The herd behavior index: A new measure for systemic risk in financial markets.

We introduce a new and easy to calculate measure for systemic risk in financial markets. This measure is baptized the Herd Behavior Index (HIX). It is model-independent and forward looking, based on observed option data. In order to determine the degree of systemic risk or herd behavior in a financial market one should compare the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The Herd Behavior Index (HIX) is defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance of this extreme market situation. Using the theory of comonotonicity, the extreme situation can easily be backed out of the observed option quotes. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days implied Herd Behavior Index for the Dow Jones Industrial Average, covering the period January 2003 to October 2009.Comonotonicity; systemic risk; correlation; VIX volatility index;

On an optimization problem related to static super-replicating strategies

In this paper, we investigate an optimization problem related to super-replicating strategies for European-type call options written on a weighted sum of asset prices, following the initial approach in Chen et al. (2008). Three issues are investigated. The first issue is the (non-)uniqueness of the optimal solution. The second issue is the generalization to an optimization problem where the weights may be random. This theory is then applied to static super-replication strategies for some exotic options in a stochastic interest rate setting. The third issue is the study of the co-existence of the comonotonicity property and the martingale property.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

FIX - The fear index. Measuring market fear.

In this paper, we propose a new fear index based on (equity) option surfaces of an index and its components. The quanti¯cation of the fear level will be solely based on option price data. The index takes into account market risk via the VIX volatility barometer, liquidity risk via the concept of implied liquidity, and systemic risk and herd-behavior via the concept of comonotonicity. It thus allows us to measure an overall level of fear (excluding credit risk) in the market as well as to identify precisely the individual importance of the distinct risk components (market, liquidity or systemic risk). As a side result we also derive an upperbound for the VIX.

Measuring the degree of herd behavior in financial markets.

The main goal of this dissertation is to measure the extent to which stock prices will move together in the future. In Part III we propose two different ways to measure herd behavior between stock prices: a model-free and a model-based approach. Measuring co-movement in a model-free way can be considered as an additional market stress indicator besides the well-known volatility estimates like the VIX. The model-based approach can be considered as an improvement of the well-known implied correlation. Part I:Comparing risks, equality in distribution and comonotonicityIn Part I of this dissertation, we investigate how the random sum of the components of a random vector can be used to reveal information about the multivariate nature of the random vector. In Part III, these results help us to construct herd behavior measures based on the random sum. Chapter 3 introduces some important integral stochastic orders. For example, we consider the convex and supermodular orders, which will play an important role throughout this dissertation. An interesting result in this chapter states that the expected utility of a random variable can be expressed as a mixture of upper and lower tail transforms of this random variable. In Chapter 6 we derive a similar result for the distorted expectation of a random variable: the distorted expectation of a random variable can be expressed as a mixture of Tail Values-at-Risk of this random variable. These representations for the expected utility and the distorted expectation will be used in Chapter 7 to show that under the appropriate conditions, one can prove an equality in distribution by comparing expected utilities or distorted expectations. The concept of comonotonicity is considered in Chapter 4. Loosely speaking, comonotonicity refers to a situation where all random variables are non-decreasing functions of only one random source, i.e. they will move in the same direction . The distribution function and the stop-loss premiums of a sum of comonotonic random variables can be determined in an analytical form, which makes the comonotonic sum an attractive random variable; see Chapter 5. We finish the first part of this dissertation with a set of characterizations of comonotonicity based on the distribution of the sum.Part II: Pricing index optionsAn index option is an option which has as underlying a basket of stocks. The dependence structure between the different stocks makes it a hard task to derive closed form solutions for the price of such an exotic option. In this dissertation we consider two different approaches. Chapter 9 derives a static super-replicating strategy for the pay-off of a basket option. This strategy uses only traded vanilla options and is model-free. We consider the finite market case, where the prices of the options on the stocks of which the index is composed are available for a finite number of strikes. We introduce a framework where the super-replicating strategy for an index call and an index put is derived at the same time. We prove that such an integrated approach gives rise to an efficient algorithm which is fast to calculate. Even if the individual stocks composing the basket can be described by the celebrated Black & Scholes model, the price of an index option is still hard to determine. In Chapter 11 we derive an upper and lower bound for this price, based on the theory of comonotonicity. A linearcombination of these two bounds results in a close approximation for the price of the index option.Part III:Measuring herd behavior in stock marketsThe last part of this dissertation deals with the problem of measuring dependence between stock prices. Chapter 12 and Chapter 13 construct a family of dependence measures, based on the theory of comonotonicity and the information content of the random sum. Measuring dependence can become cumbersome when the number of r.v. s involved becomes large. Using the information contained in the random sum results in a tractable dependence measure for the average level ofco-movement. Chapter 12 uses swap rates on the index to determine the degree of herd behavior whereas Chapter 13 is based on distorted expectations of the index. It will be shown that both the swap rate and the distorted expectation can be determined in a model-free way using the available option data. As result, the corresponding estimate for the degree of herd behavior is forward looking and model-free. Chapter 14 considers a model-based approach to measure the dependence between asset prices. More precisely, we assume that the stocks can be described by a multivariate Black & Scholes model. Inverting the pricing formula for index options results in an implied estimate for the mean level of correlation.status: publishe

The multivariate Variance Gamma model: Basket option pricing and calibration

status: publishe