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首页> 外文期刊>International journal of theoretical and applied finance >DETERMINATION OF THE LéVY EXPONENT IN ASSET PRICING MODELS
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DETERMINATION OF THE LéVY EXPONENT IN ASSET PRICING MODELS

机译：资产定价模型中的Lévy指数

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We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure ?, consists of a pricing kernel {πt}t≥0 together with one or more non-dividend-paying risky assets driven by the same Lévy process. If {St}t≥0 denotes the price process of such an asset, then {πtSt}t≥0 is a ?-martingale. The Lévy process {ξt}t≥0 is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = t?1log??(eαξt) for α in an interval A ? ? containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is HT = (ζT)q for some time T > 0, are given at time 0 for a range of values of q, where {ζt}t≥0 is the so-called benchmark portfolio defined by ζt = 1/πt, then the Lévy 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 HT = (ST)q for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation ψ(α) → ψ(α + μ) ? ψ(μ) + cα, where c and μ are constants.
机译：考虑到衍生品的价格数据，我们考虑确定Lévy模型中Lévy指数的问题。在真实世界的措施下制定的模型？由定价内核{πt}t≥0与由同一levy过程驱动的一个或多个非股息支付风险资产组成。如果{st}t≥0表示此类资产的价格过程，则{πtst}t≥0是a？-martingale。假设Lévy过程{ξt}t≥0具有指数时刻，暗示在间隔A中为α的α（α）=t≤1log??（eαξt）的存在？还将原点作为适当子集。我们表明，如果电力回报衍生品的价格为HT =（ζT）Q，则在某个时间t> 0给出，对于Q的一系列值，在Q的范围内给出，其中{ζt}t≥0是由ζt= 1 /πt定义的所谓的基准产品组合，然后levy指数决定为无关的线性术语。在这种环境中，衍生价格体现了有关价格跳跃的完整信息：特别是，价格跳跃的频谱可以从当前的衍生品价格下制定。更一般地说，如果HT =（ST）Q为一般的非股息支付风险资产，并且如果我们知道定价内核由相同的Lévy进程驱动，那么达到一定因素，那么从当前的电力收益衍生物价格，我们可以推断levy指数的结构，直到变换ψ（α）→ψ（α+μ）？ ψ（μ）+Cα，其中C和μ是常数。

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