Advanced Financial Mathematics and Structured Derivatives

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Module 1, governed by the WU Executive Academy and the Unicredit Group.

Munich, June 11th - 14th, 2008.

Last update of this page: 23.01.2009

[edit] Material

Attendants of the course may download the presentation sheets from: Sheets


[edit] Software

Computational activities of the course are based on the language R. The examples on the presentation sheets are presented as R-code which requires the package QuantLab.

R can be obtained as free software from R-homepage.

The package QuantLab of R-functions can be downloaded and installed from Package.

In order to make use of this package proceed as follows:

  • download and install R (use the self-extracting .exe-file)
  • download the .zip-file of the QuantLab package to some voluntary directory
  • start R (use the GUI under Windows)
  • use Menu: File -> Packages -> Install packages form local zip-Files, to install the package QuantLab
  • execute: library(QuantLab)

Now the package should be loaded.

  • use the menu item Help -> HTML Help
  • click: Packages
  • click: QuantLab
  • study: QuantLab-package
  • click the commands and enjoy the examples (located at the end of QuantLab-package)

In order to learn the basics of R study the [manual]


[edit] Exam

The exam will consist of eight question, sampled from the list below. The sheets of the course can be used during the exam. The answers should be given in a handwritten form.

  • Describe in words the binomial model and show that NA holds iff d < er(T − t) < u.
  • Describe the single-period lognormal model. Explain the parameters of the model.
  • Prove for a single period model: If NA is satisfied, then prices of admissible portfolios are uniquely determined.
  • Explain the concept of a forward price.
  • State and explain the put-call parity for binary options.
  • Show for the single-period binomial model: NA holds iff there exist a risk neutral probability.
  • Show for the single period binomial model: The risk neutral probability is unique.
  • For a single period model: Explain the concept of risk neutrality. Discuss the lognormal case.
  • On sheet 39 the formula for the Delta in the last line is wrong. What is the correct formula and why ?
  • Explain in words (and handwritten diagrams) why an American Call and a European Call (on a non-dividend paying stock) have the same price. What about Puts ? (Distinguish the cases r > 0 and r = 0.)
  • Discuss for a multiperiod binomial model: What is the influence of jump height and jump probability on the price of a derivative ? What does this mean for the lognormal approximation of the binomial model ?
  • Explain carefully each step of the proof on sheet 71.
  • Show that the Wiener process is a martingale.
  • Find E(W_t^2|\mathcal{F}_s).
  • Find At such that (W_t^2-A_t is a martingale.
  • Find \lim_{t\to\infty} W_t/t.
  • Find drift and diffusion of a generalized Wiener process.
  • Let (St) be a Black-Scholes asset. Find E(St) and E(S_t|\mathcal{F}_s). When is (St) a martingale ?
  • Explain the quadratic variation of a Wiener process (by a limit argument).
  • Solve the (deterministic) differential equation K(t)' = K(t)r(t) + c(t).
  • Explain (stochastic) integration by parts (for dX_t^2) intuitively by considering the discrete counterpart.
  • Evaluate \int_0^t W_s dW_s.
  • Deduce the (stochastic) integration by parts formula for dXtYt as a consequence of the corresponding formula for dX_t^2.
  • Find dW_t^3 by integration by parts.
  • Find dW_t^3 by Ito's formula.
  • Discuss intuitively the relation between Ito's formula and Taylor's formula.
  • Show that the components of an Ito process are uniquely determined.
  • Show that every smooth function of an Ito process is again an Ito process.
  • Show that the product of two Ito processes is again an Ito process.
  • Show that the stochastic exponential of a continuous semimartingale solves a linear stochastic differential equation.
  • Explain the relation between the Black-Scholes model an the corresponding stochastic differential equation.
  • Explain carefully the steps of the derivation on sheets 147 and 149.
  • Explain the properties of the Wiener integral for the case when the integrands are step functions.
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