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This unit introduces the mathematical theory of pricing financial derivatives and shows how to apply these ideas to a variety of option contracts. The course begins with a simple discrete time, discrete state formulation of risk-neutral valuation, moving through the Cox Ross Rubenstein model, through to the martingale perspective on the Black-Scholes option pricing formula. Implied volatility and volatility arbitrage may be discussed if time permits.
This unit begins with the elementary probabilistic and computational ideas of Monte Carlo methods, using iid random variables, transformation, rejection and importance sampling. Key notions from the ergodic theory of Markov chain Monte Carlo are introduced rigorously in the setting of a finite state-space. The Metropolis-Hastings algorithm, Gibbs sampling and Sequential Monte Carlo methods are demonstrated and compared.