Linear Gaussian Markov (LGM)

The multi-currency simulation engine (src/models/lgm/) is built from LGM rate components plus lognormal FX and equity components, all driven under the domestic risk-neutral measure.

`LgmRateModel`

pub struct LgmRateModel<'a, T: Scalar> {
    lambda: T,                    // mean reversion (1/years); 0 = none
    sigma_schedule: Vec<(f64, T)>,// piecewise-constant σ(t)
    discount_curve: &'a dyn InterestRatesTermStructure<T>,
}
LgmRateModel::new(lambda, sigma, &curve)
LgmRateModel::new_piecewise(lambda, schedule, &curve)?        // schedule non-empty, increasing
LgmRateModel::calibrated(lambda, &curve, &calibration_config, &store, &quotes, Level::Mid)?

calibrated runs the Hull-White caplet/swaption bootstrap (Hull-White) and transfers the sigma schedule.

State variable \(z_t\) with \(z_0 = 0\):

Method	Formula
`H(t)`	\(\frac{1-e^{-\lambda t}}{\lambda}\) (\(= t\) when \(\lambda\approx 0\))
`H_dot(t)`	\(e^{-\lambda t}\)
`alpha(t)`	\(\sigma(t)e^{\lambda t}\)
`zeta(t)`	\(\int_0^t\alpha(s)^2ds\)
`P_discount(t, T, z)`	\(\frac{P(0,T)}{P(0,t)}\exp\!\bigl(-(H(T)-H(t))z - \tfrac12(H(T)^2-H(t)^2)\zeta(t)\bigr)\)
`numeraire(t, z)`	\(\exp\!\bigl(H(t)z + \tfrac12H(t)^2\zeta(t)\bigr)/P(0,t)\)
`instantaneous_forward_rate(t, T, z)`	\(f(0,T) + H’(T)H(T)\zeta(t) + H’(T)z\)
`short_rate(t, z)`	\(f(t,t\mid z)\)
`self_drift(t)`	0 (domestic factor is driftless)
`gamma_under_domestic_measure(t, &dom, fx_vol, rho_zx, rho_zz)`	\(\rho{zz}\alpha_i\alpha_d H_d - \alpha_i^2 H_i - \rho{zx}\sigma_X\alpha_i\)
`evolve_domestic_factor_euler(t, z, dt, dw)`	\(z + \alpha(t)\,dW\)
`evolve_foreign_factor_under_domestic_measure_euler(..)`	\(z + \gamma\,dt + \alpha(t)\,dW\)

FX and equity components

LgmFxModel::new(&domestic_rates, &foreign_rates, fx_vol, spot_0, rho_zx_dom_fx)  // spot = domestic per foreign
LgmEquityModel::new(&domestic_rates, equity_vol, spot_0, dividend_yield: Option<f64>, rho_zs_dom)

`LgmMarketModel`

let mut model = LgmMarketModel::new(Currency::USD, MarketIndex::SOFR, reference_date, DayCounter::Actual365)
    .with_n_paths(2000)                  // must be even (antithetic)
    .with_seed(42)
    .with_correlation_matrix(corr);      // ordered as the state vector below
model.add_curve_model(MarketIndex::SOFR, sofr_lgm);
model.add_curve_model(MarketIndex::ICP, icp_lgm);
model.add_fx_model(Currency::CLP, clp_fx);
model.add_equity_model("AAPL".into(), aapl);
model.set_curve_driver(MarketIndex::TermSOFR3m, MarketIndex::SOFR);  // index simulated off another factor
model.set_evaluation_dates(dates);
model.set_requests(requests);

State vector \(Y(t) = [z_d, z_{f_1},\dots,z_{f_F}, \log X_1,\dots,\log X_F, \log S_1,\dots,\log S_E]\) with dynamics under the domestic measure

\[ \begin{aligned} dz_d &= \alpha_d\,dW_d, & dz_{f_i} &= \gamma_i\,dt + \alpha_i\,dW_{f_i},\ d\log X_i &= (r_d - r_i + \rho_{d,X_i}\alpha_d H_d\sigma_{X_i} - \tfrac12\sigma_{X_i}^2)dt + \sigma_{X_i}dW_{X_i},\ d\log S_j &= (r_d - q_j + \rho_{d,S_j}\alpha_d H_d\sigma_{S_j} - \tfrac12\sigma_{S_j}^2)dt + \sigma_{S_j}dW_{S_j}. \end{aligned} \]

Path generation: Owen-scrambled Sobol draws → antithetic pairs → Cholesky-correlated increments → Euler steps between consecutive evaluation dates → resolve_request answers each SimulationRequest (discount factor, forward rate, FX, spot, numeraire) from the state. Discount factors within a path are P_discount, forward rates come from instantaneous_forward_rate/P_discount ratios, and the numeraire is used to deflate cashflows in exposure and pricing engines.

JSON configuration

ModelConfiguration::Lgm { lambda, volatility } in a SimulationConfiguration, or LgmModelConfig inside XvaEngineConfig (XVA Overview):

{
  "market_index": "SOFR",
  "lambda": 0.05,
  "volatility": {
    "Calibrated": {
      "source": { "Surface": { "market_index": "SOFR" } },
      "quote_ids": [
        "CapletFloorlet_USD_SOFR_3M_1Y_Absolute_0.045_Straddle_Black"
      ],
      "strike": "Atm",
      "alpha": 0.05
    }
  }
}

Provide either sigma (constant) or volatility; driver lets an index reuse another index’s factor.

Example

cargo run -p pfe builds a USD SOFR swap and an EUR/USD FX forward, bootstraps SOFR and ESTR, loads an LGM configuration, simulates, and prints per-trade NPV, the exposure profile through PfeAggregator (quantiles by date) — see Exposure Simulation.

Keyboard shortcuts

The QuantSupport Book