This module introduces the learner to the mathematics of hedging and pricing of financial derivatives by arbitrage in a continuous-time framework by building on prior knowledge of discrete-time models. Key concepts such as conditional expectation, martingales, change-of-measure, Wiener processes, and Ito calculus are developed in the lead up to the derivation of the Black-Scholes formula and Black-Scholes equation. Monte Carlo methods are considered for solving stochastic differential equations.
Transition from discrete to continuous processes, properties of Brownian motions
Non-stochastic calculus, stochastic integration and differentials, Ito's Lemma, Ito calculus
Change of measure
Girsanov’s theorem, martingale representation theorem
Black Scholes formula and equation
Derivation, pricing, manipulation
Monte-Carlo methods for option pricing
Euler scheme, Milstein scheme, convergence
Lectures supported by problem-solving sessions and the use of mathematical software packages where applicable.
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