Module 2: Black-Scholes Theory

The next course will take place: Monday 25 February - Friday 1 March 2019 



  • Elementary stochastic differential equations, strong and weak solutions, transition density functions, Feynman-Kac formula, exit times and hitting probabilities, maximum/minimum of Brownian motion;
  • Martingale methods in continuous time: stopping times, martingales and local martingales; stochastic integrals and Ito formula, martingale representation theorem;
  • The Black-Scholes model: assumptions, perfect replication, risk-neutral valuation, the Black-Scholes PDE and solutions, discrete and continuous dividend yields, time-dependent volatility, dividends and interest rates;
  • Hedging, Greeks: delta, gamma, vega, rho, less common sensitivities;
  • Introduction to P & Q Measures: equilibrium pricing, review of physical and risk-neutral probabilities in discrete and continuous time, CAPM, paradoxes, risk premia;
  • Introduction to the term structure of interest rates, bond price equilibria, duration and convexity, caps, floors, swaps;
  • Introduction to Monte Carlo: sampling non-uniform distributions, expectation and integration, implementation of MC methods; simple variance reduction techniques: importance sampling, antithetic sampling, control variates; finite difference Greeks, likelihood ratio method, pathwise sensitivities, workshop; and
  • Explicit and implicit finite difference schemes, implementation, accuracy and stability, Greeks and smoothing schemes; workshop.


For students enrolled on the course

Course Materials - including student instructions, lecture notes, assignment and submission link