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 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.     

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.     

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.     

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

In this paper, we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete‐ and continuous‐time optimal stopping problems. In this context, we consider tractability of such problems via a useful notion of semitractability and the introduction of a tractability index for a particular numerical solution algorithm. It is shown that in the discrete‐time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by with being the dimension of the underlying Markov chain. Furthermore, we study the WSM approach in the context of continuous‐time optimal stopping problems and derive the corresponding complexity bounds. Although we cannot prove semitractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example.     

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.     

Equilibrium concepts for time‐inconsistent stopping problems in continuous time

A new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, which in this paper are called mild equilibrium and weak equilibrium, respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous‐time Markov chain and the discount function is log subadditive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.     

Optimal Stopping and the American Put

We show that the problem of pricing the American put is equivalent to solving an optimal stopping problem. the optimal stopping problem gives rise to a parabolic free-boundary problem. We show there is a unique solution to this problem which has a lower boundary. We identify an integral equation solved by the boundary and show that it is the unique solution to this equation satisfying certain natural additional conditions. the proofs also give a natural decomposition of the price of the American option as the sum of the price of the European option and an "American premium."     

General stopping behaviors of naïve and noncommitted sophisticated agents,with application to probability distortion

We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time‐inconsistent. We study stopping decisions of naïve agents who reoptimize continuously in time, as well as equilibrium strategies of sophisticated agents who anticipate but lack control over their future selves' behaviors. When the state process is one dimensional and the payoff functional satisfies some regularity conditions, we prove that any equilibrium can be obtained as a fixed point of an operator. This operator represents strategic reasoning that takes the future selves' behaviors into account. We then apply the general results to the case when the agents distort probability and the diffusion process is a geometric Brownian motion. The problem is inherently time‐inconsistent as the level of distortion of a same event changes over time. We show how the strategic reasoning may turn a naïve agent into a sophisticated one. Moreover, we derive stopping strategies of the two types of agent for various parameter specifications of the problem, illustrating rich behaviors beyond the extreme ones such as “never‐stopping” or “never‐starting.”     

DUAL REPRESENTATIONS FOR GENERAL MULTIPLE STOPPING PROBLEMS

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cash flows which are subject to volume constraints modeled by integer‐valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers ( 2012 ), Bender ( 2011a ), Bender ( 2011b ), Aleksandrov and Hambly ( 2010 ), and Meinshausen and Hambly ( 2004 ) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cash flow structures than the additive structure in the above references. For example, some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for prices of multiple exercise options and illustrate it with a numerical study on the pricing of a swing option in an electricity market.     

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.     

A STATE‐CONSTRAINED DIFFERENTIAL GAME ARISING IN OPTIMAL PORTFOLIO LIQUIDATION

We consider n risk‐averse agents who compete for liquidity in an Almgren–Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed‐form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.     

THE NUMÉRAIRE PROPERTY AND LONG‐TERM GROWTH OPTIMALITY FOR DRAWDOWN‐CONSTRAINED INVESTMENTS

We consider the portfolio choice problem for a long‐run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude toward risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numéraire property through the notion of expected relative return and prove that drawdown‐constrained numéraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time‐horizon becomes distant, the drawdown‐constrained numéraire portfolio is given explicitly through a model‐independent transformation of the unconstrained numéraire portfolio. The asymptotically growth‐optimal strategy is obtained as limit of numéraire strategies on finite horizons.     

RETHINKING DYNAMIC CAPITAL STRUCTURE MODELS WITH ROLL‐OVER DEBT

Dynamic capital structure models with roll‐over debt rely on widely accepted arguments that have never been formalized. This paper clarifies the literature and provides a rigorous formulation of the equity holders’ decision problem within a game theory framework. We spell out the linkage between default policies in a rational expectations equilibrium and optimal stopping theory. We prove that there exists a unique equilibrium in constant barrier strategies, which coincides with that derived in the literature. Furthermore, that equilibrium is the unique equilibrium when the firm loses all its value at default time. Whether the result holds when there is a recovery at default remains a conjecture.     

OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS

The connection between optimal stopping of random systems and the theory of the Snell envelop is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization of these results to the case of possible multiple stopping. We prove existence of the multiple exercise policies in a fairly general set-up. We then concentrate on the Black–Scholes model for which we give a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case. The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms based on ideas of the Malliavin calculus.     

BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.     

OPTIMAL EXECUTION OF A VWAP ORDER: A STOCHASTIC CONTROL APPROACH

We consider the optimal liquidation of a position of stock (long or short) where trading has a temporary market impact on the price. The aim is to minimize both the mean and variance of the order slippage with respect to a benchmark given by the market volume‐weighted average price (VWAP). In this setting, we introduce a new model for the relative volume curve which allows simultaneously for accurate data fit, economic justification, and mathematical tractability. Tackling the resulting optimization problem using a stochastic control approach, we derive and solve the corresponding Hamilton–Jacobi–Bellman equation to give an explicit characterization of the optimal trading rate and liquidation trajectory.     

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.     

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.