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From Stochastic processes to mathematical finance: learning to translate real-life financial challenges into mathematics and solve them applying modern probability theory

Final Activity Report Summary - MATHFINANCEOBLOJ (From stochastic processes to mathematical finance: learning to translate real-life financial challenges into mathematics and solve them)

The principal objective of Dr Jan Obloj (Fellow) was to gain complementary scientific competencies and reach a position of professional maturity and independence as a recognised researcher in the domain of mathematical finance. Coming from a Ph.D. in Probability Theory, he hoped to find natural applications for his theoretical work and continue to work in the new domain defining an appealing research agenda with peer-recognised specific areas of expertise.

We strongly feel these objectives were met and that Fellow's two years at Imperial College were extremely successful. He published seven papers, four of which were started during the Fellowship. Further three papers are available as preprints and Dr Obloj is currently working on four projects resulting from the Fellowship. Most of his work was done in collaboration. In fact, the Fellowship was extremely fruitful in establishing collaborations and research projects which led to a promising scientific programme for the upcoming years. Dr Obloj was invited to give talks in international conferences and research seminars. His expertise was recognised with numerous reviewing assignments and invited contributions. He participated actively in activities of the Mathematical Finance group at Imperial College London organising reading seminars for Ph.D. students as well as group's main research seminar. Apart from rich interactions with the scientific community, he was also exposed to direct applications as a supervisor of industry-based M.Sc. projects.

Possibly the most tangible sign of the success of the Fellowship is the fact that subsequently Dr Obloj obtained job offers from several leading universities and he decided to join University of Oxford as a University Research Lecturer in Mathematical Finance and a Member of the Oxford-Man Institute of Quantitative Finance.

It is an ongoing debate whether markets complete themselves or not, i.e. if one may hedge all risk using the assets traded in the markets. As a general modelling principle, it is often stated that market is complete if one takes as many traded assets as factors spanning the market (or the information). However this had previously been proved only for special cases. In a joint work we gave a necessary and sufficient condition for market completeness in a general setup when market is driven by a multi-dimensional diffusion process (martingale model). Further, using techniques form the theory of real analytic functions, we showed how the criterion can be greatly simplified when option prices are very regular functions. Rather surprisingly, the appropriate regularity results for solutions of parabolic partial differential equations have never been investigated. This led to a new project and a collaboration with Professor Peter Takac (University of Rostock).

With Dr Alexander Cox (University of Bath), the Fellow investigated robust pricing and hedging of double barrier options which payout conditional on the stock price reaching or not-reaching given two levels before maturity T. The payoff depends on the whole path up to time T which makes these more complicated to price and hedge than European options whose payoff depends only on the value of the underlying (stock price, exchange rate etc) at time T. Given prices of European calls with maturity T, Cox and Obloj describe the range of possible (arbitrage-free) prices of double barriers. Furthermore, they devise simple (quasi-static) strategies which super- or sub- replicate double barriers and which use liquidly traded assets such as calls or forwards. They performed extensive numerical simulations which indicate that these strategies, even though they may require additional capital to set up, can outperform standard hedging techniques (delta/vega hedging) especially in the presence of model uncertainty and transaction costs. Their results are appealing both from theoretical and practical point of view.