Portfolio selection with a drawdown constraint

When identifying optimal portfolios, practitioners often impose a drawdown constraint. This constraint is even explicit in some money management contracts such as the one recently involving Merrill Lynch’ management of Unilever’s pension fund. In this setting, we provide a characterization of optimal portfolios using mean–variance analysis. In the absence of a benchmark, we find that while the constraint typically decreases the optimal portfolio’s standard deviation, the constrained optimal portfolio can be notably mean–variance inefficient. In the presence of a benchmark such as in the Merrill Lynch–Unilever contract, we find that the constraint increases the optimal portfolio’s standard deviation and tracking error volatility. Thus, the constraint negatively affects a portfolio manager’s ability to track a benchmark.     

Optimal Debt and Profitability in the Trade‐Off Theory

I develop a dynamic model of leverage with tax deductible interest and an endogenous cost of default. The interest rate includes a premium to compensate lenders for expected losses in default. A borrowing constraint is generated by lenders' unwillingness to lend an amount that would trigger immediate default. When the borrowing constraint is not binding, the trade‐off theory of debt holds: optimal debt equates the marginal interest tax shield and the marginal expected cost of default. Contrary to conventional interpretation, but consistent with empirical findings, increases in current or future profitability reduce the optimal leverage ratio when the trade‐off theory holds.     

A unified model of entrepreneurship dynamics

We develop an incomplete-markets q-theoretic model to study entrepreneurship dynamics. Precautionary motive, borrowing constraints, and capital illiquidity lead to underinvestment, conservative debt use, under-consumption, and less risky portfolio allocation. The endogenous liquid wealth-illiquid capital ratio w measures time-varying financial constraint. The option to accumulate wealth before entry is critical for entrepreneurship. Flexible exit option is important for risk management purposes. Investment increases and the private marginal value of liquidity decreases as w decreases and exit becomes more likely, contrary to predictions of standard financial constraint models. We show that the idiosyncratic risk premium is quantitatively significant, especially for low w.     

Minimal Hellinger martingale measures of order <Emphasis Type="Italic">q</Emphasis>

This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q≠1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Föllmer and Schweizer (here q=2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in ? ^d . Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market’s exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.     

Optimal dividend policy and growth option

We analyse the interaction between the dividend policy and the decision on investment in a growth opportunity of a liquidity constrained firm. This leads us to study a mixed singular control/optimal stopping problem for a diffusion that we solve quasi-explicitly by establishing a connection with an optimal stopping problem. We characterize situations where it is optimal to postpone the distribution of dividends in order to invest at a subsequent date in the growth opportunity. We show that uncertainty and liquidity shocks have an ambiguous effect on the investment decision.      

Quadratic minimization with portfolio and terminal wealth constraints

We address a problem of stochastic optimal control drawn from the area of mathematical finance. The goal is to minimize the expected value of a general quadratic loss function of the wealth at close of trade when there is a specified convex constraint on the portfolio over the trading interval, together with a specified almost-sure lower-bound on the wealth at close of trade. We use a variational approach of Rockafellar which leads naturally to an appropriate vector space of dual variables, a dual functional on the space of dual variables such that the dual problem of maximizing the dual functional is guaranteed to have a solution (i.e. a Lagrange multiplier) when a simple and natural Slater condition holds for the terminal wealth constraint, and obtain necessary and sufficient conditions for optimality of a candidate wealth process. The dual variables are pairs, each comprising an Itô process paired with a member of the adjoint of the space of essentially bounded random variables measurable with respect to the event \(\sigma \)-algebra at close of trade. The necessary and sufficient conditions are used to construct an optimal portfolio in terms of the Lagrange multiplier. The dual problem simplifies to maximization of a concave function over the real line when the portfolio is unconstrained but the terminal wealth constraint is maintained.     

A real options perspective on the future of the Euro

A break-up of the Eurozone is no longer regarded as implausible. This will be a costly and irreversible decision in conditions of continuing uncertainty; therefore it is amenable to analysis in the real options framework. We do so by solving as an n-dimensional optimal stopping problem with country-specific shocks and “convergence” of member economies. We compare a complete break-up with individual country departures. In calibrated solutions for a symmetric case we find a non-negligible but small option value. Furthermore, we find a new theoretical result on the non-monotonicity of abandonment threshold with respect to volatility.     

Investment, Uncertainty, and Liquidity

We analyze the dynamic investment decision of a firm subject to an endogenous financing constraint. The threat of future funding shortfalls lowers the value of the firm's timing options and encourages acceleration of investment beyond the first‐best optimal level. As well as highlighting another way by which capital market frictions can distort investment behavior, this result implies that (1) the sensitivity of investment to cash flow can be greatest for high‐liquidity firms and (2) greater uncertainty has an ambiguous effect on investment.     

Contracting with private rewards

The canonical moral hazard model is extended to allow the agent to face endogenous and noncontractible uncertainty. The agent works for the principal and simultaneously pursues outside rewards. The contract offered by the principal thus manipulates the agent's work–life balance. The participation constraint is slack whenever it is optimal to distort the agent's work–life balance away from life compared to a symmetric-information benchmark. Then, the agent's expected utility is high and he faces flatter incentives. Such contracts may be optimal when the two activities are strong substitutes in the agent's cost function or when reservation utility is low.     

Least-squares Monte-Carlo methods for optimal stopping investment under CEV models

The optimal stopping investment is a kind of mixed expected utility maximization problems with optimal stopping time. The aim of this paper is to develop the least-squares Monte-Carlo methods to solve the optimal stopping investment under the constant elasticity of variance (CEV) model. Such a problem has no closed-form solutions for the value functions, optimal strategies and optimal exercise boundaries due to the early exercised feature. The dual optimal stopping problem is first derived and then the strong duality between the dual and prime problems is established. The least-squares Monte-Carlo methods based on the dual control theory are developed and numerical simulations are provided. Both the power and non-HARA utilities are studied.     

Optimal Insurance Under the Insurer's VaR Constraint

In this paper, we impose the insurer's Value at Risk (VaR) constraint on Arrow's optimal insurance model. The insured aims to maximize his expected utility of terminal wealth, under the constraint that the insurer wishes to control the VaR of his terminal wealth to be maintained below a prespecified level. It is shown that when the insurer's VaR constraint is binding, the solution to the problem is not linear, but piecewise linear deductible, and the insured's optimal expected utility will increase as the insurer becomes more risk-tolerant. Basak and Shapiro (2001) showed that VaR risk managers often choose larger risk exposures to risky assets. We draw a similar conclusion in this paper. It is shown that when the insured has an exponential utility function, optimal insurance based on VaR constraint causes the insurer to suffer larger losses than optimal insurance without insurer's risk constraint.     

Quadratic minimization with portfolio and intertemporal wealth constraints

We address a problem of stochastic optimal control motivated by portfolio optimization in mathematical finance, the goal of which is to minimize the expected value of a general quadratic loss function of the wealth at close of trade when there is a specified convex constraint on the portfolio, together with a specified almost-sure lower-bound on intertemporal wealth over the full trading interval. A precursor to the present work, by Heunis (Ann Financ 11:243–282, 2015), addressed the simpler problem of minimizing a general quadratic loss function with a convex portfolio constraint and a stipulated almost-sure lower-bound on the wealth only at close of trade. In the parlance of optimal control the problem that we shall address here exhibits the combination of a control constraint (i.e. the portfolio constraint) together with an almost-sure intertemporal state constraint (on the wealth over the full trading interval). Optimal control problems with this combination of constraints are well known to be quite challenging even in the deterministic case, and of course become still more so when one deals with these same constraints in a stochastic setting. We nevertheless find that an ingenious variational approach of Rockafellar (Conjugate duality and optimization, CBMS-NSF series no. 16, SIAM, 1974), which played a key role in the precursor work noted above, is fully equal to the challenges posed by this problem, and leads naturally to an appropriate vector space of dual variables, together with a dual functional on the space of dual variables, such that the dual problem of maximizing the dual functional is guaranteed to have a solution (or Lagrange multiplier) when the problem constraints satisfy a simple and natural Slater condition. We then establish necessary and sufficient conditions for the optimality of a candidate wealth process in terms of the Lagrange multiplier, and use these conditions to construct an optimal portfolio.     

Pricing and static hedging of American-style options under the jump to default extended CEV model

This paper prices (and hedges) American-style options through the static hedge approach (SHP) proposed by Chung and Shih (2009) and extends the literature in two directions. First, the SHP approach is generalized to the jump to default extended CEV (JDCEV) model of Carr and Linetsky (2006), and plain-vanilla American-style options on defaultable equity are priced. The robustness and efficiency of the proposed pricing solutions are compared with the optimal stopping approach offered by Nunes (2009), under both the JDCEV framework and the nested constant elasticity of variance (CEV) model of Cox (1975), using different elasticity parameter values. Second, the early exercise boundary near expiration is derived under the JDCEV model.     

