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Risk-Free Rates, Inflation, and American Option Pricing

Article Quant Q&A · Author: SSC Fan
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Summary

The document discusses which interest rate to use in option pricing and whether the rate should be adjusted for inflation. Its answer recommends nominal market swap curves matched to the option’s tenor, rather than treating Treasury yields as the default proxy. It argues that inflation need not be separately subtracted because derivative pricing uses nominal cash flows and hedging or replication with a risk-free return. It mentions public and institutional sources for obtaining rate data.

The response distinguishes European pricing from the greater complications of American options. It says Black–Scholes has no closed-form solution for American options and describes the Cox–Ross–Rubinstein binomial tree as a common approach, alongside finite-difference methods and Leisen–Reimer improvements to tree convergence. The discussion is mainly practical guidance, not a comparison of rate sources or pricing methods backed by results in the document. It also notes that dividends and volatility-surface fitting can introduce substantial additional modeling work.

Key ideas

  • Use a nominal market rate curve matched to the option’s maturity as a practical pricing input.
  • The response says not to adjust the nominal rate separately for inflation in derivative pricing.
  • American options generally require numerical methods rather than a closed-form Black–Scholes solution.
  • Binomial trees and finite-difference methods are among the described numerical approaches.
  • Dividend treatment and volatility-surface fitting can be more involved than selecting the rate.

Tags

OptionsDerivatives pricingVolatilityFixed income
Full text
# Risk free rate for Black and Scholes model: Incorporating inflation?


# Risk free rate for Black and Scholes model: Incorporating inflation?












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?

## Answer by AKdemy (score 5, accepted)

https://quant.stackexchange.com/a/77805

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.

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.