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Abstract:
This paper proposes a new simulation-based approach for optimal portfolio allocation in realistic environments with complex dynamics for the state variables and large numbers of factors and assets. A first illustration involves a choice between equity and cash with nonlinear interest rate and market price of risk dynamics. Intertemporal hedging demands significantly increase the demand for stocks and exhibit low volatility. We then analyze settings where stock returns are also predicted by dividend yields and where investors have wealth-dependent relative risk aversion. Large-scale problems with many assets, including the Nasdaq, SP500, bonds, and cash, are also examined.

Abstract:
This paper provides (i) new results on the structure of optimal portfolios, (ii) economic insights on the behavior of the hedging components and (iii) simulation-based methods for numerical implementation of allocation rules. The core of our approach relies on closed-form solutions for functionals of diffusion processes which simplify their numerical simulation and facilitate the computation and simulation of the hedging components of optimal portfolios. One of our procedures relies on a variance-stabilizing transformation of the underlying process which eliminates stochastic integrals from the representation of random variables in hedging terms and ensures the existence of an exact weak approximation scheme. This improves the performance of Monte-Carlo methods in the numerical implementation of portfolio rules derived on the basis of probabilistic arguments. Our approach is flexible and can be used even when the dimensionality of the set of underlying state variables is large. We implement the procedure for a class of bivariate and trivariate models in which the uncertainty is described by diffusion processes for the market price of risk (MPR), the interest rate (IR) and other relevant factors. After calibrating the models to the data we document the behavior of the portfolio demand and the hedging components relative to the parameters of the model such as risk aversion, investment horizon, speeds of mean-reversion, IR and MPR levels and volatilities. We show that the hedging terms are important and cannot be ignored for asset allocation purposes. Risk aversion and investment horizon emerge as the most relevant factors: they have a substantial impact on the size of the optimal portfolio and on its economic properties for realistic values of the models' parameters.

Abstract:
This article reviews recent scientific literature on consumer financial decisions over the life cycle outlining its implications for the design of pension plans. It begins with a review of advances in the theory of rational financial planning and wealth management. It then summarizes the recent empirical literature on the actual behavior of households regarding saving, investing, and insuring their consumption in old age. Finally, it briefly comments on the practical implications of the theory for the design of pension systems and outlines areas of future research.

Life cycle finance, portfolio choice, pension, consumption, leisure

Abstract:
We examine the portfolio choice problem of an investor with constant relative risk aversion in a financial market with partially hedgeable interest rate risk. The individual shadow price of the portfolio constraint is characterized as the solution of a new backward equation involving Malliavin derivatives. A generalization of this equation is studied and solved in explicit form. This result, applied to our financial model, yields closed-form solutions for the shadow price and the optimal portfolio. The effects of parameters such as risk aversion, interest rate volatility, investment horizon, and tightness of the constraint are examined. Applications of our method to a monetary economy with inflation risk and to an international setting with currency risk are also provided.

Abstract:
A dynamic asset-liability management model for defined-benefit pension plans is developed. The plan can be in surplus or deficit. The sponsor is loss averse and tolerates limited shortfalls in assets under management relative to the liability due. The optimal contribution policy, the optimal dividend policy and the associated asset allocation rule are derived and analyzed. Sound Asset-Liability Management is shown to entail future withdrawals from as well as future contributions to the pension fund, even if the current funding shortfall is large. The impact of model parameters, such as contribution capacity, shortfall ratios tolerated, risk aversion and loss aversion, is examined. Wealth effects are found to be critical for the properties of asset allocation rules.

Abstract:
This paper examines the valuation of European- and American-style volatility options based on a general equilibrium stochastic volatility framework. Properties of the optimal exercise region and of the option price are provided when volatility follows a general diffusion process. Explicit valuation formulas are derived in four particular cases. Emphasis is placed on the MRLP (mean-reverting in the log) volatility model which has received considerable empirical support. In this context we examine the properties and hedging behavior of volatility options. Unlike American options, European call options on volatility are found to display concavity at high levels of volatility.

