Early exercise European option and early termination American option pricing models

The maximum relative error between continuous-time American option pricing model and binomial tree model is very small. In order to improve the European and American options in trade course, the thesis tried to build early exercise European option and early termination American option pricing models. Firstly, the authors reviewed the characteristics of American option and European option, then there was compares between them. Base on continuous-time American option pricing model, this research analyzed the value of these options.

European Option Pricing under a Class of Fractional Market

In order to price European contingent claim in a class of fractional Black-Scholes market, where the prices of assets follow a Wick-Ito stochastic differential equation driven by the fractional Brownian motion and market coefficients are deterministic functions, the pricing formula of European call option was explicitly derived by the method of the stochastic calculus of tile fractional Brownian motion. A result about fractional Clark derivative was also obtained.

NEW METHOD TO OPTION PRICING FOR THE GENERAL BLACK-SCHOLES MODEL-AN ACTUARIAL APPROACH

Using physical probability measure of price process and the principle of fair premium, the results of Mogens Bladt and Hina Hviid Rydberg are generalized. In two cases of paying intermediate divisends and no intermediate dividends, the Black_Scholes model is generalized to the case where the risk_less asset (bond or bank account) earns a time_dependent interest rate and risk asset (stock) has time_dependent the continuously compounding expected rate of return, volatility. In these cases the accurate pricing formula and put_call parity of European option are obtained. The general approach of option pricing is given for the general Black_Scholes of the risk asset (stock) has the continuously compounding expected rate of return, volatility. The accurate pricing formula and put_call parity of European option on a stock whose price process is driven by general Ornstein_Uhlenback (O_U) process are given by actuarial approach.

The application of option pricing theory to the evaluation of mining investment

A rational evaluation on an investment project forms the basis of a right investment decision making. The discounted cash flow (DCF for short) method is usually used as a traditional evaluation method for a project investment. However, as the mining investment is influenced by many uncertainties, DCF method cannot take into account these uncertainties and often underestimates the value of an investment project. Based on the option pricing theory of the modern financial assets, the characteristics of a real project investment are discussed, and the management option of mine managers and its pricing method are described.

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.

Study on Quantum Finance Algorithm:Quantum Monte Carlo Algorithm based on European Option Pricing

As one of the major methods for the simulation of option pricing,Monte Carlo method assumes random fluctuations in the distribution of asset prices.Under certain uncertainties process,different evolution paths could be simulated so as to finally yield the expectation value of the asset price,which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations.In order to solve the above computational problem,quantum Monte Carlo(QMC)has been established and applied in the relevant systems such as European call options.In this work,both MC and QM methods are adopted to simulate European call options.Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer,the amplitude estimation(AE)algorithm is found to play a secondary role in accelerating the pricing of European options.More importantly,the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.

Application of Binomial Option Pricing Model to the Appraisal of Knowledge Management Investment

This paper views knowledge management （KM） investment from the angle of real options, and demonstrates the utility of the real options approach to KM investment analysis. First, KM project has characteristics of uncertainty, irreversibility and choice of timing, which suggests that we can appraise KM investment by real options theory. Second, the paper analyses corresponding states of real options in KM and finance options. Then, this paper sheds light on the way to the application of binomial pricing method to KM investment model, which includes modeling and conducting KM options. Finally, different results are shown of using DCF method and binomial model of option evaluation via a case.

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.

Barrier Option Pricing in Regime Switching Models with Rebates

This paper is concerned with the valuation of single and double barrier knock-out call options in a Markovian regime switching model with specific rebates.The integral formulas of the rebates are derived via matrix Wiener-Hopf factorizations and Fourier transform techniques,also,the integral representations of the option prices are constructed.Moreover,the first-passage time density functions in two-state regime model are derived.As applications,several numerical algorithms and numerical examples are presented.

Empirical Study on Option Pricing under Markov Regime Switching Economics

In this research,we summarize the results of a practical study of index options based on the option valuation model which was proposed by Siu and Yang(Acta Math.Appl.Sin.Engl.Ser.,25(3)(2009),pp.339{388),where an EMM kernel is integrated which takes into account all risk components of a regime-switching model.Further,the regime-switching risk of an economy in the options is priced using a hidden Markov regime-switching model with the risky underlying asset being modulated by a discrete-time,nite-state,hidden Markov chain whose states represent the hidden states of an economy.We apply such a model to the pricing of Hang Seng Index options based on the real-world nancial data from October 2009 to October 2010(i.e.,for the year in which the model was proposed).We employed the entropy martingale measure(EMM)approach proposed by Siu and Yang(Acta Math.Appl.Sin.Engl.Ser.,25(3)(2009),pp.339{388)to determine the optimal martingale measure for the Markov-modulated GBM.In addition,we have proposed a numerical technique called the weighted di erence method to compliment the EMM approach.We have also veri ed the extended jump-di usion model under regime-switching that we proposed recently(Int.J.Finan.Eng.,6(4)(2019),1950038)using the 50ETF options which are obtained from Shanghai Stock Exchange covering a time span from January 2018 to December 2022.Further,we have highlighted the challenges for the EMM kernel-based Markov regime-switching model for pricing the out-of-the-money index options in the real world.

