Course ObjectiveAfter the course, the student will
- be aquainted with the notion of pricing derivatives and arbitrage, and
with the basics of discrete stochastic processes, including the concepts
of martingales; be able to price derivatives in the binomial model.
- be familiar with the basics of continuous time stochastic processes,
concepts of measure theory, Brownian motion, martingales; be able to
verify whether the process is a Brownian motion and/or is a martingale.
- be familiar with the notions of stochastic integral Ito and SDE, the
Ito formula; be able to compute stochastic integrals, to use the Ito
formula, and to derive properties of stochastic integrals and the
solutions of SDEs.
- be familiar with the Black-Sholes model; be able to price options by
the no-arbitrage and Girsanov's theorem, to compute the hedging
portfolio and the Greeks, and to make connections to the corresponding
- be familiar with the extended Black-Sholes models; be able to price
derivatives in those models, to make connection with the corresponding
PDEs, and to apply the Black-Scholes machinary to the interest rates
Course ContentFinancial institutions trade in risk, and it is therefore essential to
measure and control such risks. Financial instruments such as options,
swaps, forwards, etc. play an important role in risk management, and to
handle them one needs to be able to price them. This course gives an
introduction to the mathematical tools and theory behind risk
A "stochastic process" is a collection of random variables, indexed by a
set T. In financial applications the elements of T model time, and T is
the set of natural numbers (discrete time), or an interval in the
positive real line (continuous time). "Martingales" are processes whose
increments over an interval in the future have zero expectation given
knowledge of the past history of the process. They play an important
role in financial calculus, because the price of an option (on a stock
or an interest rate) can be expressed as an expectation under a
so-called martingale measure. In this course we develop this theory in
discrete and continuous time. Most models for financial processes in
continuous time are based on a special Gaussian process, called Brownian
motion. We discuss some properties of this process and introduce
"stochastic integrals" with Brownian motion as the integrator. Financial
processes can next be modeled as solutions to "stochastic differential
equations". After developing these mathematical tools we turn to finance
by applying the concepts and results to the pricing of derivative
instruments. Foremost, we develop the theory of no-arbitrage pricing of
derivatives, which are basic tools for risk management.
Teaching MethodsLectures (13 x 2 hours) and discussion of exercises (13 x 1 hour)
Method of AssessmentAssignments (30%) and written examination/resit examination (70%).
There is no resit possibility for the assignments.
Entry RequirementsProbability (X_400622) and Analysis 1 (X_400005), or their equivalents.
Shreve, "Stochastic Calculus for Finance I: The Binomial Asset Pricing
Shreve, "Stochastic Calculus for Finance II: Continuous-time models",
Target AudiencemBA, mBA-D, mMath, mSFM, master Econometrics.
Additional InformationA significant part of the course is used to introduce mathematical
subjects and techniques like Brownian motion, stochastic integration and
Ito calculus. In view of this, the course is NOT meant for students who
already followed the master course "Stochastic Integration" or
"Stochastic differential equations". On the other hand, after completing
this course, students may be motivated to follow other courses (like the
two mentioned above) where stochastic calculus is treated in a deeper
and more rigorous way.
Recommended background knowledgeMeasure Theory.
|Language of Tuition||English|
|Faculty||Faculty of Science|
|Course Coordinator||dr. D. Dobler|
|Examiner||dr. D. Dobler|
dr. D. Dobler
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Last-minute registration is available for this course.
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