OPTIMAL STATIC–DYNAMIC HEDGES FOR BARRIER OPTIONS

We study optimal hedging of barrier options, using a combination of a static position in vanilla options and dynamic trading of the underlying asset. The problem reduces to computing the Fenchel–Legendre transform of the utility-indifference price as a function of the number of vanilla options used to hedge. Using the well-known duality between exponential utility and relative entropy, we provide a new characterization of the indifference price in terms of the minimal entropy measure, and give conditions guaranteeing differentiability and strict convexity in the hedging quantity, and hence a unique solution to the hedging problem. We discuss computational approaches within the context of Markovian stochastic volatility models.     

MARKOWITZ'S PORTFOLIO OPTIMIZATION IN AN INCOMPLETE MARKET

In this paper, for a process S , we establish a duality relation between K_p , the     - closure of the space of claims in     , which are attainable by "simple" strategies, and     , all signed martingale measures     with     , where   p ≥ 1, q ≥ 1  and     . If there exists a     with     a.s., then K_p consists precisely of the random variables     such that ϑ is predictable S -integrable and     for all     . The duality relation corresponding to the case   p = q = 2  is used to investigate the Markowitz's problem of mean–variance portfolio optimization in an incomplete market of semimartingale model via martingale/convex duality method. The duality relationship between the mean–variance efficient portfolios and the variance-optimal signed martingale measure (VSMM) is established. It turns out that the so-called market price of risk is just the standard deviation of the VSMM. An illustrative example of application to a geometric Lévy processes model is also given.     

INDIFFERENCE PRICES AND IMPLIED VOLATILITIES

We consider a general local‐stochastic volatility model and an investor with exponential utility. For a European‐style contingent claim, whose payoff may depend on either a traded or nontraded asset, we derive an explicit approximation for both the buyer's and seller's indifference prices. For European calls on a traded asset, we translate indifference prices into an explicit approximation of the buyer's and seller's implied volatility surfaces. For European claims on a nontraded asset, we establish rigorous error bounds for the indifference price approximation. Finally, we implement our indifference price and implied volatility approximations in two examples.     

A Martingale Characterization of Consumption Choices and Hedging Costs with Margin Requirements

This paper examines optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements or, more generally, of nonlinear wealth dynamics and constraints on the portfolio policies. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities' price process and the investor's preferences. As an illustration, explicit solutions are provided for an agent with ‘logarithmic’ utility. A PDE characterization of the cost of hedging a nonnegative path‐independent European contingent claim is also provided.     

PRICING IN AN INCOMPLETE MARKET WITH AN AFFINE TERM STRUCTURE

We apply the principle of equivalent utility to calculate the indifference price of the writer of a contingent claim in an incomplete market. To recognize the long-term nature of many such claims, we allow the short rate to be random in such a way that the term structure is affine. We also consider a general diffusion process for the risky stock (index) in our market. In a complete market setting, the resulting indifference price is the same as the one obtained by no-arbitrage arguments. We also show how to compute indifference prices for two types of contingent claims in an incomplete market, in the case for which the utility function is exponential. The first is a catastrophe risk bond that pays a fixed amount at a given time if a catastrophe does not occur before that time. The second is equity-indexed term life insurance which pays a death benefit that is a function of the short rate and stock price at the random time of the death of the insured. Because we assume that the occurrence of the catastrophe or the death of the insured is independent of the financial market, the markets for the catastrophe risk bond and the equity-indexed life insurance are incomplete.     

CONVEX RISK MEASURES FOR GOOD DEAL BOUNDS

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no‐arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no‐free‐lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.     

Convergence of utility indifference prices to the superreplication price in a multiple‐priors framework

This paper formulates a utility indifference pricing model for investors trading in a discrete time financial market under nondominated model uncertainty. Investor preferences are described by possibly random utility functions defined on the positive axis. We prove that when the investors's absolute risk aversion tends to infinity, the multiple‐priors utility indifference prices of a contingent claim converge to its multiple‐priors superreplication price. We also revisit the notion of certainty equivalent for multiple‐priors and establish its relation with risk aversion.     

PRICING FOR LARGE POSITIONS IN CONTINGENT CLAIMS

Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi‐martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit arises endogenously as the hedging error for the claim vanishes.     

Duality Theory and Long–Period Price Systems

This paper concerns duality between long–period price systems in Sraffian models of production and the technologies of the various industries. We present a ‘principle of duality between prices and technique’ referred to a system of production, due to Bidard and Salvadori, and compare it with the usual textbook duality between cost and production functions, referred to firms/industries. We argue that the principle of duality for the system of production can be obtained, industry by industry, on the basis of the usual duality theory and that, in so doing, both ‘dualities’ are given a formulation more appropriate to firms/industries in long–period equilibrium.     

INDIFFERENCE VALUATION OF MORTGAGE‐BACKED SECURITIES IN THE PRESENCE OF PREPAYMENT RISK

We present a utility‐based methodology for the valuation and the risk management of mortgage‐backed securities subject to totally unpredictable prepayment risk. Incompleteness stems from its embedded prepayment option which affects the security's cash flow pattern. The prepayment time is constructed via deterministic or stochastic hazard rate. The relevant indifference price consists of a linear term, corresponding to the remaining outstanding balance, and a nonlinear one that incorporates the investor's risk aversion and the interest payments generated by the mortgage contract. The indifference valuation approach is also extended to the case of homogeneous mortgage pools.     

