| With continuing development of domestic and foreign financial markets and con-stant emergence of various financial innovations, how to apply probability theory to real financial problems has become a very active research area. Main topics in this area are asset pricing, security portfolio, investment strategy, risk management and so on; which are not only very practical for the individual investor, but also play a crucial role in the security of the whole financial system. This thesis is devoted to studying the following3real and popular financial problems by means of stochastic process and stochastic analysis:1. How to make a successful investment strategy?(See Chapter2)2. How to price exotic options correctly?(See Chapters3-4)3. How to analyze empirical financial phenomenon theoretically?(See Chapter5)For Problem1, we study the pricing problem about cash flow under uncertain-ty. This kind of cash flow consists of both dividend from holding and sudden payoff when an exercise happens. For different kinds of cash flows, we study that under what conditions, there exists a corresponding optimal stopping strategy. For Problem2, we employ the methods of change of dynamic structure, backward dynamic program, to price some special exotic options. And for Problem3, we establish a structural model including issuing both bonds and stocks, to explain an important empirical financial phenomenon about relationship between dividend payout ratio and leverage, from a viewpoint of optimal capital structure.In Chapter1, we introduce some backgrounds on studied problems and corre-sponding main results of this thesis.In Chapter2, a mathematical framework of optimal entry and exit problems is pre-sented by stochastic integrals with admissible stopping times. In contrast to Mordecki and Salminen [55], and Christensen and Salminen [19], our model aims to value the cash flow generated continuously in time, rather than a single exercise payoff. We char- acter value function of the entry or exit problem as an excessive function of the supre-mum or infimum process and show that the optimal stopping times are of threshold. Unlike the representation approach with Green kernel in Mordecki and Salminen [55], we employ the EPV (expected present value) operators to calculate the value functions. This approach can be easily applied to many familiar cases:scaled Brownian motion with linear drift, diffusions with exponentially distributed jumps, and spectrally positive (or negative) processes. For those whose EPV operators are not available, such as nor-mal inverse Gaussian process, an approximation method was given by Kudryavtsev and Levendorskii [39]. Compared to Mordecki and Salminen [55], we can find the optimal stopping boundary without solving a differential equation of the value function. Iden-tically, our approach gives a quicker method to find the optimal stopping strategy, and hence acts more quickly, in response to the fluctuation of market. In Subsection2.4.2, we also find that the smooth pasting condition is not necessary for the value function due to the irregularity of the underlying process. In Section2.5, we give two numerical examples to illustrate our results, one satisfies the smooth pasting condition, and the other has a discontinuous first order derivative.In Chapter3, due to broad use of barrier options, we operate a model to price a special barrier option of type UIP, which is of American type and has a time-dependent barrier. The pricing problem of a fixed barrier option has been studied in a lot of pa-pers, see Merton [51], Boyle and Lau [11], Rich [58], Geman and Yor [29]. However, barrier option with a dynamic barrier plays a more flexible role in financial market. As in Rogers and Zane [59], value of European barrier option with a moving barrier is ap-proximated assuming that underlying process is a scaled Brownian motion with a drift. In contrast to Rogers and Zane [59], our model can be applied to a general class of L6vy processes. The key point of our approach is to reduce the moving barrier to the case of fixed barrier by transforming the dynamic structure of underlying process. In this way, we obtain an explicit expression for the value function. In addition, we apply the Carr’s randomization to pricing the UIP with a finite maturity. Numerical results are also cited in this chapter and some characteristics are consistent with financial deductions.In Chapter4, we study a new optimal multiple stopping problem different from other papers, for example, Carmona and Touzi [14], Targino, Peters, Sofronov and Shevchenko [66], Dai and Kwok [22]. Firstly, a natural generalization is to consider dividend payoff continuously in time from holding instruments. Underlying instrument in our model includes both dividend payoff and terminal payoff when it is exercised. Secondly, from a viewpoint of practice, every exercise means a signal to market. For example, if we consider a multi-exercise American put, every exercise of put can also be viewed as a signal of distress to public. Hence we assume that in our model, every ex-ercise will cause an influence on underlying process. This kind of underlying processes can also be viewed as a regime switch type but with an endogenous switch. These t-wo generalizations make our model closer to reality and more applicable to a broader range of real problems. Our approach of solving this problem is to reduce the optimal multiple stopping problem to a sequence of ordinary optimal stopping problems. An explicit expression of the value function is given and we show how to find the optimal multiple strategy by a backward dynamic method. This optimal multiple strategy d-educed by our approach has a good property of consistency:Suppose that a n-vector x(n)=(x1,…, xn) is the optimal strategy for the n-stopping problem, then (n-1)-vector x(n-1)=(x2,…,xn) is optimal for the corresponding (n-1)-stopping problem. We also present a numerical example of scaled Brownian motion with a drift, to illustrate our approach in detail.In Chapter5, motivation of study on optimal capital structure of corporations is an empirical conflict between two papers, which gave different conclusions but studied the same financial behaviors empirically. One is Kapoor [36], saying that there is a positive relationship between dividend payout rate and debt-equity ratio; the other is Asif, Ra-sool and Kamal [3], saying that this relationship is negative. Two different conclusions refer to the same thing. Why does this happen? In order to give a good explanation for this phenomenon, we need to operate a comprehensive model to including most cor-porations’ financial affairs. Compared to other papers about optimal capital structure, such as Hilberink and Rogers [32], Leland [43], Leland and Toft [44], instead of consid-ering the value of debt only, our model incorporates financing behaviors:both issuing bonds and issuing stocks, and also a dividend distribution continuously in time. Con-sider that an operator establishes a firm financing by issuing bonds and stocks. When the firm runs, a dividend payoff is distributed to shareholders continuously in time. We also consider a dynamic structure of bonds. The bond is retired and reissued constant-ly. Then the operator should choose a dividend rate and a default time, in order to maximize the total profit for the shareholders. We establish a stochastic double-control model to value total profit from holding stocks, and price the value of equity and debt (value in bonds). By numerical analysis cited in Section5.4, we find that structure of bonds is the crucial point in the above conflict. If the bonds issued are mainly long-term, then there will be a negative relationship between dividend payout rate and the debt-equity ratio; but if the bonds issued are mainly short-term, then the relationship will be positive. Note that [36] mainly studied the Indian firms, which pay dividends by borrowing from banks at subsidized rates rather than their own profit. This behavior is equal to make a short-term finance, or equivalent to issue short-term bonds.
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