Hedging an Option through the Black-Scholes model in discrete time
Abstract As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimensional integral, numerical calculation is time-consuming. In the current studies, some scholars just obtained theoretical derivation, or gave some simulation calculations. Others impose underlying assets on some strong assumptions, for example, a lot of calculations are based on the Black-Scholes model.
This thesis considers Merton jump diffusion model as the basic model to derive the pricing formula of discrete double barrier option; numerical calculation method is used to approximate the continuous convolution discrete option is calculating discrete convolution. Then we compare the results of theoretical calculation with simulation results by Monte Carlo method, to verify their efficiency and accuracy.
By comparing the results of degeneration constant parameter model with the results of previous models we verified the calculation method is correct indirectly. Compared with the Monte Carlo simulation method, the numerical results are stable.
Even if we assume the simulation results are accurate, the time consumed by the numerical method to achieve the same accuracy is much less than the Monte Carlo simulation method.
Application of Asymmetric IRT Modeling to Discrete-Option Multiple-Choice Test Items
Open Journal of Statistics, 7, Introduction Options as a kind of important financial derivatives have been developed rapidly in recent decades.
With the increase of investment demands and the progress of theoretical level, all kinds of exotic options were born. As an important path dependent exotic option, barrier option was accounted for nearly half of the share.
Their research has a very important practical significance. So, discrete option is exotic option pricing problem has become an important research topic. Black and Scholes  set up the B-S model and get the pricing formula of the call option.
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Common barrier options pricing has a long history. It was first studied by Merton  on the binary options on the internet and out call option pricing.
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After this period, Reimer and Sandmann  use the binary tree model research barriers options. Then Gao, Huang and Subrahmanyam  pushed forward the pricing analysis of the American barrier option by additional income method the decomposition technique, and Dai and Kwok  has got the pricing formula of American down and in call option.
The model of the underlying asset is also developed from the original Options trap model to a variety of more complex models.
Such as Wang, Du  obtained pricing formula in the class of barrier options under jump diffusion model, and Xie  put forward the constant elasticity of variance CEV pricing barrier option process.
However, these developments are based on the constant parameter model. Farnoosh, Sobhani, Rezazadeh  proposed a method based on the time dependent parametric B-S model of the barrier options to extend the research of this topic to another direction.
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Looking at the development of barrier discrete option is pricing, the barrier option pricing theory is from a study based on the continuous observation idealized model to the effective discrete observation model, and the stochastic model of the target stock is also developed from B-S model with constant parameters to model with time-dependent parameter, or constant parameter jump diffusion model, CEV model, SVJD model and so on.
In order to make the scope of application more extensive, the goal of this paper is the further development for the numerical algorithm of the option pricing based on Merton jump diffusion model. In the numerical calculation section, discrete option is a wide range of applications, Monte Carlo simulation method is certainly one of the alternative calculation methods.
One potentially useful practical application of such models is toward the scoring of discrete-option multiple-choice DOMC items.
But it is often accompanied by a wide range of accuracy and low computational efficiency. As traditional numerical algorithm performs badly in non- smooth solutions, Golbabai, Ballestraand Ahmadian used finite element method FEM to improve orders of convergence in . In Ahmadian and Ballestra found it also performs well under a constant elasticity of variance model with jump diffusion. For some models, the other way to get numerical method is directly starting from the analytic discrete option is.
However, this model is derived in continuous time. What happens when we use it to hedge an option in discrete-time?
In Farnoosh, Sobhani, Rezazadeh is proposed that for time dependent numerical parameters of barrier options based on B-S model an analytical solution can be obtained.
They started from solving partial differential equations.
By constructing a heat conduction equation, they finally got the analytical solution. Using the numerical solution of the Romberg extrapolation algorithm to calculate is taking time discrete option is precision into account.
A Psychometric Model for Discrete-Option Multiple-Choice Items
But consider from the efficiency of the integral calculation, it is not as good as quadrature method which been used by Discrete option is and Tagliani in general B-S model . The difference is probably cause by the influence of boundary.
The quadrature method used on general B-S model is able to extend to more complex model. Furthermore, for time dependent parameters, it also can achieve good results. In the second section, we will consider knock-out call option as an example to introduce risk neutral pricing based on Merton jump diffusion model.
Она ожидала, что полиция задержит ее при выходе, однако никто не подумал остановить. Но оказавшись снаружи, она едва сдерживала восторг. "Мама, мама, - думала Элли и слезы наполняли ее. - Я так рада за тебя".
Finally, we obtain an analytical formula of the discrete double barrier option. In the third section, we will introduce the application of quadrature method to calculate the high dimensional integrals generation generated in the second section. We also introduce Monte Carlo simulation method to compute the result for comparison.
In the fourth section, the computational efficiency of the numerical method is compared with samples in literatures and results of simulation method. In the conclusion, we note that this calculation method is significantly better in accuracy and efficiency, and there is a certain promotion potential.
Discrete Double Barrier Options Discrete option is We assume the European discrete knock-out double barrier option has the following conditions: 1 European options, i.