Brigo與Mercurio提出了三種新的資產價格過程，分別是位移CEV過程、位移對數常態過程與混合對數常態過程。在這三種過程中，資產價格的波動度不再是一個固定的常數，而是時間與資產價格的明確函數。而由這三種過程所推導出來的歐式選擇權評價公式，將會導致隱含波動度曲線呈現傾斜曲線或是微笑曲線，且提供了參數讓我們能夠配適市場的波動度結構。本文利用台指買權來實證Brigo與Mercurio所提出的三種歐式選擇權評價公式，我們發現校準結果以混合對數常態過程優於位移CEV過程，而位移CEV過程則稍優於位移對數常態過程。因此，在實務校準時，我們建議以混合對數常態過程為台指買權的評價模型，以達到較佳的校準結果。 / Brigo and Mercurio proposed three types of asset-price dynamics which are shifted-CEV process, shifted-lognormal process and mixture-of-lognormals process respectively. In these three processes, the volatility of the asset price is no more a constant but a deterministic function of time and asset price. The European option pricing formulas derived from these three processes lead respectively to skew and smile in the term structure of implied volatilities. Also, the pricing formula provides several parameters for fitting the market volatility term structure. The thesis applies Taiwan’s call option to verifying these three pricing formulas proposed by Brigo and Mercurio. We find that the calibration result of mixture-of-lognormals process is better than the result of shifted-CEV process and the calibration result of shifted-CEV process is a little better than the result of shifted-lognormal process. Therefore, we recommend applying the pricing formula derived from mixture-of-lognormals process to getting a better calibration.

本論文提出線性規劃的方法以還原隱藏於選擇權市場價格中的風險中立機率測度，並利用該機率測度計算選擇權的合理價格。模型中假設選擇權對應同一標的資產與到期日，資產價格於到期日的狀態為離散點且個數有限，當市場不具任何套利機會時，以極小化市場價格與合理價格之離差總和作為挑選風險中立機率測度的準則。最後，以臺指選擇權(TXO)的交易資料做為實證對象。實證中發現，加入平滑限制式與離差權重之線性規劃模型在評價歐式選擇權合理價格的效能最為優異。 / The thesis proposes a liner programming to recover the risk-neutral probability distribution of an underlying asset price from its associated market option prices, and we evaluate the fair prices of options via the resulting risk-neutral probability distribution. Assume that we face a series of European options with different exercise prices on the same maturity and underlying asset in this linear programming model. The criterion of choosing a risk-neutral probability distribution is minimizing the sum of total deviations subject to requiring that the fair prices of options are consistent with observed market option prices. Finally, we take the trading data of TXO as an empirical study. The empirical study indicates that the model with smooth constraints and weighted deviations has the best performance in pricing the rational price of European options.