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Option Pricing And Optimal Investment-consumption In A Double Exponential Jump-diffusion Model With Market Structure Risks

Posted on:2007-01-24Degree:DoctorType:Dissertation
Country:ChinaCandidate:G H DengFull Text:PDF
GTID:1119360182488153Subject:Basic mathematics
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Since F.Black,M.Scholes and R.Merton made a major breakthrough in the pricing of financial derivatives,rapid progress has been gained in the theory and application of mathematical finance. With the deepness of study in financial practice,especially,from the serious impact concerning recent rare financial events and many questions of financial reform,etc. the Black-Scholes model is found to be not appropriate for changes of the modern financial market.In 1976,Merton established firstly a jump-diffusion model where the jump risks are unsystematic and the jump magnitude of the log of the asset price is assumed to be a normal distribution,and consider option pricing of European option.Hereafter Merton's work,many research achievements have been gained. Despite the success of the Black-Scholes and Mer-ton model,recent empirical works indicate the inability of such two models to capture the true features of asset fluctuating.and suggest: (l)the jump risks can not be ignored,and may implicate some important economical interpretation;(2)the asset returns may represent non-Gaussian and asymmetric leptokurtic features. On the other hand,there are many uncertain factors describing market structure changes that not only fluctuate the asset prices but also bring the market structure risks to investor.Therefore,in this dissertation we investigate option pricing and optimal investment /consumption in a combined model which incorporates two ingredients:the market structure risks and a jump-diffusion model where the jump size of the asset price follows double exponential distribution. As a result,the model is capable of fitting the empirical fact and describing both the true market and asset fluctuation.Our main contributions are as follow:Chapter 1 provides an introduction to the necessity and significance of research on option pricing and optimal investment/consumption in mathematical finance.From four aspects,we survey the academic literature and the latest development of a discontinuous model of this problem.Furthermore,we provide our motivations and main study topics in this dissertation.In Chapter 2,We start by introducing a double exponential jump-diffusion model(for short DexpJ)for stock price and related preliminaries.First,the pricing formulas for European option are then derived by using martingale method on the condition that the volatility of stock price and interest rates are both constant,and the cases of im-plied volatility and pricing error on option valuations are analyzed with numerical examples, we apply also the results to price 4 types of special exotic options and an nonlinear payoffs option.Second,in Chapter 3 we extend the results in Chapter 2 to more general occasions in two kinds model where the interest rates follow Va-sicek or CIR stochastic and the volatility of stock prices evolves CIR one,and derive closed form solutions for European stock call options by applying Fourier inversion transforms and Partial Differential Equation methods. In the same way,we devote to numerical analysis of the option prices and implied volatility,and to comparisons with results from the Black-Scholes model.Finally,we consider a multi-factor CIR market structure modal and use Ricatti equation to derive a closed formula of option ,which solves and extends the problem advanced by Duffie(2000[112],§4.3).In Chapter 4,we investigate in DexpJ model an American option which the stock pays continuously a constant-dividend.First,The closed form solutions for perpetual American option and American binary option are derived and some numerical examples are given. Second,we extend this model to the volatility of the stock prices being CIR stochastic,and after an analysis of the properties of the American put option price function ,we obtain its decomposition theorem. Finally,we gain integral representations of an American put price and the optimal exercise boundary which are based on the decomposition theorem,a key lemma(Lemma 4.15,Page 66) and Fourier inversion transforms.The two representations can be applied to develop numerical schemes for American put option and figures of its exercise boundary under both stochastic and/or constant volatility. The advantage of this scheme shows in that it is built on the classical Trapezoidal,Simpson and Newton-Raphson techniques and is an easier approach for pricing a long-term option or pricing option in a multi-dimension model.The scheme develops a new solution in a jump-diffusion model and avoids some of the shortcomings of the finite difference approach to pricing American option in a multi-dimension model.Chapter 5,directly derived from Chapter 3,establishes a combined credit structural model, and discusses term-structures of the credit spreads of corporation de-faultable bond.And then we further our study on the assumptions that interest rates follow a multi-factor Vasicek stochastic,no matter whether the default premium is defined by both fixed rates or by floating rates risky debts.Moreover,we analyze thequality spread differentials of demands for the above-mentioned two types of debts. Because of considering the effects of the uncertain factors both interest rates and volatility,the fact in the combined credit structural model is more appropriate for that in Merton's.Finally,we set up a instantaneous spread rates reduce-form model with jump risks,and study the credit spread and probability of default time when interest rates and spreads are related to each other.In the 6th chapter, we consider an optimal portfolio choice problem for an investor who can invest his wealth in stock,bond and cash account in DexpJ model where the short nominal interest rates as well as the inflation uncertainty and the excess return of stock are assumed to follow correlated Vasicek mean-reverting processes.Guided by dynamic programming principle and stochastic analysis,we gain a Hamilton-Jacobi-Bellman equation which corresponds to the value function with maximum utility over terminal wealth, and the optimal portfolio strategies in implicit form.Finally,in the case of CRRA utility function,we determine approximated explicit solutions for the portfolio choice,and illustrate the behaviors of investment strategy to stock by examining the impacts of jumps,risk aversion parameters and different investment time horizons.Chapter 7 investigates an impulse optimal investment and consumption problem where the risky asset is governed by a general Levy model and there exists a transaction cost that consists of a sum of a fixed cost plus a cost proportional to the size of the transaction in the financial market.The model may be regarded as an extension of Oksendal(2002[95])and Korn( 1999[93])worked in the setting of Black-Scholes model and consumption be allowed only at the discrete times of transactions.The optimal investment and consumption strategies are in the verification theorem,which is obtained by using Quasi-variational inequalities equation when the value function satisfies some regular conditions.Finally,in view of the viscosity solution theory,the value function is characterized as the unique viscosity solution of its associated Qvi-HJB equation subject to state space constraints.The results have been basis of a numerical implementation and further study in theory.In Chapter 8,we summarize the main conclusions of the study and suggest some unresolved issues for further work.In the end.it is worth noting that all numerical results including charts are gainedwith the help of mathematica or matlab codes programmed by the author except for fewness remarked in this dissertation.
Keywords/Search Tags:Double exponential distribution, Jump-diffusion model, European/American option, Credit spread, Market structure risks, Optimal investment-consumption
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