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.     

Convergence rates of trinomial tree methods for option pricing under regime-switching models

Recently trinomial tree methods have been developed to option pricing under regime-switching models. Although these novel trinomial tree methods are shown to be accurate via numerical examples, it needs to give a rigorous proof of the accuracy which can theoretically guarantee the reliability of the computations. The aim of this paper is to prove the convergence rates (measure of the accuracy) of the trinomial tree methods for the option pricing under regime-switching models.     

Option pricing under residual risk and imperfect hedging

This paper is concerned in the option pricing in a discrete time incomplete market. We emphasize the interplay between option pricing and residual risk as well as imperfect hedging. It has been shown that the value of a European option satisfies a hyperbolic, rather than parabolic, partial differential equation. The closed-form solution for this hyperbolic equation has been obtained, which will collapse to the Black–Scholes formula as the time scaling converges to zero.     

Option pricing under regime-switching models: Novel approaches removing path-dependence

A well-known approach for the pricing of options under regime-switching models is to use the regime-switching Esscher transform (also called regime-switching mean-correcting martingale measure) to obtain risk-neutrality. One way to handle regime unobservability consists in using regime probabilities that are filtered under this risk-neutral measure to compute risk-neutral expected payoffs. The current paper shows that this natural approach creates path-dependence issues within option price dynamics. Indeed, since the underlying asset price can be embedded in a Markov process under the physical measure even when regimes are unobservable, such path-dependence behavior of vanilla option prices is puzzling and may entail non-trivial theoretical features (e.g., time non-separable preferences) in a way that is difficult to characterize. This work develops novel and intuitive risk-neutral measures that can incorporate regime risk-aversion in a simple fashion and which do not lead to such path-dependence side effects. Numerical schemes either based on dynamic programming or Monte-Carlo simulations to compute option prices under the novel risk-neutral dynamics are presented.     

Option pricing in incomplete markets

Expected utility maximization is a very useful approach for pricing options in an incomplete market. The results from this approach contain many important features observed by practitioners. However, under this approach, the option prices are determined by a set of coupled nonlinear partial differential equations in high dimensions. Thus, it represents numerous significant difficulties in both theoretical analysis and numerical computations. In this paper, we present accurate approximate solutions for this set of equations.     

Duality in option pricing based on prices of other derivatives

We clarify a financial meaning of duality in the semi-infinite programming problem which emerges in the context of determining a derivative price range based only on the no-arbitrage assumption and the observed prices of other derivatives. The interpretation links studies in the above context to studies in stochastic models.     

带基差风险和交易费用的不完全市场下的期权定价方法

本文研究了同时带有基差风险和交易费用的不安全市场中的权证定价方法。把[1]的模型推广到了考虑基差风险的情况[2]。期权的价格以一个三维自由边界问题的解给出,并含有两个相关的股票价格变量的相关系数。     

Contingent claim valuation in a market with different interest rates

The problem of contingent claim valuation in a market with a higher interest rate for borrowing than for lending is discussed. We give results which cover especially the European call and put options. The method used is based on transforming the problem to suitable auxiliary markets with only one interest rate for borrowing and lending and is adapted from a paper of Cvitanic and Karatzas (1992) where the authors study constrained portfolio problems.     

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 with a direct adaptive sparse grid approach

We present an adaptive sparse grid algorithm for the solution of the Black–Scholes equation for option pricing, using the finite element method. Sparse grids enable us to deal with higher-dimensional problems better than full grids. In contrast to common approaches that are based on the combination technique, which combines different solutions on anisotropic coarse full grids, the direct sparse grid approach allows for local adaptive refinement. When dealing with non-smooth payoff functions, this reduces the computational effort significantly. In this paper, we introduce the spatially adaptive discretization of the Black–Scholes equation with sparse grids and describe the algorithmic structure of the numerical solver. We present several strategies for adaptive refinement, evaluate them for different dimensionalities, and demonstrate their performance showing numerical results.     

Option pricing with regime switching by trinomial tree method

We present a fast and simple tree model to price simple and exotic options in Markov Regime Switching Model (MRSM) with multi-regime. We modify the trinomial tree model of Boyle (1986) [12] by controlling the risk neutral probability measure in different regime states to ensure that the tree model can accommodate the data of all different regimes at the same time preserving its combining tree structure. In MRSM, the market might not be complete, therefore we provide some ideas and discussions on managing the regime switching risk in support of our results.     

方差Gamma过程与期权定价

研究了几何Levy过程中，具有代表性的一类过程-方差Gamma过程下的等价鞅测度类问题，并且讨论了其具有的分析性质．进一步，我们也考虑了基于该过程的普通期权的定价．     

Binomial option pricing with nonidentically distributed returns and its implications

The argument of Cox, Ross, and Rubinstein for pricing options is generalized in the direction of using nonidentically distributed binomial returns as a model for the stock price process. It is found that the use of nonidentically distributed binomial returns, in the limit exhaust the class of infinitely divisible distributions. The pricing of these models are considered and it is shown that the model is a generalization of the Black-Scholes model. The use, however, of nonidentically distributed returns, it is shown, can lead to contradictions. Hence, it is argued, the models used for stock price behavior requires restrictions.     

