發(fā)布時間：2018-03-31 13:21

  本文選題：混合分數(shù)布朗運動　切入點：跳擴散過程　出處：《中國礦業(yè)大學》2017年碩士論文

【摘要】：隨著近年來中國金融市場的不斷開放、發(fā)展,國內(nèi)的期權市場有了較大的實際意義上突破。伴著2015年新年的腳步,上證50ETF期權登錄上海證券交易所。這意味著起步24年后,中國內(nèi)地股市迎來了“期權時代”。在此之后,一些其他種類的期權將會陸續(xù)進入金融市場,交易規(guī)模也將不斷增大。隨之也會出現(xiàn)場外市場,那么當期權在場外市場進行交易時,由于沒有類似清算所之類的監(jiān)督機構來監(jiān)管期權的空頭方在到期時履行相應的義務,就會導致期權的多頭方要同時承受市場風險和信用風險,而這必定會導致交易過程中違約的出現(xiàn)。到目前為止,關于市場信用違約風險的理論研究已經(jīng)相對成熟,但是對于在存在違約風險的市場定價方面的研究就相對較少,考慮到的條件也較少,不能很好的描述期權價格的改變。因此本文考慮將跳擴散過程,隨機利率和混合布朗運動引入,考慮簡約模型與結構模型相結合,在以往的基礎上對定價公式進行完善,增強定價公式的準確性。本文主要研究了以下幾個問題:(1)由于真實市場中股票價格不能完全符合幾何布朗運動,金融資產(chǎn)收益的分布具有“尖峰厚尾”的特征,且股價變化也不是隨機游走,而是不同時間呈現(xiàn)不同程度的長期相關性和自相似性。這些特征與標準的布朗運動存在一定的差距,而分數(shù)布朗運動正好具備自相似性和長期相關性,更加適合金融市場的特性。因此本文在假設股票價格服從幾何、分數(shù)布朗運動的條件下對脆弱期權的定價進行研究。(2)本文引入了跳擴散過程,但一般的Girsanov定理不能運用在這種情況下,所以本文先研究了跳過程經(jīng)測度變換后在新測度下的表達公式,在變換后則可以在新測度下應用Girsanov變換對期權進行定價。(3)在實際的金融市場中,隨著場外交易的增多,利率和違約行為都具有較強的隨機性。因此本文在隨機利率和隨機違約強度基礎上,對混合布朗運動模型下的歐式脆弱期權定價進行了相關研究,并求出解析解。本文通過鞅測度變換得到了歐式脆弱期權定價的顯式解。通過數(shù)值試驗將本文的定價公式與經(jīng)典Black-Scholes定價公式進行比較研究,結果表明本文的期權定價公式更加符合實際金融市場的特征。
[Abstract]:With the opening and development of China's financial market in recent years, the domestic options market has made a big breakthrough in practical terms. With the pace of the 2015 New year, Shanghai Stock Exchange 50ETF options entered the Shanghai Stock Exchange. This means 24 years after the start. China's mainland stock market is ushered in an "option era." after that, some other types of options will gradually enter the financial market, and the scale of transactions will continue to grow. There will also be an over-the-counter market. Well, when options are traded in the over-the-counter market, because there is no supervisory body such as clearing houses to monitor the options' short parties to fulfill their corresponding obligations at maturity, So far, the theoretical research on market credit default risk has been relatively mature, which will lead to both market risk and credit risk, which will inevitably lead to the occurrence of default in the course of trading. However, there are relatively few studies on the market pricing with default risk, and the conditions are also less, which can not describe the change of option price very well. Therefore, this paper considers the jump diffusion process. With the introduction of stochastic interest rate and mixed Brownian motion, considering the combination of the reduced model and the structural model, the pricing formula is improved on the basis of the past. To enhance the accuracy of pricing formulas, this paper mainly studies the following questions: 1) since stock prices in real markets do not fully conform to the geometric Brownian motion, the distribution of financial assets returns has the characteristics of "peak and thick tail". And the change of stock price is not random walk, but show different degree of long-term correlation and self-similarity at different time. These characteristics are different from the standard Brownian motion. The fractional Brownian motion has self-similarity and long-term correlation, which is more suitable for the characteristics of financial market. In this paper, we introduce the jump diffusion process, but the general Girsanov theorem can not be applied in this case. So this paper first studies the expression formula of the jump process under the new measure after the measure transformation. After the transformation, we can use the Girsanov transform to price options under the new measure.) in the actual financial market, with the increase of over-the-counter transactions, Both interest rate and default behavior have strong randomness. Therefore, on the basis of stochastic interest rate and stochastic default intensity, the pricing of European fragile options under mixed Brownian motion model is studied in this paper. In this paper, the explicit solution of European fragile option pricing is obtained by martingale measure transformation. The pricing formula of this paper is compared with the classical Black-Scholes pricing formula by numerical experiments. The results show that the option pricing formula is more in line with the characteristics of the actual financial market.
【學位授予單位】：中國礦業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O211.6;F830.9

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