THE REGULARITY OF THE FREE BOUNDARY FOR STRIKE RESET OPTION

A strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder. The pricing model of the option can be formulated as a parabolic variational inequality and the optimal reset strategy is the free boundary. The smoothness of the free boundary in some cases was showed in our article published in JDE. We would prove its smoothness in the other case in this paper by a generalized comparison principle for the variational inequality.     

A BLACK-SCHOLES FORMULA FOR OPTION PRICING WITH DIVIDENDS

Abstract. We obtain a Black-Scholes formula for the arbitrage-free pricing of Eu-ropean Call options with constant coefficients when the underlylng stock generatesdividends. To hedge the Call option, we will always borrow money from bank. We seethe influence of the dividend term on the option pricing via the comparison theoremof BSDE(backward stochastic di~erential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R whichis not equal to the interest rate r. The corresponding Black-Sdxoles formula is given.We notice that it is in fact the borrowing rate that plays the role in the pricing formula.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

马尔可夫交换Lévy过程模型下的期权定价及其对冲（英文）

In this paper, we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging. In this model, the market interest rate, the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process. We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure. The option price using this model is obtained by the Fourier transform method. We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

A variational inequality arising from European option pricing with transaction costs

In this paper we present a method which can transform a variational inequality with gradient constraints into a usual two obstacles problem in one dimensional case.The prototype of the problem is a parabolic variational inequality with the constraints of two first order differential inequalities arising from a two-dimensional model of European call option pricing with transaction costs.We obtain the monotonicity and smoothness of two free boundaries.     

Estimating the shareholder's terminal payoff based on insurer's solvency ratio in mixed fractional Brownian market

This paper studies the insurer's solvency ratio model in a class of mixed fractional Brownian motion(MFBM) market, where the prices of assets follow a Wick-It stochastic differential equation driven by the MFBM, by the method of the stochastic calculus of the MFBM and the pricing formula of European call option for the MFBM, the explicit formula for the expected present value of shareholders' terminal payoff is given. The model extends the existing results.     

VISCOSITY SOLUTIONS OF HJB EQUATIONS ARISING FROM THE VALUATION OF EUROPEAN PASSPORT OPTIONS

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related H J13 equation are proved. A relationship between the passport and lookback options is discussed.     

A FAST AND HIGH ACCURACY NUMERICAL SIMULATION FOR A FRACTIONAL BLACK-SCHOLES MODEL ON TWO ASSETS

In this paper, a two dimensional(2D) fractional Black-Scholes(FBS) model on two assets following independent geometric Lévy processes is solved numerically. A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established. The fractional derivative is a quasidifferential operator, whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix. In order to speed up calculation and save storage space, a fast bi-conjugate gradient stabilized(FBi-CGSTAB) method is proposed to solve the resultant linear system. Finally, one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique. The pricing of a European Call-on-Min option is showed in the other example, in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.     

AN OPTION PRICING PROBLEM WITH THEUNDERLYING STOCK PAY1NG DIVIDENDS~

In this paper, a pricing problem of European call options is considered, wbete the underlying stock generates dividends d, at some fixed future dates T, before the expiration date T .without the inappropriate assumption made in that the dlvkdeMs being payed continously.The arbitrage free pricing of the option is determined via a series of partial differential equations.which is derived at the view point of backward s‘tochasric differential ertuation (BBDE). It isshowed how the dividends affect the fair price of the call options. Some simulating results are alsogiven to illust rate the respective in fluence of parameters a.T.r,K.di and F1 on the option pricing.     

Random weighting method for Cox’s proportional hazards model

Variance of parameter estimate in Cox’s proportional hazards model is based on asymptotic variance. When sample size is small, variance can be estimated by bootstrap method. However, if censoring rate in a survival data set is high, bootstrap method may fail to work properly. This is because bootstrap samples may be even more heavily censored due to repeated sampling of the censored observations. This paper proposes a random weighting method for variance estimation and confidence interval estimation for proportional hazards model. This method, unlike the bootstrap method, does not lead to more severe censoring than the original sample does. Its large sample properties are studied and the consistency and asymptotic normality are proved under mild conditions. Simulation studies show that the random weighting method is not as sensitive to heavy censoring as bootstrap method is and can produce good variance estimates or confidence intervals.     

美式债券期权定价问题的有限元方法

The aim of this paper is to investigate the finite element methods for pricing the American put option on bonds. Based on a new variational inequality equation for the option pricing problems, both semidiscrete and fully discretized finite element approximation schemes are established. It is proved that the finite element methods are stable and convergent under L2 and H^1 norms.     

Sieve M-estimation for semiparametric varying-coefficient partially linear regression model

This article considers a semiparametric varying-coefficient partially linear regression model.The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable.A sieve M-estimation method is proposed and the asymptotic properties of the proposed estimators are discussed.Our main object is to estimate the nonparametric component and the unknown parameters simultaneously.It is easier to compute and the required computation burden is much less than the existing two-stage estimation method.Furthermore,the sieve M-estimation is robust in the presence of outliers if we choose appropriate ρ( ).Under some mild conditions,the estimators are shown to be strongly consistent;the convergence rate of the estimator for the unknown nonparametric component is obtained and the estimator for the unknown parameter is shown to be asymptotically normally distributed.Numerical experiments are carried out to investigate the performance of the proposed method.     

Study on the model of an insurer’s solvency ratio in Markov-modulated Brownian markets

In this paper the insurer’s solvency ratio model with or without jump diffusion process in the presence of financial distress cost is constructed, where an insurer’s solvency ratio is characterized by a Markov-modulated dynamics. By Girsanov’s theorem and the option pricing formula, the expected present value of shareholders’ terminal payoff is provided.     

Option Pricing for Time-Change Exponential Levy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure （MEMM） and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.     

The Error in the Approximation

In applying any numerical method such as the bisection method to determine a root, it is important to realize that the best we can usually achieve is an approximation ofthe exact root. At each iteration of the method,we obtain a better estimate of the root. Thus it becomes desirable that we be able to estimate how accurate the approximation is at each stage so that we know when to stop the process.With the bisection method,suppose we know that there is a root in some interval [a,b], where a and b are successive integers, say 2 and 3. If we select the midpoint M1 of this interval, then it is obvious that the root R is     

灰色正项几何规划研究

In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory, and using some analysis strategies, a model of grey polynomial geometric programming, a model of θpositioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem. This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.     

一类无约束离散Minimax问题的区间调节熵算法

In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C^1. The paper deals with this problem by means of taking the place of maximum-entropy function with adjustable entropy function. By constructing an interval extension of adjustable entropy function and some region deletion test rules, a new interval algorithm is presented. The relevant properties are proven, The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum-entropy algorithm. Both theoretical and numerical results show that the method is reliable and efficient.     

Time variant multi-objective linear fractional interval-valued transportation problem

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.     

基于傅里叶级数展开式的有效期权定价方法

A novel option pricing method based on Fourier-cosine series expansion was proposed by Fang and Oosterlee. Developing their idea, three new option pricing methods based on Fourier, Fourier-cosine and Fourier-sine series expansions are presented in this paper, which are more efficient when the option prices are calculated with many strike prices. A series of numerical experiments under different exp-L~vy models are also given to compare these new methods with the Fang and Oosterlee＇s method and other methods.