Risks of volatility and validation of models


Objective

In this course we study the implicit volatility smile used in the market and the option calibration and hedging options in these models. The emphasis will be on numerical methods for resolving calibration problems

Planning

  1. Presentation of derivative markets.
  2. Local volatility models and Delta risk –Implicit trees method. Dupire equation.
  3. Stochastic volatility models and Vega risk –Option valuation using Fourier transform methods.
  4. Models with jumps –Compound Poisson process; Merton model. Calibration and hedging.
  5. Hybrid models with jumps and stochastic volatility.
  6. Ill-posed linear problems and the concept of regularisation –Example of an ill-posed linear problem: the construction of the rate curve.
  7. Regularisation of calibration problems –Calibration of local volatility surfaces. Entropy regularisation.
  8. Numerical methods for model calibration –Convex criteria optimisation algorithms. Introduction to evolutionist algorithms.

Références

Références:
* P. Tankov "Surface de volatilité", polycopié
* L. Bergomi, "Stochastic volatility modeling", Chapman & Hall / CRC, 2016.
* J. Gatheral, "The volatility surface: a practitioner's guide" Wiley, 2011
* R. Rebonato, "Theory and practice of model risk management." Modern Risk Management: A History’, RiskWaters Group, London (2002): 223-248.
* R. Cont, (2006). "Model uncertainty and its impact on the pricing of derivative instruments". Mathematical Finance. 16 (3): 519–547
* Federal reserve system, Supervisory guidance on model risk management, 2011
* Bank of International Settlements, Minimal Capital Requirements for Market Risk, 2016