Gain-loss based convex risk limits in discrete-time trading

Cataloged from PDF version of article.We present an approach for pricing and hedging in incomplete markets, which encompasses other recently introduced approaches for the same purpose. In a discrete time, finite space probability framework conducive to numerical computation we introduce a gain–loss ratio based restriction controlled by a loss aversion parameter, and characterize portfolio values which can be traded in discrete time to acceptability. The new risk measure specializes to a well-known risk measure (the Carr–Geman– Madan risk measure) for a specific choice of the risk aversion parameter, and to a robust version of the gain–loss measure (the Bernardo–Ledoit proposal) for a specific choice of thresholds. The result implies potentially tighter price bounds for contingent claims than the no-arbitrage price bounds. We illustrate the price bounds through numerical examples from option pricing

A dual representation of gain-loss hedging for European claims in discrete time

Cataloged from PDF version of article.Superhedging of European claims in incomplete markets is a well-studied problem. The superhedging value of a European claim is known to yield a price too large to be interesting in some cases. In this note, an alternative hedging strategy based on an expected gain–loss criterion is studied for European claims in infinite state space, discrete time financial markets. A dual representation for the gain–loss hedging value is obtained

Mixed-integer second-order cone programming for lower hedging of American contingent claims in incomplete markets

Cataloged from PDF version of article.We describe a challenging class of large mixed-integer second-order cone programming models which arise in computing the maximum price that a buyer is willing to disburse to acquire an American contingent claim in an incomplete financial market with no arbitrage opportunity. Taking the viewpoint of an investor who is willing to allow a controlled amount of risk by replacing the classical no-arbitrage assumption with a "no good-deal assumption" defined using an arbitrage-adjusted Sharpe ratio criterion we formulate the problem of computing the pricing and hedging of an American option in a financial market described by amulti-period, discrete-time, finite-state scenario tree as a large-scale mixed-integer conic optimization problem. We report computational results with off-the-shelf mixed-integer conic optimization software

Gain-loss pricing under ambiguity of measure

Motivated by the observation that the gain-loss criterion, while offering economically meaningful prices of contingent claims, is sensitive to the reference measure governing the underlying stock price process (a situation referred to as ambiguity of measure), we propose a gain-loss pricing model robust to shifts in the reference measure. Using a dual representation property of polyhedral risk measures we obtain a one-step, gain-loss criterion based theorem of asset pricing under ambiguity of measure, and illustrate its use. © EDP Sciences, SMAI, 2008

Tactical Assets Allocation: Evidence from the Nigerian Banking Industry

The core of portfolio selection theory centers on striking a balance between risk-return trade-off of a given investment layout so as to maximize benefits. Literature reveals that portfolio selection or asset allocation problems often involve the use of mathematical programming in propounding solution. This paper uses a blend of simultaneous equation and graphical approach to linear programming algorithm to help solve investors’ problem in allocating assets among various alternatives when faced with problems associated with risk-return trade-off

Pricing and hedging of contingent claims in incomplete markets

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2010.Thesis (Ph. D.) -- Bilkent University, 2010.Includes bibliographical references leaves 94-99.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.Camcı, AhmetPh.D

Utility based pricing of contingent claims

In a discrete setting, we develop a model for pricing a contingent claim. Since the presence of hedging opportunities influences the price of a contingent claim, first we introduce the optimal hedging strategy assuming a contingent claim has been issued: a strategy implemented by investing the budget plus the selling price is optimal if it maximizes the expected utility of the agent's revenue, which is the difference between the outcome of the hedging portfolio and the payoff of the claim. Next, we introduce the `reservation price' as a subjective valuation of a contingent claim. This is defined as the minimum price to be added to the initial budget that makes the issue of the claim more preferable than optimally investing in the available securities. We define the reservation price both for a short position (reservation selling price) and for a long position (reservation buying price) in the contingent claim. When the contingent claim is redundant, both the selling and the buying price collapse in the usual Arrow-Debreu price. We develop a numerical procedure to evaluate the reservation price and two applications are provided. Different utility functions are used and some qualitative properties of the reservation price are shown.Incomplete markets, reservation price, expected utility, optimization

Finance Without Probabilistic Prior Assumptions

We develop the fundamental theorem of asset pricing in a probability-free infinite-dimensional setup. We replace the usual assumption of a prior probability by a certain continuity property in the state variable. Probabilities enter then endogenously as full support martingale measures (instead of equivalent martingale measures). A variant of the Harrison-Kreps-Theorem on viability and no arbitrage is shown. Finally, we show how to embed the superhedging problem in a classical infinite-dimensional linear programming problem

Option pricing with transaction costs using a Markov chain approximation

An efficient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (in: Dempster, M.A.H., Pliska, S.R. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, UK). A fair option price is determined by requiring that an infinitesimal diversion of funds into the purchase or sale of options has a neutral effect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the option payoff into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely specified option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general definition of an option hedging strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed