This thesis comprises two independent parts. In the first part, we develop a pathwise calculus for functionals of integer-valued measures and extend the framework of Functional Itô Calculus to functionals of integer-valued random measures by constructing a ’stochastic derivative’ operator with respect to such integer-valued random measures. This allows us to obtain weak martingale representation formulae holding beyond the class of Poisson random measures, and allowing for random and time-dependent compensators. We study the behaviour of this operator and compare it with other previous approaches in the literature, providing in passing a review of the various Malliavin approaches for jump processes. Finally, some examples of computations are provided. The second part is oriented towards nonparametric statistics, with a financial application as our main goal: we aim at recovering a surface of FX call options on a pegged currency such as the Hong Kong dollar against the U.S. dollar, based on a small number of noisy measurements (the market bid-ask quotes). Inspiring ourselves from the Compressed Sensing literature, we develop a methodology that aims at recovering an arbitrage-free call surface. We first apply this methodology, based on tensor polynomial decomposition of the surface, to a sparse set of call-option prices on the S&P500, recovering the call option prices within desired tolerance, as well as a smooth local-volatility surface. On a pegged currency such as the HKD/USD, it appears that tensor polynomials may not be an adequate way to model the smiles across maturities. Modifying the methodology in favour of structure-preserving functions, we apply the new methodology to our HKD/USD dataset, recovering the smiles, and the corresponding state-price density.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:702819 |
| Date | January 2016 |
| Creators | Blacque-Florentin, Pierre |
| Contributors | Bingham, Nicholas |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/43537 |
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