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.

Mixing Monte-Carlo and Partial Differential Equations for Pricing Options

There is a need for very fast option pricers when the financial objects are modeled by complex systems of stochastic differential equations.Here the authors investigate option pricers based on mixed Monte-Carlo partial differential solvers for stochastic volatility models such as Heston's.It is found that orders of magnitude in speed are gained on full Monte-Carlo algorithms by solving all equations but one by a Monte-Carlo method,and pricing the underlying asset by a partial differential equation with random coefficients,derived by Ito calculus.This strategy is investigated for vanilla options,barrier options and American options with stochastic volatilities and jumps optionally.

Early exercise premium method for pricing American options under the J-model

Background:This study develops a new model called J-am for pricing American options and for determining the related early exercise boundary(EEB).This model is based on a closed-form solution J-formula for pricing European options,defined in the study by Jerbi(Quantitative Finance,15:2041-2052,2015).The J-am pricing formula is a solution of the Black&Scholes(BS)PDE with an additional function called f as a second member and with limit conditions adapted to the American option context.The aforesaid function f represents the cash flows resulting from an early exercise of the option.Methods:This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called J-am,BS-am,and Heston-am models.These three models are based on the J-formula by Jerbi(Quantitative Finance,15:2041-2052,2015),BS model,and Heston(Rev Financ Stud,6:327-343,1993)model,respectively.This study performs a general algorithm leading to the EEB and to the American option price for the three models.Results:After implementing the algorithms,we compare the three aforesaid models in terms of pricing and the EEB curve.In particular,we examine the equivalence between J-am and Heston-am as an extension of the equivalence studied by Jerbi(Quantitative Finance,15:2041-2052,2015).This equivalence is interesting since it can reduce a bi-dimensional model to an equivalent uni-dimensional model.Conclusions:We deduce that our model J-am exactly fits the Heston-am one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies.The required CPU time to compute the solution is significantly less in the case of the J-am model compared with to the Heston-am model.

New Method for American Options Pricing

A new method using nonlinear regression to approximate the option price based on approximate dynamic programming is proposed. As a result a representation of the American option price is obtained as a solution to the dual minimization problem. In addition, an available Q-value iteration algorithm in practice is given.

Pricing European Options Based on a Logarithmic Truncated t-Distribution

The t-distribution has a “fat tail” feature, which is more suitable than the normal probability density function to describe the distribution characteristics of return on assets. The difficulty of using t-distribution to price European options is that a fat tail can lead to a deviation in one integral required for option pricing. We use a distribution called logarithmic truncated t-distribution to price European options. A risk neutral valuation method was used to obtain a European option pricing model with logarithmic truncated t-distribution.

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.

Investment in deepwater oil and gas exploration projects:a multi-factor analysis with a real options model

Deepwater oil and gas projects embody high risks from geology and engineering aspects, which exert substantial influence on project valuation. But the uncer- tainties may be converted to additional value to the projects in the case of flexible management. Given the flexibility of project management, this paper extends the classical real options model to a multi-factor model which contains oil price, geology, and engineering uncertainties. It then gives an application example of the new model to evaluate deepwater oil and gas projects with a numerical analytical method. Compared with other methods and models, this multi-factor real options model contains more project information. It reflects the potential value deriving not only from oil price variation but also from geology and engi- neering uncertainties, which provides more accurate and reliable valuation information for decision makers.

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.

A Power Penalty Approach to Numerical Solutions of Two-Asset American Options

This paper aims to develop a power penalty method for a linear parabolic variational inequality （VI） in two spatial dimensions governing the two-asset American option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant λ〉 1 and a power parameter k 〉 0. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order O（λ^-k/2）. A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method.

Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity

This paper considers the problem of numerically evaluating discrete barrier option prices when the underlying asset follows the jump-diffusion model with stochas-tic volatility and stochastic intensity.We derive the three-dimensional characteristic function of the log-asset price,the volatility and the jump intensity.We also provide the approximate formula of the discrete barrier option prices by the three-dimensional Fourier cosine series expansion(3D-COS)method.Numerical results show that the 3D-COS method is rather correct,fast and competent for pricing the discrete barrier options.

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.

An ETD Method for American Options under the Heston Model

A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs.Free boundary is treated by the penalty method.Transformed nonlinear partial differential equation is solved numerically by using the method of lines.For full discretization the exponential time differencing method is used.Numerical analysis establishes the stability and positivity of the proposed method.The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.

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.

Nonparametric estimation of employee stock options

We proposed a new model to price employee stock options （ESOs）. The model is based on nonparametric statistical methods with market data. It incorporates the kernel estimator and employs a three-step method to modify Black- Scholes formula. The model overcomes the limits of Black-Scholes formula in handling option prices with varied volatility. It disposes the effects of ESOs self-characteristics such as non-tradability, the longer term for expiration, the eady exercise feature, the restriction on shorting selling and the employee＇s risk aversion on risk neutral pricing condition, and can be applied to ESOs valuation with the explanatory variable in no matter the certainty case or random case.

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.

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.

Asset Pricing and Simulation Analysis Based on the New Mixture Gaussian Processes

European compound option pricing model is established by using the mixed bifractional Brownian motion. Firstly, using the principle of risk-neutral pricing, the European option pricing formulas and the parity formulas are obtained. Secondly, with the Delta hedging strategy, the corresponding compound option pricing formulas and the parity formulas are got. Finally, using the daily closing price data of “Lingang B shares” and “Yitai B shares” respectively, the results show that the mixed model is closer to the true value than the previous model.

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.

Recovering implied risk-neutral probability density function using SVR

Using support vector regression （SVR）, a novel non-parametric method for recovering implied risk-neutral probability density function （IRNPDF） is investigated by solving linear operator equations. First, the SVR principle for function approximation is introduced, and an SVR method for solving linear operator equations with knowing some values of the right-hand function and without knowing its form is depicted. Then, the principle for solving the IRNPDF based on SVR and the method for constructing cross-kernel functions are proposed. Finally, an empirical example is given to verify the validity of the method. The results show that the proposed method can overcome the shortcomings of the traditional parametric methods, which have strict restrictions on the option exercise price; meanwhile, it requires less data than other non-parametric methods, and it is a promising method for the recover of IRNPDF.