Pricing and hedging in an incomplete interest rate market: Applications of the Laplace transform.

Strom, Christopher Solon (2008) Pricing and hedging in an incomplete interest rate market: Applications of the Laplace transform. PhD thesis, London School of Economics and Political Science.

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Abstract

This thesis explores pricing models for interest rate markets. The model used to describe the short rate is based on the discontinuous shot noise process. As a consequence the market is incomplete, meaning that not all securities contingent on the short rate can be replicated perfectly with a dynamically adjusted portfolio of a bond and cash. This framework is still consistent with the absence of arbitrage as evidenced by the existence of an equivalent martingale measure. This measure is not unique, however, due to the incompleteness of the market. Two approaches to pricing contingent claims are pursued. The first, risk-neutral pricing, evaluates the expected value of the pay-off at expiration under an equivalent martingale measure. A parameterized class of martingales, based on the Esscher transform, allows for the definition of a flexible set of equivalent martingale measures and results in a formula for the conditional joint Laplace transform of the short rate and its time-integral. The pricing formula for a discount bond follows trivially from these results. A method for pricing a European call option is also proposed, requiring numerical inversion of the aforementioned Laplace transform. The second approach, mean-variance hedging, addresses the incompleteness of the market. A contingent claim is priced by forming a portfolio of a bond and cash. The portfolio is dynamically updated to mimic the pay-off of the claim at expiration. The replicating portfolio is restricted to be self-financing and predictable. This approach leads to a closed-form pricing formula for a discount bond and formulae for European call and put options, requiring the numerical Laplace inversion methods mentioned above. All this is in the context of a discrete-time model that includes as a special case a discrete-time version of the shot noise process.

Item Type:	Thesis (PhD)
Uncontrolled Keywords:	Economics, Finance
Sets:	Collections > ProQuest Etheses Departments > Statistics
URI:	http://etheses.lse.ac.uk/id/eprint/2156

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