Collateralized structured products
Mirco Mahlstedt (Technische Universität München, Germany)
Joint work with Marcos Escobar, Sven Panz and Rudi Zagst

Wednesday June 4, 14:30-15:00 | session 5.5 | Options, Futures | room G

In this paper multidimensional structured products with a collateral triggered by a default of the issuing company are studied. In the last decade, the volume of trades in structured products has increased tremendously. Particularly after the subprime and nancial crisis with the default by Lehman Brothers, the issue of default risk gained relevance worldwide. Since the early work of [Black and Cox 1978], the default risk of a corporation is known to be a barrier-type product. Here, we present closed form solutions for arbitrary collateralized structured products (CSP) in the framework of n assets and two barriers, one representing default and the second one aa market-related option. Numerical results indicate that investors don't have to struggle with additional charges by demanding a collateral component in their structured product compared with the price of a similar product where the default risk is neglected.

European and American Parisian options in a jump-diffusion model
Nikola Vasiljevic (University of Zurich, Switzerland)
Joint work with Marc Chesney

Wednesday June 4, 15:00-15:30 | session 5.5 | Options, Futures | room G

In this paper, we study the maturity randomization (Canadization) technique for pricing of European and American-style Parisian options. We consider a hyper-exponential jump-diffusion process which can approximate any Levy process with completely monotone density. The model is general and flexible enough to capture the asymmetric leptokurtic feature and the volatility smile, and one of its main advantages is the analytical tractability for pricing of path-dependent options. We follow the Gaver-Stehfest Canadization approach for solving a system of two-dimensional partial integro-differential equations describing the option price dynamics. In the first step, we analytically solve the pricing problem for Canadized European and American-style Parisian options by taking the double Laplace transform with respect to the option maturity and the Parisian window. Subsequently, we utilize the recursive algorithm for Gaver-Stehfest inversion, and present both theoretical and numerical results for the computation of option prices, Greeks, and early exercise boundaries. Our approach is presented for both up-and-out and down-and-out Parisian call options with and without already started excursion. In order to obtain prices of other types of options we provide necessary parity and symmetry relations in the hyper-exponential jump-diffusion setting. Furthermore, we examine the convergence of the obtained results for European (American) Parisian options to the European (American) plain vanilla or barrier options depending on the length of the Parisian window. Finally, the impact of jumps on the option prices, the Greeks and the early exercise boundary is discussed in the paper. Therefore, the Gaver-Stehfest Canadization method for Parisian options combines the mathematical appeal inherent to analytical approach with the ease of implementation of the Laplace numerical inversion, and provides important economic insight for pricing and hedging in the presence of jumps.

A Dividend Discount Model for Equity Derivatives
Oliver Brockhaus (MathFinance AG, Germany)

Wednesday June 4, 15:30-16:00 | session 5.5 | Options, Futures | room G

Within equity models discrete dividends are often assumed to be proportional to spot or deterministic. In order to better capture dividend dynamics practitioners also represent dividends as affine functions of spot, where near dividends are deterministic and far dividends are proportional, see Overhaus et al. [1]. The resulting model has inhomogeneous spot dynamics. This paper presents a homogenous equity model with realistic dividend dynamics. Firstly, a family of equity models is introduced allowing for dynamics such as local or stochastic volatility. This family is defined as dividend discount models where all dividends are driven by a single factor. It is shown that the family includes as special cases important discrete dividend models such as deterministic, proportional and affine dividends as well as the Korn-Rogers model [2]. Secondly, the proposed model is introduced as special case within this family. In contrast to other models the impact of the factor on a given dividend decreases with time such that, with respect to a future time, near dividends are less volatile than far dividends. Analytic approximations for Vanilla and forward starting options in a setting with deterministic volatility are given. Numerical approaches for model calibration with local and stochastic volatility are also presented. These rely on Monte Carlo simulation and fixed-point method. Finally, it is shown that the model remedies some of the shortcomings of other dividend models, in particular non-homogeneity of dividend treatment.

[1] M. Overhaus et al.: Equity Hybrid Derivatives. Wiley, 2007.
[2] R. Korn and L. C. G. Rogers: Stocks paying discrete dividends: modelling and option pricing. The Journal of Derivatives 13, 44-48, 2005.