WOLFRAM|DEMONSTRATIONS PROJECT

Geske-Johnson Method

​
time to expiry (years)
T
3
starting asset price
S
0
40
strike price
X
45
risk-free rate
r
0.07
dividend yield
δ
0
volatility
σ
0.3
Geske-Johnson approximation for the American put = 8.06486
Bermudan put
P
3
= 7.46865
Bermudan put
P
2
= 7.16061
European put
P
1
= 6.19676
This Demonstration locates the asset critical prices, denoted
*
S
, that optimize the payoff function of uniformly time-discretized Bermudan puts with one
(
P
2
)
and with two
(
P
3
) possible early exercise temporal points (lower and upper graph, respectively). While European options can be exercised only at the date of expiration, Bermudan options extend the notion of exercise to a finite number of time points during their life. American options, which can be exercised at any time up to maturity, can be approximated by considering Bermudan options, where the number of early exercise points becomes considerably dense; Geske and Johnson [1] apply Richardson extrapolation to accelerate the convergence to an American option.