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Abstract:
We study the problem of determining the minimum cost of super-replicating a non-negative contingent claim when there are convex constraints on the portfolio weights. It is shown that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a dominating claim, i.e., a claim whose payoffs are increased in an appropriate way relative to the original claim. The results hold for a wide variety of options, including standard European and American calls and puts, multi-asset options, and some path-dependent options. We also provide somewhat similar analysis when there are constraints on the gamma of the replicating portfolio.

Abstract:
This paper studies the returns from investing in index options. Previous research documents significant average option returns, large CAPM alphas, and high Sharpe ratios, and concludes that put options are mispriced. We propose an alternative approach to evaluate the significance of option returns and obtain different conclusions. Instead of using these statistical metrics, we compare historical option returns to those generated by commonly used option pricing models. We find that the most puzzling finding in the existing literature, the large returns to writing out-of-the-money puts, is not even inconsistent with the Black-Scholes model. Moreover, simple stochastic volatility models with no risk premia generate put returns across all strikes that are not inconsistent with the observed data. At-the-money straddle returns are more challenging to understand, and we find that these returns are not inconsistent with explanations such as jump risk premia, Peso problems, and estimation risk.

put pricing puzzle, option returns, jump-diffusion models, risk premia

Abstract:
This paper examines specification issues and estimates volatility and jump risk premia using the information in the cross-section of S&P futures options from 1987 to 2003. We first test for the presence of jumps in volatility by analyzing the higher moment behavior of option implied variance, and we find strong evidence supporting their presence. Based on cross-sectional fit, we find strong evidence for jumps in prices, and modest evidence for jumps in volatility. Regarding the factor risk premiums, we are not able to identify a statistically significant volatility risk premium, but are able to identify statistically significant, although modest jump risk premiums. The jump risk premiums are economically meaningful as they contribute a significant component to the equity risk premium and can explain observed put returns.

Abstract:
In this study, we show how a dynamic insurance program can be implemented within a mean-variance framework. The approach combines elements of the single period safety first idea suggested by Telser and developed by Leibowitz with multiperiod insurance strategies like CPPI and TIPP. The insurance program allows the user to set a probability of hitting a specified floor or target and also allows for changing risk attitudes through time. When the insurance strategy is tested on historical data, the insured portfolio achieves high long-term returns while mostly avoiding long bear markets. In order to understand how the insurance strategy might perform in the future, we simulate returns of the stock market and compare the insurance strategy to buy and hold strategies. An additional benefit of the safety first approach is that it specifies a strategy for underfunded portfolios as well as overfunded portfolios.

insurance, Telser, risk, insurance portfolio

Abstract:
In a contingent claims framework with a single issue of debt and full information, we show that the presence of a bankruptcy code with automatic stay, absolute priority rules, and potential debt forgiveness, can lead to significant conflicts of interest between the borrowers and lenders. In the first-best outcome, the code can add significant value to both parties by way of higher debt capacity, lower credit spreads, and improvement in the overall value of the firm. If control of the ex-ante timing of entering into bankruptcy and the ex-post decision to liquidate once the firm goes into bankruptcy is given to equity holders, most of the benefits of the code are appropriated by the equity holders at the expense of the debt holders. We show that the debt holders can restore the first-best outcome, in large measure, by seizing this control or by the ex-post transfer of control rights which allows them to decide when to liquidate the firm that has been taken to the Chapter 11 process by the equity holders. Irrespective of who is in control of the bankruptcy and liquidation decision, our model implies, based on the term structure of probabilities of default and liquidation, that firms are more likely to default on average and are less likely to liquidate on average relative to the benchmark model of Leland (1994).

Contingent Claims Approach, Default, Liquidation, Optimal Security Values, Control transfer

Abstract:
This paper studies the returns from investing in index options. Previous research documents significant average option returns, large CAPM alphas, and high Sharpe ratios, and concludes that put options are mispriced. We propose an alternative approach to evaluate the significance of option returns and obtain different conclusions. Instead of using these statistical metrics, we compare historical option returns to those generated by commonly used option pricing models. We find that the most puzzling finding in the existing literature, the large returns to writing out-of-the-money puts, is not even inconsistent with the Black-Scholes model. Moreover, simple stochastic volatility models with no risk premia generate put returns across all strikes that are not inconsistent with the observed data. At-the-money straddle returns are more challenging to understand, and we find that these returns are not inconsistent with explanations such as jump risk premia, Peso problems, and estimation risk.

jump risk premia, jump-diffusion models, options returns, put pricing puzzle

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:
In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direct methods, the information from a single simulation can be used to estimate multiple derivatives along with a security's price. The main advantage of the direct methods over re-simulation is increased computational speed. Another advantage is that the direct methods give unbiased estimates of derivatives, whereas the estimates obtained by re-simulation are biased. Computational results are given for both direct methods and comparisons are made to the standard method of re-simulation to estimate derivatives. The methods are illustrated for a path independent model (European options), a path dependent model (Asian options), and a model with multiple state variables (options with stochastic volatility).

Abstract:
This paper develops methods for relating the prices of discrete- and continuous-time versions of path-dependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpreted as shifting a barrier, a strike, or an extremal price. These correction terms enable us to use closed-form solutions for continuous option prices to approximate their discrete counterparts. We also develop discrete-time discrete-state lattice methods for determining accurate prices of discrete and continuous path-dependent options. In several cases, the lattice methods use correction terms based on the connection between discrete- and continuous-time prices which dramatically improve convergence to the accurate price.

Abstract:
Monte Carlo simulation has trouble with American options because the exercise decision at a given date must compare the option's immediate exercise value against its continuation value. The option value if it is not exercised is a function of its value along all possible future price paths from that point on, and each path will present further exercise decisions with the same difficulty in resolving them. The authors propose a hybrid valuation technique that bridges Monte Carlo simulation and lattice methods. Instead of simulating price paths, they simulate whole price trees. The tree emanating from each point is used to assess the option continuation value for that date and stock price. While the results are accurate, inevitably the procedure requires a large number of computations. The authors then offer a variety of techniques that substantially increase efficiency.

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.

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.