# Essays on mathematical finance: Applications of moment expansions and filtering theory.

Yamada, Takeshi (2010) Essays on mathematical finance: Applications of moment expansions and filtering theory. PhD thesis, London School of Economics and Political Science.

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## Abstract

This thesis contains three essays on mathematical finance. The first discusses approximation methods for pricing swaptions based on moment expansions with multi-factor affine jump-diffusion models in Chapter 2. Two methods are examined. One is based on a Gram-Charlier expansion and the other is based on a generalized Edgeworth expansion. The density function of the forward swap is replaced with more tractable functions and their moments. Numerical simulations are conducted to confirm their accuracy. Models with a Gaussian-type or CIR-type volatility with an exponentially, normally or truncated normally distributed jump size are employed. The second essay proposes a framework to study the spot and forward relationship in carbon allowances markets and the third deals with the same problem in a different setting. The framework is based on the no-arbitrage principle. The value of the spot price depends on two underlying variables: the forward price and the net position of the zone defined as the difference between allocated carbon allowances and emissions. In Chapter 3, the net position of the market is modelled as a Markov chain and in Chapter 4, as a linear diffusion. Two kinds of filtration used in pricing are considered: complete information where market participants observe both processes continuously and incomplete information where they observe the forward price continuously and the net position of the zone periodically. Pricing problems occur in an incomplete market, since the net position of the zone is not tradable. A locally risk-minimization approach is used to fix the martingale measure. Under the complete information setting, The analytical spot price is obtained. Under the incomplete information setting, a filtered process is used for pricing, leading to the use of filtering theory. The spot price is computed numerically. Chapter 5 concludes.

Item Type: Thesis (PhD) Economics, Finance, Applied Mathematics Collections > ProQuest EthesesDepartments > Statistics http://etheses.lse.ac.uk/id/eprint/2201