Mathematical investigation of recall-type options with underlying asset whose value undergoes jumps
Kees VAN SCHAIK, AXA Research Fellow
Year of selection
Support by the AXA Research Fund
TYPE OF SUPPORT
Kees van Schaik is taming the exotic. Classic models for widely traded exotic financial derivatives such as recall options are based on financial index models (such as stock prices) that do not to fit real market data, particularly in rare—but potentially devastating—cases such as sudden shocks. Using a rigorous mathematical approach, van Schaik has developed improved methods for assessing the risks of these options by extending the class of objects used to model the underlying financial index, yielding a much better fit to market data. This gives risk managers a more market-consistent analysis of such derivatives and provides traders with a better buy/sell strategy. “While mathematics’ reputation may have been damaged by the financial crises, it only provides the tools,” he says. “It is the user's responsibility to employ them wisely.”
Kees van Schaik was born in the Netherlands in 1977. After obtaining his masters degree in mathematics (with distinction) from the University of Utrecht, he moved to Frankfurt am Main, Germany, to start his Ph.D. studies at the Goethe University under supervision of Prof. Dr. Christoph Kühn. Currently Kees is employed at REAAL Insurances in the Netherlands.
I think most of us agree the recent financial crisis has taught us a.o. this: truly understanding and managing the financial risks aboard is crucial for a financial institution to be healthy and stable. One of the difficulties is that products traded on financial markets can be quite complex. How to get a grip on such complicated financial constructions?
Think e.g. of a product like a callable convertible bond. It is more complex than a classic American option as it allows also the writer (i.e. the seller) to terminate the contract, at the expense of paying the amount the holder (i.e. the buyer) could claim at that time -- which is typically some function of an underlying stock -- plus an extra penalty. Products of such nature are traded in large volumes on markets around the world. The optimal exercise strategies of both parties involved turn out to correspond to the Nash equilibrium (in a stochastic sense) of a corresponding optimal stopping game, and the fair price of this product thus equals the value of this game.
Furthermore, since over the years empirical research has shown that the classic Black & Scholes model has serious limitations in its ability to be fitted to actual market data, in particular with the (rare, but potentially devastating) tail events, models based on Lévy processes have become increasingly popular.
My current research involves studying optimal stopping games driven by certain Lévy processes. Not only do they have an important application as pointed out above, also they are challenging and interesting objects from a mathematical point of view. Distinctive properties arise that are not found with such games in a setting with a continuous driving process.
The grant provided by the AXA Research Fund, for which I am very grateful, enables me to continue the research described above, as a lot is yet to be discovered, in the stimulating company of world leading experts and close to one of the worlds main financial centres.
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