Pricing American Interest Rate Options in a Heath-Jarrow-Morton Framework Using Method of Lines

We consider the pricing of American bond options in a Heath-Jarrow-Morton framework in which the forward rate volatility is a function of time to maturity and the instantaneous spot rate of interest. We have shown in Chiarella and El-Hassan (1996) that the resulting pricing partial differential operators are two dimensional in the spatial variables. In this paper we investigate an efficientnumerical method to solve there partial differential equations for American option prices and the corresponding free exercise surface. We consider in particular the method of lines which other investigators (eg Carr and Faguet (1994) and Van der Hoek and Meyer (1997)) have found to be efficient for American option pricing when there is one spatial variable. In extending this method for the two dimensional case, we solve the pricing equation by discretising the time variable and one state varialbe and using the spot rate of interest as a continuous variable. We compare our method with the lattice method of Li, Ritchken and Sankarasubramanian (1995).

Tracking Error and Active Portfolio Management

Persistent bear market conditions have led to a shift of focus in the tracking error literature. Until recently the portfolio allocation literature focused on tracking error minimization as a consequence of passive benckmark management under portfolio weights, transaction costs and short selling constraints. Abysmal benchmark performance shifted the literature's focus towards active portfolio strategies that aim at beating the benchmark while keeping tracking error within acceptable bounds. We investigate an active (dynamic) portfolio allocation strategy that exploits the predictability in the conditional variance-covariance matrix of asset returns. To illustrate our procedure we use Jorion's (2002) tracking error frontier methodology. We apply our model to a representative portfolio of Australian stocks over the period January 1999 through November 2002.

Hedging Diffusion Processes by Local Risk-Minimisation with Applications to Index Tracking

The solution to the problem of hedging contingent claims by local risk-minimisation has been considered in detail in Follmer and Sondermann (1986), Follmer and Schweizer (1991) and Schweizer (1991). However, given a stochastic process Xt and tau1 tau2, the strategy that is locally risk-minimising for Xtau1 is in general not locally risk-minimising for Xtau2. In the case of diffusion processes, this paper considers the problem of determining a strategy that is simultaneously locally risk-minimising for Xtau for all tau. That is, a strategy that is locally risk-minimising for the entire process Xt. The necessary and sufficient conditions under which this is possible are obtained, and applied to the problem of index tracking. In particular, a close connection between the local risk-minimising and the tracking error variance minimising strategies for index tracking is established, and leads to a simple criterion for the selection of optimal set of assets from which to form a tracker portfolio, as well as a value-at-risk type measure for the set of assets used.minimal martingale measure; local risk-minimisation; hedging; incomplete market; index tracking; portfolio selection

Evaluation of Derivative Security Prices in the Heath-Jarrow-Morton Framework as Path Integrals Using Fast Fourier Transform Techniques

This paper considers the evaluation of derivative security prices within the Heath-Jarrow-Morton framework of stochastic interest rates, such as bond options. Within this framework, the stochastic dynamics driving prices are in general non-Markovian. Hence, in principle the partial differential equations governing prices require an infinite dimensinal state space. We discuss a class of forward rate volatility functions which allow the stochastic dynamics to be expressed in Markovian form and hence obtain a finite dimensional state space for the partial differential equations governing prices. By applying to the Markovian form, the transformed suggested by Eydeland (1994), the pricing problem can be set up as a path integral in function space. These integrals are evaluated using fast fourier transform techniques. We apply the technique to the pricing of American bond options and compare the computational time with other methods currently employed such as the method of lines and more traditional partial differential equation solution techniques.

The Reduction of Forward Rate Dependent Volatility HJM Models to Markovian Form: Pricing European Bond Option

We consider a single factor Heath-Jarrow-Morton model with a forward rate volatility function depending upon a function of time to maturity, the instantaneous spot rate of interest and a forward rate to a fixed maturity. With this specification the stochastic dynamics determining the prices of interest rate derivatives may be reduced to Markovian form. Furthermore, the evolution of the forward rate curve is completely determined by the two rates specified in the volatility function and it is thus possible to obtain a closed form expression for bond prices. The prices of bond options are determined by a partial differential equation involving two spatial variables. We discuss the evaluation of European bond options in this framework by use of the ADI method.

The Calibration of Stock Option Pricing Models Using Inverse Problem Methodology

We analyse the procedure for determining volatility presented by Lagnado and Osher, and explain in some detail where the scheme comes from. We present an alternative scheme which avoids some of the technical complications arising in Lagnado and Osher's approach. An algorithm for solving the resulting equations is given, along with a selection of numerical examples.

A Preference Free Partial Differential Equation for the Term Stucture of Interest Rates

The objectives of this paper are twofold: the first is the reconciliation of the differences between the Vasicek and the Heath-Jarrow-Morton approaches to the modelling of term structure of interest rates. We demonstrate that under certain (not empirically unreasonable) assumptions prices of interest-rate sensitive claims within the Heath-Jarrow-Morton framework can be expressed as a partial differential equation which both is preference-free and matches the currently observed yield curve. This partial differential equation is shown to be equivalent to the extended Vasicek model of Hull and White. The second is the pricing of interest rate claims in this framework. The preference free partial differential equation that we obtain has the added advantage that it allows us to bring to bear on the problem of evaluating American style contingent claims in a stochastic interest rate environment the various numerical techniques for solving free boundary value problems which have been developed in recent years such as the method of lines.