Optimal Trading Strategies Under Arbitrage

Title:

Optimal Trading Strategies Under Arbitrage

Author(s):

Ruf, Johannes Karl Dominik

Thesis Advisor(s):

Karatzas, Ioannis

Date:

2011

Type:

Dissertations

Department:

Statistics

Permanent URL:

http://hdl.handle.net/10022/AC:P:10250

Notes:

Ph.D., Columbia University.

Abstract:

This thesis analyzes models of financial markets that incorporate the possibility of arbitrage opportunities. The first part demonstrates how explicit formulas for optimal trading strategies in terms of minimal required initial capital can be derived in order to replicate a given terminal wealth in a continuoustime Markovian context. Towards this end, only the existence of a squareintegrable market price of risk (rather than the existence of an equivalent local martingale measure) is assumed. A new measure under which the dynamics of the stock price processes simplify is constructed. It is shown that delta hedging does not depend on the "no free lunch with vanishing risk" assumption. However, in the presence of arbitrage opportunities, finding an optimal strategy is directly linked to the nonuniqueness of the partial differential equation corresponding to the BlackScholes equation. In order to apply these analytic tools, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The phenomenon of "bubbles," which has been a popular topic in the recent academic literature, appears as a special case of the setting in the first part of this thesis. Several examples at the end of the first part illustrate the techniques contained therein. In the second part, a more general point of view is taken. The stock price processes, which again allow for the possibility of arbitrage, are no longer assumed to be Markovian, but rather only It^o processes. We then prove the Second Fundamental Theorem of Asset Pricing for these markets: A market is complete, meaning that any bounded contingent claim is replicable, if and only if the stochastic discount factor is unique. Conditions under which a contingent claim can be perfectly replicated in an incomplete market are established. Then, precise conditions under which relative arbitrage and strong relative arbitrage with respect to a given trading strategy exist are explicated. In addition, it is shown that if the market is quasicomplete, meaning that any bounded contingent claim measurable with respect to the stock price filtration is replicable, relative arbitrage implies strong relative arbitrage. It is further demonstrated that markets are quasicomplete, subject to the condition that the drift and diffusion coefficients are measurable with respect to the stock price filtration.

Subject(s):

Mathematics
Finance
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