The Upper Bound on an ATM Swaption Premium in Black-76

Summary

The note explains why an at-the-money call swaption priced under Black-76 has an upper bound, an issue that can arise when inferring implied volatilities from market premiums. In Black-76, the discounted forward swap rate plays a role analogous to the stock price in the standard Black–Scholes call option argument. A call payoff cannot exceed the underlying value, so its discounted premium cannot exceed the discounted forward rate.

The question arose after implied-volatility gaps appeared at both short and long maturities. Negative rates were identified as a limitation for a lognormal model at short maturities, while the document does not explain the high-maturity gaps in detail. It confirms the payoff-bound intuition but does not derive numerical thresholds or show how to diagnose data or implementation issues. The bound limits admissible premiums for inversion; it does not by itself resolve why particular market observations exceed it.

Key ideas

• A Black-76 call premium is bounded by the discounted forward rate.
• The bound follows because the call payoff cannot exceed the value of the underlying forward-rate exposure.
• A premium above the bound cannot be inverted to a Black-76 implied volatility.
• The note mentions negative rates as a short-maturity limitation of the lognormal model but leaves high-maturity gaps unexplained.

Tags

# Why is there an upper limit on the premium of an ATM (!) call swaption in the Black76 model?


# Why is there an upper limit on the premium of an ATM (!) call swaption in the Black76 model?












Trying to imply Black76 (where the forward swap rate is log-normal) volatilities as Bloomberg does in their VCUB screen we see holes at two regions:

- at short maturities due to negative rates which can not be captured in a log-normal model - that is clear

- and also at high maturities - which is suprising for me.

The documentation says that in the Black76 model there is an upper bound for ATM-call prices on swap-rates. If the premium is higher, then one can not imply the volatility. But how can there be an upper bound?

EDIT: Is the upper bound just the discounted rate? If we recall B76 $$ C= \exp(-r T)[F N(d_1) - K N(d_2)], $$ then ATM means $F=K$ and thus $$ C= \exp(-r T)F [N(d_1) - N(d_2)], $$ which is $$ \exp(-r T)F [N(\sigma \sqrt{T}/2) - N(-\sigma \sqrt{T}/2)], $$ and using symmetry we arrive at $$ \exp(-r T)F [1-2N(-\sigma \sqrt{T}/2)] \le \exp(-r T)F, $$ which would be a rather high bound ....

## Answer by Mark Joshi (score 3, accepted)

https://quant.stackexchange.com/a/24390

In the BS model there is the upper bound of the stock price, which can be proven by the fact the stock price bounds the call option pay-off. Here we are seeing a similar effect: the discounted rate corresponds to the stock price.

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.