Ann De Schepper

Are blue chip stock market indices good proxies for all-shares market indices? The case of the Brussels Stock Exchange 1833-2005

In this article, we calculate a market-weighted return index for the 20 largest stocks listed on the Brussels Stock Exchange over the period 1833–2005, based on a new, unique and high-quality database. We find that this index captures the most important stylised facts of the value-weighted return of all shares listed on the Brussels Stock Exchange in this period. Our results support the empirical practice of concentrating on just the largest stocks. The indices we construct are based on one of the longest Belgian time series available. The indices take into account the exact dividends, the timing of the dividend cash flows and all capital operations. We are therefore able to decompose total returns into capital gain returns and dividend returns, which is not possible with most historical return series. We show that, to construct a credible return index, it is crucial to fully take into account dividends.

Measuring Comonotonicity in M-Dimensional Vectors

In this contribution, a new measure of comonotonicity for m-dimensional vectors is introduced, with values between zero, representing the independent situation, and one, reflecting a completely comonotonic situation. The main characteristics of this coefficient are examined, and the relations with common dependence measures are analysed. A sample-based version of the comonotonicity coefficient is also derived. Special attention is paid to the explanation of the accuracy of the convex order bound method of Goovaerts, Dhaene et al. in the case of cash flows with Gaussian discounting processes.

A new graphical tool for copula selection

The selection of copulas is an important aspect of dependence modeling. In many practical applications, only a limited number of copulas is tested, and the modeling applications usually are restricted to the bivariate case. One explanation is the fact that no graphical copula tool exists that allows us to assess the goodness-of-fit of a large set of (possible higher-dimensional) copula functions at once. This article seeks to overcome this problem by developing a new graphical tool for the copula selection, based on a statistical analysis technique called “principal coordinate analysis.” The advantage is three-fold. First, when projecting the empirical copula of a modeling application on a two-dimensional (2D) copula space, it allows us to visualize the fit of a whole collection of multivariate copulas at once. Second, the visual tool allows us to identify “search” directions for potential fit improvements (e.g., through the use of copula transforms). Finally, the tool makes it also possible to give a 2D visual overview of a large number of known copula families, leading to a better understanding and a more efficient use of the different copula families. The robustness of the new graphical tool is investigated by means of a small simulation study, and the practical use of the tool is demonstrated for two 2D and two 3D (three-dimensional) fitting examples. MATLAB code through the examples is available online in the supplementary materials.

A New Method for the Construction of Bivariate Archimedean Copulas Based on the λ Function

We introduce and discuss a general method for constructing bivariate Archimedean copula families. The central item in our method is the function (t ∈ [0, 1]), where ϕ is the generator of the Archimedean copula. The construction of new copulas by means of λ has several advantages. The most important one is the straightforward relationship between the λ function and Kendalls τ and the coefficients of upper and lower tail dependence λ L and λ U , as defined in Joe (1997), which makes it possible to use these quantities as copula parameters and to control them independently of each other. Furthermore, the λ-method allows to construct multi-parameter families in a clear and organized way. The methodology is explained and illustrated by two- and three-parameter copula families.

Stable laws and the present value of fixed cash flows

Abstract In this paper, the authors consider the present value of a series of fixed cash flows under stochastic interest rates. To model these interest rates, they don’t use the common lognormal model, but stable laws, which better fit in with reality. Their main intention is to derive a result for the distribution function of such a present value. However, due to the dependencies between successive discounted payments, the calculation of an exact analytical distribution is impossible. Therefore, use is made of the methodology of comonotonic random variables and the convex ordering of risks, introduced by the same authors in some previous papers. The present paper starts with a brief overview of properties and features of stable laws, and of the possible application of the concept of convex ordering to sums of risks, which is also the situation for a present value of future payments. Afterwards, the authors show how, for the present value under investigation, an approximation in the form of a convex upper bound can be derived. This upper bound has an easier structure than the original present value, and they derive elegant calculation formulas for the distribution of this bound. Finally, they provide some numerical examples that illustrate the precision of the approximation. Due to the design of the present value and the construction of the upper bound, these illustrations show great promise concerning the accuracy of the approximation.

The GARCH(1,1)-M model: results for the densities of the variance and the mean

Abstract This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307–327], and investigates the limit diffusion form as it is presented in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38]. The distribution for the conditional variance process is derived, and in the limit for t going to infinity is shown to coincide with the stationary distribution given in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38]. In addition it is shown how the distribution for the complete model can be arrived at; explicit calculations are given in case the conditional variance is a martingale.

Atomic Implied Volatilities

In this note, we present a novel approach to derive asymptotics for Black implied volatilities under the same generic model as proposed in Antonov and Misirpashaev (2009). We perform a time substitution as used by Duru and Kleinert (1979) to calculate the path integral formulation of the H-atom. We demonstrate that the method provides asymptotic implied volatility formula comparable to the result of Hagan and Woodward (1999) for local volatility models and Hagan et al. (2001) for stochastic volatility models. We also discuss possible application to the pricing of basket options. The method is presented as an alternative to Markov projection as introduced by Piterbarg (2006) and is claimed to be applicable to a wide range of numerical problems arising in finance.

IBNR reserves under stochastic interest rates

Abstract This paper intends to evaluate the present value of incurred but not reported (IBNR) reserves, when future interest rates are unknown. We first derive a result for the Laplace transform of the present value, when it is assumed that the interest rates are stochastic and can be modelled by means of a stochastic process which is similar to the model of Cox et al. (1985). Starting from this Laplace transform, it is shown how the probability distribution for the quantity under investigation can be found. The results are illustrated numerically.

Edgeworth expansions of stochastic trading time

Under most local and stochastic volatility models the underlying forward is assumed to be a positive function of a time-changed Brownian motion. It relates nicely the implied volatility smile to the so-called activity rate in the market. Following Young and DeWitt-Morette (1986), we propose to apply the Duru-Kleinert process-cum-time transformation in path integral to formulate the transition density of the forward. The method leads to asymptotic expansions of the transition density around a Gaussian kernel corresponding to the average activity in the market conditional on the forward value. The approximation is numerically illustrated for pricing vanilla options under the CEV model and the popular normal SABR model. The asymptotics can also be used for Monte Carlo simulations or backward integration schemes.

Duru-Kleinert Asymptotic Expansions for Long-Term Foreign Exchange and Swaptions Implied Volatility Smile

In this paper, we develop asymptotic formulas for long-dated Foreign Exchange (FX) and swaptions implied volatilities. We extend the method exposed in Decamps and De Schepper (2009b) to a generic model with time-dependent parameters. Imposing a condition on the skew, we derive averaging formulas for the parameters. The method is applied to the pricing of FX options when the domestic and foreign interest rate curves are driven by Gaussian short-term rate models and to the pricing of swaptions in the Libor market model.