American options, early exercise premium, European options, hedging, optimal exercise, stochastic volatility, viability

Abstract:
This paper addresses the problem of valuing American call options with caps on dividend paying assets. Since early exercise is allowed, the valuation problem requires the determination of optimal exercise policies. Options with two types of caps are analyzed: constant caps and caps with a constant growth rate. For constant caps the optimal exercise policy is to exercise at the first time at which the underlying asset's price equals or exceeds the minimum of the cap and the optimal exercise boundary for the corresponding uncapped option. For caps that grow at a constant rate the optimal exercise strategy can be specified by three endogenous parameters.

Abstract:
In this paper we provide valuation formulas for several types of American options on two or more assets. First we characterize the optimal exercise regions and provide valuation formulas for a number of American option contracts on multiple underlying assets with convex payoff functions. Examples include options on the maximum of two assets, dual strike options, spread options, exchange options, options on the product and powers of the product, and options on the arithmetic average of two assets. Second, we also consider a class of contracts with non-convex payoffs, such as American capped exchange options. For this option we explicitly identify the optimal exercise boundary and provide a decomposition of the price in terms of a capped exchange option with automatic exercise at the cap and an early exercise premium involving the benefits of exercising prior to reaching the cap.

Abstract:
We provide a simple binomial framework to value American-style derivatives subject to trading restrictions. The optimal investment of liquid wealth is solved simultaneously with the early exercise decision of the non-traded derivative. No-short-sales constraints on the underlying asset manifest themselves in the form of an implicit dividend yield in the risk neutralized process for the underlying asset. One consequence is that American call options may be optimally exercised prior to maturity even when the underlying asset pays no dividends. Applications to Executive Stock Options (ESO) are presented: it is shown that the value of an ESO could be substantially lower than that computed using the Black-Scholes model. We also analyze non-traded payoffs based on a price that is imperfectly correlated with the price of a traded asset.

Abstract:
We develop lower and upper bounds on the prices of American call and put options written on a dividend paying asset. We provide two option price approximations, one based on the lower bound (term LBA) and one based on both bounds (termed LUBA). The LUBA approximation has an average accuracy comparable to a 1000-step binomial tree with a computation speed comparable to a 50-step binomial tree. We introduce a modification of the binomial method (termed BBSR) which is very simple to implement and performs remarkably well. We also conduct a careful large-scale evaluation of many recent methods for computing American option prices.

Abstract:
In this paper we provide lower and upper bounds on the prices of American call and put options written on a dividend paying asset. Based on the bounds, we provide two option price approximations. Our second approximation, which uses both lower and upper bound information, has an average accuracy comparable to a 1000-step binomial tree with a computation speed comparable to a 50-step binomial tree. Put another way, our second approximation is 6 times more accurate than a 200-step binomial tree and is about 15 times faster than a 200-step binomial tree. Furthermore, the approximations are sufficiently simple that they can be computed in a spreadsheet. In addition, we conduct a careful large-scale evaluation of many recent methods for computing American option prices. Comparisons are made on the basis of accuracy and speed of computation and lead to some surprising results.

Abstract:
In this paper, we consider American option contracts when the underlying asset has stochastic dividends and stochastic volatility. We provide a full discussion of the theoretical foundations of American option valuation and exercise boundaries. We show how they depend on the various sources of uncertainty which drive dividend rates and volatility, and derive equilibrium asset prices, derivative prices and optimal exercise boundaries in a general equilibrium model. The theoretical models yield fairly complex expressions which are difficult to estimate. We therefore adopt a nonparametric approach which enables us to investigate reduced forms. Indeed, we use nonparametric methods to estimate call prices and exercise boundaries conditional on dividends and volatility. Since the latter is a latent process, we propose several approaches, notably using EGARCH filtered estimates, implied and historical volatilities. The nonparametric approach allows us to test whether call prices and exercise decisions are primarily driven by dividends, as has been advocated by Harvey and Whaley (1992a,b) and Fleming and Whaley (1994) for the OEX contract, or whether stochastic volatility complements dividend uncertainty. We find that dividends alone do not account for all aspects of call option pricing and exercise decisions, suggesting a need to include stochastic volatility.