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 ....

  • $\begingroup$ Writing down the equations I start to get a feeling that for ATM there might exist a bound ... $\endgroup$ – Ric Feb 18 '16 at 14:04
  • $\begingroup$ That appears fine with me. Note that $F$ is the forward swap rate, which can not be too large. $\endgroup$ – Gordon Feb 18 '16 at 15:46
  • $\begingroup$ Yes .. maybe that's all. It was not clear to me until today when I saw that this is the reason why BB does not deliver any implie vola (B76) for long maturities. $\endgroup$ – Ric Feb 18 '16 at 15:59

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.