11th Winter school on Mathematical Finance


Tom Hurd: Introduction to financial networks and systemic risk

The study of contagion in financial systems is very topical in light of recent events in the global markets. "Contagion" refers to the spread of defaults through a system of financial institutions, with each successive default causing increasing pressure on the remaining components of the system. The term "systemic risk" refers to the contagion-induced threat to the financial system as a whole, due to the default of one (or more) of its component institutions. The ultimate question for me is how mathematical models can help us understand systemic risk. (handout of lectures 1-3, handout of lectures 4,5)

More specific topics and questions I intend to address in this minicourse include:
1. What is a good working definition of systemic risk?
2. How do banks fail?
3. What is a random graph?
4. What is the basic economic picture of the financial system as a random graph?
5. What conclusions have resulted from various economic studies of the financial systems of different countries?
6. What important effects did pre-crisis models miss that we can now learn from?
7. What is the mathematics of cascading defaults in a random graph?
8. What does the theory of random graphs hint about the nature of the financial system?
9. Are there some useful "deliberately simplified models of systemic risk"?


Alexander Lipton: Applications of classical mathematical methods in finance

Recently, most fundamental and long-considered solved problems of financial engineering, such as construction of yield curves and calibration of implied volatility surfaces, have recently turned out to be more complex than previously thought. In particular, it has become apparent that one of the main challenges of options pricing and risk management is the sparseness of market data for model calibration, especially in severe conditions. Market quotes can be very sparse in both strike and maturity. As the spot price moves, options that were close to at-the-money at inception become illiquid, so that one has to find ways to interpolate and extrapolate the implied volatilities of liquid options to mark them to market. Moreover, for certain asset classes the concept of implied volatility surface is badly defined. For instance, for commodities it is not uncommon to have market prices of options for only a single maturity, while for foreign exchange it is customary to quote option prices with no more than five values of delta and very few maturities.
The calibration of a model to sparse market data is needed not only for the consistent pricing of illiquid vanilla options, but also for the valuation of exotic options. The latter is particularly demanding since it requires the construction of implied and local volatility surfaces across a wide range of option strikes and maturities.
In this mini-course we shall discuss a universal volatility model (UVM) and discuss its applications to pricing of financial derivatives. First, we describe three sources of UVM, namely, local volatility model; stochastic volatility model; and jump-diffusion model. Second, we describe three component parts of UVM, namely, calibration of the model to the market; pricing of vanilla and first-generation exotic options; pricing of second-generation exotics. Third, we discuss main analytical, semi-analytical, and numerical techniques needed for efficient implementation of UVM from a practical standpoint with a particular emphasis on the Lewis-Lipton formula and its applications.

Lecture 1: Market overview. How mathematical finance is used in practice.
Lecture 2: Three sources and three component parts of the universal volatility model.
Lecture 3: Local volatility model.
Lecture 4: Stochastic volatility model and the Lewis-Lipton formula.
Lecture 5: Universal volatility model.

[1] Alexander Lipton, Mathematical Methods for Foreign Exchange, World Scientific, 2001
[2] Alexander Lipton, Applications of Classical Mathematical Methods in Finance, Theory and Practice (slides of the lectures)

Special invited lectures

Elyès Jouini: Financial markets equilibrium with heterogeneous agents

This paper presents an equilibrium model in a pure exchange economy when investors have three possible sources of heterogeneity. Investors may differ in their beliefs, in their level of risk aversion and in their time preference rate. We study the impact of investors heterogeneity on the properties of the equilibrium. In particular, we analyze the consumption shares, the market price of risk, the risk free rate, the bond prices at different maturities, the stock price and volatility as well as the stock's cumulative returns, and optimal portfolio strategies. We relate the heterogeneous economy with the family of associated homogeneous economies with only one class of investors. We consider cross sectional as well as asymptotic properties. (joint work with Jaksa Cvitanic, Semyon Malamud, Clotilde Napp)

[1] Cvitanic, Jouini, Napp and Malamud, 2011, Financial Markets Equilibrium with Heterogeneous Agents, with J. Cvitanic, E. Jouini and S. Malamud, Forthcoming, Review of Finance. Online First July 2011. (SSRN preprint)
[2] Jouini and Napp, 2010, Unbiased Disagreement in Financial Markets, Waves of Pessimism and the Risk-Return Tradeoff, with E. Jouini, Review of Finance, 15 (3), 575-601. (SSRN preprint)

Yuri Kabanov: On local martingale deflators and market portfolios

Local martingale deflator is a multiplicator, transforming value processes into local martingales. If the inverse of such a deflator is the value process of portfolio, the latter is called market portfolio. We provide some conditions for the existence of the mentioned objects.

