Introduction to Implied, Local and Stochastic Volatility
Implied Volatility - Ito's Lemma
Applying Ito to the Hedging Portfolio
Risk-Neutralization and No-Arbitrage
Implied Volatility (Smiles and Skews)
Coupled SDEs for Stochastic Volatility
Risk-Neutralization and No-Arbitrage
Exact Solution for Heston Volatility
Introduction to Implied, Local and Stochastic Volatility
References
[1] Bera, Anil K. and Higgins, Matthew L., (1998). "A Survey of Arch Models: Properties, Estimation and Testing". Risk Books, Volatility, June−98, 23-59.
[2] Björk, Thomas, (1998). "Arbitrage Theory in Continuous Time". Oxford University Press Inc., New York.
[3] Dupire, Bruno, (1994). "Pricing with a Smile". Risk Magazine, 7, 18-20.
[4] Elder, John, (2002). "Hedging for Financial Derivatives". University of Oxford, Ph.D. Thesis.
[5] Ghomrasni, Raouf, (2003). "On Distributions Associated with the Generalized Lévy’s Stochastic Area Formula". University of Aarhus, Centre for Mathematical Physics and Stochastics (MaPhySto) [MPS]; RR 2003/4.
[6] Heston, Steven L., (1993). "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options". The Review of Financial Studies, Volume 6, Issue 2, 327-343.
[7] Hull, John, (1993). "Options, Futures, and other Derivation Securities". Prentice Hall, Inc.
[8] Kloeden, Peter E. and Platen, Eckhard, (1999). "Numerical Solution of Stochastic Differential Equations". Springer.
[9] Kloeden, Peter E., (2002). "The Systematic Derivation of Higher Order Numerical Schemes for Stochastic Differential Equations". Milan Journal of Mathematics.
[10] Lévy, P., (1950). "Wiener’s Random Function, and other Laplacian Random Functions". Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 171-187.
[11] Lewis, Alan L., (2000). "Option Valuation under Stochastic Volatility: with Mathematica Code". Finance Press.
[12] Schmitz, Klaus, (2004). "Introduction to Implied, Local and Stochastic Volatility". http://www.maths.ox.ac.uk/~schmitz/project1.htm
[13] Schmitz, Klaus, (2004). "Strong Taylor Schemes for Stochastic Volatility". http://www.maths.ox.ac.uk/~schmitz/project2.htm
[14] Shaw, William, (2000). "Instability of Implied Volatility, Fictitious Skews and Smiles and Hazards of Exotics". AIP Conference Proceedings, Vol. 553, 309-314.
[15] Shaw, William, (1999). "Modelling Financial Derivatives with Mathematica". Cambridge University Press.
[16] Shaw, William, (2003). "Stochastic Volatility, Models of Heston Type". University of Oxford, Course Notes.
[17] Wilmott, Paul, Howison, Sam and Dewynne, Jeff, (1995). "The Mathematics of Financial Derivatives". Cambridge University Press.
[18] Wilmott, Paul, (1998). "Derivatives: The Theory and Practice of Financial Engineering". John Wiley and Sons.
Prof. Klaus Schmitz
Introduction to Implied, Local and Stochastic Volatility
The purpose of this document is to introduce implied, local and stochastic volatility, to review evidence of non-constant volatility, and to consider the implications for option pricing of alternative random or stochastic volatility models. We focus on continuous time diffusion models for the volatility, but we also briefly discuss certain classes of discrete time models, such as ARV or ARCH.
By Prof. Klaus Erich Schmitz Abe