MATHS517-23A (HAM)

Stochastic Differential Equations with Applications to Finance

15 Points

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Division of Health Engineering Computing & Science
School of Computing and Mathematical Sciences
Department of Mathematics and Statistics

Staff

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Convenor(s)

Yuri Litvinenko

Yuri Litvinenko
8363
G.3.10
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yuri.litvinenko@waikato.ac.nz

Lecturer(s)

Administrator(s)

: maria.admiraal@waikato.ac.nz

Placement/WIL Coordinator(s)

Tutor(s)

Student Representative(s)

Lab Technician(s)

Librarian(s)

: alistair.lamb@waikato.ac.nz

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What this paper is about

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The paper provides an informal introduction to stochastic calculus and its applications in finance. The following topics are covered.

Random variables, Brownian motion
Stochastic integration and differentiation
Stochastic differential equations
Applications in finance: stocks, bonds, interest rates, options
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How this paper will be taught

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2 lectures a week, plus a tutorial each week. Depending on the number of enrolled students, the paper may be offered as a reading course.
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Required Readings

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O. Calin. An introduction to stochastic calculus with applications to finance
D.J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations

Recommended Readings

C.W. Gardiner. Stochastic methods
A.J. Roberts. Elementary calculus of financial mathematics
U.F. Wiersema. Brownian motion calculus
P. Wilmott, S. Howison, J. Dewynne. The mathematics of financial derivatives
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Learning Outcomes

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Students who successfully complete the course should be able to:

  • Solve different types of problems arising in the theory and application of stochastic differential equations

    Find probability densities of stochastic processes related to the Wiener process.

    Find means and variances of stochastic processes related to the Wiener process.

    Evaluate Wiener integrals.

    Use Ito's formula to evaluate the Ito integrals.

    Determine whether a stochastic process is a martingale.

    Change variables using Ito's lemma.

    Solve exact stochastic differential equations.

    Solve homogeneous linear stochastic differential equations.

    Solve inhomogeneous linear stochastic differential equations.

    Use the Stratonovich calculus to solve the Ito differential equations.

    Derive the Fokker-Planck equation for a stochastic process.

    Use a geometric Brownian motion to describe the behaviour of a financial asset.

    Find the distribution moments of a financial asset.

    Use the Feynman-Kac formula to evaluate distribution moments.

    Use approximations of the Black–Scholes model (arithmetic Brownian motion, vanishing volatility) to estimate the value of standard European call and put options.

    Use put-call parity to find the value of European call and put options.
    Linked to the following assessments:
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Assessments

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How you will be assessed

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The paper is internally assessed using several assignments worth a total of 40% and a compulsory final two-hour test worth 60% of the final grade. Assignments are to be submitted to the lecturer. All assignments and notices about this paper will be posted on moodle.
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The internal assessment/exam ratio (as stated in the University Calendar) is 100:0. There is no final exam. The final exam makes up 0% of the overall mark.

The internal assessment/exam ratio (as stated in the University Calendar) is 100:0 or 0:0, whichever is more favourable for the student. The final exam makes up either 0% or 0% of the overall mark.

Component DescriptionDue Date TimePercentage of overall markSubmission MethodCompulsory
1. assignments
40
2. final test
60
✔
Assessment Total:     100    
Failing to complete a compulsory assessment component of a paper will result in an IC grade
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