Seminar: Spring 2008

Patrick Cheridito (Princeton University)

Coherent, convex and monetary risk measures were introduced in a setup where uncertain outcomes are modelled by bounded random variables. We study such risk measures on Orlicz hearts. This includes Lp-spaces for $1\leq p \leq \infty$
and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is automatically real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm
Lipschitz-continuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. We then give dual conditions for monetary risk measures to have properties like Gateaux-differentiability, strict monotonicity with respect to almost sure inequality, strict convexity modulo translation, strict convexity modulo comonotonicity, or monotonicity with respect to different stochastic orders. The theoretical results are applied to analyze various specific examples of risk measures. Some of them have appeared in earlier papers, others are new.

Joint work with Tianhui Li.
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Alexander Schied (Cornell University)

A variety of circumstances can force a market participant to liquidate an asset position that is so large that selling it will significantly impact the underlying asset price. In this talk, we will review some of the mathematical models that have been proposed to deal with this often nonlinear price impact. We will then discuss the problem of constructing optimized liquidation algorithms that minimize a cost
functional or maximize the expected utility of the seller. In the latter case, the optimal strategy is characterized by fully nonlinear PDEs. Sensitivity analysis for solutions of this PDE yields qualitative properties of the strategy depending on the absolute risk aversion of the utility function. A particularly interesting situation
occurs when competing traders become aware of the seller's intention and try to make a profit out of it. We show by an equilibrium analysis that the optimal strategies of seller and competitors are strongly dependent on the liquidity characteristics of the market and the number of competitors. (Based on joint papers with Aurelien Alfonsi, Torsten Schoeneborn, and Antje Schulz).
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Terry Benzschawel (Citigroup)

Available prices on corporate loans are mostly stale and thinly quoted. By analyzing price changes for only liquidly quoted loans, diffusion processes for loans can be deduced and these can be used to estimate expected returns and losses on portfolios of loans. Another aspect of loan valuation is that there is no generally accepted method for computing credit spreads on loans that accounts for their embedded credit dependent prepayment options. I will describe methods for risk-neutral pricing of loans and for computing meaningful measures of loans’ credit spreads and durations. Based on those measures, I derive a function for evaluating relative value among corporate loans. I also examined prices of loan credit default swaps (LCDS) relative to their reference loans and the price value of the LCDS cancellability feature. Finally, I will discuss implications of these results for current market prices of loans, LCDS and bond CDS.
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Jean-Pierre Fouque (University of California Santa Barbara)

The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. In the case of many names we discuss various ways of creating correlation of defaults. In highly-
dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.

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Xunyu Zhou (Oxford University)

An investor holding a stock needs to decide when to sell the stock. It is tempting to think that he should sell at the maximum price over a given investment horizon -- which is what investors always dream of. Unfortunately this is a ``mission impossible". A close yet realistic goal is to sell the stock at the time when the expected relative error between the sold price and the maximum price is minimized. This problem is thoroughly investigated for a geometric Brownian motion model, and it is shown that when the stock is good enough -- which we specify explicitly --the optimal strategy is to sell at the end of the horizon. Moreover, the resulting expected relative error diminishes to zero when the Sharpe ratio of the stock tends to infinity. This result justifies the conventional wisdom that one should buy and hold a stock -- if it is good, that is.
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Liuren Wu (Baruch College)

We develop a simple robust linkage between credit default swaps (CDS) and
American put stock options on the same reference company. Assuming that the
stock price stays above a barrier B before default but drops and remains below a lower barrier A<B after default, we show that the spread between two co-terminal American put options struck within the default corridor [A,B] scaled by their strike
difference replicates a standardized credit insurance contract that pays one dollar at default whenever the company defaults prior to the option expiry and zero otherwise. As long as the default corridor exists, this simple replicating strategy is robust to the details of pre- and post-default stock price dynamics, interest rate
movements, and default risk fluctuations. We use the American put spread to infer risk-neutral default probabilities and compare them to those estimated from the CDS spreads. Collecting data on several companies, we identify strong co-movements between the risk-neutral default probabilities inferred from the two markets. We also find that deviations between the two estimates predict future movements in both markets.
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Tanya Styblo Beder (SBCC)

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Jun Pan (MIT)

This paper examines the connection among corporate bonds, stocks, and Treasury bonds under the Merton model with stochastic interest rate, focusing in particular on the volatility of corporate bonds and its connection to the equity volatility of the same firm and the Treasury bond volatility. For a broad cross-section of corporate bonds from 2002 through 2006, empirical measures of bond volatility are constructed using bond returns over daily, weekly, and monthly horizons. Comparing the empirical volatility with its model-implied counterpart, we find an overwhelming degree of excess volatility that is difficult to be explained by a default-based model. This excess volatility is found to be the strongest at the daily and weekly horizons, indicating a more pronounced liquidity component in corporate bonds at short horizons. At the monthly horizon, the excess volatility tapers off but remains significant. Moreover, we find that variables known to be linked to bond liquidity are important in explaining the cross-sectional variations in excess volatility, providing further evidence of a liquidity problem in corporate bonds. Finally, subtracting the equity and Treasury exposures from corporate bond returns, we find a non-trivial systematic component in the bond residuals that give rise to the excess volatility.

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Kay Giesecke (Stanford)

The ongoing credit crisis highlights the need for portfolio credit risk estimates that account for the feedback from credit events. We introduce and estimate from U.S. corporate default data spanning 1970 to 2006 a new model of self-exciting defaults that incorporates the feedback from events to arrival rates. Our filtered point process likelihood estimators indicate that a default has a significant contagious impact on the surviving firms, after controlling for firms' exposure to a common Feller diffusion risk factor that may be a frailty. Contagion is found to be an important source of the default clustering in the data. Goodness-of-fit tests indicate the statistical importance of incorporating the effects of contagion into a model of correlated default timing. These findings have significant implications for the risk management of corporate debt portfolios and the risk analysis of securities exposed to correlated default risk, such as collateralized debt obligations.
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