What Is the Bjerksund-Stensland Model and How Does It Work?
Learn how the Bjerksund-Stensland model estimates American option prices by incorporating early exercise value and dividend considerations.
Options traders and financial analysts rely on mathematical models to determine fair prices for options contracts. While the Black-Scholes model is widely known, it does not account for early exercise of American options. The Bjerksund-Stensland Model addresses this limitation by offering a more accurate pricing method.
Developed in 1993 and later refined, this model efficiently estimates option values while considering dividends and early execution potential. Unlike numerical methods that require extensive computations, it provides a closed-form approximation, making it useful for traders needing quick valuations.
Model Assumptions
The Bjerksund-Stensland Model assumes the underlying asset follows a geometric Brownian motion, meaning its price evolves unpredictably within a probabilistic framework. This assumption aligns with real market behavior.
A constant risk-free interest rate is also assumed, representing the return on a theoretical risk-free investment like U.S. Treasury securities. While real-world interest rates fluctuate, using a fixed rate simplifies calculations without significantly reducing accuracy.
Unlike the Black-Scholes model, which assumes options can be exercised at any time, the Bjerksund-Stensland Model limits exercise to specific points. This better reflects how traders make decisions, particularly when dividends are involved.
Mathematical Components
The model provides an analytical solution for pricing American-style options, avoiding the computational complexity of numerical methods like binomial trees or finite differences. This efficiency is especially valuable in fast-moving markets where traders need immediate pricing information.
A key feature is its approximation of the early exercise boundary—the price level at which exercising the option becomes more beneficial than holding it. Instead of solving a complex free boundary problem directly, the model uses a piecewise exponential function to estimate this threshold, maintaining accuracy while keeping calculations manageable.
It is particularly effective for options with short maturities, where early exercise decisions are more predictable. It also accounts for dividend payments and volatility levels, ensuring a more realistic valuation.
Calculating Early Exercise Value
Deciding whether to exercise an American-style option before expiration depends on the balance between intrinsic and time value. Intrinsic value is the immediate profit from exercising, while time value reflects the potential for future gains. If time value is low or negative, early exercise becomes more attractive.
Market volatility plays a key role. High volatility increases time value, making early exercise less appealing, while low volatility reduces time value, making early exercise more likely. This is especially relevant for deep in-the-money call options, where further price appreciation may not justify waiting.
Interest rates also affect this decision. A higher risk-free rate lowers the present value of the strike price, making early exercise more appealing for call options. For put options, higher rates discourage early exercise, as holding the option allows the investor to sell at a fixed price while earning interest elsewhere.
Considering Dividends
Dividends impact American-style options, particularly call options, since stock prices drop by the dividend amount on the ex-dividend date. This reduction in price can make early exercise advantageous for in-the-money call holders who want to capture the dividend.
The Bjerksund-Stensland Model incorporates discrete dividend payments rather than assuming a continuous yield, making it more accurate for real-world dividend schedules. It adjusts the early exercise boundary to account for expected stock price drops, which is especially useful for stocks with high dividend yields.
Equity Market Relevance
The Bjerksund-Stensland Model is widely used in equity markets, particularly for pricing American-style stock and index options. Its ability to account for early exercise and discrete dividends makes it more effective than traditional models.
Market makers and proprietary trading firms benefit from its computational efficiency, allowing them to quickly adjust pricing strategies as market conditions change. Portfolio managers also use the model to assess fair option values and optimize hedging strategies.
By providing a fast and reliable pricing method, the Bjerksund-Stensland Model helps traders and institutions make informed decisions in dynamic markets.