Estimating Callable Swap Exercise Probability with Normal Volatility

Summary

The document explains how to estimate the exercise probability for a European callable interest rate swap with one call date. In a normal volatility model, the probability is framed as the chance that the forward swap rate for the remaining swap life falls below the strike rate on the call date. It gives a closed-form expression using the cumulative normal distribution, the difference between strike and forward rate, normal volatility, and the square root of time to the call date.

This provides an analytical counterpart to a Monte Carlo estimate based on the share of simulated paths in which the embedded swaption has positive value. The expression assumes a single exercise date and a normal model for the relevant swap rate. The brief answer does not discuss calibration, discounting conventions, model validation, or extensions to multiple exercise dates, so those aspects would need separate treatment in practical pricing.

Key ideas

• For a single call date, exercise probability is the probability that the relevant forward swap rate is below the strike.
• A normal volatility model expresses this probability through the cumulative normal distribution.
• The calculation depends on the forward rate, strike, normal volatility, and time until the call date.
• Monte Carlo path frequencies and the analytical probability address the same single-date exercise event under their respective modeling assumptions.

Tags

# Call Probability of European callable IRS


# Call Probability of European callable IRS












When pricing a callable IRS (say only one call date) with a diffusion model (e.g. HW 1F) with a Montecarlo resolution, one can get the call probability on the call date versus maturing the date (which is the percentage of paths where the embedded swaption end up with a value > 0)

What would be an analytical way of implying a call probability using a closed-form of Black & Scholes (normal vol) when we only have one call date?

Thanks,

## Answer by dm63 (score 0, accepted)

https://quant.stackexchange.com/a/53477

The call probability is just the probability that the swap rate for the remaining life of the swap is below the strike rate. This is easily obtainable in a normal vol model, it is $$N((Strike-Forward Rate)/NormVol*Sqrt(T))$$ where T is time from now until call date, where $N$ is the cumulative Normal distribution.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.