發(fā)布時(shí)間：2018-06-23 05:03

  本文選題：鞅 + 測度變換��； 參考：《電子科技大學(xué)》2015年博士論文

【摘要】：隨機(jī)分析是研究金融市場中隨機(jī)利率下的歐式期權(quán)定價(jià)問題和復(fù)雜動(dòng)態(tài)網(wǎng)絡(luò)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(jī)性,使得隨機(jī)分析成為重要的研究工具。本文利用隨機(jī)分析的理論、方法和技巧研究了隨機(jī)影響下復(fù)雜動(dòng)態(tài)網(wǎng)絡(luò)的同步和隨機(jī)利率下的歐式期權(quán)定價(jià)問題,取得了如下研究成果。一.研究了隨機(jī)Markov切換的擾動(dòng)復(fù)雜網(wǎng)絡(luò)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(shè)計(jì)了控制方案,使隨機(jī)切換拓?fù)浜蛿_動(dòng)情況下的復(fù)雜動(dòng)態(tài)網(wǎng)絡(luò)實(shí)現(xiàn)了均方意義下的函數(shù)投影同步,進(jìn)一步給出了驅(qū)動(dòng)網(wǎng)絡(luò)和響應(yīng)網(wǎng)絡(luò)幾乎必然同步的白適應(yīng)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(dòng)率是一個(gè)常數(shù)的條件下,給出了隨機(jī)利率下的災(zāi)難期權(quán)的顯式閉形式公式。盡管價(jià)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(jī)利率下的價(jià)差期權(quán)定價(jià)問題。本文給出隨機(jī)利率下的兩個(gè)新穎的價(jià)差期權(quán)定價(jià)模型。這研究假設(shè)貼現(xiàn)的債券的波動(dòng)率是時(shí)間t的函數(shù)而不是一個(gè)常數(shù)。本文不僅提出一個(gè)好的方法去構(gòu)造隨機(jī)利率下的價(jià)差期權(quán)定價(jià)模型,而且提供了新的實(shí)驗(yàn)臺去理解隨機(jī)利率下的、各式各樣的資產(chǎn)定價(jià)模型中的期權(quán)價(jià)格動(dòng)態(tài)。在一些資產(chǎn)定價(jià)模型中,討論了這模型和測度變換的優(yōu)點(diǎn)。在一些資產(chǎn)定價(jià)模型中,這測度變換是有用的。在隨機(jī)利率下,本文給出價(jià)差期權(quán)定價(jià)公式,一般化的Black-Scholes-Merton期權(quán)定價(jià)公式,一個(gè)交換期權(quán)定價(jià)公式以及在跳模型下的一個(gè)歐式期權(quán)定價(jià)公式。最后,給出了價(jià)差期權(quán)的一些敏感性分析。對于標(biāo)的股票收益連續(xù)條件下的價(jià)差期權(quán)定價(jià)公式,運(yùn)用了規(guī)則網(wǎng)格的計(jì)算方法。對于標(biāo)的股票收益不連續(xù)條件下的價(jià)差期權(quán)定價(jià)公式,運(yùn)用了規(guī)則網(wǎng)格和Monte Carlo計(jì)算方法。通過數(shù)值試驗(yàn)和仿真,演示了隨機(jī)利率是影響期權(quán)價(jià)格的重要因素。數(shù)值試驗(yàn)表明標(biāo)的資產(chǎn)價(jià)格的波動(dòng)率顯著地影響期權(quán)的價(jià)值。三.給出了隨機(jī)利率下的三個(gè)新穎的一籃子期權(quán)定價(jià)公式。對于標(biāo)的股票收益連續(xù)條件下的、低維一籃子期權(quán)定價(jià)問題,給出了非常有效的計(jì)算技巧。進(jìn)一步,這研究也獲得了兩個(gè)新穎的隨機(jī)利率下的脆弱期權(quán)定價(jià)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(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：F830.9;F224
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本文編號：2055898