Jump-diffusion (Merton 1976 / Zhou 1997)

Merton’s 1976 option-pricing extension adds a compound-Poisson jump to the asset process; Zhou (1997) repurposed it for structural credit modelling. The resulting default probability has a clean Poisson-weighted series form that subsumes the standard Merton model.

Setup

Under the risk-neutral measure:

\[dV/V = (r - q - \lambda\kappa)\,dt + \sigma\,dW + (Y - 1)\,dN,\]

with

  • N a Poisson process of intensity \lambda (mean jumps per year),

  • Y \sim \text{lognormal}(\mu_J, \sigma_J^2) the jump multiplier,

  • \kappa = e^{\mu_J + \sigma_J^2/2} - 1 = E[Y] - 1 chosen so the total drift matches r - q.

Conditional Gaussian

Conditional on exactly n jumps in [0, T], the asset log-return is Gaussian:

\[\log(V_T / V_0)\,\bigl|\,N_T = n \sim \mathcal{N}(m_n,\,s_n^2),\]

with

\[m_n = (r - q - \lambda\kappa - \tfrac{1}{2}\sigma^2)T + n\mu_J,\qquad s_n^2 = \sigma^2 T + n\sigma_J^2.\]

PD formula

Unconditioning on N_T:

\[\mathrm{PD} = \sum_{n=0}^\infty \frac{e^{-\lambda T}(\lambda T)^n}{n!}\,\Phi(-d_2^{(n)}),\]

where

\[d_2^{(n)} = \frac{\log(V_0/D) + m_n}{s_n}.\]

The implementation truncates the series at \lambda T + 6\sqrt{\lambda T} jumps, which captures \geq 1 - 10^{-12} of the Poisson mass.

Example

from merton import Firm
from merton.extensions import JumpDiffusionModel, jump_diffusion_pd

# Direct PD: Merton baseline + 0.5 expected jumps/year, downward bias
pd = jump_diffusion_pd(
    asset_value=100, asset_vol=0.30, debt=60, rf=0.04, T=1.0,
    jump_intensity=0.5, jump_mean=-0.05, jump_std=0.15,
)

# Model usage
firm = Firm(equity=50, debt_short=30, debt_long=50, equity_vol=0.5, horizon=1.0)
result = JumpDiffusionModel(
    jump_intensity=1.0, jump_mean=-0.10, jump_std=0.20,
).fit(firm)
print(result.summary())

When to prefer jump-diffusion

  • Crisis-period calibration. Crash risk that Merton can’t capture (LTCM 1998, 2008) shows up as a fat negative tail in equity returns. Adding negative-mean jumps fits observed credit spreads at short horizons much better.

  • Emerging-market sovereign analytics. Currency-crisis “jumps” are the dominant source of default risk; modelling them explicitly matters.

  • Long-horizon LGD assumptions. Jumps push the expected loss higher even when the average drift is unchanged.

The trade-off is parameter estimation: \lambda, \mu_J, and \sigma_J require equity-options data or judgement-based priors. For firms without listed options, fixing them to industry medians is a reasonable default.