考慮市場流動性不完全下之選擇權

Abstract

本篇論文建構在Frey與Patie 2002 年的模型上，考慮流動性為股價所控制的函數。 Frey在標準的Black-Scholes偏微分方程式中，加入流動性變數，用此求取選擇權價格。本篇論文的目的為，改善Frey偏微分方程式中的人造條件，使得此非線性偏微分方程更加穩定。因此，利用拔靴法求取波動度的上界，用其取代原本不合理的波動度上界。數值理論部份，改善了希臘字母(Greeks)在流動性不佳的情況下，不穩定的狀態。透過希臘字母的趨勢變化，可以幫助交易者更了解回饋效果在不同流動性市場下的變化。實證部份，我們挑選了CBOE交易量前50的股票選擇權當作標的物，且用上述的非線性偏微分方程去計算個別選擇權的價格。結果顯示在不完全流動的市場下，改善後的偏微分方程，可提供更精確的選擇權價格。

In this paper we build on Frey and Patie’s literature (2002), where liquidity is a deterministic function of stock price. Frey implements an important factor, liquidity, into the standard Black-Scholes partial differential equation (PDE) to calculate the option price. The objective of our model is to improve an artificial pattern of Frey PDE to make the nonlinear PDE more reliable. Therefore, we choose bootstrap method to obtain the upper bound of volatility to replace the unreasonable setting. In numerical research, Greeks become smoother than before while using the bigger liquidity parameter. It helps traders to realize the variation of Feedback effect under different liquid markets. In empirical study, we choose the top 50 stock options of CBOE as underlying assets and use the PDE which contains liquidity parameter to solve each option price. The result shows that using the improved PDE offers more precise option prices in illiquid market.