Pricing of contingent claims under the real-world measure
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The aim of this thesis is to price contingent claims under the real-world probability measure. Real-world pricing results naturally by selecting the numeraire as the growth optimal portfolio (GOP). Under this approach, the existence of an equivalent risk-neutral probability measure is not required. Furthermore, the GOP can be used to define other basic contingent claims, such as exchange prices, primary security accounts, and even zero-coupon bonds. We begin with application of the real-world pricing formula to derive forward prices for each of these financial quantities. The obtained formulae are model independent, yet reveal important differences between the real-world arid classical risk-neutral approaches. Real-world prices are systematically derived under each of the models studied within this thesis for the following contingent claims: zero-coupon bonds; options on the GOP: options on exchange prices; and interest rate caps and floors via options on zero-coupon bonds. We start with the classic Black-Scholes-Merton model, where the GOP follows a geometric Brownian motion. Under this model, real-world pricing recovers the results of classical risk-neutral pricing, since the corresponding Radon-Nikodym derivative is a martingale. For each of the remaining models studied, the GOP is based on a time-transformed squared Bessel process. In each case, real-world prices may differ from classical risk-neutral prices because the candidate Radon-Nikodym derivative is a strict supermartingale. The second model considered proposes a modified form of the constant elasticity of variance model for the GOP. New analytic results for zero- coupon bonds and options on the GOP are derived that were previously analysed using numerical methods. Real-world prices for options on exchange prices and interest rate derivatives are also provided. Three versions of the minimal market model are also examined. This model class overcomes some of the deficiencies of the aforementioned approaches since the dynamics for the GOP better reflect empirical market features, such as leptokurtic returns, the leverage effect and a stochastic yet stationary volatility structure. Under a stylised version of the minimal market model with a constant short rate, we derive analytic solutions to the complete suite of contingent claims examined within the thesis. We subsequently allow the short rate to be stochastic in order to accurately model the term structure of interest rates, with a focus on low interest rate environments. The proposed model provides a very good fit to interest rate.
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