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.     

INCORPORATING RISK AND AMBIGUITY AVERSION INTO A HYBRID MODEL OF DEFAULT

It is well known that purely structural models of default cannot explain short‐term credit spreads, while purely intensity‐based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short‐term spreads.     

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.     

Shortfall aversion

Shortfall aversion reflects the higher utility loss of spending cuts from a reference than the utility gain from similar spending increases. Inspired by Prospect Theory's loss aversion and the peak‐end rule, this paper posits a model of utility from spending scaled by past peak spending. In contrast to traditional models, which call for spending rates proportional to wealth, the optimal policy in this model implies a constant spending rate equal to the historical peak when wealth is relatively large. The spending rate increases when wealth reaches a model‐determined multiple of peak spending. In 1926–2015, shortfall‐averse spending is smooth and typically increasing.     

INDIFFERENCE PRICE WITH GENERAL SEMIMARTINGALES

Using duality methods, we prove several key properties of the indifference price π for contingent claims. The underlying market model is very general and the mathematical formulation is based on a duality naturally induced by the problem. In particular, the indifference price π turns out to be a convex risk measure on the Orlicz space induced by the utility function.     

Risk aversion,disappointment aversion,and futures hedging

This article examines the effect of disappointment aversion on futures hedging. We incorporated a constant‐absolute‐risk‐aversion (CARA) utility function into the disappointment‐aversion framework of Gul (1991). It is shown that a more disappointment‐averse hedger will choose an optimal futures position closer to the minimum‐variance hedge than will a less‐disappointment‐averse hedger. The effect of disappointment aversion is stronger when the hedger is less risk averse. A small disappointment aversion will cause a near‐risk neutral hedger to take a drastically different position. In addition, a more‐risk‐averse or disappointment‐averse hedger will have a lower reference point. Numerical results indicate that the reference point of a disappointment‐averse hedger tends to be lower than that of a conventional loss‐averse hedger. Consequently, the disappointment‐averse hedger will act more conservatively, not exploiting profitable opportunities as much as the conventional loss averse hedger will. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:123–141, 2002     

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.     

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.     

PORTFOLIOS OF AMERICAN OPTIONS UNDER GENERAL PREFERENCES: RESULTS AND COUNTEREXAMPLES

We consider the optimal exercise of a portfolio of American call options in an incomplete market. Options are written on a single underlying asset but may have different characteristics of strikes, maturities, and vesting dates. Our motivation is to model the decision faced by an employee who is granted options periodically on the stock of her company, and who is not permitted to trade this stock. The first part of our study considers the optimal exercise of single options. We prove results under minimal assumptions and give several counterexamples where these assumptions fail—describing the shape and nesting properties of the exercise regions. The second part of the study considers portfolios of options with differing characteristics. The main result is that options with comonotonic strike, maturity, and vesting date should be exercised in order of increasing strike. It is true under weak assumptions on preferences and requires no assumptions on prices. Potentially the exercise ordering result can significantly reduce the complexity of computations in a particular example. This is illustrated by solving the resulting dynamic programming problem in a constant absolute risk aversion utility indifference model.     

A Note on Loss Aversion and Futures Hedging

This note examines the effect of loss aversion on the futures trading behavior of a short hedger. Using a modified constant‐absolute‐risk‐aversion utility function, I show that loss aversion has no effect in an unbiased futures market. It has different, predictable impacts when the futures market is in backwardation or contango. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 681–692, 2001     

LONG HORIZONS,HIGH RISK AVERSION,AND ENDOGENOUS SPREADS

For an investor with constant absolute risk aversion and a long horizon, who trades in a market with constant investment opportunities and small proportional transaction costs, we obtain explicitly the optimal investment policy, its implied welfare, liquidity premium, and trading volume. We identify these quantities as the limits of their isoelastic counterparts for high levels of risk aversion. The results are robust with respect to finite horizons, and extend to multiple uncorrelated risky assets. In this setting, we study a Stackelberg equilibrium, led by a risk‐neutral, monopolistic market maker who sets the spread as to maximize profits. The resulting endogenous spread depends on investment opportunities only, and is of the order of a few percentage points for realistic parameter values.     

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 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.     

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.     

THE DEMAND FOR A RISKY ASSET: SIGNING,JOINTLY AND SEPARATELY,THE EFFECTS OF THREE DISTRIBUTIONAL SHIFTS

We show that, if an individual's utility function exhibits a degree of relative temperance smaller than one, the individual will react, in a plausible way, to each of three common shifts in the stochastic distribution of his wealth, namely to FSD shifts, mean‐preserving spreads and increases in downside risk. First, we derive, in a unified setting, necessary and sufficient conditions for signing the comparative‐static effects of each of these shifts separately, and, second, we invoke implications of the property of mixed risk aversion to merge these separate conditions into a single sufficient condition for jointly signing all comparative‐static effects.     

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.     

Nonlinear price impact and portfolio choice

In a market with price impact proportional to a power of the order flow, we find optimal trading policies and their implied performance for long‐term investors who have constant relative risk aversion and trade a safe asset and a risky asset following geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the curvature of the price impact function. Trading rates are finite as with linear impact, but are lower near the target portfolio, and higher away from the target. The model nests the square‐root impact law and, as extreme cases, linear impact and proportional transaction costs.     

Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear price impact. In our model, cross‐impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second‐order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.     

When to sell an asset amid anxiety about drawdowns

We consider risk‐averse investors with different levels of anxiety about asset price drawdowns. The latter is defined as the distance of the current price away from its best performance since inception. These drawdowns can increase either continuously or by jumps, and will contribute toward the investor's overall impatience when breaching the investor's private tolerance level. We investigate the unusual reactions of investors when aiming to sell an asset under such adverse market conditions. Mathematically, we study the optimal stopping of the utility of an asset sale with a random discounting that captures the investor's overall impatience. The random discounting is given by the cumulative amount of time spent by the drawdowns in an undesirable high region, fine‐tuned by the investor's personal tolerance and anxiety about drawdowns. We prove that in addition to the traditional take‐profit sales, the real‐life employed stop‐loss orders and trailing stops may become part of the optimal selling strategy, depending on different personal characteristics. This paper thus provides insights on the effect of anxiety and its distinction with traditional risk aversion on decision making.     

OPTIMAL RISK SHARING FOR LAW INVARIANT MONETARY UTILITY FUNCTIONS

We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents' utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk.