Module 2: Black-Scholes Theory

26 February - 2 March 2018

Timetable for Module 2 

Course Materials



  • Elementary stochastic differential equations, strong and weak solutions, transition density functions, Feynman-Kac formula, exit times and hitting probabilities, maximum/minimum of Brownian motion;
  • Change of measure: Girsanov theorem, exponential martingales, change of numeraire;
  • 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 payments, time-dependent volatility, dividends and interest rates;
  • Hedging, Greeks: delta, gamma, vega, rho, less common sensitivities;
  • 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.



Assignment due: 12 noon UK time, 2 April 2018

The Module 2 assignment is available under Course Materials during, or shortly after, the week you are in Oxford. Although not a compulsory assignment, it is highly recommended that you do submit an assignment to give an indication of how you are progressing on the course, ready for the examinations. If you do not plan to submit the assignment then please let the Course Administrator know.

Submit assignment here

Keep a record of the submission ID for future reference. If you encounter any problems please contact the Course Administrator as soon as possible.

Please remember to put your candidate number, rather than your name, on the assignment. You can find your candidate number on the Student Self Service System.