Bjerksund-Stensland Model
Calculate American option prices using the Bjerksund-Stensland (2002) closed-form analytical approximation.
Unlike European options, American options can be exercised at any time up to and including expiration, which normally requires a numerical approach such as the Binomial Tree model (see get_binomial_model). The Bjerksund-Stensland model instead derives a closed-form approximation by assuming the early-exercise boundary is flat in steps: once the stock price crosses the trigger price, immediate exercise is assumed optimal. The option’s life is split at t1 = ½(√5 - 1) * T into two periods, each with its own flat trigger price, which brings the price closer to the exact American value than a single flat trigger.
The formula is as follows (for a call, with cost of carry b = r - q smaller than r):
\[\beta = (0.5 - b / \sigma ^{2}) + \sqrt{(b / \sigma ^{2} - 0.5) ^{2} + 2r / \sigma ^{2}}\] \[\text{B∞} = \beta / (\beta - 1) \cdot K\] \[B_{0} = \max(K,\; r / (r - b) \cdot K)\] \[h(t) = - (b \cdot t + \operatorname{2sigmasqrt}(t)) \cdot K ^{2} / ((\text{B∞} - B_{0}) \cdot B_{0})\] \[I(t) = B_{0} + (\text{B∞} - B_{0}) \cdot (1 - e ^{h} (t)),\;\; \text{with}\; I_{1} = I(t_{1}) \text{and}\; I_{2} = I(T)\] \[\text{Call} = \max(\text{Approximation},\; \text{European Black-Scholes Call},\; \max(S - K,\; 0)),\;\; \text{where} \;\; \text{the Approximation is}\; S - K \;\text{when}\; S \geq I_{2} \;\text{and otherwise combines univariate} (\text{first period}) \text{and bivariate} (\text{both periods}) \text{normal terms},\;\; \text{see black\_scholes\_model.get\_bjerksund\_stensland}\] \[\text{Put} (S,\; K,\; T,\; r,\; b,\; \sigma) = \text{Call} (K,\; S,\; T,\; r - b,\; - b,\; \sigma),\;\; \text{the put-call transformation}\]Where S is the stock price (the traded close on the pricing date), K is the strike price, r is the continuously compounded risk-free rate, q is the continuous dividend yield (ln(1 + dividend yield)), σ is the annualized volatility (the standard deviation of the daily returns over the trailing 252 trading days times √252) and T is the time to expiration in years (days / 365).
When b ≥ r early exercise of a call is not optimal and the value is the Black-Scholes price (floored at the exercise value). Where the boundary I2 falls to or below the strike, a 1,000-step binomial tree prices the option instead. At zero days to expiration the value is the exercise value.
See: Bjerksund, P., & Stensland, G. (2002), “Closed Form Valuation of American Options”, Working paper, NHH, as presented in Haug, E. G. (2007), “The Complete Guide to Option Pricing Formulas” (2nd ed.), section 3.1.4.
Also known as: BS2002, Bjerksund-Stensland approximation, American option approximation.
Bjerksund-Stensland Model in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_bjerksund_stensland as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["VZ"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
bjerksund_stensland = toolkit.options.get_bjerksund_stensland(
strike_price_range=0.15, strike_step_size=1, expiration_time_range=91
)
bjerksund_stensland.loc["VZ", ["2026-01-30", "2026-03-01", "2026-03-31"]]
Or, after setting up the Finance Toolkit MCP, ask:
What were American calls on Verizon worth under the Bjerksund-Stensland model at the end of 2025 for the expiries of 30 January, 1 March and 31 March 2026, for strikes within 15% of the share price in $1 steps?
Which returns:
| Strike Price | 2026-01-30 | 2026-03-01 | 2026-03-31 |
|---|---|---|---|
| 34 | 6.73 | 6.73 | 6.73 |
| 35 | 5.73 | 5.73 | 5.7417 |
| 36 | 4.73 | 4.753 | 4.8234 |
| 37 | 3.7409 | 3.8477 | 3.9719 |
| 38 | 2.8269 | 3.0217 | 3.202 |
| 39 | 2.0118 | 2.2958 | 2.5242 |
| 40 | 1.3361 | 1.6834 | 1.9439 |
| 41 | 0.8214 | 1.189 | 1.4615 |
| 42 | 0.4646 | 0.8079 | 1.0723 |
| 43 | 0.241 | 0.5278 | 0.7678 |
| 44 | 0.1144 | 0.3314 | 0.5365 |
| 45 | 0.0497 | 0.2001 | 0.366 |
Parameters
get_bjerksund_stensland accepts the following parameters.
| Parameter | Type | Description | Example |
|---|---|---|---|
start_date | str | None | The date whose price, volatility and rates the values start from; a date without prices (a weekend or a holiday) uses the last trading day before it, and a ticker without a price on that day is left out with a warning. Defaults to None, which uses the last date of the historical data. | "2021-01-01" |
put_option | bool | Whether to calculate the put option price. Defaults to False which means it will calculate the call option price. | True |
strike_price_range | float | The percentage range to use for the strike prices. Defaults to 0.25 which equals 25% and thus results in strike prices from 75 to 125 if the current stock price is 100. | 0.1 |
strike_step_size | int | The step size to use for the strike prices. Defaults to 5 which means that the strike prices will be 75, 80, 85, 90, 95, 100, 105, 110, 115 and 120 if the current stock price is 100. | 1 |
expiration_time_range | int | The number of days to use for the time to expiration: the columns are the expirations 1 up to expiration_time_range - 1 days after the start date. Defaults to 30, which gives expirations of 1 to 29 days. | 90 |
risk_free_rate | float | The risk free rate to use for the calculation. Defaults to None, which uses the Toolkit’s risk-free rate on the pricing date: the 10-year Treasury yield unless the Toolkit’s risk_free_rate parameter selects ‘13w’, ‘5y’ or ‘30y’, converted from its semi-annual bond-equivalent quote to continuous compounding. A rate passed here is used as given, as a continuously compounded rate. | 0.04 |
dividend_yield | float | The dividend yield to use for the calculation. Defaults to None, which uses the dividend yield on the pricing date (the dividends of the 365 days up to it relative to its close, as the continuously compounded ln(1 + q)). A given value is used as a continuous yield. | 0.02 |
show_input_info | bool | Whether to show the input information. Defaults to False. | True |
rounding | int | None | The number of decimals to round the results to. Defaults to None, which uses the rounding of the Toolkit (4 unless set otherwise). | 2 |
standardize | bool | Whether to standardize (Z-Score) the result across the time to expiration columns for each ticker and strike price. Defaults to False. | True |