第１回デリバティブ部会セミナー

投稿日: 2011/07/06 5:58:39

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日時： 8月8日19時～20時30分

場所： 立命館大学 東京キャンパス (サピアタワー8階)

講師： Professor Lane P. Hughston（Department of Mathematics Imperial College London）

題目： General Theory of Geometric Lévy Models for Dynamic Asset Pricing

概要： The theory of Lévy models for asset pricing simplifies considerably if we take a pricing kernel approach, which enables one to bypass market incompleteness issues. The special case of a geometric Lévy model (GLM) with constant parameters can be regarded as a natural generalisation of the standard geometric Brownian motion model used in the Black-Scholes theory. In the one-dimensional situation, for any choice of the underlying Lévy process the associated GLM model is characterised by four parameters: the initial asset price, the interest rate, the volatility, and a risk aversion factor. The pricing kernel is given by the product of a discount factor and the Esscher martingale associated with the risk aversion parameter. The model is fixed by the requirement that for each asset the product of the asset price and the pricing kernel should be a martingale. In the GBM case, the risk aversion factor is the so-called market price of risk. In the GLM case, this interpretation is no longer valid as such, but instead one finds that the excess rate of return is given by a non-linear function of the the volatility and the risk aversion factor. We show that for positive values of the volatility and the risk aversion factor the excess rate of return above the interest rate is positive, and is monotonically increasing in the volatility and in the risk aversion factor. In the case of foreign exchange, we know from Siegel's paradox that it should be possible to construct FX models for which the excess rate of return (above the interest rate differential) is positive both for the exchange rate and the inverse exchange rate. We show that this condition holds for any GLM for which the volatility exceeds the risk aversion factor. Similar results are shown to hold for multiple-asset markets driven by vectorial Lévy processes, and for market models based on certain more general classes of Lévy martingales. (Work with D. Brody, E. Mackie, F. Mina, and M. Pistorius.)