Sup-Inf/Inf-Sup Problem on Choice of a Probability Measure by Forward–Backward Stochastic Differential Equation Approach

This paper presents a new asset pricing model incorporating fundamental uncertainties by choice of a probability measure. This approach is novel in that we incorporate uncertainties on Brownian motions describing risks into the existing asset pricing model. Particularly, we show extensions of interest rate models to the ones with uncertainties on the Brownian motions, which make the yield curve reflect not only economic factors but also views of the market participants on the Brownian motions. Such yield curve models are especially important in yield curve trading of hedge funds as well as monetary policy making of central banks under low interest rate environments observed after the global financial crisis, in which yield curves are less affected by economic factors since they are controlled by the central banks, but are driven mainly by sentiments of market participants. 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, by choice of a probability measure. 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. Particularly, 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.