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.     

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.     

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.     

ON AGENT’S AGREEMENT AND PARTIAL‐EQUILIBRIUM PRICING IN INCOMPLETE MARKETS

We consider two risk‐averse financial agents who negotiate the price of an illiquid indivisible contingent claim in an incomplete semimartingale market environment. Under the assumption that the agents are exponential utility maximizers with nontraded random endowments, we provide necessary and sufficient conditions for negotiation to be successful, i.e., for the trade to occur. We also study the asymptotic case where the size of the claim is small compared to the random endowments and we give a full characterization in this case. Finally, we study a partial‐equilibrium problem for a bundle of divisible claims and establish existence and uniqueness. A number of technical results on conditional indifference prices is provided.     

Optimal investment,derivative demand,and arbitrage under price impact

This paper studies the optimal investment problem with random endowment in an inventory‐based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading and nonlinear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, nonlinear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage‐inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.     

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.     

OPTIMAL INVESTMENT WITH AN UNBOUNDED RANDOM ENDOWMENT AND UTILITY-BASED PRICING

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies—those strategies whose wealth process is a super-martingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.     

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.     

LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY

We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power‐law and exponential‐decay order book models explicitly and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.     

Hedging and Portfolio Optimization in Financial Markets with a Large Trader

We introduce a general continuous-time model for an illiquid financial market where the trades of a single large investor can move market prices. The model is specified in terms of parameter-dependent semimartingales, and its mathematical analysis relies on the nonlinear integration theory of such semimartingale families. The Itô–Wentzell formula is used to prove absence of arbitrage for the large investor, and, using approximation results for stochastic integrals, we characterize the set of approximately attainable claims. We furthermore show how to compute superreplication prices and discuss the large investor's utility maximization problem.     

Do market prices correspond with consumer demands? Combining market valuation and consumer utility for extra virgin olive oil quality attributes in a traditional producing country

The objective of this research is to examine whether there is a relationship between the value of attributes based on the market price and on consumer utilities. To address this objective, the results from a hedonic price (HP) approach are combined with the actual consumer utilities from a real choice experiment (RCE) for extra virgin olive oil (EVOO) attributes. The results indicate that the origin of production attribute positively influences consumer utility and it is also positively related to market EVOO prices. Conversely, the PDO quality certification positively influences consumer utility and willingness to pay, although it is not related to EVOO prices in the real market.     

DO ARBITRAGE‐FREE PRICES COME FROM UTILITY MAXIMIZATION?

In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage‐free prices at which a utility‐maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers all initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show, however, how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.     

Credit portfolio selection with decaying contagion intensities

We develop a fixed‐income portfolio framework capturing the exponential decay of contagious intensities between successive default events. We show that the value function of the control problem is the classical solution to a recursive system of second‐order uniformly parabolic Hamilton–Jacobi–Bellman partial differential equations. We analyze the interplay between risk premia, decay of default intensities, and their volatilities. Our comparative statics analysis finds that the investor chooses to go long only if he is capturing enough risk premia. If the default intensities deteriorate faster, the investor increases the size of his position if he goes short, or reduces the size of his position if he goes long.     

Equilibrium investment with random risk aversion

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation.     

Endogenous Random Asset Prices in Overlapping Generations Economies

This paper derives a general explicit sequential asset price process for an economy with overlapping generations of consumers. They maximize expected utility with respect to subjective transition probabilities given by Markov kernels. The process is determined primarily by the interaction of exogenous random dividends and the characteristics of consumers, given by arbitrary preferences and expectations, yielding an explicit random dynamical system with expectations feedback. The paper studies asset prices and equity premia for a parametrized class of examples with CARA utilities and exponential distributions. It provides a complete analysis of the role of risk aversion and of subjective as well as rational beliefs.     

OPTIMAL CONSUMPTION AND PORTFOLIO DECISIONS WITH PARTIALLY OBSERVED REAL PRICES

We consider optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth subject to a random inflation in the consumption basket price over time. We consider two cases: (i) when the investor observes the basket price and (ii) when he receives only noisy observations on the basket price. We derive the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The compositions of the funds in the two cases are the same, but in general the investor's allocations of his wealth into these funds will differ. However, in the particular case when the investor has constant relative risk-aversion (CRRA) utility, his optimal investment allocations into these funds are also the same in both cases.     

STABILITY OF THE EXPONENTIAL UTILITY MAXIMIZATION PROBLEM WITH RESPECT TO PREFERENCES

This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs, and optimal investment strategies are obtained, their rate of convergence is also determined. Stability of utility‐based pricing is studied as an application. Second, a sequence of utilities defined on converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in [Nutz, M. (2012): Risk aversion asymptotics for power utility maximization. Probab. Theory & Relat. Fields 152, 703–749], which establishes the convergence for a sequence of power utilities.     

PRICING AND HEDGING OF DERIVATIVES BASED ON NONTRADABLE UNDERLYINGS

This paper is concerned with the study of insurance related derivatives on financial markets that are based on nontradable underlyings, but are correlated with tradable assets. We calculate exponential utility‐based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward‐backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical “delta hedge” in complete markets.     

OPTION PRICING AND HEDGING WITH SMALL TRANSACTION COSTS

An investor with constant absolute risk aversion trades a risky asset with general Itô‐dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading‐order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.     

The Value of Imputation Tax Credits on Australian Hybrid Securities

Hybrid securities are becoming an increasingly important component of the capital structure of Australian firms. While displaying characteristics of both debt and equity, one principal equity attribute of hybrids is their ability to pay franked dividends. This enables resident domestic investors to claim corporate tax payments as a credit against personal tax obligations under Australia's dividend imputation tax system. This paper estimates a value for the ‘franking credits’ that attach to hybrid securities by examining stock price changes around ex‐dividend dates. We add to the literature that examines the ex‐day price changes of ordinary shares (OS) in that the hybrid securities we examine have high dividend yields and are relatively insensitive to market movements. Therefore the signal‐to‐noise ratio is much higher than for OS. Our analysis reveals that cum‐dividend day prices on hybrid securities do not include any value for franking credits. This result is consistent with the notion that the price‐setting investor in the Australian market is a foreign investor who places no value on franking credits.