Bermudan Option Valuation by Backward Continuation Values

Summary

The document introduces Bermudan options, which can be exercised only on specified dates, and corrects the idea that their value is simply the sum of several discounted European option values. Instead, valuation proceeds backward through the exercise dates. At each date, the option’s value is determined by comparing the immediate exercise payoff with the value of continuing to hold the option. The continuation value is an expectation of the next-date option value under the pricing measure, conditional on the current state.

The answer sketches a Fourier cosine expansion method for approximating that conditional expectation. It represents the next-date value through cosine coefficients over a bounded state interval and combines them with the conditional characteristic function of the modeled state. The state is often log asset price, but the document presents this as a modeling choice. The equations indicate a numerical approach rather than a complete implementation: interval selection, discretization, model specification, and error controls are not explained, and the answer refers readers elsewhere for further details.

Key ideas

• A Bermudan option can be exercised only on predefined dates.
• Its value is computed backward across exercise dates rather than by summing European option values.
• At each exercise date, immediate payoff is compared with the continuation value.
• A Fourier cosine expansion can approximate the conditional continuation value using characteristic functions.

Tags

# How can one value a Bermudan option?


# How can one value a Bermudan option?












A Bermudan 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?

## Answer by user16651 (score 2, accepted)

https://quant.stackexchange.com/a/18569

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.

and \begin{align} V_k(t_m)=\frac{2}{b-a}\int_{a}^{b}v(y,t_m) cos\left(k\pi\frac{y-a}{b-a}\right)dy \end{align}

for more details, you can see this article.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.