arXiv 1111.21692011-11q-fin.PRmath.PR

动态资产定价几何Lévy模型的一般理论

General Theory of Geometric Lévy Models for Dynamic Asset Pricing

Dorje C. Brody, Lane P. Hughston, Ewan Mackie

备注：: 20 pages, version to appear in Proceedings of the Royal Society London A
发表信息：: Proc. R. Soc. A June 8, 2012 468 1778-1798
DOI：: 10.1098/rspa.2011.0670

摘要：

几何Lévy模型（GLM）是用于推导Black-Scholes公式的几何布朗运动模型（GBM）的自然推广。如果采用定价核方法，这种模型的理论将大大简化。在一维中，一旦基本的莱维过程被指定，GLM就有四个参数：初始价格、利率、波动率和风险厌恶。定价核是一个贴现因子和一个风险厌恶鞅的乘积。对于GBM，风险厌恶参数是风险的市场价格。对于GLM，这种解释是不成立的：超额收益率是波动率和风险厌恶的非线性函数。结果表明，对于正的波动率和风险厌恶的超额收益率以上的利率是积极的，并增加这些变量。在外汇的情况下，西格尔的悖论意味着，人们可以构建外汇模型，其中超额收益率对于汇率和逆汇率都是正的。这一条件被证明适用于任何几何Lévy模型的外汇的波动性超过风险厌恶。

英文摘要：

The geometric Lévy model (GLM) is a natural generalisation of the geometric Brownian motion model (GBM) used in the derivation of the Black-Scholes formula. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility, and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. For a GLM, this interpretation is not valid: the excess rate of return is a nonlinear function of the volatility and the risk aversion. It is shown that for positive volatility and risk aversion the excess rate of return above the interest rate is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel's paradox implies that one can construct foreign exchange models for which the excess rate of return is positive both for the exchange rate and the inverse exchange rate. This condition is shown to hold for any geometric Lévy model for foreign exchange in which volatility exceeds risk aversion.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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