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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 ....

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  • $\begingroup$ Writing down the equations I start to get a feeling that for ATM there might exist a bound ... $\endgroup$ – Richard 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. $\endgroup$ – Richard Feb 18 '16 at 15:59
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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.

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