merton.extensions.longstaff_schwartz

Longstaff-Schwartz (1995) two-factor structural model.

Generalises Merton by letting the risk-free rate evolve stochastically under a Vasicek process. Asset value V_t follows GBM and the short rate r_t follows mean-reverting Ornstein-Uhlenbeck:

\[\begin{split}dV/V &= (r_t - q)\,dt + \sigma_V\,dW^V, \\ dr &= \kappa(\theta - r)\,dt + \eta\,dW^r, \\ \mathrm{Corr}(dW^V, dW^r) &= \rho.\end{split}\]

Default occurs at the first time V_t hits a constant barrier K in [0, T]. Allowing stochastic rates widens credit spreads in regimes where \mathrm{Corr}(\Delta r, \Delta V) < 0 — when rates spike, asset values fall, pushing the firm closer to its barrier.

Implementation

Closed-form PDs only exist for special parameter combinations. The practical path is Monte Carlo simulation of correlated (V_t, r_t) paths with first-passage tracking. We expose longstaff_schwartz_pd_mc() for the MC estimator and LongstaffSchwartzModel for a calibrated single-firm fit.

References

Longstaff, F. A., Schwartz, E. S. (1995). A Simple Approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finance 50 (3), 789-819.

Classes

VasicekParams	Vasicek short-rate parametrisation: dr = κ(θ - r) dt + η dW.
LongstaffSchwartzModel	Calibrated single-firm Longstaff-Schwartz model.

Functions

longstaff_schwartz_pd_mc(→ tuple[float, float])	Monte Carlo first-passage PD over [0, T].
longstaff_schwartz_pd_analytic(→ float)	Closed-form first-passage PD under the zero-correlation special

Module Contents

class merton.extensions.longstaff_schwartz.VasicekParams

Vasicek short-rate parametrisation: dr = κ(θ - r) dt + η dW.

kappa: float

theta: float

eta: float

r0: float

merton.extensions.longstaff_schwartz.longstaff_schwartz_pd_mc(*, asset_value: float, asset_vol: float, barrier: float, T: float, vasicek: VasicekParams, correlation: float = 0.0, dividend_yield: float = 0.0, n_paths: int = 50000, n_steps: int = 252, seed: int | None = None) → tuple[float, float]

Monte Carlo first-passage PD over [0, T].

Returns:

The MC point estimate and its standard error (≈ \sqrt{p(1-p)/N}).

Return type:

pd, stderr

merton.extensions.longstaff_schwartz.longstaff_schwartz_pd_analytic(*, asset_value: float, asset_vol: float, barrier: float, T: float, vasicek: VasicekParams, dividend_yield: float = 0.0) → float

Closed-form first-passage PD under the zero-correlation special case: rates and asset value are independent, so the Vasicek rate process only matters for discounting (not for the default trigger). This reduces to the standard Black-Cox formula with a deterministic drift equal to the unconditional mean of r_t.

class merton.extensions.longstaff_schwartz.LongstaffSchwartzModel(*, vasicek: VasicekParams, correlation: float = 0.0, mc_paths: int = 20000, mc_steps: int = 252, mc_seed: int | None = None, tol: float = 1e-08, max_iter: int = 200)

Bases: merton.extensions.base.StructuralModel

Calibrated single-firm Longstaff-Schwartz model.

Inputs the user supplies on top of the Firm:

• vasicek — short-rate parametrisation.

• correlation — between asset and rate shocks.

• mc_paths, mc_steps, mc_seed — Monte Carlo controls.

method = 'longstaff_schwartz'

vasicek

correlation

mc_paths = 20000

mc_steps = 252

mc_seed = None

tol = 1e-08

max_iter = 200

fit(firm: merton.core.firm.Firm) → merton.extensions.base.StructuralResult

Return a StructuralResult for firm.