Calibrating the state price densities using TAIEX options

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吳諭昕(Yu-hsin Wu) 查詢紙本館藏

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統計研究所

論文名稱

(Calibrating the state price densities using TAIEX options)

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摘要(中)

狀態價格密度（State price densities， SPD）為一建構於風險中立測度之下用於選擇權定價之機率密度函數。實務上，狀態價格密度經常應用於定價以及風險管理。本文將以Black-Scholes模型為基礎與CRR樹狀結構方式、Derman Kani樹狀結構方式及Teng與Liechty之貝氏估計方式，以2008年及2012年台灣加權指數選擇權去測定狀態價格密度。
實證分析顯示，因CRR樹狀結構為Black-Scholes模型之離散化模型，故其二者結果相似。此外，為了建構Derman Kani樹狀結構，我們以內插法建構隱含波動率曲面，用以求算Derman Kani樹狀結構每一節點所需之買權價值。在2008年與2012年之實證分析，除Derman Kani樹狀結構易受市場變動而影響，其餘三者並未因2008年之市場劇烈變動而影響其訂價能力。整體而言，無論合約期間長短為何， Teng與Liechty方法最為穩健，並且對於長天期選擇權契約的配適較其他方法為優。

摘要(英)

State price densities are the probability densities to price financial options under the risk-neutral measures, and are particularly important for pricing and risk management. In this thesis, we compare four different methods to calibrate the state price densities using Taiwan stock index options data for years 2008 and 2012. The celebrated Black-Scholes model serves as our benchmark model. In addition, we consider its discretized version, the Cox, Ross and Rubinstein model (1979). However, to avoid model misspecification, we further study method proposed by the Derman and Kani (1994) which relaxes the assumptions on constant volatility, and a Bayesian nonparametric approach by Teng and Liechty (2009).
Our empirical results indicate that Cox, Ross and Rubinstein tree performs quite similar as the benchmark model because it is a discretized version of the Black-Scholes model. However, their pricing ability are not worse when the market changes dramatically, for example, during the 2008 financial crisis. Only Derman and Kani method perform worse during the period. Here, we use a standard interpolation to smooth the implied volatility. Derman and Kani tree does not produce good model fit, and appears to be unstable when the market changes significantly. Teng and Liechty method appears to be robust for options with all maturities. It outperforms all the other methods for options with longer time to maturities. The same conclusion holds for both 2008 and 2012.

關鍵字(中)

★ 狀態價格密度
★ 台指選擇權
★ Derman Kani樹狀結構
★ 貝氏正交方法

關鍵字(英)

★ State-Price Density
★ TAIEX Options
★ Derman Kani method
★ Bayesia
★ Quadrature

論文目次

Contents
摘要 i
Abstract ii
致謝 iv
List of Figures vii
List of Tables x
1 Introduction 1
2 Literature review 3
2.1 Black-Scholes model 5
2.2 Cox, Ross and Rubinstein tree 5
2.3 Derman and Kani tree 7
2.4 Bayesian Quadrature method 12
3 Simulation studies 17
3.1 Black-Scholes model 17
3.2 Derman and Kani tree 18
4 Empirical analysis 26
4.1 The data 26
4.2 Model Comparisons 32
4.3 Estimate with the fixed time to maturity 43
5 Conclusion 50
6 Discussion 51
Reference 52

參考文獻

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指導教授

鄧惠文(Huei-wen Teng)

審核日期

2013-6-25

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