# How can one value a Bermuda option?

A Bermuda option allows early exercise at predefined dates, e.g. at maturity equal to $t_1$, $t_2$, $t_3$,...;

hence , would its value be the sum of 3 discounted European options with 1-year maturity?

• The easiest way for Bermudan option pricing is the Binomial tree method. eprints.maths.ox.ac.uk/789/1/Thom.pdf – user16651 Jun 29 '15 at 11:37
• @BehrouzMaleki This is a general numerical method similar to MC Simulation, but I was looking for an analytic approach. – emcor Jun 29 '15 at 11:41
• No, since it can only be exercised once. – ocstl Jun 29 '15 at 11:44
• @emcor , you can use Fourier-Cosine Method for Pricing Bermudan Options.link – user16651 Jun 29 '15 at 14:48

we should first define some notation before discussing pricing. Let $t_0$ be initial time and $t_1, . . . , t_M$ be pre-specified exercise dates with $t_0 < t_1 < · · · < t_M = T$ , the final maturity, and $Δt = t_m−t_{m−1}$. Without a loss of generality it is assumed exercise dates are equidistant. To price a Bermudan option, its value is split into two parts, the continuation value and the immediate exercise payoff. At time $t_{m−1}$, the value of $v(x, t_{m−1})$ consists of the continuation value and the early exercise payoff value.An approximated continuation value, assuming the option is not exercised in the current period, is (look article) \begin{align} c(x,t_{m-1})=\sum_{k=0}^{N-1} Re\left[\phi\left(\frac{k\pi}{b-a};y|x\right)exp\left(-ik\pi\frac{a}{b-a}\right)\right]V_k(t_m) \end{align} where
$x:$ be the modeled quantity at t, often the log asset price.
$y:$ be the modeled quantity at T, often the log asset price.
$f(y|x):$ be the probability density function under the pricing measure.