Bermudan Option Pricing

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This page demonstrates pricing of Bermudan Options using either Cox-Ross-Rubinstein Binomial Trees or Least-Squares Monte Carlo.

The Least-Squares Monte Carlo offers an extended model with bias correction to improve the accuracy, as presented in Bias-Corrected Least Squares Monte Carlo for Utility Based Optimal Stochastic Control Problems. All methods generate numerical solutions, hence the accuracy of the price depends on the number of time steps used and simulation paths. Hover over the input box for further explanations of the inputs.

The underlying asset is assumed to follow a geometric Brownian motion with process \[ dS_t = (r - \delta) S_t dt + \sigma S_t dW_t, \] where \(r\) is the riskless rate of return, \(\delta\) is a continuously paid dividend, \(\sigma\) is the standard deviation and \(W_t\) is the Brownian motion. At each predetermined exercise date, the holder can exercise the option or continue to hold it. The objective is therefore to maximize the expected payoff of the option, which equals the price and is defined as \[ P(K, S_{t_0}, T) = \sup_{\tau} \mathbb{E}\left[ \phi \, e^{-r (\tau - t_0) } \left (S_\tau - K \right )^+ \Bigm| S_{t_0}\right] \] where \(\tau \in \{ t_0, t_1, t_2, ... T \}\) is the predetermined exercise dates, \( \phi = 1\) for call options and \( \phi = -1 \) for put options, \(K\) is the strike price and \(S\) is the underlying asset price. The superscript '+' represents that the payoff cannot be negative.

 

Pricing method		Binomial Tree LSMC LSMC + Bias
Option type		Put Call
Stock price ($)	\(S_{t_0} = \)		Please enter a value in range (0, 1 000 000] The input cannot be empty
Strike price ($)	\(K = \)		Please enter a value in range (0, 1 000 000] The input cannot be empty
Maturity (years)	\(T = \)		Please enter a value in range (0, 100] The input cannot be empty
Interest rate (decimal)	\(r = \)		Please enter a value in range [-1, 1] The input cannot be empty
Dividend rate (decimal)	\(\delta = \)		Please enter a value in range [-1, 1] The input cannot be empty
Volatility (decimal)	\(\sigma = \)		Please enter a value in range [0, 1] The input cannot be empty
Exercise dates			Please enter an integer in range [1, 1000] The input cannot be empty
Time steps			Please enter an integer in range [10, 1 000 000] The input cannot be empty
Degree of polynomials			Please enter an integer in range [1, 20] The input cannot be empty

Important! This model is for online demonstration purposes only. Calculations can take a long time if the number of time steps and exercise dates are high. The time-out for calculations has been set to 1 hour, after which the webpage will return an error. We suggest you start with low values and then increase them if better accuracy is needed.

Click button to calculate Option price