Maximizing the probability of a perfect hedge

In the framework of continuous-time, Itô processes models for financial markets, we study the problem of maximizing the probability of an agent's wealth at time T being no less than the value C of a contingent claim with expiration time T. The solution to the problem has been known in the context of complete markets and recently also for incomplete markets; we rederive the complete markets solution using a powerful and simple duality method, developed in utility maximization literature. We then show how to modify this approach to solve the problem in a market with partial information, the one in which we have only a prior distribution on the vector of return rates of the risky assets. Finally, the same problem is solved in markets in which the wealth process of the agent has a nonlinear drift. These include the case of different borrowing and lending rates, as well as "large investor" models. We also provide a number of explicitly solved examples

Testing composite hypotheses via convex duality

We study the problem of testing composite hypotheses versus composite alternatives, using a convex duality approach. In contrast to classical results obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289--302), where sufficient optimality conditions are derived via Lagrange duality, we obtain necessary and sufficient optimality conditions via Fenchel duality under compactness assumptions. This approach also differs from the methodology developed in Cvitani\'{c} and Karatzas (Bernoulli 7 (2001) 79--97).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ249 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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

Utility Maximization in Imperfected Markets

We analyze a problem of maximization of expected terminal wealth and consumption in markets with some ``imperfection'', such as constraints on the permitted portfolios, labor income, or/and nonlinearity of portfolio dynamics. By using general optional decomposition under constraints in multiplicative form, we develop a dual formulation. Then, under some conditions imposed on the model setting and the utility functions, we are able to prove an existence and uniqueness of an optimal solution to primal and to the corresponding dual problem by convex duality.Stochastic Optimization, Utility Optimization, Duality Theory, Convex and State Constraints, Optional Decomposition, Optimal Stopping

Optimal Dynamic Portfolio with Mean-CVaR Criterion

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of Neyman-Pearson type binary solution. We add a constraint on expected return to investigate the Mean-CVaR portfolio selection problem in a dynamic setting: the investor is faced with a Markowitz type of risk reward problem at final horizon where variance as a measure of risk is replaced by CVaR. Based on the complete market assumption, we give an analytical solution in general. The novelty of our solution is that it is no longer Neyman-Pearson type where the final optimal portfolio takes only two values. Instead, in the case where the portfolio value is required to be bounded from above, the optimal solution takes three values; while in the case where there is no upper bound, the optimal investment portfolio does not exist, though a three-level portfolio still provides a sub-optimal solution

Efficient Hedging and Pricing of Equity-Linked Life Insurance Contracts on Several Risky Assets

The authors use the efficient hedging methodology for optimal pricing and hedging of equitylinked life insurance contracts whose payoff depends on the performance of several risky assets. In particular, they consider a policy that pays the maximum of the values of n risky assets at some maturity date T , provided that the policyholder survives to T . Such contracts incorporate financial risk, which stems from the uncertainty about future prices of the underlying financial assets, and insurance risk, which arises from the policyholder's mortality. The authors show how efficient hedging can be used to minimize expected losses from imperfect hedging under a particular risk preference of the hedger. They also prove a probabilistic result, which allows one to calculate analytic pricing formulas for equity-linked payoffs with n risky assets. To illustrate its use, explicit formulas are derived for optimal prices and expected hedging losses for payoffs with two risky assets. Numerical examples highlighting the implications of efficient hedging for the management of financial and insurance risks of equity-linked life insurance policies are also provided.Financial markets;

Actuarial versus Financial Pricing of Insurance

This paper grew out of various recent discussions with academics and practitioners around the theme of the interplay between insurance and finance. Some issues were: The increasing collaboration between insurance companies and banks The emergence of finance related insurance products, as there are catastrophy futures and options, PCS options, indexed linked policies... The deregulation of various (national) insurance markets The discussion around risk management methodology for financial institutions The evolution from a more liability modelling oriented industry (insurance) to a more global financial industry involving asset-liability and risk-capital based modelling The emergence of financial engineering as a new profession, its interplay with actuarial training and research. Rather than aiming at giving a complete overview of the issue at hand, the author concentrates on some recent (and not so recent) developments which from a methodological point of view offer new insight into the comparison of pricing mechanisms between insurance and finance. The author views this paper very much as work in progress. This paper was presented at the Financial Institutions Center's May 1996 conference on "

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.