We unify and extend a number of approaches related to constructing multivariate Madan-Seneta Variance-Gamma model for option pricing. Complementing Grigelionis' (2007) class, an overarching model is derived by subordinating multivariate Brownian motion to a subordinator from class of generalised Gamma convolutions. Multivariate classes developed by Pérez-Abreu and Stelzer (2014), Semeraro (2008) and Guillaume (2013) are submodels. The classes are shown to be invariant under Esscher transforms, and quite explicit *
This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.
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AbstractWe study the valuation of unit-linked life insurance contracts with surrender guarantees. Instead of solving an optimal stopping problem, we propose a more realistic approach accounting for policyholders' rationality in exercising their surrender option.The valuation is conducted at the portfolio level by assuming the surrender intensity to be bounded from below and from above. The lower bound corresponds to purely exogenous surrender and the upper bound represents the limited rationality of the policyholders. The valuation problem is formulated by a valuation PDE and solved with the finite difference method. We show that the rationality of the policyholders has a significant effect on average contract value and hence on the fair contract design.We also present the separating boundary between purely exogenous surrender and endogenous surrender. This provides implications on the predicted surrender activity of the policyholders.
This paper surveys a class of Generalised Ornstein-Uhlenbeck (GOU) processes associated with Lévy processes, which has been recently much analysed in view of its applications in the financial modelling area, among others. We motivate the Lévy GOU by reviewing the framework already well understood for the "ordinary" (Gaussian) Ornstein-Uhlenbeck process, driven by Brownian motion; thus, defining it in terms of a stochastic differential equation (SDE), as the solution of this SDE, or as a time changed Brownian motion. Each of these approaches has an analogue for the GOU. Only the second approach, where the process is defined in terms of a stochastic integral, has been at all closely studied, and we take this as our definition of the GOU (see Eq. (12) below). The stationarity of the GOU, thus defined, is related to the convergence of a class of "Lévy integrals", which we also briefly review. The statistical properties of processes related to or derived from the GOU are also currently of great interest, and we mention some of the research in this area. In practise, we can only observe a discrete sample over a finite time interval, and we devote some attention to the associated issues, touching briefly on such topics as an autoregressive representation connected with a discretely sampled GOU, discrete-time perpetuities, self-decomposability, self-similarity, and the Lamperti transform. Some new statistical methodology, derived from a discrete approximation procedure, is applied to a set of financial data, to illustrate the possibilities.
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