30-OCT-2023
We study finite-maturity Israeli put options in the Black-Scholes framework when the writer faces Knightian uncertainty about the risk-neutral drift. Modelling this ambiguity as a bounded drift distortion controlled adversarially by the writer, the valuation problem becomes a three-player zero-sum game whose value satisfies a Hamilton-Jacobi-Isaacs (HJI) equation coupled with a double obstacle. The bang-bang structure of the optimal drift control is established in closed form, and the resulting nonlinear PDE is solved numerically via an explicit upwind finite-difference scheme. We find that drift ambiguity strictly lowers the option value, contracts the early-exercise boundary, and generates a positive ambiguity discount $V^0 – V^\kappa$ that peaks near the exercise boundary and vanishes throughout the cancellation interval centered at $s = K$. The cancellation region is a bounded interval $[s_l(\tau), s_r(\tau)]$ around $K$ that is absent near expiry and grows with time-to-maturity, shrinking and eventually disappearing as $\kappa$ increases.