In the following you find three possible thesis projects. The first is on Fourier Transform methods and option pricing, the second is about the Libor Market Model with SABR stochastic volatility and the third is about Constant Maturity Spread Options valuation.
For questions or comments please contact:
Dr. Jörg Kienitz
Head of Quantitative Analyis
Treasury TR OB
Deutsche Postbank AG
Friedrich-Ebert-Allee 114-126
53113 Bonn
Germany
+49 (0) 228 920 53320
joerg.kienitz@postbank.de
Project 1
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(Fourier Transform Methods)
Description:
We consider the discrete version of the Fourier Transform, DFT. It is used to value several types of options. Starting from the pioneering work of Carr and Madan, [1], the numerical methods have been improved with respect to stability and speed, see for instance [2], [3] and [4].
The aim of this project is to study the relevant literature and compare the approaches, especially, CM, [1], CONV, [2], COS, [4], Lewis [3], in terms of runtime and stability with respect to different pricing models.
To this end all the methods have to be implemented using the FFT algorithm and numerical integration techniques.
As financial relevant models we consider:
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VG-GOU (Variance Gamma with Gamma Ornstein-Uhlenbeck clock)
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VG-CIR (Variance Gamma with Cox-Ingersoll-Ross clock)
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NIG (Normal Inverse Gaussian)
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CGMY model (Carr-Geman-Madan-Yor)
Further research can be done by considering the extension of the method to multi-dimensional problems. An example for using 2D FFT to price options on two underlyings or more underlyings in the 2 factor Variance Gamma model for instance, see [5].
! Coding should be done in Matlab, VB or C++!
[1] Carr,P. and Madan, D., Option Valuation Using the Fast Fourier Transform
[2] Lord, R., et al., A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes
[3] Lewis, A., A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes
[4] Fang, F., Oosterlee, K., A novel pricing method for European options based on Fourier-Cosine series expansions
[5] Eberlein, E., Glau, K. and Papapantoleon, A., Analysis of Fourier Transform Valuation Formulas and Applications
Project 2
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(SABR Libor Market Models – Comparison of different Approaches / Calibration / Simulation)
Description:
The aim of this project is to consider the Libor Market Model, see [4], which is nowadays a market standard but we focus on introducing smile into the model. We wish to consider the LMM-SABR model.
When introduced, see [5], the SABR model was only considered to model a single forward rate for a given fixed maturity T and the forward rate dynamics is a martingale under the T-forward measure. To this end extending the Libor Market model - which is a term structure model - using a SABR style approach means that the joint dynamic has to be specified.
In particular there are three approaches, see [1], [2] and [3]. The aim of the project is to review all three approaches. We wish to compare the proposed methodology of all three approaches in terms of the proposed parametrisation, approximation formulae for calibration instruments (especially swaptions), approximation for the swap dynamic and for fast drift approximation.
Ideally, a numerical example will be implemented in matlab, VB or C++. This example should cover calibration to given data and a comparison in terms of goodness of fit and computational effort.
A final discussion should give pros and cons of each approach to the LMM-SABR model.
[1] Rebonato, The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives: Pricing, Calibrating and Hedging
[2] Hagan, P. and Lesniewski, A., Libor Market Model with SABR style stochastic volatility
[3] Mercurio, F. and Morini, M., No-Arbitrage dynamics for a tractable SABR term structure
Libor Model
[4] Brigo, D. and Mercurio, F. Interest Rate Models – Theory and Practice (2nd edition), Springer
[5] Hagan, Kumar, Lesinewsky and Woodward "Managing Smile Risk"
Project 3
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(Constant Maturity Spread Options)
Description:
The aim of this project is to consider the valuation of Constant Maturity Swap (CMS) Spread Options.
CMS Spread are interest rate derivatives linked to a difference (spread) of two constant maturity swap rates such as the 2 year and 10 year rate. They are liquid instruments quoted in the broker market for different underlyings and maturities. CMS Spreads allow taking positions on the slope of an interest rate curve. The CMS Spread options we consider are the CMS Cap and the CMS Floor.
In particular we wish to examine three approaches
- Markovian Projection, [4]
- Copula and Local Time Approach, [3]
- Term Structure Modelling, [1] and [2]
First, the approaches are compared in terms of assumptions and the taken mathematical approaches.
Second, the suggested approaches to valuation of CMS Spread option should be implemented in matlab, VB or C++. All models should be compared in terms of fitting the observed market prices and the performance in terms of runtime and stability.
[1] Antonov, A. and Matthieu, A., "Analytical formulas for pricing CMS products in the Libor Market Model with the stochastic volatility"
[2] Kiesel, R. und Lutz, T. "Efficient Pricing of CMS Spread Options in a Stochastic Volatility LMM2
[3] Benhamou, E. and Croissant, O., "Local Time for the SABR Model: Connection with the 'Complex' Black Scholes and Application to CMS and Spread Options"
[4] Kienitz, J. and Wittke, M., "Option Valuation in Multivariate SABR Models (with an application to the CMS spread)"
