Raphael Douady

CLOSED FORM FORMULAS FOR EXOTIC OPTIONS AND THEIR LIFETIME DISTRIBUTION

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge likee-α n2.

Financial Crisis Dynamics: Attempt to Define a Market Instability Indicator

The impact of increasing leverage in the economy produces hyperreaction of market participants to variations of their revenues. If the income of banks decreases, they mass-reduce their lendings; if corporations sales drop, and they cannot adjust their liquidities by further borrowing due to existing debt, then they must immediately reduce their expenses, lay off staff, and cancel investments. This hyperreaction produces a bifurcation mechanism, and eventually a strong dynamical instability in capital markets that is commonly called systemic risk. In this article, we show that this instability can be monitored by measuring the highest eigenvalue of a matrix of elasticities. These elasticities measure the reaction of each sector of the economy to a drop in its revenues from another sector. This highest eigenvalue—the spectral radius—of the elasticity matrix can be used as an early indicator of market instability and potential crisis. Grandmont and subsequent research showed the possibility that the ‘invisible hand’ of markets becomes chaotic, opening the door to uncontrolled swings. Our contribution is to provide an actual way of measuring how close to chaos the market is. Estimating elasticities and actually generating the indicators of instability will be the topic of forthcoming research.

On the Super-Additivity and Estimation Biases of Quantile Contributions

Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to sample size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of the exponent. These estimators can vary over time and increase with the population size, as shown in this article, thus providing the illusion of structural changes in concentration. They are also inconsistent under aggregation and mixing distributions, as the weighted average of concentration measures for A and B will tend to be lower than that from A U B. In addition, it can be shown that under such fat tails, increases in the total sum need to be accompanied by increased sample size of the concentration measurement. We examine the estimation superadditivity and bias under homogeneous and mixed distributions.

On Measuring Nonlinear Risk with Scarce Observations

We propose a methodology for estimating the risk of portfolios that exhibit nonlinear dependence on the risk driving factors and have scarce observations, which is typical for portfolios of investments in hedge funds. The methodology consists of two steps: first, regressing the portfolio return on nonlinear functions of each single risk driving factor and second, merging together the obtained estimates taking into account the dependence between different factors. Performing the second step leads us to a certain probabilistic problem, for which we propose an analytic and computationally feasible solution for the case where the joint law of the factors is a Gaussian copula. A typical practical application can be to estimate the risk of a hedge fund or a portfolio of hedge funds. As a theoretical consequence of our results, we propose a new definition of the factor risk, i.e., the risk of a portfolio brought by a given factor.

Repurchase Agreements and the European Sovereign Debt Crises: The Role of European Clearinghouses

This article investigates the European repo market and its role as an amplification channel for sovereign-debt crises. We focus on the centrally cleared segment, representing the majority of European repos. A novel data set on repo and margin haircuts applied to sovereign bonds by central clearing counter parties (CCPs) is gathered, allowing us to assess the haircut methodologies used by the major European CCPs. We document that following increases in sovereign risk, haircuts set bymajor CCPs on peripheral sovereign bonds increased significantly. The procyclicality of haircuts and the concentration of bilateral repos raise concerns about the CCP-intermediated repo market as a source of systemic risk in the Eurozone. This is however mitigated by the counter cyclical monetary policy of the European Central Bank (ECB).

Extreme Risk, Excess Return and Leverage: The LP Formula

The LP formula is based upon the substitution of the exogenous risk aversion hypothesis by a credit equilibrium hypothesis. This leads to a trade-off between expected blue-sky return – the expected return excluding default scenarios – and extreme risk estimated from scenarios leading to default. An empirical study on the past 90 years shows that this trade-off curve is almost identical across asset classes. In equilibrium, an asset expected blue-sky return is proportional to its contribution to extreme risk. Assuming normal returns, we obtain CAPM as a sub-case of the LP relation. This relationship makes extreme risk underestimation a strong driver of asset price bubbles.