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Negative interest rates: why and how?

Jozef Kisel’ák , Philipp Hermann and Milan Stehlík EMAIL logo
From the journal Mathematica Slovaca

Abstract

Interest rates (or nominal yields) can be negative, this is an unavoidable fact which has already been visible during the Great Depression (1929–39). Nowadays we can find negative rates easily by e.g. auditing. Several theoretical and practical ideas how to model and eventually overcome empirical negative rates can be suggested, however, they are far beyond a simple practical realization. In this paper we discuss the dynamical reasons why negative interest rates can happen in the second order differential dynamics and how they can influence the variance and expectation of the interest rate process. Such issues are highly practical, involving e.g. the banking sector and pension securities.

MSC 2010: Primary 34F05; 91B70

(Communicated by Gejza Wimmer)


Acknowledgement

We are indebted to Felix Fuders for his valuable comments.

First author was partially supported by grant VEGA MŠ SR 1/0344/14. Second author gratefully acknowledges support of ANR project Desire FWF I 833-N18. Corresponding author acknowledges Fondecyt Proyecto Regular No 1151441. We acknowledge also project LIT-2016-1-SEE-023.

Last but not the least, the authors are very grateful to the editor and the reviewers for their valuable comments.

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Appendix A. Appendix

We write |Z|2=i,j|Zij|2 for matrix Z.

Theorem A.1

([21: Theorem 5.2.1] LetT > 0 andb(⋅, ⋅): [0, T] × ℝn → ℝn, Σ(⋅,⋅): [0, T] × ℝn→ ℝn×mbe measurable functions satisfying

|b(t,x)|+|Σ(t,x)|C(1+|x|);

(x, t) ∈ ℝn × [0, T] for some constant C, and such that

|b(t,x)b(t,y)|+|Σ(t,x)Σ(t,y)|D|xy|;

x, y ∈ ℝn, t ∈ [0, T] for some constant D. Let Z be a random variable which is independent of theσ-algebraF(m)generated byWs(⋅), s ≥ 0 and such that

E|Z|2<.

Then the stochastic differential equation

dXt=b(t,Xt)dt+Σ(t,Xt)dWt,0tT,X0=Z

has a unique t-continuous solutionXt(ω) with the property thatXt(ω) is adapted to the filtrationFtZgenerated by Z andWs(⋅); stand

E[0T|Xt|2dt<].

Theorem A.2

(Itôs lemma). LetXtbe a process given by

dXt=f(Xt,t)dt+σ(Xt,t)dWt.(A.1)

Letψ(Xt, t) ∈ C(ℝn × [0, ∞)), then for a transformationZt = [ψ1 (Xt, t), …, ψn (Xt, t)], Ztis again an Itô process given by

dZt=ψt(Xt,t)dt+i=1nψxi(Xt,t)dXi,t+12i=1nj=1n2ψxixj(Xt,t)dXj,tdXi,t,(A.2)

where dXj, tdXi, tis calculated according to the standard rules (dt)2 = dt dWi, t = 0, dWj, t dWi, t = 0, ji, dWi, t dWi, t = dt.

Lemma A.1

(Higher moments of Wiener process, see e.g. [21]). For standard Brownian motion (Wiener process) the following formula for higher moments holds:

βk(t):=E[Wtk],k=0,,t0βk(t)=k(k1)20tβk2(s)ds,k2,whichimpliesE[Wtk]=0kisodd,k!2k2(k2)!tk2,kiseven.

Lemma A.2

Let z be bivariate normal with zero mean ands1, s2be nonnegative integers, then

E[Z1s1Z2s2]=0,ifs1+s2isodd,σ1s1σ2s2s1!s2!2s1+s22j=0min(s1,s2)2(2cor(Z1,Z2))2j(2j)!s12j!s22j!,ifs1,s2areeven,σ1s1σ2s2s1!s2!2s1+s222j=0min(s11,s21)2(2cor(Z1,Z2))2j+1(2j+1)!s112j!s212j!,ifs1,s2areodd.
Received: 2015-6-29
Accepted: 2016-5-12
Published Online: 2017-9-22
Published in Print: 2017-10-26

© 2017 Mathematical Institute Slovak Academy of Sciences

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