The main purpose of this thesis is to derive basic properties and present closed form solutions to the pricing problem of exotic options with two free boundaries. These include American strangle, American chooser, British strangle and American eagle options. These options are designed for an underlying asset with high volatility. Pricing an option with the early exercise feature is equivalent to find an optimal stopping time which will maximise the expected options payoff. Due to the Markovian nature of the underlying process, the optimal stopping problem is linked in a onetoone way with the freeboundary problem consisting of a parabolic partial differential equation (PDE) satisfying suitable boundary conditions. By the local timespace formula on curves,the closed form solution to the options value can be derived from the freeboundary problem and we characterize two optimal stopping boundaries (free boundaries) as the unique solution pair to the system of two nonlinear integral equations.In Chapter 2 and 3, we study the American strangle option and the American chooser option respectively. The payoff of strangle options is the maximum between the payoff of American call options and the payoff of American put options. The payoff of chooser options is the maximum between the value of American call options and the value of American put options. The major undertaking in this context is to split the optimal stopping region into two disjoint closed sets and to analyse basic properties of the value function and the free boundaries. After performing the required financial analysis, we find that the returns of American strangle options outperform the returns of 'traditional' American strangle options. The returns of American chooser options underperform the returns of American strangle options if exercised before the maturity.In Chapter 4, we design the British strangle option which enjoys the early exercise feature. To inherit higher returns from the Britishtype option, we insert two contract drifts into the payoff function. The 'tolerance contract drift' minimises the loss and the 'preference contract drift' maximises the gain. We prove the existence of a function of time in the continuation region that implies the two free boundaries never intersect.In Chapter 5, we provide another example of an option with two free boundaries named the American eagle option. Compared with the American strangle option, the eagle option avoids the unlimited loss for the option seller and reduces the option premium for the option buyer. We show that the smooth fit is not satisfied when the free boundaries are equal to the constant caps so that the early exercise premium representation of American eagle options contains the local time term. Using basic properties of standard Brownian motion, we transfer the expectation of the local time term into a computational form. By classifying eagle options into eagle options with balanced wings and eagle options with unbalanced wings, we analyse the properties of the value function and the free boundaries, respectively.Chapter 6 gives a brief analysis of other options with two free boundaries. By changing the underlying measure and 'method of scaling strike', we reduce the high dimensional optimal stopping problem into a lower dimension. Examples of this include Asian strangle options and American lookback strangle options. Combining the payoff of vanilla options, we create American condor and American calendar options.
Date of Award  1 Aug 2016 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Goran Peskir (Supervisor) & John Moriarty (Supervisor) 

EARLY EXERCISE OPTIONS WITH TWO FREE BOUNDARIES
Qiu, S. (Author). 1 Aug 2016
Student thesis: Phd