Asian Option
Calculate the closed-form price of a geometric-average Asian option using the Kemna & Vorst (1990) formula.
An Asian option’s payoff depends on the average price of the underlying stock over the option’s life, rather than the price at a single point in time, which typically makes it cheaper than the equivalent vanilla option (the averaging reduces variance). This prices the continuously sampled geometric-average Asian option only; the arithmetic-average version has no closed form and is not offered. The geometric-average version has a closed-form solution based on an adjusted volatility and cost of carry.
The formula is as follows:
\[\sigma_{A} = \sigma / \sqrt{3}\] \[b_{A} = 0.5 \cdot (b - \sigma ^{2} / 6),\;\; \text{where} \;\; b = r - q \;\text{is the cost of carry}\] \[d_{1} = (\ln(S / K) + (b_{A} + \sigma_{A} ^{2} / 2) \cdot t) / (\sigma_{A} \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma_{A} \cdot \sqrt{t}\] \[\text{Call Price} = S \cdot e ^{(b_{A} - r) \cdot t} \cdot N(d_{1}) - K \cdot e ^{- r \cdot t} \cdot N(d_{2})\] \[\text{Put Price} = K \cdot e ^{- r \cdot t} \cdot N(- d_{2}) - S \cdot e ^{(b_{A} - r) \cdot t} \cdot N(- d_{1})\]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), t is the time to expiration in years (days / 365) and N is the cumulative standard normal distribution.
Also known as: geometric Asian option, average rate option, average price option.
Asian Option in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_asian_option as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["XOM"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
asian_option = toolkit.options.get_asian_option(
strike_price_range=0.1, strike_step_size=2, expiration_time_range=91
)
asian_option.loc["XOM", ["2026-01-30", "2026-03-01", "2026-03-31"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was a geometric-average Asian call on ExxonMobil worth at the end of 2025, for the expiries of 30 January, 1 March and 31 March 2026 and strikes within 10% of the share price in $2 steps?
Which returns:
| Strike Price | 2026-01-30 | 2026-03-01 | 2026-03-31 |
|---|---|---|---|
| 108 | 12.2984 | 12.3072 | 12.3784 |
| 110 | 10.3173 | 10.3976 | 10.5491 |
| 112 | 8.365 | 8.5609 | 8.8116 |
| 114 | 6.4839 | 6.8383 | 7.197 |
| 116 | 4.7442 | 5.2756 | 5.7352 |
| 118 | 3.2336 | 3.9144 | 4.4502 |
| 120 | 2.0276 | 2.7832 | 3.3568 |
| 122 | 1.1573 | 1.8907 | 2.4581 |
| 124 | 0.5964 | 1.2243 | 1.7458 |
| 126 | 0.276 | 0.7545 | 1.2016 |
| 128 | 0.1143 | 0.4421 | 0.8013 |
| 130 | 0.0423 | 0.2462 | 0.5176 |
Parameters
get_asian_option 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 |