# A simulation method for skewness correction

This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set…

Contents

1 Introduction
1.1 Skewness
1.2 Setting and notation
2 The Edgeworth expansion
2.1 Deﬁnition and formal conditions
2.2 Derivation
2.2.1 Edgeworth expansion for Sn
2.2.2 Edgeworth expansions for more general statistics
2.2.3 Edgeworth expansion for Tn
2.2.4 Some remarks; skewness correction
2.3 Cornish-Fisher expansions for quantiles
3 Methods for skewness correction
3.1 Coverages of conﬁdence intervals
3.2 Edgeworth correction
3.3 The bootstrap
3.3.1 The bootstrap and Sn
3.3.2 Bootstrap conﬁdence intervals
3.4 A new simulation method
3.4.1 Skewness correction through addition of a random variable
3.4.2 Simulation procedure
4 Comparison
4.1 Coverages of conﬁdence intervals
4.2 Criteria for the comparison
4.3 Comparisons of the upper limits
4.4 Simulation results
4.5 Discussion
A Appendix: Skewness and kurtosis
A.1 The skewness of a sum of random variables
A.2 The kurtosis of a sum of random variables
B Appendix: P(ˆ θnew ≤ ˆ θEcorr)
C Appendix: Simulation results

Author: Eriksson, Måns

Source: Uppsala University Library