ANNA BATTAUZ

The critical price of an American put option is the underlying stock price level that triggers its immediate optimal exercise. We provide a new perspective on the determination of the critical price near the option maturity T when the jump-adjusted dividend yield of the underlying stock is either greater than or weakly smaller than the riskfree rate. Firstly, we prove that the critical price coincides with the critical price of the covered American put (a portfolio that is long in the put as well as in the stock). Secondly, we show that the stock price that represents the indifference point between exercising the covered put and waiting until T is the European-put critical price, at which the European put is worth its intrinsic value. Finally, we prove that the indifference point's behavior at T equals the critical price's behavior at T
when the stock price is either a geometric Brownian motion or a jump-diffusion. Our results provide a thorough economic analysis of the critical price and rigorously show the correspondence of an American option problem to an easier European option problem at maturity.

• Critical_price2021 (478 Kb)

We provide a comprehensive description of the optimal exercise policies associated with American quanto options. We show that a non-standard exercise policy characterized by a double continuation region may be optimal in the presence of non–positive domestic interest rates. We study empirical examples of fi…nite-maturity American quanto options for which a double continuation region surrounding a non-empty early exercise region exists even if the infinite-maturity early exercise region is empty and the value of the in…finite-maturity option is unbounded. For such empirical examples, we carefully characterize the existence, the monotonicity properties and the asymptotics of the upper and lower critical prices at maturity.

• Battauz-DeDonno-Sbuelz Quanto options (292 Kb)

We study infinite-maturity American equity options in a stochastic interest rate framework of Vasicek type (Vasicek (1977)). We allow for a non-zero correlation between the innovations driving the equity price and the interest rate. Importantly, we also allow for the interest rate to assume negative values, which is the case for some investment grade government bonds in Europe in recent years. In this setting we focus on American equity call and put options and characterize analytically their two-dimensional free boundary, i.e. the underlying equity and the interest rate values that trigger the optimal exercise of the option before maturity. We show that non-standard double continuation regions may appear, extending the findings obtained by Battauz et al. (2015) in a constant interest rate framework. Moreover, we contribute by developing a bivariate discretization of the equity price and interest rate processes that converges in distribution as the time step shrinks. The discretization, described by a recombining quadrinomial tree, allows us to compute American equity options' prices and their related free boundaries. In particular, we document the existence of non-standard optimal exercise policies for American call options on a non-dividend-paying equity. We also verify the existence of a non-standard double continuation region for American equity options, and provide a detailed analysis of the associated free boundaries with respect to the time and the current interest rate variables.

• AmericanEquityOptionsStochasticInterestRates (993 Kb)
• Slides-Presentation-ADA2019 (1.316 Kb)

Earnout agreements link part of the payment of an acquisition to the future performances of the acquired company. They are structured as real options on the future value of the target, and should be valued as such. However, the models used so far do not take into consideration two peculiar sources of risk that affect these contracts: the risk of the bidder's default before the earnout expiration (counterparty risk) and the risk of litigation that might arise in connection to these contracts (litigation risk). We develop an option pricing model that fills this gap. The performed sensitivity analysis and the  presented case study show that counterparty risk and litigation risk are significant, because they might have a remarkable impact on the earnout values. The relevance of the model is also given by recently issued accounting standards, which now require contingent payments to be valued at fair value.

• Earnouts-Battauz-alii2016 (260 Kb)