被廣泛應用在選擇權定價的Black-Scholes 模型[3] 時常在深價內與深價外 的選擇權價格有錯價的現象，也就是理論價格估計實際市場價格的偏差。藉由 Black-Scholes 評價公式所反推出的隱含波動度往往不像我們所期待的在不同履約價格具有一致性，這種現象被稱為波動度的微笑曲線。在這份論文裡，我們參考Jarrow and Rudd [13] 提出的方法，將Edgeworth展開式套用在Black-Scholes模型作延伸應用，進而推導出偏態峰態修正後的的評價公式，再利用台指選擇權的市場資料作實證分析並與Filho and Rosenfeld [1] 的研究作比較。我們發現從台指選擇權的實證結果得到非常態分配的隱含偏態和隱含峰態。此外，理論價格的估計偏誤比例顯著的被新的模型改善且隱含波動度的微笑曲線也變的較為平坦，這個方法提供我們一個有效的方法，利用標的資產的偏態峰態得到該資產的近似分配。 / The Black-Scholes [3] option pricing model widely applied in option contracts frequently misprices deep-in-the-money and deep-out-of-the-money options. The implied volatilities computed by the Black-Scholes formula are not identical on each strike price as we expect. This phenomenon is called the volatility smile or skew. In this thesis, we derived a skewness- and kurtosis-adjusted option pricing model using an Edgeworth expansion constructed by Jarrow and Rudd [13] to an investigation of TAIEX option prices and compare the results with those in Filho and Rosenfeld [1]. We found that non-normal skewness and kurtosis are implied by TAIEX option returns. Moreover, the magnitude of price deviations were signicantly corrected and the volatility skew is attened. This approach provides an useful way to derive an approximate distribution of a underlying security with its skewness and kurtosis.

本論文主要討論貝氏漸近的比較，推導出參數的聯合後驗分配與利用圖形來診斷馬可夫鏈蒙地卡羅的收斂。Johnson (1970)利用泰勒展開式得到個別後驗分配的展開式，此展開式是根據概似函數與先驗分配。 Weng (2010b) 和 Weng and Hsu (2011) 利用 Stein’s 等式且由概似函數與先驗分配估計後驗動差；將這些後驗動差代入Edgeworth 展開式得到近似後驗分配，此近似分配的誤差可精確到大O的負3/2次方與Johnson’s 相同。另外Weng and Hsu (2011)發現Weng (2010b) 和Johnson (1970)的近似展開式各別項誤差到大O的負1次方不一致，由模擬結果得到Weng’s 在此項表現比Johnson’s 好。另外由Weng (2010b)得到一維參數 的Edgeworth 近似後驗分配延伸到二維參數的聯合後驗分配；並應用二維參數的聯合後驗分配於多階段資料。本論文我們提出利用圖形來診斷馬可夫鏈蒙地卡羅收斂的方法，並且應用一般化線性模型與混合常態模型做為模擬。 關鍵字: Edgeworth 展開式；馬可夫鏈蒙地卡羅；個別後驗分配；Stein’s 等式 / Johnson (1970) obtained expansions for marginal posterior distributions through Taylor expansions. The expansion in Johnson (1970) is expressed in terms of the likelihood and the prior. Weng (2010b) and Weng and Hsu (2011) showed that by using Stein's identity we can approximate the posterior moments in terms of the likelihood and the prior; then substituting these approximations into an Edgeworth series one can obtain an expansion which is correct to O(t{-3/2}), similar to Johnson's. Weng and Hsu (2011) found that the O(t{-1}) terms in Weng (2010b) and Johnson (1970) do not agree and further compared these two expansions by simulation study. The simulations confirmed this finding and revealed that our O(t{-1}) term gives better performance than Johnson's. In addition to the comparison of Bayesian asymptotics, we try to extend Weng (2010a)'s Edgeworth series for the distribution of a single parameter to the joint distribution of all parameters. Since the calculation is quite complicated, we only derive expansions for the two-parameter case and apply it to the experiment of multi-stage data. Markov Chain Monte Carlo (MCMC) is a popular method for making Bayesian inference. However, convergence of the chain is always an issue. Most of convergence diagnosis in the literature is based solely on the simulation output. In this dissertation, we proposed a graphical method for convergence diagnosis of the MCMC sequence. We used some generalized linear models and mixture normal models for simulation study. In summary, the goals of this dissertation are threefold: to compare some results in Bayesian asymptotics, to study the expansion for the joint distribution of the parameters and its applications, and to propose a method for convergence diagnosis of the MCMC sequence. Key words: Edgeworth expansion; Markov Chain Monte Carlo; marginal posterior distribution; Stein's identity.