Seminars (DQF)

Modelling the default performance of a large heterogeneous portfolio is a major topic in credit risk. One approach is to derive analytic or partly analytic approximations based on the law of large numbers and/or central limit theorem; examples are Vasicek’s large homogeneous portfolio model or the saddle point approximations used in CreditRisk+. Here we introduce an approach based on ideas from stochastic networks. The portfolio members are thought of as particles that move around a number of credit risk states (credit ratings) before eventually defaulting. The transition rates are supposed to depend on an external ‘environment’ process, thus introducing dependence between the particles. We study the limiting behaviour of this system as the number of particles increases, obtaining conditional fluid and diffusion limits from which portfolio performance can be predicted.

We discuss recent theoretical progress in hedging and managing credit risk together with issues of practical implementation with respect to specific products.

We propose a dynamic multi-name credit model framework based on time changed point processes. At the center of our approach is the sequence of unpredictable defaults and losses, which we represent as a rescaled marked Poisson process. We construct the stochastic time change through the compensator of the default counting process. This yields algorithms for the simulation of dependent defaults and losses that start with a simple Poisson sequence. The dynamics of dependent defaults are governed by the evolution of observable information. Specific information structures lead to the known multi-name models and a great deal more. We characterize a new class of flexible self-exciting default processes as time-changed Poisson processes. Applications include the pricing and risk management of multi-name credit products such as basket CDS, CDO's and tranches.

We discuss aspects of the correlation skew in portfolio credit derivatives, in particular the relationship between implied and base correlation for tranches. We present a model which generates correlation skews by mixing copulae and introducing stochastic loss given default variables. This allows us to present a whole range of arbitrage-free base correlation curves.

With the dual pourpose of investigating short-comings of the Gaussian copula model and of modelling the correlation "skew" observed in the CDO market, we describe extensions to the Gaussian copula model which incorporate random recovery and random (level dependent) factor loadings, respectively. We discuss the calibration of these new models and their respective impact on CDO tranche prices. The main conclusion is that when properly calibrated, the random recovery extension does not give rise to a significant skew, whereas the random factor loading model can generate a wide range of skews, including those observed in the market.

Our objective in this paper is to determine and analyze the trading strategy that would allow an investor such as a hedge fund to take advantage of the excessive stock price volatility that has been documented in the empirical literature on asset pricing. To achieve our objective, we first construct a general equilibrium model where stock prices are excessively volatile. We do this using the same device as in Scheinkman and Xiong (2003) where there are two classes of agents and one class is overconfident about the value of the signal. We then analyze the trading strategy of the rational investors who is not overconfident about the the signal. We find that the portfolio of rational investors consists of three components: a static (i.e., Markowitz) component based only on current expected stock returns and risk, a component that hedges the investor against future revisions in the market's expected dividend growth, and a component that hedges against future disagreement in revisions of expected dividend growth. That is, while rational risk-arbitrageurs find it beneficial to trade on their belief that the market is being foolish, when doing so they must hedge future fluctuations in the market's foolishness. Thus, our analysis illustrates that risk arbitrage cannot be based on just a current price divergence; the risk arbitrage must include also a protection in case there is a deviation from that prediction. We also find that the presence of a few rational traders is not sufficient to eliminate the effect of overconfident investors on excess volatility. Moreover, overconfident investors may survive for a long time before being driven out of the market by rational investors.

Multi-factor risk modelling is well established within the equity world. With its theoretical foundations in Arbitrage Pricing Theory, the practical implementation has either relied upon investment practice (fundamental factor models) or statistical data analysis (factor analysis). Academic research so far has amply proven that systematic risk factors are also present in hedge funds. However the identification of these factors has been hampered by - lack of reliable and high frequency return data - a lack of transparency of the underlying investment strategy - the widespread presence of derivative based (sub)-strategies that are harder to capture In our talk we will briefly review which 'factors' have so far been identified within the various hedge fund strategies. We will review their (in and out of sample) explanatory power and draw inferences for hedge fund portfolio construction.