Josef Teichmann: Finite dimensional realizations for the CNKK-volatility surface model

We show that parametrizations of volatility surfaces (and even more involved multivariate objects) by time-dependent Lévy processes (as proposed by Carmona-Nadtochiy-Kallsen-Krühner) lead to quite tractable term structure problems. In this context we can then ask whether the corresponding term structure equations allow for (regular) finite dimensional realization, which necessarily leads to models driven by an affine factor process. This is another confirmation that affine processes play a particular role in mathematical finance. The analysis is based on a careful geometric analysis of the term structure equations by methods from foliation theory. (slides)

Short lectures

Paul Gruntjes: Modeling dynamic default correlation in a Lévy world with applications to CDO pricing

In this talk I will present an intuitive, practical, Lévy-based, dynamic default correlation model, with applications to CDO pricing. More specifically, we first model marginal default probabilities using a dynamic structural Variance Gamma (VG) model. Then we propose a dynamic correlation structure between the individual obligors that is based on these marginal probabilities. The key advantage of our correlation structure is that it is intuitive, dynamic, and allows for easy calibration on the market since the underlyings are market observables, for which there is ample data available. In case of a homogeneous basket of obligors, our model is as easy as the copula approach. In case of a non-homogeneous basket, the evaluation of our model is fast and accurate, and therefore does not impose any restrictions on numerical computations. In this case our model relies on the computation of convolutions to calculate a recursion, which can be efficiently and accurately done using numerical Laplace transform inversion. The complexity of the algorithm is low.

Verena Hagspiel: Optimal investment strategies for product-flexible and dedicated production systems under demand uncertainty

This paper studies the optimal investment strategy of a firm having the managerial freedom to acquire either flexible or dedicated production capacity. Flexible capacity is more expensive but allows the firm to switch costlessly between products and handle changes in relative volumes among products in a given product mix. Dedicated capacities restrict to manufacture one specific product but for lower acquisition costs. Specifically, I model the investment decision of a monopolist selling two products in a market characterized by price-dependent and uncertain demand, in a continuous time setting. I find that flexibility especially pays off when uncertainty is high, substitutability low, and profit levels of the two products are substantially different. In the flexible case, the firm just produces the most profitable product under high demand, while if demand is low the firm produces both products to make total market demand bigger. In the dedicated case the firm invests in both capacities only if the substitutability rate is low and profitability of both products high enough. Otherwise, it restricts investment to one dedicated capacity for the more profitable product. Considering a firm's decision to change from dedicated to flexible capacity, it is shown that despite perfectly positively correlated demand the firm will undertake this switch even for very low demand cases if the profitability of the products is substantially different. The option to increase total capacity accelerates investment in flexible capacity when the profit levels of both products are high enough.

Verena Hagspiel, Optimal Investment Strategies for Product-Flexible and Dedicated Production Systems Under Demand Uncertainty

Kolja Loebnitz: Liquidity risk meets economic capital and RAROC

A bank's liquidity risk lays in the intersection of funding risk and market liquidity risk. We offer a mathematical framework to make Economic Capital and RAROC sensitive to illiquidity. We introduce the concept of a liquidity cost profile as a quantification of a bank's illiquidity at balance sheet level. This leads to the concept of liquidity-adjusted risk measures defined on the vector space of asset and liability pairs. We show that convexity and positive super-homogeneity of risk measures is preserved under the liquidity adjustment, while coherence is not, reflecting the common idea that size does matter. We indicate how liquidity cost profiles can be used to determine whether combining positions is beneficial or harmful. Finally, we address the liquidity-adjustment of the well known Euler allocation principle. Our framework may be a useful addition to the toolbox of bank managers and regulators to manage liquidity risk.
Loebnitz, K. and Roorda, B. (2011), Liquidity Risk meets Economic Capital and RAROC (SSRN preprint)

Bowen Zhang: An efficient pricing method for Asian options based on Fourier cosine expansions

I will give a talk on our pricing method for Asian options. It is an efficient pricing method for arithmetic and geometric Asian options under Lévy processes, based on Fourier-cosine expansions and Clenshaw-Curtis quadrature. The pricing method works for both European-style and American-style Asian options, and for both discretely and continuously monitored versions. The exponential convergence rate of cosine expansions and Clenshaw-Curtis quadrature reduces the computational time of the method to less than one millisecond for geometric Asian options and to 2 seconds for arithmetic Asian options. Our method can be seen as an alternative for existing pricing methods for Asian options. The algorithm for arithmetic Asian options gives the prices in a robust way when the number of monitoring dates increases, for example, for Lévy processes, like CGMY or NIG processes. The method's accuracy is illustrated by a detailed error analysis and its performance is further demonstrated through various numerical examples. (slides)

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