英文标题：
《Determination of the L\\\'evy Exponent in Asset Pricing Models》
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作者：
George Bouzianis, Lane Hughston
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最新提交年份：
2019
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英文摘要：
We consider the problem of determining the L\\\'evy exponent in a L\\\'evy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure $\\mathbb P$, consists of a pricing kernel $\\{\\pi_t\\}_{t\\geq0}$ together with one or more non-dividend-paying risky assets driven by the same L\\\'evy process. If $\\{S_t\\}_{t\\geq0}$ denotes the price process of such an asset then $\\{\\pi_t S_t\\}_{t\\geq0}$ is a $\\mathbb P$-martingale. The L\\\'evy process $\\{ \\xi_t \\}_{t\\geq0}$ is assumed to have exponential moments, implying the existence of a L\\\'evy exponent $\\psi(\\alpha) = t^{-1}\\log \\mathbb E(\\rm e^{\\alpha \\xi_t})$ for $\\alpha$ in an interval $A \\subset \\mathbb R$ containing the origin as a proper subset. We show that if the initial prices of power-payoff derivatives, for which the payoff is $H_T = (\\zeta_T)^q$ for some time $T>0$, are given for a range of values of $q$, where $\\{\\zeta_t\\}_{t\\geq0}$ is the so-called benchmark portfolio defined by $\\zeta_t = 1/\\pi_t$, then the L\\\'evy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if $H_T = (S_T)^q$ for a general non-dividend-paying risky asset driven by a L\\\'evy process, and if we know that the pricing kernel is driven by the same L\\\'evy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the L\\\'evy exponent up to a transformation $\\psi(\\alpha) \\rightarrow \\psi(\\alpha + \\mu) - \\psi(\\mu) + c \\alpha$, where $c$ and $\\mu$ are constants.
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中文摘要：
我们考虑在给定衍生品价格数据的情况下，在资产价格的利维模型中确定利维指数的问题。该模型是根据现实世界的度量值$\\mathbb P$制定的，由定价内核$\\{\\pi\\U t}\\UU{t\\geq0}$和一个或多个由相同的列维过程驱动的非股息支付风险资产组成。如果${S\\u t\\}u{t\\geq0}$表示此类资产的价格过程，则${\\pi\\u t S\\u t}u{t\\geq0}$是$\\mathbb P$-鞅。假设L砦vy过程$\\{\\xi\\u t}{\\t\\geq0}$具有指数矩，这意味着L砦vy指数$\\psi（\\alpha）=t ^{-1}\\log\\mathb E（\\rm E ^{\\alpha\\xi\\u t}）$for$\\alpha$在包含原点作为适当子集的区间$\\subset\\mathb R$中存在。我们表明，如果给定了一系列价值为$q$的幂收益衍生工具的初始价格，其中$H\\u T=（\\zeta\\u T）^q$，一段时间$T>0$，其中${\\zeta\\u T}{T\\geq0}$是由$\\zeta\\u T=1/\\pi\\u T$定义的所谓基准投资组合，则L25vy指数确定为一个不相关的线性项。在这种情况下，衍生产品价格体现了价格跳跃的完整信息：特别是，价格跳跃的范围可以从衍生产品的当前市场价格计算出来。更一般地说，如果由Levy过程驱动的一般非股息支付风险资产为$H\\u T=（S\\T）^q$，并且如果我们知道定价内核由相同的Levy过程驱动，达到比例因子，然后，从电力收益衍生工具的当前价格中，我们可以推断出L’evy指数的结构，直至转换$\\ psi（\\ alpha）\\rightarrow\\psi（\\ alpha+\\ mu）-\\ psi（\\ mu）+c \\ alpha$，其中，$c$和$\\ mu$是常数。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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