Daniël Linders

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 quantification 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 an additional result, we also derive an upperbound for the VIX.

A multivariate dependence measure for aggregating risks

To evaluate the aggregate risk in a financial or insurance portfolio, a risk analyst has to calculate the distribution function of a sum of random variables. As the individual risk factors are often positively dependent, the classical convolution technique will not be sufficient. On the other hand, assuming a comonotonic dependence structure will likely overrate the real aggregate risk. In order to choose between the two approximations, or perhaps use a weighted average, we should have an indication of the accuracy. Clearly this accuracy will depend on the copula between the individual risk factors, but it is also influenced by the marginal distributions. In this paper we introduce a multivariate dependence measure that takes both aspects into account. This new measure differs from other multivariate dependence measures, as it focuses on the aggregate risk rather than on the copula or the joint distribution function itself. We prove several interesting properties of this new measure and discuss its relation to other dependence measures. We also give some comments on the estimation and conclude with examples and numerical results.

Ordered Random Vectors and Equality in Distribution

Abstract In this paper we show that under appropriate moment conditions, the supermodular ordered random vectors and with equal expected utilities (or distorted expectations) of the sums and for an appropriate utility (or distortion) function, must necessarily be equal in distribution, that is . The results in this paper can be considered as generalizations of some recent results on comonotonicity, where necessary conditions related to the distribution of are presented for the random vector to be comonotonic.

A Framework for Robust Measurement of Implied Correlation

In this paper we consider the problem of deriving correlation estimates from observed option data. An implied correlation estimate arises when we match the observed index option price with a corresponding model price. The underlying model assumes that stock prices can be described using a lognormal distribution, while a Gaussian copula describes the dependence structure. Within this multivariate stock price model, the index option price is not given in closed form and has to be approximated. Different methods exist and each choice leads to another implied correlation estimate. We show that the traditional approach for determining implied correlations is a member of our more general framework. It turns out that the traditional implied correlation underestimates the real correlation. This error is more pronounced when some stock volatilities are large compared to the other volatility levels. We propose a new approach to measure implied correlation which does not has this drawback. However, our numerical illustrations show that determining implied correlations with the traditional approach may be justified for strike prices which are close to the at-the-money strike price. We also show that implied correlation estimates can be used to define an index, called the Implied Correlation Index (ICX), which reflects the markets perception about future (short term) co-movement between stock prices. Using a volatility index together with the ICX gives an accurate description of the future level of market fear.

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. We investigate three issues related to super-replicating strategies for options written on a weighted sum of asset prices.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.The third issue is the study of the co-existence of the comonotonicity property and the martingale property.

The Multivariate Black & Scholes Market: Conditions for Completeness and No-Arbitrage

In order to price multivariate derivatives, there is need for a multivariate stock price model. To keep the simplicity and attractiveness of the one-dimensional Black & Scholes model, one often considers a multivariate model where each individual stock follows a Black & Scholes model, but the underlying Brownian motions might be correlated. Although the classical one-dimensional Black & Scholes model is always arbitrage-free and complete, this statement does not hold true in a multivariate setting. In this paper, we derive conditions under which the the multivariate Black & Scholes model is arbitrage-free and complete.

Index Options: A Model-Free Approach

This paper contains an overview and an extension of the theory on comonotonicitybased model-free upper bounds and super-replicating strategies for stock index options, as presented in Hobson et al. (2005) and Chen et al. (2008). Whereas these authors only consider index call options, here a uni…ed approach for call and put options is presented. Considering a uni…ed framework gives rise to an e¢ cient algorithm for calculating upper bounds and for determining the corresponding superhedging strategies for both cases. The uni…ed framework also allows to extend several existing results, in particular on the optimality of the superhedging strategies. Several practical issues concerning the implementation of the results are discussed. In particular, a simpli…ed algorithm is presented for the situation where for some of the constituent stock in the index there are no options available.

The Multivariate Variance Gamma Model: Basket Option Pricing and Calibration

In this paper, we propose a methodology for pricing basket options in the multivariate Variance Gamma model introduced in Luciano and Schoutens [Quant. Finance 6(5), 385–402]. The stock prices composing the basket are modelled by time-changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we express the basket option price as a linear combination of Black & Scholes prices. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way. As an illustration, we show that even in the constrained situation where the pairwise correlations between the Brownian motions are assumed to be equal, the multivariate VG model can closely match the observed Dow Jones index options.

Option prices and model-free measurement of implied herd behavior in stock markets

In this paper, we introduce two classes of indices which can be used to measure the market perception concerning the degree of dependency that exists between a set of random variables, representing different stock prices at a fixed future date. The construction of these measures is based on the theory of comonotonicity. Both types of herd behavior indices are model-free and risk-neutral, derived from available option data. Depending on its particular definition, each index represents a particular aspect of the market sentiment concerning future co-movement of the underlying stock prices.

Basket Option Pricing and Implied Correlation in a Lévy Copula Model

In this paper we employ the Lcopula model to determine basket option prices. More precisely, basket option prices are determined by replacing the distribution of the real basket with an appropriate approximation. For the approximate basket we determine the underlying characteristic function and hence we can derive the related basket option prices by using the Carr-Madan formula. Two approaches are considered. In the first approach, we replace the arithmetic sum by an appropriate geometric average, whereas the second approach can be considered as a three-moments-matching method. Numerical examples illustrate the accuracy of our approximations; several L´ evy models are calibrated to market data and basket option prices are determined. In a last part we show how our newly designed basket option pricing formula can be used to define implied L´ evy correlation by matching model and market prices for basket op-