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### Black vs. Bachelier revisited

Last week  we realized that one of the strengths of the Black76 model - interest rates must not  become negative - is one of its weaknesses at the same time - interest rates must not become negative and that the Bachelier model may be a possibility to handle interest rates that are close to zero.
Black:       dF= sigma F dW
Bachelier: dF= sigma W
Note that the Bachelier volatility is an absolute volatility not depending on the actual level of the underlying, whereas the Black volatility is mulitplied by the value of the underlying.

Black76
For the practitioner, the Black76 model is widely used to valuate vanilla interest rate options like caps, floors or swaption. The formula to be applied is
 Black option formulae for call (C) and put (P) options
Here, F is the forward rate (e.g. of the floating rate to be capped), K is the strike level (the cap rate), sigma is the annualized Black volatility and T is the time, when the floating rate is set. If sigma tends to infinity, then the call value converges to exp(-rT) F, independent of the strike level.

 Cap value for at the money caplet as a function of sigma. (F = K = r = 0.5 % = 50 bp, T= 1)
Bachelier
The same can be done for the Bachelier model.
 Bachelier option formulae for call (C) and put (P) options.

Here, if sigma tends to infinity, then d1 becomes zero, and the call value (and also the put value) grows unboundedly.
 Cap value using the Bachlier model. Instrument data as above.
Can the Black and the Bachelier volatilities be translated into one another?
For very small sigma (sigma = 0), the Black and the Bachelier option values coincide. For growing sigma, the Bachlier value grows unboundedly, whereas the Black value goes into saturation. This means that for every Black volatility, there is a corresponding Bachelier volatility delivering the same price for the specific vanilla instrument; this implied Bachelier volatility depends on the specific instrument. The following figure shows the mapping for various at the money caplets.

The other way round, i.e. obtaining the implied Black volatility from the Bachelier value, is more complicated, because the existence of a solution is not guaranteed.

Note that if the translation is possible at all, moderate Bachelier volatilities of 2 percent are translated into Black volatilities of 800 percent and more for small at the money rates.

Again, we have found a classical ill-posed problem. The solution need not exist and if it does exist, small perturbations in the data (Bachelier vola, termsheet data) may lead to arbitrarily large perturbations in the solution (the Black volatility).