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I am new to quantitative finance and I am trying to create a model for option pricing. Naturally the Black and Scholes equation is front and center for this sort of thing, but that raises the question of the r term. If I'm trying to build a model for options pricing, what exactly ought I use for the risk free rate term?

Doing some digging around this site I see a lot of different things mentioned, and most of it I can wrap my head around. However, doing some background reading on the risk free rate I read that:

The so-called "real" risk-free rate can be calculated by subtracting the current inflation rate from the yield of the Treasury bond matching your investment duration.

But when I read answers like this one or this one inflation is not mentioned. When inputting values into the Black and Scholes equation to create a reliable model, ought one include inflation? It seems like it should be a big deal, but all these mentions of risk free rate keep eliding it. And if it should be included, where can someone find up to date inflation data to use for their models?

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    $\begingroup$ Treasury bonds aren't good and usually not used. E.g. Bloomberg wouldn't even allow you to select them on all their derivatives pricers. See here for some details. There is no need for inflation. Finance is about nominal values and option pricing is all about hedging and replication, leaving only the risk free return. $\endgroup$
    – AKdemy
    Commented Dec 23, 2023 at 8:29
  • $\begingroup$ @AKdemy Ok so based on your comment and the link you posted, it seems what I should do is this: Get Latest SOFR rates (because I am using dollars) from a publicly available source such as this page for the New York Fed, Select the tenor that matches closest to the option maturity, calculate continuous compount interest if applicable, then use that? Is that right? One thing that confuses me about the link you posted is it says to use the exact tenor term, but where can you find SOFR rates for every possible term? $\endgroup$
    – SSC Fan
    Commented Dec 23, 2023 at 21:54
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    $\begingroup$ The link you posted are just average rates of daily SOFR. You need swap rates but I do not think you can get SOFR swaps anywhere for free. It is the smallest issues when pricing options though. I assume you look at equities? These will be American, frequently pay dividends, you will not have reliable implied vol surfaces... $\endgroup$
    – AKdemy
    Commented Dec 23, 2023 at 22:23
  • $\begingroup$ @AKdemy ok gotcha. So you're saying to be the most accurate you need to get real time values of the rates being exchanged in SOFR swaps at the time the options are being priced. And if I'm reading you right you're also saying, even though this data is not free, it's a small price to pay for having accurate implied volatility. Am I reading that right? If so, do you know where I can find SOFR swaps then? $\endgroup$
    – SSC Fan
    Commented Dec 24, 2023 at 4:23

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I initially voted to close this question because I think asking about risk free rates is off topic on quant se (and answered several times). However, I do not think there is a good alternative where one can ask questions that are deemed to be too basic here. Therefore, I will try to answer and go beyond just interest rates inputs.

Based on the question and your comments, I suspect you ask about listed equity option?

As mentioned in a comment, Treasury bonds aren't good proxy for a risk-free rate and usually not used. E.g. Bloomberg wouldn't even allow you to select them on all their derivatives pricers. See here for some details. You can find an example of a SOFR swap in this quant se answer. Vendors like Bloomberg will have market quoted swaps for all sorts of tenors and stripped curves to allow you to get the exact quote needed for any option tenor. If you do not have BBG (or anything similar) at work, you can try a public library or university library. Chathamfinancial has some basic tenors which should be sufficient to get reasonable values if you don't have access to a more complete data source.

There is no need to adjust for inflation. Finance is about nominal values and option pricing is all about hedging and replication, leaving only the risk-free return.

However, the risk-free rate is only a minor input detail when pricing American options. Generally, these options are price quoted anyways. You cannot simply input values in the Black Scholes equation because there exists no closed form solution for American options. Frequently, the Cox-Ross-Rubenstein Model - CRR is used, which is a recombining binomial tree. It is a particular case of an explicit FDM scheme.

There are many papers discussing alternatives to CRR - see for example Andreasen, American Option Pricing in a Tick from Saxo Bank or Bank of America Merrill Lynch where the author claims to be able to improve the efficiency of American option pricing algorithms by at least 4 orders of magnitude. This Risk.net article also discusses fast and stable American option pricing with FDM.

CRR itself has also been modified because the convergence of the binomial tree-based value to the limit is not monotone but rather oscillatory. This observation was the basis for a method developed by Leisen and Reimer (LR) (1996) to compute accurate results with a "minimum" number of time steps.

Voladynamics also uses LR. The link has a presentation that provides an excellent summary of the complexities of dividend modelling that arise when pricing equity options. Voladynamics comes in places where things get really difficult, namely trying to fit vol surfaces. You can find a description of the commonly used steps in this quant se answer.

My point was really that I would not spend too much time worrying about an appropriate risk free rate if you are just trying to get something working on your own for educational purposes. There are many more challenging and complex issues that need to be solved before you need to worry about the exact risk free rate to be used in the model.

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