Michèle Vanmaele

An Overview of Comonotonicity and Its Applications in Finance and Insurance

Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this chapter, we give an extensive bibliographic overview—without claiming to be complete—of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements in the period 2004–2010. These applications range from pricing and hedging of derivatives over risk management to life insurance.

Some results in lumped mass finite-element approximation of eigenvalue problems using numerical quadrature formulas

In this paper we establish the convergence and the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a lumped mass finite-element approximation. Various aspects of lumped mass techniques have been discussed for such eigenvalue problems by Fix (1972), Ishihara (1977), Strang and Fix (1973) and Tong et al. (1971), among others. In our approach the lumping of the mass matrix results from the use of a Lobatto quadrature formula for the integrals over rectangular Lagrange finite elements of degree k.

Moment matching approximation of Asian basket option prices

In this paper we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of [M. Curran, Valuing Asian and portfolio by conditioning on the geometric mean price, Management Science 40 (1994) 1705-1711] and of [G. Deelstra, J. Liinev, M. Vanmaele, Pricing of arithmetic basket options by conditioning, Insurance: Mathematics & Economics 34 (2004) 55-57] in several ways. We create a framework that allows for a whole class of conditioning random variables which are normally distributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also improve the bounds of [G. Deelstra, I. Diallo, M. Vanmaele, Bounds for Asian basket options, Journal of Computational and Applied Mathematics 218 (2008) 215-228]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.

Pricing and hedging Asian basket spread options

Asian options, basket options and spread options have been extensively studied in the literature. However, few papers deal with the problem of pricing general Asian basket spread options. This paper aims to fill this gap. In order to obtain prices and Greeks in a short computation time, we develop approximation formulae based on comonotonicity theory and moment matching methods. We compare their relative performances and explain how to choose the best approximation technique as a function of the Asian basket spread characteristics. We also give explicitly the Greeks for our proposed methods. In the last section we extend our results to options denominated in foreign currency.

External finite-element approximations of eigenfunctions in the case of multiple eigenvalues

Abstract The paper deals with the finite-element analysis of second-order elliptic eigenvalue problems when the approximate domains Ωh are not subdomains of the original domain Ω⊂ R 2 . The considerations are restricted to piecewise linear approximations. Special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues. As yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h. We avoid this disadvantage.

Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.

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.

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.

Explicit portfolio for unit-linked life insurance contracts with surrender option

Introducing a surrender option in unit-linked life insurance contracts leads to a dependence between the surrender time and the financial market. [J. Barbarin, Risk minimizing strategies for life insurance contracts with surrender option, Tech. rep., University of Louvain-La-Neuve, 2007] used a lot of concepts from credit risk to describe the surrender time in order to hedge such types of contracts. The basic assumption made by Barbarin is that the surrender time is not a stopping time with respect to the financial market. The goal of this article is to make the hedging strategies more explicit by introducing concrete processes for the risky asset and by restricting the hazard process to an absolutely continuous process. First, we assume that the risky asset follows a geometric Brownian motion. This extends the theory of [T. Moller, Risk-minimizing hedging strategies for insurance payment processes, Finance and Stochastics 5 (2001) 419-446], in that the random times of payment are not independent of the financial market. Second, the risky asset follows a Levy process. For both cases, we assume the payment process contains a continuous payment stream until surrender or maturity and a payment at surrender or at maturity, whichever comes first.

On a variational approximation method for a class of elliptic eigenvalue problems in composite structures

We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in M nonoverlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. We present a general finite element method to obtain a numerical solution of the eigenvalue problem, starting from a nonstandard formally equivalent variational formulation in an abstract setting in product Hilbert spaces. We use standard Lagrange finite element spaces on the subdomains. Moreover, the bilinear forms are approximated by suitable numerical quadrature formulas. We obtain error estimates for both the eigenfunctions and the eigenvalues, allowing for the case of multiple exact eigenvalues, by a pure variational method.