考量跳躍風險下的一籃子選擇權評價

Valuation of Basket Options within the Bernoulli Jump Diffusion Process

本研究假設資產價格動態服從伯努立跳躍擴散過程(Bernoulli Jump Diffusion Process)，並在此架構下推導一籃子選擇權的近似評價公式。採用伯努立跳躍擴散過程除了可以捕捉資產價格跳躍與報酬的偏態與峰態等市場特性，亦可簡化波松跳躍擴散過程為評價模型帶來的複雜性，使金融從業人員可以更容易操作評價模型。此外，採用Johnson (1949)提出的Johnson Distribution Family 作為近似分配，可以捕捉一籃子資產報酬的偏態與厚尾特性，使得近似評價公式可以提供更精準的評價結果。本研究的數值分析顯示，近似評價公式同時兼具有準確性與計算效率，避免蒙地卡羅模擬法計算耗時的缺點。最後提出敏感度分析與避險比率分析，以供金融從業人員了解一籃子選擇權近似評價公式中各項參數的意涵與避險參考。

This article derives an approximate pricing formula of the basket options within the framework that the dynamics of their asset prices follow the Bernoulli jump diffusion process. The Bernoulli jump diffusion process captures not only the empirical features: the price jump and the skewness and kurtosis of the asset return, but also simplifies the complexity of the pricing formula based on the Poisson jump diffusion process. These features enhance the feasibility and the tractability of the presented pricing formula and thus makes it much easier for the practitioners to price basket options. In addition, this article employs the LS system of the Johnson Distribution Family, proposed in Johnson (1949), to approximate the distribution of the basket of underlying assets. The Johnson Distribution Family can capture the skewness and the fat-tail features of the basket of asset returns, which makes the approximate pricing formula provides more accurate pricing results. The numerical analysis shows that our approximate pricing formula is accurate and computationally efficient, which avoids the time-consuming property that stems from the Monte Carlo simulation methods. Sensitivity analysis and hedge ratio analysis are also proposed for financial market practitioners to understand the economic intuitions of various parameters in the pricing formula of basket options.