Nontraded Asset Valuation with Portfolio Constraints: A Binomial Approach

TLDR: Detemple et al. as mentioned in this paper provided a simple binomial framework to value American-style derivatives subject to trading restrictions, where the optimal investment of liquid wealth is solved simultaneously with the early exercise decision of the nontraded derivative.

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 nontraded derivative. Noshort-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 nontraded payoffs based on a price that is imperfectly correlated with the price of a traded asset. The economics of asset pricing when one or more of the assets in the opportunity set are either subject to trading restrictions or entirely nontraded is a matter of great interest. Viewed from a practical perspective, we have several important examples of such assets that are subject to trading restrictions. Pensions, which represent perhaps the most significant of assets held by individual households, are subject to trading restrictions. It is typically the case that assets in pensions are not available for immediate consumption. Borrowing against pension assets is subject to significant direct and indirect costs by way of taxes and early withdrawal penalties. Human capital is another example. Housing investment is also illiquid and subject to significant transaction costs. Together, pensions, human capital, and housing constitute a substantial part of a typical household’s assets. The significance of such nontraded assets for risk premia has already been noted by Bewley (1982). There are other circumstances where lack of unrestricted trading plays an important role. Executive compensation plans usually take the This article was presented at Boston University. We would like to thank the editor, Bernard Dumas, and two anonymous referees for very helpful comments. We also thank Nalin Kulatilaka and seminar participants for useful suggestions. Ganlin Chang, Fei Guan, and Carlton Osakwe provided valuable research assistance. J. Detemple acknowledges support from the Social Sciences and Humanities Research ‐