Stochastics Seminar

Speaker: Dr. Ming Liao

Title: Stochastic finance

Abstract: We will describe a stock market model based on semimartingales (as stock prices).  This includes the more traditional model based on stochastic differential equations driven by Brownian motion, but is much more general.  There are two basic theorems.  The first theorem says that the assumption of no arbitrage (that is, no opportunity for riskless profit after discounting interest), is essentially equivalent to the existence of a new probability measure under which the discounted stock price process is a martingale.  This allows the powerful tool of martingales to be applied.  The second theorem deals with the completeness of the stock market, which guarantees that any stock option can be exactly fulfilled by a suitably chosen investment portfolio.  This is a consequence of the martingale representation theorem, and provides a precise way to price stock options, and leads to a rigorous treatment of the famous Black-Scholes formula.