Minicourses
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 contagioninduced 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 13, 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 precrisis 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"?
References
 A. The 2009 perspective
 Andrew G Haldane,
Rethinking the financial network
 Steven L. Schwarcz, Systemic Risk Duke Law School Legal
Studies, Research Paper Series Research Paper No. 163
 Darrell Duffie,
How Big Banks Fail and What to Do about It
 Tobias Adrian and Hyun Song Shin,
Liquidity and Leverage, 2009.
 B. General Models of Systemic Risk

Franklin Allen and Douglas Gale,
Financial Contagion
 L. Eisenberg and T. H. Noe, Systemic risk in financial
systems, Management Science, 47(2):236 249, 2001.
 Erlend Nier, Jing Yang, Tanju Yorulmazer, Amadeo Alentorn,
Network Models and Financial Stability, Bank of England Working
Paper No. 346
 Hyun Song Shin,
Securitisation and Financial Stability, Economic Journal, 119,
309332 (2009)
 Tobias Adrian and Hyun Song Shin,
Liquidity and Leverage, 2009.
 Prasanna Gai, Sujit Kapadia,
Contagion in Financial Networks
 J. Gleeson, T. Hurd, S. Melnik, A. Hackett,
Systemic Risk in banking networks without Monte Carlo
simulation, working paper, 2011.
 T. Hurd, J. Gleeson,
A framework for analyzing contagion in banking networks,
submitted paper, 2011.
 P. Gai, A. Haldane, S. Kapadia,
Complexity, Concentration and Contagion, Journal of Monetary
Economics 58, 2011.
 C. Studies of Specific Financial Systems
 Alfred Lehar,
Measuring Systemic Risk: A Risk Management Approach
 Helmut Elsinger, Alfred Lehar, and Martin Summerc,
Using Market Information for Banking System Risk Assessment
 Rama Cont, Amal Moussa, Edson Bastose Santos,
Network structure and systemic risk in banking systems, 2010
 C. Upper,
Simulation methods to assess the danger of contagion in interbank
markets, Journal of Financial Stability, forthcoming, 2011.
 D. Random Graph Theory
 Watts DJ, Strogatz SH, Collective dynamics of 'smallworld'
networks, Nature Jun 4;393(6684):40910, 1998.
 Duncan J. Watts, A simple model of global cascades on random
networks
 M.E.J. Newman, Networks: An Introduction, Oxford University
Press, 2010.
Alexander Lipton: Applications of classical mathematical methods in finance
Recently, most fundamental and longconsidered 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 atthemoney 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 minicourse 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 jumpdiffusion model. Second, we describe three component parts of UVM, namely, calibration of the model to the market; pricing of vanilla and firstgeneration exotic options; pricing of secondgeneration exotics. Third, we discuss main analytical, semianalytical, and numerical techniques needed for efficient implementation of UVM from a practical standpoint with a particular emphasis on the LewisLipton 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 LewisLipton formula.
Lecture 5: Universal volatility model.
References
[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)
References
[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 RiskReturn Tradeoff, with E. Jouini, Review of
Finance, 15 (3), 575601.
(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 CNKKvolatility surface
model
We show that parametrizations of volatility surfaces (and
even more involved multivariate objects) by timedependent Lévy
processes (as proposed by CarmonaNadtochiyKallsenKrühner) 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, Lévybased, 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 nonhomogeneous 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 productflexible 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 pricedependent 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.
Reference
Verena Hagspiel, Optimal Investment Strategies for ProductFlexible 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 liquidityadjusted risk measures defined on the vector space of asset and liability pairs. We show that convexity and positive superhomogeneity 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 liquidityadjustment 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.
Reference
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 Fouriercosine expansions and ClenshawCurtis quadrature. The pricing method works for both Europeanstyle and Americanstyle Asian options, and for both discretely and continuously monitored versions.
The exponential convergence rate of cosine expansions and ClenshawCurtis 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)
