本論文利用市場觀測的選擇權買價與賣價，將市場的交易行為描述為兩人零合賽局，其中參賽者為投資人與市場機制，分別建立雙方的最佳策略模型。假設標的資產到期日的價格為離散點且個數有限，當市場不存在套利機會，也就是投資人最佳策略時報償為零時，可利用賽局線性規劃模型導出隱含於市場價格的風險中立機率測度。此模型不須對標的資產價格的機率分配做任何假設，也不須計算波動度，就可利用資產價格的平賭性質，以還原的風險中立機率測度為選擇權作合理的定價。最後，以台指選擇權(TXO)為例，驗證本模型的評價能力，且再次證實資產價格的風險中立機率分佈與一般常假設的對數常態分佈有落差。

本論文提出利用機率平賭性質由選擇權市場價格還原風險中立機率測度的雙目標規劃模型。假設對應同一標的資產且不同履約價的選擇權均為歐式選擇權，到期時標的資產的狀態為離散點且個數有限。若市場不存在套利機會時，建構出最小化離差總和及最大化平滑的雙目標規劃模型。將此雙目標規劃模型利用權重法轉換成單一目標之非線性模型，即可還原風險中立機率測度，並利用此風險中立機率測度評價選擇權的公平價格。最後，我們以台指選擇權(TXO)為例，驗證此模型的評價能力。 / This thesis proposes a biobjective nonlinear programming model to derive risk-neutral probability distribution of underlying asset. The method are used to choose probabilities that minimize the deviation between the observed price and the theoretical price as well as maximize the smoothness of the resulting probabilities. A weighting method is used to covert the model into a single objective model. Given a non-arbitrage observed option price, a risk-neutral probability distribution consistent with the observed option can be recovered by the model. This risk-neutral probability is then utilized to evaluate the fair price of options. Finally, an empirical study applying to Taiwan’s market is given to verify the pricing ability of this model.

本論文研究如何由觀測的選擇權市場價格還原風險中立機率測度（等價平賭測度）。首先建構選擇權投資組合的套利模型，其中假設選擇權為單期，到期日時的狀態為離散點且個數有限，並且對應同一標的資產且不同履約價格。若市場不存在套利機會時，可使用拉格朗日乘數法則將選擇權套利模型導出拉格朗日乘子的可行性問題。將可行性問題作為限制式重新建構線性規劃模型以還原風險中立機率測度，並且利用此風險中立機率測度評價選擇權的公正價格。最後，我們以台指選擇權（TXO）為例，驗證此模型的評價能力。 / This thesis investigates how to recover the risk-neutral probability (equivalent martingale measure) from observed market prices of options. It starts with building an arbitrage model of options portfolio in which the options are assumed to be in one-period time, finite discrete-states, and corresponding to the same underlying asset with different strike prices. If there is no arbitrage opportunity in the market, we can use Lagrangian multiplier method to obtain a Lagrangian multiplier feasibility problem from the arbitrage model. We employ the feasibility problem as the constraints to construct a linear programming model to recover the risk-neutral probability, and utilize this risk-neutral probability to evaluate the fair price of options. Finally, we take TXO as an example to verify the pricing ability of this model.

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.