A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING: CLASSICAL SOLUTIONS

In a companion paper, we studied a control problem related to swing option pricing in a general non‐Markovian setting. The main result there shows that the value process of this control problem can uniquely be characterized in terms of a first‐order backward stochastic partial differential equation (BSPDE) and a pathwise differential inclusion. In this paper, we additionally assume that the cash flow process of the swing option is left‐continuous in expectation. Under this assumption, we show that the value process is continuously differentiable in the space variable that represents the volume in which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding BSPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.     

OPTIMAL STOCK SELLING/BUYING STRATEGY WITH REFERENCE TO THE ULTIMATE AVERAGE

We are concerned with the optimal decision to sell or buy a stock in a given period with reference to the ultimate average of the stock price. More precisely, we aim to determine an optimal selling (buying) time to maximize (minimize) the expectation of the ratio of the selling (buying) price to the ultimate average price over the period. This is an optimal stopping time problem which can be formulated as a variational inequality problem. The problem gives rise to a free boundary that corresponds to the optimal selling (buying) strategy. We provide a partial differential equation approach to characterize the free boundary (or equivalently, the optimal selling (buying) region). It turns out that the optimal selling strategy is bang‐bang, which is the same as that obtained by Shiryaev, Xu, and Zhou taking the ultimate maximum of the stock price as benchmark, whereas the optimal buying strategy can be a feedback one subject to the type of averaging and parameter values. Moreover, by a thorough characterization of free boundary, we reveal that the bang‐bang optimal selling strategy heavily depends on the assumption that no time‐vesting restrictions are imposed. If a time‐vested stock is considered, then the optimal selling strategy can also be a feedback one. In terms of a similar analysis developed by the present paper, the same phenomenon can be proved when taking the ultimate maximum as benchmark.     

Multidimensional Variance‐Optimal Hedging in Discrete‐Time Model—A General Approach

One of the well‐known approaches to the problem of option pricing is a minimization of the global risk, considered as the expected quadratic net loss. In the paper, a multidimensional variant of the problem is studied. To obtain the existence of the variance‐optimal hedging strategy in a model without transaction costs, we can apply the result of Monat and Stricker. Another possibility is a generalization of the nondegeneracy condition that appeared in a paper of Schweizer, in which a one‐dimensional problem is solved. The relationship between the two approaches is shown. A more difficult problem is the existence of an optimal solution in the model with transaction costs. A sufficient condition in a multidimensional case is formulated.     

Optimal consumption and investment under transaction costs*

In this paper, we consider the Merton problem in a market with a single risky asset and proportional transaction costs. We give a complete solution of the problem up to the solution of a first‐crossing problem for a first‐order differential equation. We find that the characteristics of the solution (e.g., well‐posedness) can be related to some simple properties of a univariate quadratic whose coefficients are functions of the parameters of the problem. Our solution to the problem via the value function includes expressions for the boundaries of the no‐transaction wedge. Using these expressions, we prove a precise condition for when leverage occurs. One new and unexpected result is that when the solution to the Merton problem (without transaction costs) involves a leveraged position, and when transaction costs are large, the location of the boundary at which sales of the risky asset occur is independent of the transaction cost on purchases.     

A direct solution method for pricing options in regime‐switching models

Pricing financial or real options with arbitrary payoffs in regime‐switching models is an important problem in finance. Mathematically, it is to solve, under certain standard assumptions, a general form of optimal stopping problems in regime‐switching models. In this article, we reduce an optimal stopping problem with an arbitrary value function in a two‐regime environment to a pair of optimal stopping problems without regime switching. We then propose a method for finding optimal stopping rules using the techniques available for nonswitching problems. In contrast to other methods, our systematic solution procedure is more direct as we first obtain the explicit form of the value functions. In the end, we discuss an option pricing problem, which may not be dealt with by the conventional methods, demonstrating the simplicity of our approach.     

Optimal dividend policies with random profitability

We study an optimal dividend problem under a bankruptcy constraint. Firms face a trade‐off between potential bankruptcy and extraction of profits. In contrast to previous works, general cash flow drifts, including Ornstein–Uhlenbeck and CIR processes, are considered. We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as comparison between discontinuous sub‐ and supersolutions of the Hamilton–Jacobi–Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution. The value function is given by a nonlinear partial differential equation (PDE) with a gradient constraint from below in one direction. We find that the optimal strategy is both a barrier and a band strategy and that it includes voluntary liquidation in parts of the state space. Finally, we present and numerically study extensions of the model, including equity issuance and gambling for resurrection.     

