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American Options on Assets with Dividends Near Expiry

​
option
Call
Put
time to expiry (years)
T - t
0.001
strike price
X
5
risk-free rate
r
0.03
dividend yield
δ
0.01
volatility
σ
0.3
This Demonstration shows an explicit expression for the optimal exercise boundary of American options on assets with dividends. The function
S
f
(t)
of the optimal exercise boundary is only valid near expiry. The graph shows the
S
f
(t)
curve (green line) near expiry for a three-year maturity option. For the option's holder, it is not optimal to exercise the option while the asset price
S
t
moves within the colored area.
The optimal exercise boundary is singular at expiry, and its behavior is a classical problem of mathematical finance. This type of singularity is common in similar free boundary problems in physics, for example, the movement of the boundary between melting ice and water.
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