本文選題：鞅 + 測度變換��； 參考：《電子科技大學》2015年博士論文

【摘要】：隨機分析是研究金融市場中隨機利率下的歐式期權(quán)定價問題和復雜動態(tài)網(wǎng)絡同步問題的重要工具。特別是金融市場中的相關(guān)問題,由于其內(nèi)在的隨機性,使得隨機分析成為重要的研究工具。本文利用隨機分析的理論、方法和技巧研究了隨機影響下復雜動態(tài)網(wǎng)絡的同步和隨機利率下的歐式期權(quán)定價問題,取得了如下研究成果。一.研究了隨機Markov切換的擾動復雜網(wǎng)絡系統(tǒng)的函數(shù)投影同步問題。通過構(gòu)造合適的Lyapunov-Krasovskii泛函,有效的采用了不等式分析技巧和Ito公式,設計了控制方案,使隨機切換拓撲和擾動情況下的復雜動態(tài)網(wǎng)絡實現(xiàn)了均方意義下的函數(shù)投影同步,進一步給出了驅(qū)動網(wǎng)絡和響應網(wǎng)絡幾乎必然同步的白適應控制方案。通過數(shù)值仿真,說明了本文獲得的結(jié)論的可行性與有效性。二.在貼現(xiàn)的零息債券的波動率是一個常數(shù)的條件下,給出了隨機利率下的災難期權(quán)的顯式閉形式公式。盡管價差期權(quán)己經(jīng)被廣泛的研究,但是幾乎沒有論文討論隨機利率下的價差期權(quán)定價問題。本文給出隨機利率下的兩個新穎的價差期權(quán)定價模型。這研究假設貼現(xiàn)的債券的波動率是時間t的函數(shù)而不是一個常數(shù)。本文不僅提出一個好的方法去構(gòu)造隨機利率下的價差期權(quán)定價模型,而且提供了新的實驗臺去理解隨機利率下的、各式各樣的資產(chǎn)定價模型中的期權(quán)價格動態(tài)。在一些資產(chǎn)定價模型中,討論了這模型和測度變換的優(yōu)點。在一些資產(chǎn)定價模型中,這測度變換是有用的。在隨機利率下,本文給出價差期權(quán)定價公式,一般化的Black-Scholes-Merton期權(quán)定價公式,一個交換期權(quán)定價公式以及在跳模型下的一個歐式期權(quán)定價公式。最后,給出了價差期權(quán)的一些敏感性分析。對于標的股票收益連續(xù)條件下的價差期權(quán)定價公式,運用了規(guī)則網(wǎng)格的計算方法。對于標的股票收益不連續(xù)條件下的價差期權(quán)定價公式,運用了規(guī)則網(wǎng)格和Monte Carlo計算方法。通過數(shù)值試驗和仿真,演示了隨機利率是影響期權(quán)價格的重要因素。數(shù)值試驗表明標的資產(chǎn)價格的波動率顯著地影響期權(quán)的價值。三.給出了隨機利率下的三個新穎的一籃子期權(quán)定價公式。對于標的股票收益連續(xù)條件下的、低維一籃子期權(quán)定價問題,給出了非常有效的計算技巧。進一步,這研究也獲得了兩個新穎的隨機利率下的脆弱期權(quán)定價公式。
[Abstract]:Stochastic analysis is an important tool to study the European option pricing problem and the synchronization problem of complex dynamic network under the stochastic interest rate in the financial market. Because of its inherent randomness, stochastic analysis has become an important research tool. In this paper, we use the theory, method and technique of stochastic analysis to study the synchronization of complex dynamic networks under random influence and the pricing of European options at random interest rates. The research results are as follows. I. The problem of function projection synchronization for perturbed complex network systems with stochastic Markov switching is studied. By constructing an appropriate Lyapunov-Krasovskii functional, the inequality analysis technique and Ito formula are used effectively, and the control scheme is designed to synchronize the function projection in the mean-square sense of the complex dynamic network with random switching topology and disturbance. Furthermore, the white adaptive control scheme, which is almost necessarily synchronous between the drive network and the response network, is presented. The feasibility and validity of the conclusions obtained in this paper are illustrated by numerical simulation. II. Under the condition that the volatility of the discounted zero interest bond is a constant, the explicit closed form formula of disaster option at random interest rate is given. Although spread options have been widely studied, there are almost no papers to discuss the pricing of spread options at random interest rates. In this paper, we present two novel pricing models of spread options at random interest rates. This study assumes that the volatility of discounted bonds is a function of time t rather than a constant. This paper not only proposes a good method to construct the pricing model of spread options under stochastic interest rate, but also provides a new experimental platform to understand the option price dynamics in various asset pricing models under stochastic interest rate. In some asset pricing models, the advantages of this model and measure transformation are discussed. In some asset pricing models, this measure transformation is useful. Under random interest rate, the pricing formula of spread option, the generalized Black-Scholes-Merton option pricing formula, an exchange option pricing formula and a European option pricing formula under jump model are given in this paper. Finally, some sensitivity analysis of spread option is given. The regular grid method is used to calculate the pricing formula of the spread option under the continuous return of the underlying stock. In this paper, the regular grid and Monte Carlo method are used to calculate the pricing formula of the price difference option under the condition of discontinuous return of the underlying stock. Through numerical experiments and simulations, it is demonstrated that stochastic interest rate is an important factor affecting option price. Numerical tests show that the volatility of underlying asset prices significantly affects the value of options. III. Three novel basket option pricing formulas under stochastic interest rate are given. For the low dimensional basket option pricing problem under the continuous return of underlying stock, a very effective calculation technique is given. Furthermore, two novel pricing formulas of fragile options under stochastic interest rate are obtained.
【學位授予單位】：電子科技大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：F830.9;F224
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本文編號：2055898