Vicky Henderson

The impact of the market portfolio on the valuation, incentives and optimality of executive stock options

This paper examines the effect on valuation and incentives of allowing executives receiving options to trade on the market portfolio. We propose a continuous time utility maximization model to value stock and option compensation from the executives perspective. The executive may invest non-option wealth in the market and riskless asset but not in the company stock itself, leaving them subject to firm-specific risk for incentive purposes. Since the executive is risk averse, this unhedgeable firm risk leads them to place less value on the options than their cost to the company. By distinguishing between these two types of risks, we are able to examine the effect of stock volatility, firm-specific risk and market risk on the value to the executive. In particular, options do not give incentive to increase total risk, but rather to increase the proportion of market relative to firm-specific risk, so executives prefer high beta companies. The paper also examines the relationship between risk and incentives, and finds firm-specific risk decreases incentives whilst market risk may decrease incentives depending on other parameters. The model supports the use of stock rather than options if the company can adjust cash pay when granting stock-based compensation.This paper examines the effect on valuation and incentives of allowing executives receiving options to trade on the market portfolio. We propose a continuous time utility maximization model to value stock and option compensation from the executives perspective. The executive may invest non-option wealth in the market and riskless asset but not in the company stock itself, leaving them subject to firm-specific risk for incentive purposes. Since the executive is risk averse, this unhedgeable firm risk leads them to place less value on the options than their cost to the company. Hay Group Study 2002 cited in Skapinker (2003). By distinguishing between these two types of risks, we are able to examine the effect of stock volatility, firm-specific risk and market risk on the value to the executive. In particular, options do not give incentive to increase total risk, but rather to increase the proportion of market relative to firm-specific risk, so executives prefer high beta companies. The paper also examines the relationship between risk and incentives, and finds firm-specific risk decreases incentives whilst market risk may decrease incentives depending on other parameters. The model supports the use of stock rather than options if the company can adjust cash pay when granting stock-based compensation.

Prospect Theory, Liquidation, and the Disposition Effect

There is a well-known intuition linking prospect theory with the disposition effect, the tendency of investors to sell assets that have risen in value rather than fallen. Recently, several authors have studied rigorous models in an attempt to formalize the intuition. However, some have found it difficult to predict a disposition effect while others produce a more extreme prediction where investors never voluntarily sell at a loss. We solve a model of asset liquidation where investors realize utility over gains and losses, and utility is concave over gains and convex over losses. Under the preferences of Tversky and Kahneman (Tversky, A., D. Kahneman. 1992. Advances in prospect theory: Cumulative representation of uncertainty. J. Risk Uncertainty5(4) 297--323) and lognormal asset prices, investors exhibit a disposition effect as gains are realized at a greater rate than losses. Nonetheless, in contrast to the extant literature, we find that the investor will “give up” and sell at a loss when the asset has a sufficiently low Sharpe ratio. This paper was accepted by Brad Barber, Teck Ho, and Terrance Odean, special issue editors.

Local time, coupling and the passport option

Abstract. A passport option, as introduced and marketed by Bankers Trust, is a call option on the balance of a trading account. The strategy that this account follows is chosen by the option holder, subject to position limits.We derive a simplified form for the price of the passport option using local time. A key insight is that Tanakas formula and the Skorokhod Lemma allow us to prove a direct relationship between the prices of passport and lookback options. Explicit calculations are provided in the case where the underlying is an exponential Brownian motion.A further issue in the analysis of passport options is the identification of the optimal strategy. The second contribution of this article is to extend existing results on the form of the optimal strategy from the exponential Brownian motion model to a wide class of alternative price processes. We achieve this using coupling arguments.

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 agents trading wealth. Further, it is possible to characterize this proportion as the solution to a transcendental equation.

Coupling and Option Price Comparisons in a Jump-Diffusion model

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.

An explicit solution for an optimal stopping/optimal control problem which models an asset sale

In this article we study an optimal stopping/optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function. The interesting features of the solution are that it is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the setup is based on continuous diffusions, the optimal martingale may involve a jump process. One interpretation of the solution is that it is optimal for the risk-averse agent to gamble.

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.

Bounds for in-progress floating-strike Asian options using symmetry

This paper studies symmetries between fixed and floating-strike Asian options and exploits this symmetry to derive an upper bound for the price of a floating-strike Asian. This bound only involves fixed-strike Asians and vanillas, and can be computed simply given one of the many efficient methods for pricing fixed-strike Asian options. The bound coincides with the true price until after the averaging has begun and again at maturity. The bound is compared to benchmark prices obtained via Monte Carlo simulation in numerical examples.

Risk Aversion, Indivisible Timing Options, and Gambling

In this paper we model the behavior of a risk-averse agent who seeks to maximize expected utility and who has an indivisible asset and a timing option over when to sell this asset. Our main contribution is to show that, contrary to intuition, optimal behavior for such a risk-averse agent can include risk-increasing gambles. For example, a manager with a choice over when to disinvest from a project, a private homeowner with a property to sell, or an employee with a grant of American-style stock options may be better off taking positions in other assets with zero Sharpe ratio that are uncorrelated with the underlying project, house, or stock price risk. The results have wider implications for the modeling and interpretation of portfolio optimization problems involving American-style timing decisions.

Is there an informationally passive benchmark for option pricing incorporating maturity

Figlewski proposed testing the incremental contribution of the Black–Scholes model by comparing its performance against an “informationally passive” benchmark, which was defined to be an option pricing formula satisfying static no-arbitrage constraints. In this paper we extend Figlewskis analysis to include options of more than one maturity. Once maturity has been included in the model, any “informationally passive” call pricing function is consistent with some “active” model. In this sense, the notion of a passive model cannot be extended to pricing formulas incorporating option maturity. We derive the index dynamics of the active model implicit in Figlewskis implied G example. These dynamics are far more complicated than the dynamics of the Samuelson–Black–Scholes or Bachelier models. The main implication of our analysis is that an appropriate benchmark for assessing option pricing models should in fact have simple dynamics, such as those of Bachelier or the Black–Scholes models. This is despite the fact that the maturity extension of Figlewskis model gives as good a fit as the Black–Scholes model.