The effect of negative interest rates on derivative pricing

Question

I am trying to get an overview of the impact on negative interest rates on financial products (in general). For the time being I distinguished the following products

• Vanilla options
• Exotic options
• Forwards
• Interst rate swaps
• Currency swaps

For each of these I would like to pinpoint the general issues. If someone has one of these products I would immediately like to tell them, "hey, have you looked at these possible issues?". For now I have seperated the issues into three categories:

1) Modeling issues 2) Legal issues (think of CSA contracts that are still fairly ambiguous) 3) System issues (e.g. banking systems that are not capable of implementing negative rates).

This being a forum on quantative finance, I am here to ask you about the modeling issues.

The modelling issues of which I am aware are:

• For vanilla options on the interest rate (e.g. swaptions), one frequently uses Black's formula or the SABR model. Both of these assume lognormal distributions, such that negative rates would nog be accepted.
• For exotic options it is common to apply numerical methods for pricing. To this end a proper interest rate model should be applied (e.g. lognormal models are no longer accepted).

I am sure that there are many more issues, would you like to help me make a more complete list? Thank you for your help!

Answer 1

Independently if it makes economically sense or not, negative interest rates have become a reality for Europe which can no longer be neglected. (Even LIBOR became negative in the last months.) One common but wrong solution was to set the rate simply to zero. (One must - by the way take care - that this "solution" is not automatically applied by correction algorithms in systems.) Concerning modeling issues, the main impact is on products that require implicit lognormal distributions. Here, two solutions are possible:

• Use a model that implies just normal distributions. The disadvantage is, however, that the "real" distribution is probably not symmetric since it's very improbable that interest rates ever become e.g. -5%. Hence, this approach makes sense only for short horizons.
• Another possible solution is by using shifted lognormal models. Here, the distribution is "shifted" by a certain parameter. This solution is normally used by financial institutes today. However, the main disadvantage is that a new variable - the shift parameter - is introduced.

A description of the models can be found here:

http://www.d-fine.com/fileadmin/d-fine/hochgeladen/Fachartikel/WhitePaper_Vols_NegIR_V1_1_en.pdf

For exotics, some models probably have still to be developed...

Finally, negative interest rates do also affect volas which were traditionally quoted as lognormal (Black76) but today more as normal or shifted lognormal. Additional problems can also occur with time series (which often don't provide negative interest rates) or stress tests (which are often no longer real "stress" tests since reality became more extreme).