ON OPTIMAL INVESTMENT FOR A BEHAVIORAL INVESTOR IN MULTIPERIOD INCOMPLETE MARKET MODELS

We study the optimal investment problem for a behavioral investor in an incomplete discrete‐time multiperiod financial market model. For the first time in the literature, we provide easily verifiable and interpretable conditions for well‐posedness. Under two different sets of assumptions, we also establish the existence of optimal strategies.     

Convex duality for Epstein–Zin stochastic differential utility

This paper introduces a dual problem to study a continuous‐time consumption and investment problem with incomplete markets and Epstein–Zin stochastic differential utilities. Duality between the primal and dual problems is established. Consequently, the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market models, utility specifications, and agent's admissible strategies. Meanwhile, the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the “least favorable” completion of the market.     

ROBUST UTILITY MAXIMIZATION IN NONDOMINATED MODELS WITH 2BSDE: THE UNCERTAIN VOLATILITY MODEL

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is nondominated. We propose studying this problem in the framework of second‐order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power, and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally, several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models.     

LIQUIDATION OF AN INDIVISIBLE ASSET WITH INDEPENDENT INVESTMENT

We provide an extension of the explicit solution of a mixed optimal stopping–optimal stochastic control problem introduced by Henderson and Hobson. The problem examines whether the optimal investment problem on a local martingale financial market is affected by the optimal liquidation of an independent indivisible asset. The indivisible asset process is defined by a homogeneous scalar stochastic differential equation, and the investor's preferences are defined by a general expected utility function. The value function is obtained in explicit form, and we prove the existence of an optimal stopping–investment strategy characterized as the limit of an explicit maximizing strategy. Our approach is based on the standard dynamic programming approach.     

Optimal consumption and investment with liquid and illiquid assets

I consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio that consists of one bond, one liquid risky asset (no transaction costs), and one illiquid risky asset (proportional transaction costs). I fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary ordinary differential equation (ODE) with an integral constraint. I find an explicit characterization of model parameters for the well‐posedness of the problem, and show that the problem is well posed if and only if there exists a shadow price process. Finally, I describe how the investor's optimal strategy is affected by the additional opportunity of trading the liquid risky asset, compared to the simpler model with one bond and one illiquid risky asset.     

OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

BUY‐LOW AND SELL‐HIGH INVESTMENT STRATEGIES

Buy‐low and sell‐high investment strategies are a recurrent theme in the considerations of many investors. In this paper, we consider an investor who aims at maximizing the expected discounted cash‐flow that can be generated by sequentially buying and selling one share of a given asset at fixed transaction costs. We model the underlying asset price by means of a general one‐dimensional Itô diffusion X , we solve the resulting stochastic control problem in a closed analytic form, and we completely characterize the optimal strategy. In particular, we show that, if 0 is a natural boundary point of X , e.g., if X is a geometric Brownian motion, then it is never optimal to sequentially buy and sell. On the other hand, we prove that, if 0 is an entrance point of X , e.g., if X is a mean‐reverting constant elasticity of variance (CEV) process, then it may be optimal to sequentially buy and sell, depending on the problem data.     

OPTIMAL TIMING FOR AN INDIVISIBLE ASSET SALE

In this paper, we investigate the pricing via utility indifference of the right to sell a non‐traded asset. Consider an agent with power utility who owns a single unit of an indivisible, non‐traded asset, and who wishes to choose the optimum time to sell this asset. Suppose that this right to sell forms just part of the wealth of the agent, and that other wealth may be invested in a complete frictionless market. We formulate the problem as a mixed stochastic control/optimal stopping problem, which we then solve. We determine the optimal behavior of the agent, including the optimal criteria for the timing of the sale. It turns out that the optimal strategy is to sell the non‐traded asset the first time that its value exceeds a certain proportion of the agent's trading wealth. Further, it is possible to characterize this proportion as the solution to a transcendental equation.     

A NOTE ON THE QUANTILE FORMULATION

Many investment models in discrete or continuous‐time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change‐of‐variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank‐dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well‐posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law‐invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.     

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.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

Optimal equilibria for time‐inconsistent stopping problems in continuous time

For an infinite‐horizon continuous‐time optimal stopping problem under nonexponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log subadditive and the state process is one‐dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. Although there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, a comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.     

TRANSIENT LINEAR PRICE IMPACT AND FREDHOLM INTEGRAL EQUATIONS

We consider the linear‐impact case in the continuous‐time market impact model with transient price impact proposed by Gatheral. In this model, the absence of price manipulation in the sense of Huberman and Stanzl can easily be characterized by means of Bochner’s theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure‐valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko on the nonexistence of transaction‐triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.