American Option Pricing, Implied Volatility, and Market Inputs
Summary
The document raises practical questions about modeling American options with Black–Scholes–Merton, including early exercise, changing bid–ask spreads, intraday time to maturity, volatility estimation, historical risk-free rates, strike listings, and expiration-specific implied volatility. The reply clarifies that American is an exercise-style label rather than a reference to U.S. geography, and distinguishes implied volatility from historical volatility. It also notes that listed options offer standardized strike choices and points to volatility smiles as relevant background.
The exchange is an introductory pointer rather than a pricing guide: it does not provide formulas, data sources, calibration steps, or a method for simulating spreads. The answer recommends further reading but leaves most of the original questions unresolved. It is useful chiefly for identifying conceptual distinctions, especially between historical and implied volatility and between American-style exercise and geographic markets.
Key ideas
- American describes an option’s exercise style, not its geographic market.
- Implied volatility is distinct from historical volatility.
- Implied volatility can vary across expirations, and volatility smiles are relevant to interpreting that variation.
- Listed options provide standardized strike choices, while customized needs may be met through over-the-counter products.
- The exchange does not give a complete method for pricing or simulating American options.
Tags
Full text
# Black Sholes Options Pricing Clarification Questions # Black Sholes Options Pricing Clarification Questions I am interested in pricing American Call and Put Options using BSM and I am new to exploring options prcing. I have some questions here that would really remove the confusion I have on how to more accurately model option prices over time if I were to simulate taking on positions: - Where would I find the correct formula for pricing U.S. Options? It is always pointed out in the literature that the European and U.S. Pricing formulas are different in that U.S. Options can be exercised at anytime and I seem to only encounter the european version. - How would you adjust bid/ask price to simulate the varying spreads of the market maker? - Does the Time to maturity in the BSM formula change with changing intraday time? Does it include weekends? I think time is represented as the number of days until expiration so I am wondering if this accounts for hours and minutes. - Is the small sigma representative of 1 annual standard deviation and is always the Implied Volatility? If so, is it based on roughly 250 trading days per year? - How does the small sigma adjust as tick data of the underylying is coming in? Is it the closing price of the perious 249 days and the price at the moment? I may be completed wrong about how to solve for sigma here. - What would I use as the risk-free interest rate in modeling historical data and where would I get this reference rate? - I can see different available Strike Prices for different Expirations for the same Symbol on a live option chain. How would market makers decide the varying strikes to use? - Lastly, I can see the Implied Volatility listed on live Option Chains being different on different expiration dates. If all known prices are only up until this very moment in time, how could I.V. be different and how is this calculated based on different expiration times? Thank you so so much. ## Answer by AKdemy (score 3) https://quant.stackexchange.com/a/63688 It seems you have not done any basic research. Reading this wikipedia page should answer a lot of your questions. American is not related to U.S (no geographical relation). It is a naming convention that may have some historical explanations. By large, equity stocks are listed products - not OTC traded (though this obviously also exists). Implied vol is not directly related to historical vol (you will find plenty of question and answers about this in the forum here). Same applies to IVOL for different expiration dates and the wikipedia link also discusses the vol smile. Different strikes are available for pretty much the same reason different shampoos can be bought in a store. There is no one size fits all, and offering choices ensures that most personal preferences are met. Since listed products are highly standardized, not everyone is happy with the limited choices that are available on exchanges. That's one major reason OTC products exits.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.