City Research Online

Abstract

This thesis is divided into three chapters, each dealing with a different aspect of market incompleteness and its consequences on quadratic hedging strategies and hedging
errors.

The first chapter studies the effects of market incompleteness due to discrete time trading. We derive the asymptotics (in trading frequency) of the quadratic hedging error of a digital option and obtain a correction to the classical granularity formula, showing that for discontinuous payoffs, the second order term driven by the Cash Gamma remains highly significant. We also show that the discrete-time quadratic hedging strategy generates the same asymptotic error as a continuous-time Black-Scholes delta-hedging strategy used on a discrete set of times.

The second chapter studies the effects of market incompleteness due to jumps in cases when the discretization error from Chapter 1 is predictable. We compute the hedging error under an exponential L´evy model for a general ’L´evy contract’ that encompasses log contracts, variance swaps and higher order moment swaps. We compare two utility-based pricing approaches for incomplete markets: quadratic hedging (corresponding to quadratic utility) and exponential utility. We show that for small jumps, numerically difficult exponential utility results are well-estimated via closedform
quadratic hedging formulas. We use our results on hedging errors to obtain 'good-deal bounds' for variance and skewness swaps.

The third chapter studies the effects of market incompleteness due to uncertainty in the exact specification of the data generating process. We conduct quadratic hedging under a regime-switching L´evy model, which switches between a finite set of distributions based on the value of a (hidden) state variable. We solve the quadratic hedging problem in two steps. First we compute a stochastic differential equation for the filtered estimate of the hidden state. We then use it to solve the quadratic hedging problem with this additional observable variable via classic techniques. We provide Fourier Transform formulas for the mean-value process and hedging strategy, and a recursive scheme for the hedging error.

Publication Type:	Thesis (Doctoral)
Subjects:	H Social Sciences > HG Finance
Departments:	Bayes Business School > Finance Doctoral Theses Bayes Business School > Bayes Business School Doctoral Theses

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