Valuing the option to invest in an incomplete market

Abstract

This paper considers the impact of entrepreneurial risk aversion and incompleteness on investment timing and the value of the option to invest. A risk averse entrepreneur faces the irreversible decision of when to pay a cost in order to receive a one-off investment payoff. The uncertainty associated with the investment payoff can be partly offset by hedging, but the remaining unhedgeable risk is idiosyncratic. Nested within our incomplete set-up is the complete model of McDonald and Siegel (Q J Econ 101:707-727, 1986) which assumes investment payoffs are perfectly spanned by traded assets. We find risk aversion and idiosyncratic risk erode option value and lower the investment threshold. Our main finding is that there is a parameter region within which the complete and incomplete models give differing investment signals. In this region, the option is never exercised (and investment never occurs) in the complete model, whereas the entrepreneur exercises the option in the incomplete setting. Strikingly, this parameter region corresponds to a negative implicit dividend yield on the payoff, and so this exercise behavior contrasts with conventional wisdom of Merton (Bell J Econ Manage 4:141-183, 1973) for complete markets. Finally, in this parameter region, increased volatility speeds-up investment and option values are not strictly convex in project value, in sharp contrast to the conclusion of standard real options models.

Getting Real with Real Options: A Utility–Based Approach for Finite–Time Investment in Incomplete Markets

“…Exceptions to this adherence to a 'near completion' assumption, but still in the context of an infinite time horizon, are Hugonnier and Morellec (2007) and Henderson (2007). In the first paper, a risk averse manager facing an investment decision tries to maximize his expected utility considering the effect that shareholders' external control will have on his personal wealth.…”

Portfolios of American Options Under General Preferences: Results and Counterexamples

“…Models based on utility‐indifference pricing of American options capture many of the important aspects of the employee's situation. Indeed, there are a number of papers in this vein, notably Carpenter et al (), Grasselli and Henderson (), Henderson (), Leung and Sircar (), and Rogers and Scheinkman (), all of whom study American option pricing under utility indifference for single or multiple identical options. (There is also a long literature in finance focussing on one‐period or binomial models typically without partial hedging; for a discussion and references, see the survey of Henderson and Sun .…”

“…The above papers make particular choices of utility function (CARA or constant relative risk aversion [CRRA]) and all assume exponential Brownian motions for the stock price and the correlated traded asset, and solve the resulting free boundary problem. This is done explicitly in some special cases (Henderson 2007;Grasselli and Henderson 2009) or via numerical solutions.…”

A Note on Irreversible Investment, Hedging and Optimal Consumption Problems

“…Note this is the original formulation in Miao and Wang [21]. There is an alternative formulation (see Henderson [6] for the case without consumption and also Miao and Wang [21] and Monoyios [22] for the case with consumption) in which the agent receives a one-off payment of magnitude Y t so that the modified wealth dynamics become…”

“…Henderson [6] solves the timing problem where a lump-sum payoff is obtained at option exercise and the agent can hedge in a traded asset. This approach has the advantage that the solution is in closed form; however, she does not consider consumption.…”