Department of Statistics
University of California
Berkeley, California
Spring Semester, 2008
Statistics 251
Stochastic Analysis with Applications to Mathematical Finance
Units: 3
Instructor: Steven N. Evans
Mon Wed 5:00-6:30, 334 EVANS
Office Hours: 329 Evans by appointment
email: evans@stat.berkeley.edu
Course Outline
The course will be an introduction to the basic concepts of stochastic
calculus, particularly those that are most relevant in mathematical
finance
and ``financial engineering''. The probability theory to be covered
will
include: Brownian motion and continuous parameter martingales,
quadratic
variations, stochastic integration, Ito's formula, representation of
martingales,
Girsanov's theorem, stochastic differential equations, and diffusion
processes.
The mathematical development will be motivated and accompanied by
examples
from the Black-Scholes-Merton theory of pricing and hedging contingent
claims, including the following finance topics: European options,
foreign
market derivatives (e.g. currency forwards, options and quantos),
American
options, exotic options, and interest rate related contracts (e.g.
Vasicek,
Cox-Ingersoll-Ross, Heath-Jarrow-Morton, Brace-Gatarek-Musiela models).
Prerequisites
Exposure to probability theory equivalent to Stat 204 or 205A. Students
should
be comfortable with the "gestalt" of the measure-theoretic approach to
probability,
conditional
expectation, discrete-time martingales, and elementary properties of
Brownian
motion. The first two chapters of the text are a good
refresher. No prior knowledge of finance will be assumed.
Text
- Stochastic Calculus for Finance
II : Continuous-Time Models, Steven E. Shreve, Springer
Lecture
slides in PDF
- don't print these off all at once, as I will probably revise them as
we go along
Recommended Reading
Stochastic calculus reading:
- Brownian Motion and Stochastic
Calculus, I. Karatzas and S.E.
Shreve,
Springer
- Diffusions, Markov Processes,
and Martingales, vol II, L.C.G.
Rogers
and
D. Williams, Wiley
- Continuous Martingales and
Brownian Motion, D. Revuz and M. Yor,
Springer
- Stochastic Integration and
Differential Equations, P. Protter,
Springer
Elementary finance reading:
- Introduction to Stochastic
Calculus Applied to Finance, D.
Lamberton
and
B. Lapeyre (translated by N. Rabeau and N. Mantion), Chapman and Hall /
CRC
- Mathematics of Financial Markets,
R. J. Elliott and P. E. Kopp,
Springer.
- Financial calculus,
Martin Baxter and Andrew Rennie, Cambridge
University
Press.
- An Introduction to the
Mathematics of Financial Derivatives,
Salih N.
Neftci,
Academic Press.
Advanced finance reading:
- Martingale methods in financial
modelling, M. Musiela and M.
Rutkowski,
Springer.
- Methods of mathematical finance,
I. Karatzas and S.E. Shreve,
Springer.
Grading
Final
project
Syllabus
Week 1+2: Brownian motion, continuous time martingales, quadratic
variation,
construction of the stochastic integral.
Week 3+4: Ito's lemma, stochastic differential equations, strong
solutions,
stochastic flows, Markov properties, the Ornstein-Uhlenbeck process.
Week 5+6: stochastic integral representations of martingales,
Girsanov's
formula and change of measure, Levy's characterization of Brownian
motion.
Week 7: the Black-Scholes model, self-financing strategies, hedging
and pricing.
Week 8: infinitesimal generator of a diffusion, the Feynman-Kac
formula,
connections between diffusions and partial differential equations,
applications
to pricing of various contingent claims.
Week 9: interest rate models.
Week 10: local time, Tanaka's formula, Trotter's theorem, Skorohod's
equation, Levy's equivalence.
Week 11: excursion theory, Poisson point processes, Ito's
decomposition.
Week 12+13: one-dimensional diffusions, transformation by scale and
speed,
classification of boundaries.