Small-time asymptotics for multi-asset options

An option is a financial contract between two parties where one party has the right, but not the obligation, to make a specified transaction at or before a predetermined date at a predetermined price. A multi-asset option is an option where the transaction involves two or more underlying assets. Closed form solutions for a majority of multi-asset options have yet to be found, resulting in a concentration of efforts towards the derivation of asymptotic and numerical approximations for their true values.; The early exercise feature of American options mathematically gives rise to free boundary problems. The continuation region, where the option is not exercised, and the exercise region are separated by the free boundary, which is unknown a priori and must be solved along with the option price. The characterization of the singular behavior of the free boundary at expiry and the evolution of the option price from a non-smooth payoff have the potential to be used in a fast and accurate numerical valuation scheme.; In this thesis I establish small-time asymptotic results for both European and American style two-asset spread options, which have payoff max( S2 - S1 - K, 0) for asset values S1 and S2 at expiry and strike price K. The method of matched asymptotics results in a single layer structure in the European case. For time tau = T - t << 1 until expiry outside of an O( t ) region below the kink of the payoff, the option is O( t exp (- (S1 + K - S2)2/tau)), and above it is the payoff plus a term of O(tau). Inside an O( t ) layer about the kink, the option is O( t ). The approximation presented is more accurate than other approximations in the recent literature and if S1 = 0, it agrees with the expansion of the exact solution for the corresponding European call on asset S2.; Asymptotic analysis of the early exercise premium representation (or EEPR) combined with the method of matched asymptotics results in a two-layer structure for American spread options. The EEPR decomposes the American spread option into a sum of its European counterpart and the expected return from early exercise, or premium. Because the premium is lower order outside of an O( t ) region about the initial value of the free boundary, the EEPR allows the single layer structure of the corresponding European spread option to be used in place of the American spread in this region, at least to leading order. Finally, the method of matched asymptotics results in the American spread option being the payoff plus a term of O( t32 ) in an O( t ) initial free boundary layer and also gives the expansion of the free boundary by imposing the smooth pasting conditions.; An introduction to derivative pricing and a detailed outline of this thesis is presented in chapter 1. A review of the Black-Scholes option pricing theory from the probability and PDE approaches in the single asset setting is provided in chapter 2. The literature, problem formulation, and asymptotic analysis of American call options on dividend paying assets are presented in chapter 3. Chapter 4 contains the Black-Scholes framework for two-asset options, a well-posedness result, and some results from the literature. The main results of this thesis are found in chapter 5, where asymptotic expansions for European and American spread options are derived. Finally, chapter 6 provides a summary, conclusion, and future projects.