Pricing and hedging of contingent claims in incomplete markets
Author
Camcı, Ahmet
Advisor
Pınar, Mustafa Ç.
Date
2010Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Abstract
In this thesis, we analyze the problem of pricing and hedging contingent claims in
the multi-period, discrete time, discrete state case. We work on both European
and American type contingent claims.
For European contingent claims, we analyze the problem using the concept of
a “λ gain-loss ratio opportunity”. Pricing results which are somewhat different
from, but reminiscent of, the arbitrage pricing theorems of mathematical finance
are obtained. Our analysis provides tighter price bounds on the contingent claim
in an incomplete market, which may converge to a unique price for a specific value
of a gain-loss preference parameter imposed by the market while the hedging
policies may be different for different sides of the same trade. The results are
obtained in the simpler framework of stochastic linear programming in a multiperiod
setting. They also extend to markets with transaction costs.
Until now, determining the buyer’s price for American contingent claims
(ACC) required solving an integer program unlike European contingent claims
for which solving a linear program is sufficient. We show that a relaxation of
the integer programming problem which is a linear program, can be used to get
the buyer’s price for an ACC. We also study the problem of computing the lower
hedging price of an American contingent claim in a market where proportional
transaction costs exist. We derive a new mixed-integer linear programming formulation
for calculating the lower hedging price. We also present and discuss an
alternative, aggregate formulation with similar properties. Our results imply that
it might be optimal for the holder of several identical American claims to exercise
portions of the portfolio at different time points in the presence of proportional
transaction costs while this incentive disappears in their absence.
We also exhibit some counterexamples for some new ideas based on our work.
We believe that these counterexamples are important in determining the direction
of research on the subject.
Keywords
Contingent ClaimMixed Integer Programming
Stochastic Linear Programming
Transaction Cost
Arbitrage
Hedging
Option Pricing