Geske compound options

When a firm has multiple debt tranches maturing at different dates t_1 < t_2 < … < t_n, equity is a call on a call on … on assets. Geske (1977) gives the closed-form value for the two-period case using the bivariate normal CDF.

Two-period formula

With D_1 due at t_1 and D_2 at t_2:

\[ E_0 = A_0 e^{-q t_2}\, \Phi_2(d_1^{(1)}, d_1^{(2)}; \rho) - D_2 e^{-r t_2}\, \Phi_2(d_2^{(1)}, d_2^{(2)}; \rho) - D_1 e^{-r t_1}\, \Phi(d_2^{(1)}), \]

with \Phi_2(\cdot,\cdot;\rho) the bivariate-normal CDF, \rho = \sqrt{t_1/t_2}, and A^* the asset value at t_1 that makes the residual call exactly worth D_1.

API

from merton import Firm
from merton.extensions import GeskeModel

firm = Firm(equity=100, debt_short=20, debt_long=30, equity_vol=0.30,
            rf=0.04, horizon=3.0)
result = GeskeModel(t_short=1.0).fit(firm)
print(result.summary())

The horizon firm.horizon is interpreted as the long-tranche maturity t_long; t_short is passed to the model constructor.

When to use it

  • Firms with separately-dated debt tranches (revolver due at year 1, term loan at year 5).

  • Comparing roll-over risk: the inner-strike A^* tells you the asset value at which the firm can no longer refinance the short tranche.

n > 2 tranches require the n-variate normal CDF; we ship the two-period case in v0.4 and document n-period as future work (v1.x).