Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning

Abstract

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge. This model may be used under special market conditions, in particular we focus on what in the literature is known as the “pinning effect”, that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary. Finally, we compare our method with the optimal exercise time based on a geometric Brownian motion by using real data exhibiting pinning.

On the monotonicity of the stopping boundary for time-inhomogeneous optimal stopping problems

“…An example of time-inhomogeneous diffusions which perfectly fits the weaker monotonicity assumption (of Theorem 4.2) on t → µ(t, x) is given by Brownian bridges. Several works have investigated optimal stopping problems involving Brownian bridges and we cite, among others, [29], [15], [14], [12], [6], [17] and [3]. Both Theorem 4.1 and Theorem 4.2 lead to the monotonicity of the optimal stopping boundary t → b(t) (see Corollary 4.6).…”