Prof. Dr. Erich Walter Farkas
- Past Thesis Supervision
- Students who are planning to write a Bachelor or Master thesis under my direction and with my team are advised to contact me well ahead of time. In more concrete terms, this means approximately one semester ahead of the planned starting date.
- An interesting material on how to write a thesis can be found at the home page of Prof. Martin Schweizer, follow this Link.
In the area of Quantitative Finance These topics are more appropriate for students having a solid background in quantitative techniques and finance.
- Cointegration in commodity markets
Cointegration is a long-run relationship where a linear combination of two or more non-stationary time series is stationary. In particular, evidence of cointegration is found in commodity markets. In general, a cointegrated system has an Error Correction Model (ECM) representation that allows for capturing the dynamics of the short-run deviations from the long-run (cointegration) relations. Several Gaussian continuous-time models have been developed in the literature to account for cointegration in commodity markets and to price commodity derivatives. Interesting themes for a thesis are: a) continuous time modeling of a cointegrated system of commodities including features such as stochastic volatility in the short-run equations, jumps in the short-run equations, structural breaks in the long-run equations, stochastic switching of regimes in the long-run equations; b) pricing methods for spread/basket options or for structured products on cointegrated commodities; c) development of strategies for hedging exposures on a commodity using other cointegrated commodities; d) portfolio management of cointegrated commodities.
- Duan, J.-C. and Pliska, S. R. (2004). Option valuation with co-integrated asset prices. Journal of Economic Dynamics and Control, 28(4):727-754.
- Farkas, W., E. Gourier, R. Huitema and C. Necula (2014). A Two-Factor Cointegrated Commodity Price Model with an Application to Spread Option Pricing. working paper.
- Nakajima, K. and Ohashi, K. (2012). A cointegrated commodity pricing model. Journal of Futures Markets, 32(11):995-1033.
- Paschke, R. and Prokopczuk, M. (2009). Integrating multiple commodities in a model of stochastic price dynamics. Journal of Energy Markets, 2(3):47-82.
- Expansion-based methods for pricing derivative products
Expansion-based methods are employed in quantitative finance in order to obtain modified option pricing formulas that account for deviations from the assumption of Gaussian log-returns in the classic Black-Scholes-Merton model. The idea consists in expanding the density function of the risk-neutral probability measure around the Gaussian probability density function, using a set of orthogonal polynomials. One such approach showing promising results is the Gauss-Hermite expansion, based on the set of "physicist" Hermite polynomials. The Gauss-Hermite expansion also allows for obtaining closed form option pricing formulas. Interesting themes for a thesis are: a) conducting an empirical analysis of the pricing and hedging performance of existing expansion-based methods; b) investigating the properties of expansion methods based on other sets of orthogonal polynomials; c) extending the Gauss-Hermite expansion method to price derivatives on multiple underlying; d) using expansion-based methods to obtain closed form option pricing methods for non-affine stochastic volatility models.
- Corrado, C. J., and T. Su. (1996). S&P 500 index option tests of Jarrow and Rudd's approximate option valuation formula. Journal of Futures Markets, 16, 611-629.
- Corrado, C. J., and T. Su. (1997). Implied volatility skews and stock index skewness and kurtosis implied by S&P 500 index option prices. Journal of Derivatives, 4, 8-19.
- Jurczenko, E., B. Maillet, and B. Negrea. (2002). Revisited multi-moment approximate option pricing models: A general comparison (Discussion Paper of the LSE-FMG, No. 430).
- Necula, C., G., Drimus, and W., Farkas. (2013). A General Closed Form Option Pricing Formula, Link.
- Microfoundations of stochastic volatility models
A popular approach for modeling volatility consists in using exogenous stochastic volatility models. In this respect, one departs from the central hypothesis of the Black-Scholes-Merton model, concerning a constant level of volatility, by specifying exogenously a dynamics (in general, affine) of the underlying and its instantaneous variance. However, there are, at micro level, various sources, such as the heterogeneity of traders (in beliefs, in preferences or in behavior) and their interaction generating herding phenomena that could influence the dynamics of volatility. It is, therefore, much more appealing to investigate the implications of such micro-foundations that could lead, in a partial or general equilibrium framework, to an endogenously determined (in general, non-affine) dynamics of the underlying-volatility system. Interesting themes for a thesis are: a) conducting an empirical analysis of the option pricing performance of existing non-affine stochastic volatility models; b) modeling heterogeneity and interaction in financial markets; c) modeling the dynamics of heterogeneity in beliefs; d) analyzing the implications of the heterogeneity and interaction of traders on the dynamics of volatility.
- Alfarano, S., T. Lux, and F. Wagner (2008). Time-variation of higher moments in a financial market with heterogeneous agents: An analytical approach. Journal of Economic Dynamics and Control, 32(1):101-136.
- Follmer, H. and M. Schweizer (1993). A microeconomic approach to diffusion models for stock prices. Mathematical Finance, 3, 1-23.
- Horst, U. (2005). Financial price fluctuation in a stock market model with many interacting agents. Economic Theory, 25, 917-932.
- Kaeck, A., and C. Alexander (2012) Volatility dynamics for the S&P 500 Further evidence from non-affine, multi-factor jump diffusions, Journal of Banking & Finance, 36, 3110-3121
In the area of Mathematical Finance / Quantitative Methods for Risk Management These topics are more appropriate for students having a solid mathematical background.
- Risk Measures
We are interested in studying measures to quantify the amount of capital that a financial institution should be required to raise and use for regulatory purposes. The theory of risk measures has been having a considerable influence on the debate about regulatory capital as witnessed by the current solvency regimes, such as the Basel regimes for banks and Solvency II and the Swiss Solvency Test for insurance companies.
Possible topics include:
- Risk measures in the absence of risk-free securities
- Liquidity-adjusted risk measures
- Risk functionals based on utility profiles
- Risk measures under model uncertainty
- Risk measures in a dynamic setting
A key requirement for financial agents is the ability to value financial assets. Risk measures can be fruitfully used for valuation and pricing purposes, from the single instrument level to the company level.
Possible topics include:
- Multi-asset risk measures
- Good deal valuation based on acceptability criteria
- Valuation in CAPM-like equilibrium models
- Capital Allocation
Once asset valuation has been performed, financial agents face the problem of portfolio selection. Risk measures typically appear as constraints in the corresponding optimization programs.
Possible topics include:
- Optimal risk sharing
- Differentiability properties of risk measures
- Convex duality and optimization applied to portfolio selection
In this context we recommend the reading of the following papers and a preliminary contact also with Ludovic Mathys:
- Farkas, W., P. Koch-Medina, C. Munari (2014). Beyond cash-additive risk measures: when changing the numeraire fails. Finance and Stochastics, 18(1):145-173, [Link to the Journal] [NCCR FinRisk] [SSRN] [arXiv]
- Farkas, W., P. Koch-Medina, C. Munari (2014). Capital requirements with defaultable securities. Insurance: Mathematics and Economics, 55, 58-67, [Link to the Journal] [NCCR FinRisk] [SSRN] [arXiv]
- Farkas, W., P. Koch-Medina, C. Munari (2015). Measuring risk with multiple eligible assets. Mathematics and Financial Economics, 9 (1), 3-27, [Link to the Journal] [SSRN] [arXiv]