在不同平賭測度下比較選擇權的價值

Abstract

在這篇文章中，我們考慮如何定價一個或有請求權的問題；而其股價是由一個幾何布朗運動模型的更一般形式幾何Lévy過程所建構。這是個不完備的市場並且沒有唯一的等價平賭測度，主要是因為股價的隨機過程有隨機的跳點。我們研究三種不同的方法來訂定歐式選擇權的價值：Föllmer-Schweizer (1990)最小測度，Black-Scholes (1973)測度，Esscher轉換。它們利用等價平賭測度從不同觀念來貼近Black-Scholes模型之下的平賭測度。我們將比較在不同平賭測度下的歐式選擇權價值並且討論波動率對其價值的影響。

In this paper, we consider the problem of pricing a contingent claim on a stock whose price process is modelled by a geometric Lévy process, a generalization of geometric Brownian motion model. The market is incomplete and there is no unique equivalent martingale measure due to the random jumps of the stock process. We study three approaches to pricing European option:the Föllmer-Schweizer[1990] minimal measure, the Black-Scholes [1973] measure and Esscher transform. They make use of equivalent martingale measures, in different senses closest to the martingale measure of classical Black-Scholes equation. We will compare the European option prices under different martingale measures and discuss the influence of the volatility on the price.