Hysteresis effects under CIR interest rates

Most decision making research in real options focuses on revenue uncertainty assuming discount rates remain constant. However, for many decisions revenue or cost streams are relatively static and investment is driven by interest rate uncertainty, for example the decision to invest in durable machinery and equipment. Using interest rate models from Cox et al. (1985b), we generalize the work of Ingersoll and Ross (1992) in two ways. Firstly, we include real options on perpetuities (in addition to zero coupon cash flows). Secondly, we incorporate abandonment or disinvestment as well as investment options, and thus model interest rate hysteresis (parallel to revenue uncertainty in Dixit (1989a)). Under stochastic interest rates, economic hysteresis is found to be significant, even for small sunk costs.     

Option pricing with mean reversion and stochastic volatility

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier transform. The proposed model allows the option pricing formula to capture both the term structure of futures prices and the market implied volatility smile within a unified framework. A bivariate trinomial lattice approach is introduced to value path-dependent options with the proposed model. Numerical examples using European options, American options and barrier options demonstrate the use of the model and the quality of the numerical scheme.     

A systematic approach to pricing and hedging international derivatives with interest rate risk: analysis of international derivatives under stochastic interest rates

This paper deals with the valuation and the hedging of non-path-dependent European options on one or several underlying assets in a model of an international economy allowing for both, interest rate risk and exchange rate risk. Using martingale theory and, in particular, the change of numeraire technique we provide a unified and easily applicable approach to pricing and hedging exchange options on stocks, bonds, futures, interest rates and exchange rates. We also cover the pricing and hedging of compound exchange options.     

The pricing of Asian options under stochastic interest rates

The purpose of this paper is to analyse the effect of stochastic interest rates on the pricing of Asian options. It is shown that a stochastic, in contrast to a deterministic, development of the term structure of interest rates has a significant influence. The price of the underlying asset, e.g. a stock or oil, and the prices of bonds are assumed to follow correlated two-dimensional Itô processes. The averages considered in the Asian options are calculated on a discrete time grid, e.g. all closing prices on Wednesdays during the lifetime of the contract. The value of an Asian option will be obtained through the application of Monte Carlo simulation, and for this purpose the stochastic processes for the basic assets need not be severely restricted. However, to make comparison with published results originating from models with deterministic interest rates, we will stay within the setting of a Gaussian framework.     

A cyclical square-root model for the term structure of interest rates

This paper presents a cyclical square-root model for the term structure of interest rates assuming that the spot rate converges to a certain time-dependent long-term level. This model incorporates the fact that the interest rate volatility depends on the interest rate level and specifies the mean reversion level and the interest rate volatility using harmonic oscillators. In this way, we incorporate a good deal of flexibility and provide a high analytical tractability. Under these assumptions, we compute closed-form expressions for the values of different fixed income and interest rate derivatives. Finally, we analyze the empirical performance of the cyclical model versus that proposed in Cox et al. (1985) and show that it outperforms this benchmark, providing a better fitting to market data.     

An efficient and accurate lattice for pricing derivatives under a jump-diffusion process

Derivatives are popular financial instruments whose values depend on other more fundamental financial assets (called the underlying assets). As they play essential roles in financial markets, evaluating them efficiently and accurately is critical. Most derivatives have no simple valuation formulas; as a result, they must be priced by numerical methods such as lattice methods. In a lattice, the prices of the derivatives converge to theoretical values when the number of time steps increases. Unfortunately, the nonlinearity error introduced by the nonlinearity of the option value function may cause the pricing results to converge slowly or even oscillate significantly. The lognormal diffusion process, which has been widely used to model the underlying asset’s price dynamics, does not capture the empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process. This paper proposes an accurate and efficient lattice for the jump-diffusion process. Our lattice is accurate because its structure can suit the derivatives’ specifications so that the pricing results converge smoothly. To our knowledge, no other lattices for the jump-diffusion process have successfully solved the oscillation problem. In addition, the time complexity of our lattice is lower than those of existing lattice methods by at least half an order. Numerous numerical calculations confirm the superior performance of our lattice to existing methods in terms of accuracy, speed, and generality.     

Option pricing and hedging in incomplete market driven by Normal Tempered Stable process with stochastic volatility

This paper develops a distribution class, termed Normal Tempered Stable, by subordinating a drifted Brownian motion through a strictly increasing Tempered Stable process that generalizes the Variance Gamma and the Normal Inverse Gaussian and is used to model the logarithm asset returns. The newly added parameter is to create subclasses for all the distributions discovered in financial market. The empirical test suggests that time series of Technology stock returns in US market reject both the Variance Gamma distribution and the Normal Inverse Gaussian distribution and admit instead another subclass of the Normal Tempered Stable distribution. Furthermore, we introduce stochastic volatilities into the Normal Tempered Stable process and derive explicit formulae for option pricing and hedging by means of the characteristic function based methods. To answer the question of how well different models work in practice, we investigate four models adopting data on daily equity option prices and obtain several findings from the numerical results. To sum up, the Normal Tempered Stable process with stochastic volatility is able to adequately capture implied volatility dynamics and seen as a superior model relative to the jump-diffusion stochastic volatility model, based on the construction methodology that incorporates more sophisticated and flexible jump structure and the systematic and realistic treatment of volatility dynamics. The Normal Tempered Stable model turns out to have the competitive performance in an efficient manner given that it only requires three parameters.