Optimal insurance contract under a value-at-risk constraint

This study develops an optimal insurance contract endogenously under a value-at-risk (VaR) constraint. Although Wang et al. [2005] had examined this problem, their assumption implied that the insured is risk neutral. Consequently, this study extends Wang et al. [2005] and further considers a more realistic situation where the insured is risk averse. The study derives the optimal insurance contract as a single deductible insurance when the VaR constraint is redundant or as a double deductible insurance when the VaR constraint is binding. Finally, this study discusses the optimal coverage level from common forms of insurances, including deductible insurance, upper-limit insurance, and proportional coinsurance. JEL Classification G22     

Pricing and static hedging of American-style knock-in options on defaultable stocks

This paper applies the static hedge portfolio approach (SHP) of Chung et al. (2013) in two new directions. First, the SHP approach is generalized from the constant elasticity of variance (CEV) model of Cox (1975) to the jump to default extended CEV (JDCEV) framework of Carr and Linetsky (2006). For this purpose, the recovery value of the American-style down-and-in put is hedged through the one attached to a European-style plain-vanilla contract whereas for an up-and-in put it is necessary to use the recovery component of the corresponding European-style up-and-in option. Second, the SHP methodology is adapted from single to double barrier American-style knock-in options by matching the value of the hedging portfolio along both lower and upper barriers. Finally, and to benchmark the accuracy of the novel SHP pricing solutions, the optimal stopping approach of Nunes (2009) is also extended to price American-style double knock-in options under the JDCEV framework. Such extension highlights the relevant credit derivative component embedded in American-style knock-in equity puts.     

Partial liquidation under reference-dependent preferences

We propose a multiple optimal stopping model where an investor can sell a divisible asset position at times of her choosing. Investors have $S$-shaped reference-dependent preferences, whereby utility is defined over gains and losses relative to a reference level and is concave over gains and convex over losses. For a price process following a time-homogeneous diffusion, we employ the constructive potential-theoretic solution method developed by Dayanik and Karatzas (Stoch. Process. Appl. 107:173–212, 2003). As an example, we revisit the single optimal stopping model of Kyle et al. (J. Econ. Theory 129:273–288, 2006) to allow partial liquidation. In contrast to the extant literature, we find that the investor may partially liquidate the asset at distinct price thresholds above the reference level. Under other parameter combinations, the investor sells the asset in a block, either at or above the reference level.     

Optimal mean-reverting spread trading: nonlinear integral equation approach

We study several optimal stopping problems that arise from trading a mean-reverting price spread over a finite horizon. Modeling the spread by the Ornstein–Uhlenbeck process, we analyze three different trading strategies: (i) the long-short strategy; (ii) the short-long strategy, and (iii) the chooser strategy, i.e. the trader can enter into the spread by taking either long or short position. In each of these cases, we solve an optimal double stopping problem to determine the optimal timing for starting and subsequently closing the position. We utilize the local time-space calculus of Peskir (J Theor Probab 18:499–535, 2005a) and derive the nonlinear integral equations of Volterra-type that uniquely characterize the boundaries associated with the optimal timing decisions in all three problems. These integral equations are used to numerically compute the optimal boundaries.     

Unemployment insurance with a hidden labor market

We consider the problem of optimal unemployment insurance (UI) in a repeated moral hazard framework. Unlike existing literature, unemployed individuals can secretly participate in a hidden labor market. This extension modifies the standard problem in three dimensions. First, it imposes an endogenous lower bound for the lifetime utility that a contract can deliver. Second, it breaks the identity between unemployment payments and consumption. And third, it hardens the encouragement of search effort. The optimal unemployment insurance system in an economy with a hidden labor market is simple, with an initial phase in which payments are relatively flat during unemployment and with no payments for long-term unemployed individuals. This scheme differs substantially from the one prescribed without a hidden labor market and resembles unemployment protection programs in many countries.     

Monetary conservatism and fiscal policy

Does an inflation conservative central bank à la Rogoff (1985) remain desirable in a setting with endogenous fiscal policy? To provide an answer we study monetary and fiscal policy games without commitment in a dynamic, stochastic sticky-price economy with monopolistic distortions. Monetary policy determines nominal interest rates and fiscal policy provides public goods generating private utility. We find that lack of fiscal commitment gives rise to excessive public spending. The optimal inflation rate internalizing this distortion is positive, but lack of monetary commitment generates too much inflation. A conservative monetary authority thus remains desirable. When fiscal policy is determined before monetary policy each period, the monetary authority should focus exclusively on stabilizing inflation. Monetary conservatism then eliminates the steady state biases associated with lack of monetary and fiscal commitment and leads to stabilization policy that is close to optimal.     

A framework for the study of expansion options,loan commitments and agency costs

We consider a firm that operates a single plant and has an expansion option to invest in a new plant. This setup leads to two-sided optimal stopping problems. We analyze optimal expansion timing and quantify the value of the loan commitment that the equityholder obtained from the lender and associated agency costs incurred on the lender's side. Moreover, we incorporate construction period for the new plant, which throws another layer of uncertainty into the model: the parties cannot tell price level of the firm's product when the construction completes. This analysis contrasts with the conventional one-sided stopping models in corporate finance literature. We can study expansion options by viewing a firm's existing operation, bankruptcy threat, and financing decisions all together.