# Models crumbling down due to negative (nominal) interest rates

Given that the negative interest rates on a lot of sovereign bonds with maturity under 10 years are trading in the negative (nominal) interest rate territory (recently also the short term EURIBOR has dropped below zero), which are the most striking applications for the models in financial economics/quant finance field?

By that I mean which of the so called "stylized facts" and standard models of modern finance are becoming highly controversial or just plain useless? As a couple of examples which spring to mind are the following (do not necessarily have to do with sovereign bond yields, but the concept of negative (nominal) interest rates as such):

• The CIR interest rates model completely breaks down due to the square root term
• The proof that an American call option written on a non-dividend paying underlying will not be exercised before the maturity is false
• Markowitz selection obviously encounters difficulties incorporating negative yields

What are the other consequences, on let us say, CAPM, APT, M&M or any other model in finance? Which long held beliefs are hurt the most by negative yields?

• Crumbling down? Don't get too excited. – Kiwiakos May 2 '15 at 10:29
• Models now have to incorporate the fact that you cannot borrow and lend at the same rate. Previously that may have been close to a rounding error but not anymore. IMO, this is going to be the biggest set of upgrades to quant models to adapt to negative rates. – user25064 Jul 31 '15 at 13:07

For the most part there is no serious difficulty in modelling with negative interest rates. Some of the earliest and most widely-used interest rate models are Gaussian, so admit the possibility of negative interest rates.

Other models, like CIR, as you point out do not allow negative rates. These are unsurprisingly less-preferred now. There has been some effort to make extensions of models to allow negative rates. One approach is to "shift" the model, replacing rate $r$ with $r-c$ for some constant $c$. That has the effect of making a new lower bound $-c$ for a model which formerly had $0$ as the lower bound for the interest rate. Another possibility is to modify the dynamics to allow negative rates. For example there is a "Free boundary" variation of SABR [1].

Your example of the American call price is trivial -- the result was only an academic curiosity anyway. Real stocks have dividends and borrow costs and the theorem wouldn't typically apply anyway.

I also don't believe there is any problem with Markowitz theory if the risk-free rate of return is negative. The efficient frontier analysis is all relative, why does it matter?

Here is a related previous StackExchange question:

Modelling with negative interest rates

Also, it seems that Black-Scholes option pricing breaks down.

• Well, BS on a stock does not break down. Black 76 when the underlying is a negative rate itself does. – Ric Jun 1 '15 at 12:00