Negative Interest Rates and the Limits of Standard Finance Models
Summary
The discussion considers which standard finance models become problematic when nominal interest rates turn negative. It distinguishes models that impose a nonnegative rate floor, such as the basic CIR specification, from Gaussian models that can already represent negative rates. Suggested adaptations include shifting a model’s rate level or changing its dynamics; a free-boundary version of SABR is mentioned as one example. The answers also argue that some theoretical results may lose practical relevance even where their assumptions are affected.
The text questions whether negative risk-free returns invalidate the Markowitz efficient-frontier framework and suggests that they do not necessarily do so, since the analysis is relative. It also raises the American-call exercise result and Black–Scholes as possible concerns, but gives little analysis of the latter. These are short forum responses, not a comprehensive treatment: the claims are not supported with derivations, and conclusions depend on each model’s assumptions and intended use.
Key ideas
- Gaussian interest-rate models can accommodate negative rates, while the basic CIR model cannot.
- A shifted rate specification can move a model’s lower bound below zero.
- Alternative dynamics can be designed to allow negative rates, including a cited SABR variation.
- The responses argue that negative risk-free returns do not inherently invalidate Markowitz analysis.
- The discussion raises questions about option-pricing results but does not analyze all of them in depth.
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Full text
# Models crumbling down due to negative (nominal) interest rates # 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? ## Answer by q.t.f. (score 1) https://quant.stackexchange.com/a/29858 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? [1] http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2557046 ## Answer by eraoul (score 0) https://quant.stackexchange.com/a/17614 Here is a related previous StackExchange question: Modelling with negative interest rates Also, it seems that Black-Scholes option pricing breaks down.
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