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The Bjerksund-Stensland model simply explained

The Bjerksund-Stensland model is an extension of the Black-Scholes-Merton model designed to tackle American-style options on stocks that pay dividends.

At its core, this model begins with the Black-Scholes framework, a tool for calculating the theoretical price of European options. This framework takes into account factors like the current stock price, strike price, time to expiration, volatility, and risk-free interest rate.

Unlike the Black-Scholes model, which assumes a continuous dividend stream, the Bjerksund-Stensland model accommodates discrete dividend payments. It considers not only when but also how much is expected to be paid in dividends over the option's lifespan.

American options differ from their European counterparts in that they can be exercised at any point before they reach expiration. The model handles this by assessing whether it's in the option holder's best interest to exercise early.

The model establishes a pivotal stock price level known as the "early exercise boundary" or "optimal exercise boundary." If the stock price drops below this critical point, it makes sense for the option holder to exercise the option early. Conversely, if the stock price exceeds this level, holding onto the option is typically the better course of action.

Now, let's illustrate this process with a simplified numerical example:

- Current stock price (S0): $100
- Strike price (K): $105
- Time to expiration (T): 1 year
- Volatility (σ): 20%
- Risk-free interest rate (r): 5%
- Dividend yield (q): 3%
- Dividend payment date: At t = 0.5 years, a dividend of $2 will be paid.

1: Calculate the Critical Stock Price (Early Exercise Boundary):
- By applying the Bjerksund-Stensland model's formula, we determine the critical stock price, which approximates $105.715.

2: Determine the American Call Option Value at Different Stock Prices:

- Scenario 1: If S = $100 (currently below the critical price):
- In this situation, as S is beneath the critical price, we utilize the Bjerksund-Stensland model to account for potential early exercise.

- Scenario 2: If S = $110 (above the critical price):
- Here, the Black-Scholes formula comes into play because S surpasses the critical price. Early exercise tends to be less favorable in such cases.

The actions we take at each stage depend on whether the stock price falls above or below the critical value, leveraging either the Bjerksund-Stensland or Black-Scholes model. This approach carefully considers the early exercise boundary and the impact of dividends when valuing American call options.

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