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.