Applications of Malliavin calculus and white noise analysis in interest rate markets, and convertible bonds with and without symmetric informaiton

Abstract

The Applications of Malliavin calculus and White noise analysis in stock markets have been well-known in the Mathematical Finance literature. But its application to interest rate markets has been minimal. We will demonstrate how Malliavin Calculus can been applied to interest rate derivative products. The interest rate processes quite often have long range dependence self similarity properties. As a consequence, we suggest that the dynamics of the bond prices can be modeled as processes driven by Fractional Brownian. Then we can apply the multi-dimensional Wick-Ito integral as it can preclude arbitrage opportunities. This framework is particular useful if the market is illiquid as the trader cannot really observe the true market priceA convertible bond has many of the same characteristics as an ordinary bond but with the additional feature that the bond may,at any time of the owner's choosing, be exchanged for a specified asset. Moreover, it is also common that the issuer may have the right to terminate the contract. Hence it can be considered as a game option. We are going to show some of the properties of this derivative. We will also consider the case when there is default risk involved. Finally we suggest a method how to price this derivative when there is insider information.