臺灣博碩士論文加值系統

Black and Scholes(1973)在股價服從幾何布朗運動之假設下，提出歐式選擇權定價理論，但由於其理論之完美市場過於嚴格，且資產價格不一定符合幾何布朗運動，此後之學者因而針對其模型中的各種假設進行修正。本研究主要探討修正波動固定假設下的選擇權定價方法，並提出合理的模擬方式分析Black-Scholes模型相對跳躍擴散(Merton(1976))、連續隨機波動(Heston(1993))與離散GARCH選擇權定價模型(Duan(1995))之評價差異，比較其理論價格以瞭解Black-Scholes模型的穩健性。模擬分析結果發現，當存續期間較短且選擇權趨於價外，Black-Scholes模型相對其他波動修正模型之評價誤差較大；而當選擇權到期日增加，抑或標的資產之波動過程受非預期性干擾的成分降低，Black-Scholes模型對價內歐式買權之評價仍具有其穩健性。除此之外，前述修正波動假設下的三種定價模型，對於市場笑狀波幅的現象均具有解釋能力。

Black and Scholes(1973) derived European option pricing model under the assumption of stock prices following the Geometric Brownian motion. But many empirical analyses show that Geometric Brownian motion may not explain the real stock data well. From then, researchers propose some modified models. In this study, we review the option pricing theory of modified volatility models. Also, we propose a simulation method to analyze the differences in pricing of European call between Jump-diffusion model(Merton(1976)), Stochastic-volatility model(Heston(1993)) and GARCH option pricing model(Duan(1995)) with Black-Scholes formula. That means we discuss the robust properties of Black-Scholes framework.
The mainly results are follows. (1) A relatively short time to expiration for out-of-the-money options is indicative of a relatively high pricing differences. (2) As the expiration dates of options are longer or the unexpected shocks in the underlying asset’s volatility processes are lower, Black-Scholes model is still robust for in-the-money options. Otherwise, all the modified volatility models mentioned could explain the phenomenon of volatility smile.

第一章緒論……………………………………………………………1

第二章定價理論與文獻回顧…………………………………………4
第一節無套利定價理論………………………………………………4
第二節Black-Scholes選擇權定價模…………………………………10
第三節波動性估計方法與其他修正模型…………………………16
(一)波動性估計方法…………………………………………………16
(二)波動修正模型……………………………………………………18
1.固定彈性變異模型…………………………………………………18
2.純粹跳躍/跳躍擴散模型…………………………………………19
3.複合選擇權模型……………………………………………………20
4.Displaced擴散模型…………………………………………………20
5.連續隨機波動模型…………………………………………………21
6.離散隨機波動模型…………………………………………………22
7.連續隨機波動、隨機利率與跳躍擴散定價模型…………………23

第三章 波動修正模型之定價理論…………………………………27
第一節 跳躍擴散之選擇權定價模型………………………………27
第二節 連續隨機波動之選擇權定價模型…………………………32
(一)幾何布朗運動之隨機波動過程…………………………………32
(二)算術Ornstein-Uhlenbeck之隨機波動過程………………………34
第三節 離散隨機波動之選擇權定價模型…………………………39
(一)GARCH過程……………………………………………………39
(二)GARCH選擇權定價模型………………………………………40

第四章 選擇權評價之模擬分析……………………………………45
第一節 模擬分析方法………………………………………………45
(一)不同定價模型之隨機過程………………………………………45
1.跳躍擴散過程(JD)…………………………………………………46
2.連續隨機波動過程(SV)……………………………………………46
3.離散隨機波動過程(GARCH)………………………………………46
(二)模擬分析步驟……………………………………………………47
第二節 選擇權評價之模擬結果與分析……………………………49
(一)跳躍擴散過程-JD模擬結果……………………………………49
1.不同存續期間之評價誤差…………………………………………49
2.不同平均跳躍規模之評價誤差……………………………………51
3.不同跳躍波動性之評價誤差………………………………………53
(二)連續隨機波動過程-SV模擬結果………………………………54
1.不同存續期間之評價誤差…………………………………………55
2.不同復歸比率與長期平均變異之評價誤差………………………57
3.不同相關係數之評價誤差…………………………………………59
(三)離散隨機波動過程-GARCH模擬結果…………………………60
1.不同存續期間之評價誤差…………………………………………61
2.不同單位風險貼水之評價誤差……………………………………62
3.不同條件變異結構之評價誤差……………………………………64
模擬分析數據結果…………………………………………………66

第五章結論與建議…………………………………………………73

參考文獻……………………………………………………………76

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