Module 3: Extensions of the Black-Scholes Framework

The next course will take place: Monday 29 April - Friday 3 May 2019



  • Derivative pricing: equivalent measures, risk-neutral pricing, fundamental theorems of asset pricing in discrete time;
  • American options: early exercise, linear complementarity problem, perpetual put, free boundary formulation, smooth pasting;
  • Finite differences for American options: explicit methods, projected iterations, a penalty method; workshop;
  • Variance swaps and the log contract, replication, the VIX index;
  • Implied and local volatility; Breeden-Litzenberger formula, Dupire's formula; hedging and pricing under stochastic volatility models;
  • Change of measure: Girsanov's theorem, exponential martingales, change of numeraire;
  • Overview of instruments in rates markets, mechanisms and conventions; interpolation of yield curves and volatility term structures, bootstrapping, funding instruments and discount curves for banks; basic derivatives, sensitivities;
  • Yield curve modelling; short rate models, use and calibration: Vasicek, CIR, Hull-White, market models (HJM, LMM), and
  • Value at risk and expected shortfall; Convex and coherent risk measures, Euler principle and capital allocation.


For students enrolled on course

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