Strategic Exercise Of Options On Non-traded Assets And Stochastic Volatility In An Incomplete Market: Indifference Pricing And Entropy Methods
The first study explores optimal investment policies for strategic option exercise when the underlying project is not traded. A duopoly model captures strategic interactions, while a partial spanning asset models market incompleteness. The option value to invest is obtained through indifference pricing, i.e., certainty equivalent value. I find that incompleteness narrows the gap between leader and follower entry dates. The follower enters much sooner, and the leader delays slightly compared to classic real options models. Modeling investment income stream as an Arithmetic Brownian motion is a better fit than Geometric Brownian motion, while reducing the necessary numerical approximations for obtaining the results in the incomplete market situation. As a byproduct of modeling two different stochastic income streams, I investigate the impact of market share and uncertainty on the relative investment trigger as well as the option value to invest. Results are sensitive to these factors; thus, it is important to model stochastic processes to accurately reflect the real world circumstances.
The second study explores the valuation consequences of incompleteness resulting from stochastic volatility in a real options setting. The optimal policy is obtained through q-optimal measures as well as indifference pricing. I examine the efficacy of different approaches to finding and justifying a particular martingale measure. Stochastic volatility induced market incompleteness affects the investment/abandonment decision in several important ways. In addition, I demonstrate that indifference prices for the option value to invest and the abandonment option solve quasilinear variational inequalities with obstacle terms. With the exponential utility function, the utility-based indifference price admits a new pricing measure, which is the minimal relative entropy martingale measure minimizing the relative entropy between the historical measure and the Q martingale measure. I also show that the indifference price is non-increasing with respect to risk aversion. As the risk aversion parameter converges to zero, the indifference price converges to the unique bounded viscosity solution of the linear variational inequality with obstacle term.