Multilayer heat equations: Application to finance

<p style='text-indent:20px;'>In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.</p>

Martingale Models for the Short Rate

This chapter is devoted to an overview and analysis of the most common short rate models, such as the Vasiček, Dothan, Hull–White, and CIR models. These models are analyzed and classified from the point of view of positive short rates, normal distribution, mean reversion, and computability. In particular we present the theory of affine term structures, and discuss the inversion of the yield curve. Analytical results for bond prices and bond options are presented for all the affine models.

Short Rate Models

The simplest Markovian short rate model is analyzed using classical and martingale methods, and the term structure equation for the determination of zero coupon bond prices is derived.

Models of the Yield Curve and Term Structure

This chapter presents an overview of the modern state of term structure modeling techniques. It provides an analytical framework that is applicable to all short rate models and considers them from the point of view of the classic approach of pricing by replication. The market price of risk and its relation to the drift of a short rate model are important considerations in modeling the term structure. The notable short rate models used in the industry for relative value pricing are introduced with a brief description of the class of affine short rate models employed for forecasting the real-world dynamics of bond prices. The chapter also includes a description of the Heath-Jarrow-Morton derivative pricing framework and an analysis of the LIBOR market model.