{-a(T-t)})/a\). Model calibration converts caplet or swaption prices into a piecewise-constant sigma schedule and retains instrument-level quality records. Selected-source domain and calibration-Jacobian limitations are described in Volatility markets.

The low-level materialized Hull–White path generator is Euler-discretized:

\[ x_{i+1}=x_i-a x_i\Delta_i+ \sigma(t_{i+1})\sqrt{\Delta_i}Z_i. \]

The generator updates a variance estimate and adds a curve-derived shift to the state. Its Euler discretization differs from the exact OU transitions used in callable-bond valuation. Refine the grid to assess discretization bias and increase path count to assess sampling precision.

Several low-level Hull–White formulas divide by mean reversion and do not implement a zero-alpha limit. Do not infer zero-alpha support from the formal continuous mathematical limit. LGM, described below, explicitly handles zero lambda.

LGM rates and numeraire

LgmRateModel expresses a rate factor through loading \(H\), volatility loading \(\alpha\), and accumulated variance \(\zeta\):

\[ H(t)=\frac{1-e^{-\lambda t}}{\lambda},\quad H(t)=t\ \text{when }\lambda=0,\qquad \alpha(t)=\sigma(t)e^{\lambda t},\quad \zeta(t)=\int_0^t\alpha(u)^2\,du. \]

The zero-lambda limit applies to direct LGM dynamics. Its calibrated construction delegates to the low-level Hull–White calibrator, whose zero-alpha singularities remain; zero-lambda calibration is not implied.

For its Gaussian state \(z\), the implemented discount factor and reference numeraire are

\[ P(t,T;z)=\frac{P(0,T)}{P(0,t)} \exp\left[-(H(T)-H(t))z -\tfrac12(H(T)^2-H(t)^2)\zeta(t)\right], \] \[ N(t,z)=\frac{\exp(H(t)z+\tfrac12H(t)^2\zeta(t))}{P(0,t)}. \]

The reference numeraire defines the measure under which the domestic factor is driftless. Foreign rate factors receive measure-change drift terms involving domestic-rate and FX correlations.

LgmMarketModel combines rate factors with supported FX/equity components. FX component spot is domestic currency per foreign unit. Curve-driver mappings let another curve reuse a factor without claiming that the two initial curves are identical. The state vector orders rates, FX log spots and equity log spots; correlation input must match that ordering.

Path construction uses Owen-scrambled Sobol draws, antithetic pairing and Cholesky correlation. Consecutive paths share opposite-sign normal draws. Even counts preserve complete pairs; odd counts, including one path, are also supported. Reproducing a run requires matching the seed, grid, model and indices. Path count controls sampling, while volatility and model inputs determine whether the dynamics are stochastic.

Model-backed observations

MarketModel<T> configures evaluation dates and typed simulation requests, then returns fallible path scenarios. Responses contain requested discounts, forwards, FX, spots, path-dependent observations and numeraires. Its generate_path and resolve_request return Result, so failures retain their market/model categories instead of becoming silently absent paths.

Scripting and XVA operate on these model responses in DualFwd. They can propagate through registered model/curve inputs while rewinding temporary path nodes. The resulting sensitivities remain subject to fixed correlations, partial calibration Jacobians and payoff smoothing. Stored numerical simulations retain the more limited risk dependencies described above.