Heat Kernel Interest Rate Models With Time-Inhomogeneous Markov Processes

Abstract

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the timeinhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

Consistent Valuation Across Curves Using Pricing Kernels

“…, continues to be the correct forward IBOR process, analagous to the FRA result. As with the FRA, the fair IRS rate process is model-implied 2 Assuming that one insists on maintaining measurability of the payoff at the IBOR reset time T i−1 . 3 If the y-tenored IBOR corresponds to the most liquid and tradable tenor, i.e.…”

“…We thus prove that (a) the LRTS models belong to the class of models developed in Macrina [34] when an infinite-time horizon is considered, and (b) that the pricing kernel generating the LRTS is a weighted heat kernel (WHK). Pricing kernels generated by WHKs in an infinite time horizon setting are introduced in Akahori et al [1] and developed in Akahori & Macrina [2] in the case tha the WHK is driven by a time-inhomogeneous Markov process. In particular, we shall show that the LRTS models produce bond price processes (P t T ) 0≤t≤T of the form…”

“…Next we consider weighted heat kernel processes over an infinite-time horizon, see Akahori et al [1], and in particular the case where the propagator is a conditional expectation, as in Akahori & Macrina [2] and Macrina [34]. Such weighted heat kernels are used to generate (explicit) pricing kernel processes.…”

Heat Kernel Framework for Asset Pricing in Finite Time

“…In the next section, we introduce weighted heat kernels to define the class of pricing kernels, and thus give rise to the asset pricing framework treated in this paper. Heat kernel models for the development of stochastic price systems have been proposed by Akahori et al [1] in an infinite-time setting, and more recently by Akahori & Macrina [2] in a finite-time context.…”

“…At first, we summarise the construction of the pricing kernel presented in Akahori & Macrina [2], and at the same time, we extend the approach so that automatic partial calibration can be accommodated. Then we write the pricing kernel models and the resulting discount bond price processes in a concise formalism, which we show remains unchanged if one were to apply a different probability measure.…”