The FX market place is the largest but least studied area of mathematical finance, primarily as the vast majority of trades are over-the-counter, but also because of the arcanery of FX quoting conventions. After demystifying the FX market it will be apparent that there is a rich source of option pricing information which can be used for model fitting. This talk will describe some of the in vogue approaches to solving the pricing problem for vanilla and exotics options and discuss some of the real-world issues facing FX quants. Additionally the future directions of FX modelling are pondered.

We are required to mark-to-market non-plain (exotic) products in a way that is consistent with the observed market prices of liquid vanilla products. This means that for each exotic we must have a one-to-one mapping between vanilla prices and the exotic's price. Such mapping is called the mark-to-market model as it produces mark-to-market price and risk exposure for each exotic. Risk management policies (risk limits, desire to minimise volatility of the mark-to-market P&L) typically compel traders to hedge exotics with vanillas such that the combined risk exposure, measured by the mark-to-market model, is close to zero.

In the traditional approach we set the exotics price equal to its' value given by a valuation model that assumes a certain stochastic evolution of the relevant risk factors. In order to fit vanilla prices practitioners use, are forced to use, over-parameterised models (models with local volatility surfaces is one example) whose resulting risk factor dynamics could be counter-intuitive. Does this make a good model, i.e., does hedging to such model’s risk exposure result in realised replication cost (derivative’s actual “manufacturing cost”) that is close to the initial exotic’s price the model produces? We cannot be sure of that!!!

What are the alternatives? Can we start with a price of an exotic produced by a standard derivatives valuation model, with risk factors dynamics that makes sense (who is to judge?), and somehow, externally, adjust this price to reflect the difference between market and model prices of relevant vanilla options? Would the resulting mapping produce a hedging model that is better than the one based on the above traditional approach?

In this presentation we provide an example of such an alternative mapping based on “external price adjustors”. We show how external price adjustors modifiy the risk exposure produced by the underlying derivatives pricing model. It is likely that the simple external-adjustors method is used by some practitioners (see Pat Hagan’s paper in Wilmott Magazine entitled “Adjusters: Turning Good Prices into Great Prices”).

This work is an extension of earlier joint work with Thierry Bollier and Craig Fithian (“Marking-to-Market Non-Plain Products”, Citigroup, June 2000).

Many smile consistent volatility models are scale invariant, including jump diffusions, standard stochastic volatility models, mixture models and sticky delta local volatility models. Sticky tree local volatility models and the SABR model are not scale invariant. The short-comings of scale invariant models motivates the specification of a general parametric stochastic local volatility model which we show is equivalent to the market model of implied volatilities introduced by Schönbucher (1999).

When volatility is scale invariant the price sensitivities are model free, the only differences between the models being their quality of fit to the market. In stochastic volatility models where price-volatility correlation is non-zero we show how this model free price sensitivity is adjusted to obtain the correct delta. Similar adjustments to obtain the delta for sticky tree and stochastic local volatility models are derived. Our theoretical and empirical results illustrate the inferior hedging performance of mixture models and sticky delta local volatility in equity index markets, even compared with the Black-Scholes model. The best hedging results are obtained with stochastic (local) volatility models.

The last part of the talk introduces the GARCH Jump model as the continuous limit of normal mixture GARCH, a discrete time model that provides the most flexible and intuitive view of skew dynamics and the closest fit to historical data in both equity and FX markets. This is a stochastic local volatility model, but not one with parameter diffusions. The parameters simply jump (occasionally, and simultaneously) between two states.

This highlights the fact that the hedging failure of mixture models can be attributed to the fixed parameters that are commonly applied. By introducing parameter uncertainty the GARCH Jump model provides a tractable, flexible and intuitive tool for capturing regime specific mean-reversion and leverage mechanisms and a skew term structure that persists into long maturities. However, its hedging performance has yet to be studied.