Abstracts Category : Other

EARLY EXERCISE OPTIONS WITH DISCONTINUOUSPAYOFF

by Min Gao

The main contribution of this thesis is to examinebinary options within the British payoff mechanism introduced byPeskir and Samee. This includes British cash-or-nothing put,British asset-or-nothing put, British binary call and Americanbarrier binary options. We assume the geometric Brownian motionmodel and reduce the optimal stopping problems to free-boundaryproblems under the Markovian nature of the underlying process. Withthe help of the local time-space formula on curves, we derive aclosed form expression for the arbitrage-free price in terms of therational exercise boundary and show that the rational exerciseboundary itself can be characterised as the unique solution to anon-linear integral equation. We begin by investigating the binaryoptions of American-type which are also called `one-touch' binaryoptions. Then we move on to examine the British binary options.Chapter~2 reviews the existing work on all different types of thebinary options and sets the background for the British binaryoptions. We price and analyse the American-type (one-touch) binaryoptions using the risk-neutral probability method. In Chapters~3 ~4and ~5, we present the British binary options where the holderenjoys the early exercise feature of American binary optionswhereupon his payoff is the `best prediction' of the Europeanbinary options payoff under the hypothesis that the true driftequals a contract drift. Based on the observed price movements, ifthe option holder finds that the true drift of the stock price isunfavourable then he can substitute it with the contract drift andminimise his losses. The key to the British binary option is theprotection feature as not only can the option holder exercise atunfavourable stock price to a substantial reimbursement of theoriginal option price (covering the ability to sell in a liquidoption market completely endogenously) but also when the stockprice movements are favourable he will generally receive highreturns. Chapters~3 and~4 focus on the British binary put optionsand Chapter~5 on call options. We also analyse the financialmeaning of the British binary options and show that with thecontract drift properly selected the British binary options becomevery attractive alternatives to the classic European/Americanoptions. Chapter~6 extends the binary options into barrier binaryoptions and discusses the application of the optimal structurewithout a smooth-fit condition in the option pricing. We firstreview the existing work for the knock-in options and present themain results from the literature. Then we examine the method in in the application to the knock-in binaryoptions. For the American knock-out binary options, the smooth-fitproperty does not hold when we apply the local time-space formulaon curves. We transfer the expectation of the local time term intoa computational form under the basic properties of Brownian motion.Using standard arguments based on Markov processes, we analyse theproperties of the value function.Advisors/Committee Members: LOEFFEN, RONNIE R, Peskir, Goran, Loeffen, Ronnie.