Discussion Papers 2019

CIRJE-F-1131	"A Novel Approach to Asset Pricing with Choice of Probability Measures"
Author Name	Saito, Taiga and Akihiko Takahashi
Date	November 2019
Full Paper	PDF File
Remarks	Revised in December 2019 and February 2021.

Abstract
This paper presents a foundation and concrete examples of yield curve models incorporating fundamental uncertainties, that is, uncertainties about Brownian motions representing fundamental market risks. Firstly, to model aggressive (positive)/conservative (cautious) attitudes towards such fundamental uncertainties, we consider a sup-inf/inf-sup problem on the utility of a representative agent with respect to uncertainties over Brownian motions, i.e. fundamental market risks. Secondly, we show that the problem is solved via a backward-stochastic differential equations (BSDEs) approach. Then, under a probability measure determined by solving the sup-inf/inf-sup problem, we propose interest rate models with those uncertainties and explicitly obtain their term structures of interest rates. Particu- larly, we present two approaches to solving the relevant coupled forward-backward stochastic differential equations (FBSDEs) to obtain expressions of the equilibrium interest rate and the term structure of interest rates. In detail, the first approach is by comparison theorems, and the second approach is to predetermine the signs of the volatilities of the BSDE in the coupled system and confirm them by explicitly solving the separated BSDE. Finally, we present concrete examples with numerical experiments.

Abstract

This paper presents a foundation and concrete examples of yield curve models incorporating fundamental uncertainties, that is, uncertainties about Brownian motions representing fundamental market risks. Firstly, to model aggressive (positive)/conservative (cautious) attitudes towards such fundamental uncertainties, we consider a sup-inf/inf-sup problem on the utility of a representative agent with respect to uncertainties over Brownian motions, i.e. fundamental market risks. Secondly, we show that the problem is solved via a backward-stochastic differential equations (BSDEs) approach. Then, under a probability measure determined by solving the sup-inf/inf-sup problem, we propose interest rate models with those uncertainties and explicitly obtain their term structures of interest rates. Particu- larly, we present two approaches to solving the relevant coupled forward-backward stochastic differential equations (FBSDEs) to obtain expressions of the equilibrium interest rate and the term structure of interest rates. In detail, the first approach is by comparison theorems, and the second approach is to predetermine the signs of the volatilities of the BSDE in the coupled system and confirm them by explicitly solving the separated BSDE. Finally, we present concrete examples with numerical experiments.