# Universität AugsburgInstitut für Mathematik

Oberseminar Stochastik

Dr. Nikolai Giesbrecht
Bonn

spricht am

Dienstag, 6. Februar 2018

um

17:30 Uhr

im

Raum 2004 (L1)

über das Thema:

## »New Markov-Process-Model for risk-neutral calibration of interest-rates«

 Abstract: Nowadays there are two types of interest-rate-models used in practice: the Short-Rate- (Exponential Vasicek, Black-Karasinski,CIR++ …) and Libor-Market-Models (LMM). Models of both types follow continuous-time differential equations and satisfy certain martingale properties. Both classes of models meet today a big problem, because of unstable property they have a too high ratio of ESG-trails with unrealistic too-high-yields and with unrealistic too-low-yields. All calibration problems of Short-Rate- and Libor-Market-Models are the following of only one matter: the high time dependence for the forward rate developing. Today we can fit the market swaption prices, if only the forward rates have at least 98% annual time dependence. Remark that we find the same high dependence by historical governments bonds developing. In this paper I propose a new model-type Markov-Process-Model (MPM), which has the characteristic that discount bond prices and forward rates are at any time a function over a Markov-Process. It is based on a discrete Chapman-Kolmogorov equation. But this equation is not sufficient for the theoretical stable property of the model, that has high time dependence. For the sufficient condition we need a Markov-Process with good ergodic property. By conform map of the infinity room into limited room with ergodic-limit-control parameters of realistic forward rate values we can build the needed ergodic Markov-Process. The high time dependence of forward rates stochastically mean, that every analytical calibration can not be stable and can not build good approximation. Because by high time dependence and non-symmetrical distributed values every analytical approximation is biased and the bias error increase rapidity with the number of time steps. I solve this numerical problem with high dependence by using the numerical unbiased Monte Carlo valuations based on symmetrical-conform-maps. And I present on the end of this presentation an example of the EURO market at Q4 2015. With this example of generalized Hull-White type of MPM calibration I present, that we can exclude all possible negative yields and at the same time MPM has no problem with unrealistic too-high-yields. I present also, that the build model match market option-prices and that the build model has better risk-neutral property as one model ever had. So the MPM is the first model, that solve the problem with high time dependents for the forward rate developing. But it is not the first time, that such a numerical problem is solved. The team from Enrico Fermi and the team from John von Neumann were the first by solving of similar problem for high time dependence of neutron transport developing for atom reactor and for nuclear exploding's valuations. They were the first by using in 1947-1948 the numerical unbiased Monte Carlo valuations based on spherical-random-work.

 Hierzu ergeht herzliche Einladung. Prof. Dr. Gernot Müller

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