OPTIMAL PORTFOLIOS WITH LOWER PARTIAL MOMENT CONSTRAINTS AND LPM-RISK-OPTIMAL MARTINGALE MEASURES

We optimize the ratio     over an (arbitrage-free) linear sub-space     of attainable returns in an incomplete market model. If a solution exists for  1 < r < ∞  , then the 1st order optimality condition allows to construct an equivalent martingale measure for     , which is shown to be the solution of an appropriate dual minimization problem over the set of all equivalent martingale measures for     . The dual minimization problem admits a solution iff there exists an equivalent martingale measure for     and its optimal value     equals the lowest upper bound     of all α-ratios over     . This new type of non-concave duality also provides an indifference pricing method. The duality result can be extended to the case     and leads to a new no (approximate) arbitrage condition: "no great expectations with vanishing risk."     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

Development of Finns’ price knowledge after the changeover to the euro

This longitudinal study examines the 2002 changeover to the euro in Finland in order to find out how long it takes consumers to learn grocery prices in the new currency. We are especially interested in the influence of age on the process. We used price estimate data and market price data, with six data-collection points over 5 years, the first one just before the changeover and the last one 5 years after it. As expected, consumer price knowledge was clearly weaker immediately after the changeover than during the era of national currency. However, it has not improved significantly in 5 years. The probable reasons for the decreased price knowledge are the diminution of nominal values brought by the changeover (1 euro equals approximately 6 Finnish marks) and general indifference to prices, especially low ones, brought by the increasing standard of living.     

RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS

We study the risk indifference pricing principle in incomplete markets: The (seller's)  risk indifference price        is the initial payment that makes the  risk  involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to find a formula for       and similarly for      . In particular, we prove that  where    p _low   and    p _up   are the lower and upper hedging prices, respectively.     

VALUATION OF CLAIMS ON NONTRADED ASSETS USING UTILITY MAXIMIZATION

A topical problem is how to price and hedge claims on nontraded assets. A natural approach is to use for hedging purposes another similar asset or index which is traded. To model this situation, we introduce a second nontraded log Brownian asset into the well-known Merton investment model with power law and exponential utilities. The investor has an option on units of the nontraded asset and the question is how to price and hedge this random payoff. The presence of the second Brownian motion means that we are in the situation of incomplete markets. Employing utility maximization and duality methods we obtain a series approximation to the optimal hedge and reservation price using the power utility. The problem is simpler for the exponential utility, and in this case we derive an explicit representation for the price. Price and hedging strategy are computed for some example options and the results for the utilities are compared.     

MONOTONICITY AND CONVEXITY OF OPTION PRICES REVISITED

The Black-Scholes option price is increasing and convex with respect to the initial stock price. increasing with respect to volatility and instantaneous interest rate, and decreasing and convex with respect to the strike price. These results have been extended in various directions. In particular, when the underlying stock price follows a one-dimensional diffusion and interest rates are deterministic, it is well known that a European contingent claim's price written on the stock with a convex (concave. respectively) payoff function is also convex (concave) with respect to the initial stock price. This paper discusses extensions of such results under more general settings by simple arguments.     

DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES

The (subjective) indifference value of a payoff in an incomplete financial market is that monetary amount which leaves an agent indifferent between buying or not buying the payoff when she always optimally exploits her trading opportunities. We study these values over time when they are defined with respect to a dynamic monetary concave utility functional, that is, minus a dynamic convex risk measure. For that purpose, we prove some new results about families of conditional convex risk measures. We study the convolution of abstract conditional convex risk measures and show that it preserves the dynamic property of time-consistency. Moreover, we construct a dynamic risk measure (or utility functional) associated to superreplication in a market with trading constraints and prove that it is time-consistent. By combining these results, we deduce that the corresponding indifference valuation functional is again time-consistent. As an auxiliary tool, we establish a variant of the representation theorem for conditional convex risk measures in terms of equivalent probability measures.     

VALUATION OF BARRIER OPTIONS VIA A GENERAL SELF‐DUALITY

Classical put–call symmetry relates the price of puts and calls under a suitable dual market transform. One well‐known application is the semistatic hedging of path‐dependent barrier options with European options. This, however, in its classical form requires the price process to observe rather stringent and unrealistic symmetry properties. In this paper, we develop a general self‐duality theorem to develop valuation schemes for barrier options in stochastic volatility models with correlation.     

The role of warehouse club membership fee in retail competition

There are several explanations as to why warehouse clubs charge membership fees and how the fees play a role in the competitive landscape of the retail grocery market. We provide another insight into the nature of the membership fee using a model of price competition between a warehouse club and a supermarket. We show that the warehouse club's membership fee is an optimal competitive reaction to the supermarket's promotional activity. The more frequent the promotion is, the lower is the membership fee. However, the larger the promotion depth is, the higher is the fee. We show that the cherry-picker segment plays a key role behind these results. Our analysis not only provides a justification of warehouse club membership fees by discovering its duality with the cherry-picker segment but also gives managers several guidelines on yearly fee and retail price decisions.     

EFFICIENT HEDGING OF EUROPEAN OPTIONS WITH ROBUST CONVEX LOSS FUNCTIONALS: A DUAL‐REPRESENTATION FORMULA

Motivated by numerical representations of robust utility functionals, due to Maccheroni et al., we study the problem of partially hedging a European option H when a hedging strategy is selected through a robust convex loss functional L(·) involving a penalization term γ(·) and a class of absolutely continuous probability measures . We present three results. An optimization problem is defined in a space of stochastic integrals with value function EH(·) . Extending the method of Föllmer and Leukerte, it is shown how to construct an optimal strategy. The optimization problem EH(·) as criterion to select a hedge, is of a “minimax” type. In the second, and main result of this paper, a dual‐representation formula for this value is presented, which is of a “maxmax” type. This leads us to a dual optimization problem. In the third result of this paper, we apply some key arguments in the robust convex‐duality theory developed by Schied to construct optimal solutions to the dual problem, if the loss functional L(·) has an associated convex risk measure ρ^L(·) which is continuous from below, and if the European option H is essentially bounded.