Basic notions of risky asset, option, replicating portfolio, no-arbitrage, and risk-neutral pricing will be introduced on a very simple one-period example. We will then briefly look at multi-period tree models before introducing the continuous time geometric Brownian motion model and the associated stochastic calculus. The famous Black-Scholes PDE will be derived by a no-arbitrage self-financing replicating portfolio argument. The Black-Scholes formula will also be explained in terms of risk-neutral expectations. Still in the context of constant volatility more complicated derivative contracts will be presented. In the second part of the mini-course we will explain why varying volatilities are considered in order to match return distributions and observed option prices. Local volatility and stochastic volatility models with their associated mathematical challenges will be discussed. Finally in the last part, models for other markets, fixed income and credit in particular, will be presented.
An optional buffet lunch follows. Please register for the course and lunch by Friday March 18.