Swaption In-the-Money Probabilities Under Volatility Skew

Summary

The document considers how to estimate the probability that a forward swap rate finishes above or below a swaption strike. A simple approximation uses the cumulative normal distribution of a standardized forward-minus-strike distance under a Bachelier model, analogous to using delta as a probability proxy in equity options. The reply cautions that this relies on constant normal volatility; a lognormal model would likewise assume constant lognormal volatility.

When market volatility varies by strike, the resulting skew means these constant-volatility assumptions can misstate the probability. The suggested alternative is to fit a model that captures the market skew, such as SABR, derive its risk-neutral density, and calculate the probability from that distribution. The document gives conceptual guidance only: it does not provide calibration details, empirical comparisons, or evidence that a particular model is suitable in every market. The probability derived from a risk-neutral density is also a pricing measure quantity, not necessarily a real-world forecast.

Key ideas

• A normal cumulative distribution approximation for swaption exercise probability assumes constant normal volatility.
• A lognormal delta approximation similarly depends on a constant-volatility assumption.
• Market volatility skew can make constant-volatility probability estimates inaccurate.
• A skew-fitting model such as SABR can be used to derive a risk-neutral density and calculate probabilities.
• Risk-neutral probabilities need not match real-world frequencies.

Tags

# Is $N(d_1)$ a good approximation that a swap enters in the money?


# Is $N(d_1)$ a good approximation that a swap enters in the money?












I'm looking for an easy method to approximate the probability of the forward swap rate that is implied by the swpation market. One possibility would be to fit a certain model, e.g. SABR, and extract the risk neutral density.

On the other hand I know from the equity case that $N(d_1)$, the delta, is used an approximation that the underlying ends in the money. Is there a similar approximation in the swaption case? I.e. could we use normal vol (bachelier model), and use $N(d_1)$, where $d_1 = \frac{F-K}{\sigma \sqrt{T}}$ for forward swap rate $F$, strike $K$, and implied normal vol $\sigma$ and time to maturity $T$. Or is there any other approximation used frequently?

## Answer by dm63 (score 4, accepted)

https://quant.stackexchange.com/a/75668

It is best to , as you say, extract the risk neutral density from a model that fits the market skew , such as sabr. Then you can compute the probability directly. The problem with N(d) is that you are assuming constant volatility , either constant normalized volatility in the Bachelier model or constant lognormal volatility in the BS model. In a market where there is significant skew , this will give you the wrong answer.

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