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Quantitative Methods in Financial and Insurance Mathematics
Aim and description of the workshop
In the recent years, the originally distinct fields of Financial Mathematics and Actuarial Science are merging. Notably, stochastic rather than deterministic processes are used nowadays as the key mathematical modelling tool in both research fields. Along with this development, the computational complexity of the employed mathematical models has witnessed a tremendous growth and clever numerical techniques are imperative for most present-day applications. This workshop is aimed at bringing together researchers from Financial and Insurance Mathematics and Computational Science, so as to exchange and discuss current insights and ideas, and to lay groundwork for future collaborations. Besides a series of internationally recognized researchers from academics, leading quantitative analysts from the financial industry also take part in this workshop.
Financial and actuarial models
At the workshop contemporary mathematical models from the financial and actuarial worlds for pricing and hedging products will be presented and discussed. The credit crisis changed the dynamics of the financial markets, both in insurance and in finance. Products with a long lifetime to maturity are, for example, traded more often. This has an impact on the mathematical models for the assets. Products that have a lifetime of, say, 30 years need to be modelled with stochastic rather than with deterministic interest rates. Also, in equity and foreign exchange (FX) markets volatility is now modelled in a stochastic way.
Whereas the public demand is for simpler financial and insurance products, this does not seem to come about in financial practice. Option products are commonly used and priced in finance. They are often also incorporated today in insurance and pension products to cover risks related to, for example, falling stock prices, inflation, or even global economic periods of depression.
During the workshop, recent insights in accurate models, from a financial and from an insurance point-of-view, will be communicated. These stochastic models may be based on systems of stochastic differential equations (SDEs), and/or on LÚvy jump processes of finite or infinite activity. The aim is to understand and look through, from a mathematical perspective, the strong and weak points of the various models advocated.
When valuing and risk-managing financial or insurance products, practitioners require fast, accurate and robust computation of prices and sensitivities. The choice of stochastic models for the underlying assets has a crucial impact on the techniques required for calibration and for the pricing of the exotic financial products. As both the financial models and products in practice become increasingly complex, a strong demand exists for advanced simulation methods to cope with these models.
The mathematical modelling of the dynamics of the underlying financial products is done nowadays by means of systems of stochastic differential equations. The resulting pricing equations for the valuation of corresponding financial derivative products, set up with the help of It˘'s Lemma, are typically high-dimensional partial differential equations. Their efficient numerical solution requires advanced numerical solution methods.
In option pricing, it is the famous Feynman-Kac theorem that connects the conditional expectation of the value of a contract payoff function under the risk-neutral measure to the solution of a partial differential equation. Starting from this representation one can apply several numerical techniques to calculate the option price itself. Broadly speaking, one can distinguish three types of computational methods:
(a) Monte Carlo simulation.
(b) Numerical solution of the partial-(integro) differential equation or P(I)DE.
(c) Numerical integration.
The distinction between the PIDE and the integral representations is subtle. Given the option pricing PIDE, one can formally write down the solution as a Green's function integral. When the Fourier transform of the Green's function is known, the problem reduces to evaluating an integral. This integral is the point of departure for Monte Carlo simulation as well as for integration methods.
In practice all three computational techniques (a), (b), (c) are employed. They are often implemented at the same time, to provide a validation of the prices and sensitivities computed. A primary aim of research in Computational Finance is to further enhance the performance, the robustness and the range of applicability of the pricing methods, so that they can effectively deal with the contemporary models. It is intended to extensively discuss current and innovative computational approaches during the workshop. Considering the techniques (a), (b), (c) simultaneously has, in particular, the important benefit of cross-fertilisation.
Goals of the workshop
The workshop has different purposes: