M5MF6 Advanced Methods in Derivatives Pricing, with application to Volatility Modelling
MSc Mathematics and Finance, Spring term 2017


In this course, we shall endeavour to cover the following topics:
      Review of basics of stochastic analysis (local martingales, stopping times, Martingale Representation, Girsanov theorem, Ito's formula)
      The super-replication paradigm
      General arbitrage-free properties of option prices
      Implied volatility: existence and asymptotic properties
      Stochastic differential equations (existence and uniqueness)
      From SDEs to PDEs: Feynmac-Kac formula
      Local and stochastic volatility


Time: Mondays, 9am-1am (Huxley 130) and Thursdays, 11am-1pm (Huxley 139)

Course Material
    [PDF] Lecture Notes (this version: 23/2/2017)
    [PDF] First Set of Exercises
    [PDF] Second Set of Exercises
    [PDF] Revision Exercises
[PDF] Revision Checklist
        [PDF] Apple Options Data implied volatility smile
    [PDF] Problem Class: SSVI local volatility

    [XLS] Project Groups
    [XLS] Heston IV surface

Python code
    Link to the IPython / Jupyter platform
    [IPynb, PDF] Black-Scholes and hedging
    [IPynb, PDF] CEV / local martingale
    [IPynb, PDF] SSVI
    [IPynb, PDF] Testing Roger Lee's bounds on Apple option data
    [IPynb, PDF] SABR

Useful websites
    CBOE
    Oxford-Man - Realisd volatility data

Related literature
    Stochastic analysis
    [PDF] G. Johnson and L. L. Helms. Class D supermartingales. Bulletin of the American Mathematical Society (1963).
    [PDF] R. Jarrow and P. Protter. A short history of stochastic integration and mathematical finance: The early years, 1880-1970. A Festschrift for Herman Rubin Institute of Mathematical Statistics (2004).

    Louis Bachelier's contribution to Mathematical Finance
    [PDF] L. Bachelier, Théorie de la spéculation. Annales Scientifiques de l'ENS (1900).
    [PDF] M. Davis. Louis Bachelier Theory of Speculation (2006).
    [Book] M. Davis and A. Etheridge. Louis Bachelier's Theory of Speculation: The Origins of Modern Finance (2006).
    [PDF] J-M. Courtault, Y. Kabanov, B. Bru, P. Crepel, I. Lebon, A. Le Marchand. Louis Bachelier On the centenary of Théorie de la Spéculation. Math. Finance (2000).
    [PDF] M. Taqqu. Bachelier and his Times: A Conversation with Bernard Bru. Preprint, Boston University (2001).

    The Black-Scholes model
    [PDF] F. Black and M. Scholes. The Pricing of Options and Corporate Liabilities. Journal of Political Economy (1973).
    [PDF] R. Merton. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science (1973).

    Miscellaneous
    [PDF] R. Ahmad and P. Wilmott, Which Free Lunch Would You Like Today, Sir. Wilmott Magazine November 2005.
    [PDF] Andre Ribeiro and Rolf Poulsen, Approximation Behooves Calibration. (corresponding data available here) Quant Finance Letters, 2013.
    [PDF] R. Mansuy, The origin of the word `martingale'. Electronic Journal for History of Probability and Statistics, 2009.

    Project Papers (2016-2017)
    [PDF] Andersen, Andreasen, Eliezer - Static replication of barrier options
    [PDF] and [PDF] Bisesti, Castagna, Mercurio - Consistent pricing and hedging of an FX options book AND Bossens et al. - Vanna-Volga methods applied to FX derivatives
    [PDF] Carr, Lee - Robust Replication of Volatility Derivatives
    [PDF] De Marco, Martini - The Term Structure of Implied Volatility in Symmetric Models with applications to Heston
    [PDF] Deng, Dulaney, McCann, Yan - Leveraged ETF
    [PDF] Skstrom - Properties of American option prices
    [PDF] Follmer, Schweizer - A Microeconomic Approach to diffusion models for stock prices
    [PDF] Fukasawa - Normalization for Implied Volatility
    [PDF] Goncu - Statistical arbitrage in the Black-Scholes model
    [PDF] Itkin - Volatility smile parameterisation
    [PDF] Jansson, Tysk - Volatility time and properties of option prices
    [PDF] Jena, Tankov - Mis-specification of stochastic volatility models
    [PDF] Tsuzuki - No-Arbitrage Bounds on Two One-Touch Options
    [PDF] Rheinlander, Schmutz - Self-dual continuous processes

    Past Exam Papers
    [PDF] Exam 2015-2016
    [PDF] Solution to Exam 2015-2016