June 14-15, 2006: Option-pricing with Stochastic Volatility and Jumps.

Fri Jun 2 16:10:47 CEST 2006: Literature links will appear later.

Local volatilty/implied modelling (Wednesday)
Textbook presentation: Wilmott; especially Section 22.6. To derive eqn. 22.6 (called the Dupire equation) we'll need the Kolmogorov forward or Fokker-Plank eqn. from Prop. 5.12 in Bjørk. The gory details of eqn. (22.7) are here.
The ideas were developed simultaneously and independently in
• Dupire, "Pricing with a Smile", RISK Magazine 1994
• Derman & Kani, "Riding on a Smile", RISK Magazine 1994
• Rubinstein, "Implied Binomial Trees", Jo Finance 1994.
Useful stregthenings were given in Andersen & Brotherton-Ratchliffe, "The implicit equity option volatility smile: an implicit finite difference approach", Jo Computational Finance 1997

The idea is to make (local) volatilty a function of time and stock-price itself, sigma = sigma(S(t),t) and then back out the functional form from observed option prices. Like interest rate model calibration (ie. same conceptual inconsistencies) but "taken up a notch" (ie. harder/better math!).

Stochastic volatility. (Wednesday)
A nice semi-specific case of Bjørk's Chapter 15 (or "like 21") on non-traded underlyings. Follows the idea of this exercise 10.2. Explict models; semi-closed solutions.
The idea is using a conditioning argument based on "The Useful Rule": If X (a random variable) is independent of F (a sigma-algebra), and Y (another random variable) is F-measurable, then E(g(X,Y)| F) = E(g(X,y))|_(y=Y)
A benchmark model w/ closed-form solution: The Heston Model Or sort of closed-form solution: Up to integral that must be found numerically; the inversion af a Fourier transform or characteristic function. The trick to proving this is very similar to what we did for affine term structure models. My R-implementation of Hestons call-price formula. (This code is actually more general and includes jumps. Just put the J-parameters and lambda = 0.) And here is a little code to plot implied volatility smiles/skews etc.
• Duffie, Pan & Singleton show that this connection between affine characteristic functions and their use for option pricing is valid in a fairly general class of models (merging term structure, stochastiv volatility and jumps).
• Lee, "Option pricing by transform methods: extensions, unification and error control", Jo Computational Finance 2004 points out the inversion may cause numerical difficulties, and suggest way to remedy these.

Jumps. (Thursday from 11 'till we can't take it anymore.)
The fundamental buliding block is the Poisson process. The is a stochastic process, {N(t)}_t that may be characterized by any of the follwing properties
• (Levy) It has independent increments and N(t) - N(s) ~ Po(lambda (t-s) )
• (Counting process) It is a process that jumps with increments of 1 and the time between jumps are independent and ~ Exp (lambda)
• (Intensity) It is a process that has probability 1-lambda*dt of staying constant over any small interval, probability lambda*dt of increasing 1 unit, and (thus) probability o(dt) of more than 1 jump.
• Poisson process simulation.
Other issues
• (Compensation) The fundamental Poisson martingale.
• (Compound) Processes of doubly stochatic Poisson-type nature.
• (Cox) Po-processes with non-constant intesity.
• (Intergrals) Stochastic integrals of functions/processes wrt. Po-process: Weigh increments with (function) values and sum over jump times.
• Ito-formula for jump-diffusions. Finding the martingale part
Literature: Rama Cont & Peter Tankov (2004) Financial Modelling with Jumps Sections 2.5-6, Ch. 8. and much more. Highly recommended book. Buy it.

Option pricing: The original is Merton 76; the "quick and dirty" (but effective -- as almost always) is Wilmott.
• The hedge-argument and where it breaks down.
• Non-priced jump risk and partial integro differential equation.
• Merton's formula
• Equivalent measure changes for compuond Po-process and "fear of jumps".
Option pricing and transform methods: The whole thing put together in one neat package! The Bakshi/Cao/Chen or the Bates model. .