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.

Option Pricing for Time-Change Exponential Lévy 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.

Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

This paper presents two new versions of uncertain market models for valuing vulnerable European call option.The dynamics of underlying asset,counterparty asset,and corporate liability are formulated on the basis of uncertain differential equations and uncertain fractional differential equations of Caputo type,respectively,and the solution to an uncertain fractional differential equation of Caputo type is presented by employing the Mittag-Leffler function andα-path.Then,the pricing formulas of vulnerable European call option based on the proposed models are investigated as well as some algorithms.Some numerical experiments are performed to verify the effectiveness of the results.

Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

This paper establishes a lattice Boltzmann method （LBM） with two amending functions for solving partial differential equations （PDEs） arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O（N）, where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.

Option Pricing under the Double Exponential Jump-Diffusion Model with Stochastic Volatility and Interest Rate

This paper proposes an efficient option pricing model that incorporates stochastic interest rate(SIR),stochastic volatility(SV),and double exponential jump into the jump-diffusion settings.The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns,rare events,and an SIR.Using the model,we deduce the pricing characteristic function and pricing formula of a European option.Then,we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump-diffusion model with SIR and SV.For verification purposes,we conduct time efficiency analysis,goodness of fit analysis,and jump/drift term analysis of the proposed model.In addition,we compare the pricing accuracy of the proposed model with those of the Black-Scholes and the Kou(2002)models.The empirical results show that the proposed option pricing model has high time efficiency,and the goodness of fit and pricing accuracy are significantly higher than those of the other two models.

Pricing Stochastic Barrier Options under Hull-White Interest Rate Model

A barrier option valuation model with stochastic barrier which was regarded as the main feature of the model was developed under the Hull-White interest rate model.The purpose of this study was to deal with the stochastic barrier by means of partial differential equation methods and then derive the exact analytical solutions of the barrier options.Furthermore,a numerical example was given to show how to apply this model to pricing one structured product in realistic market.Therefore,this model can provide new insight for future research on structured products involving barrier options.

American Barrier Option Pricing Formulas for Currency Model in Uncertain Environment

Option pricing problem is one of the central issue in the theory of modern finance.Uncertain currency model has been put forward under the foundation of uncertainty theory as a tool to portray the foreign exchange rate in uncertain finance market.This paper uses uncertain differential equation involved by Liu process to dispose of the foreign exchange rate.Then an American barrier option of currency model in uncertain environment is investigated.Most important of all,the authors deduce the formulas to price four types of American barrier options for this currency model in uncertain environment by rigorous derivation.

A Fixed Point Method for the Linear Complementarity Problem Arising from American Option Pricing

For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over Relaxation （PSOR） has been widely applied to solve the resulted LCP. In this paper, we propose a fixed point iterative method to solve this type of LCPs, where the splitting technique of the matrix is used. We show that the proposed method is globally convergent under mild assumptions. The preliminary numerical results are reported, which demonstrate that the proposed method is more accurate than the PSOR for the problems we tested.

Quasi-Monte Carlo simulation of Brownian sheet with application to option pricing

Monte Carlo and quasi-Monte Carlomethods arewidely used in scientific studies.As quasi-Monte Carlo simulations have advantage over ordinaryMonte Carlomethods,this paper proposes a new quasi-Monte Carlo method to simulate Brownian sheet via its Karhunen–Loéve expansion.The proposed new approach allocates quasi-random sequences for the simulation of random components of the Karhunen–Loéve expansion by maximum reducing its variability.We apply the quasi-MonteCarlo approach to an option pricing problem for a class of interest rate models whose instantaneous forward rate driven by a different stochastic shock through Brownian sheet andwe demonstrate the application with an empirical problem.

Option Pricing by Mean Correcting Method for Non-Gaussian Lvy Processes

For a non-Gaussian Levy model, it is shown that if the model exists a trivial arbitrage-free interval, option pricing by mean correcting method is always arbitrage-free, and if the arbitrage-free interval is non-trivial, this pricing method may lead to arbitrage in some cases. In the latter case, some necessary and sufficient conditions under which option price is arbitrage-free are obtained.