# Binomial binary options.

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Under the binomial model, current value of an option equals the present value of the probability-weighted future payoffs from the options. It is different from the Black-Scholes-Merton model which is most appropriate for valuing path-independent options. At any point of time, the underlying can have two price movements: either an up move or a down move. Similarly, in case of a down move, the ratio of the new price S- to S is called the down-factor d. The call option is in-the-money when the spot price of the underlying is higher than the exercise price of the option.

Published12 Mar Abstract Randomized binomial tree and methods for pricing American options were studied. Firstly, both the completeness and the no-arbitrage conditions in the randomized binomial tree market were proved.

Secondly, the description of the node was given, and the cubic polynomial relationship between the number of nodes and the time steps was also obtained. Then, the characteristics of paths and storage structure of the randomized binomial tree were depicted.

Then, the procedure and method for pricing American-style options were given in a random binomial tree market. Finally, a numerical example pricing the American option was illustrated, and the sensitivity analysis of parameter was carried out. The results show that the impact of the occurrence probability of the random binomial tree environment on American option prices is very significant.

This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software including a spreadsheet. Although computationally slower than the Black—Scholes formulait is more accurate, particularly for longer-dated options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets.

With the traditional complete market characteristics of random binary and a stronger ability to describe, at the same time, maintaining a computational feasibility, randomized binomial tree is a kind of promising method for pricing financial derivatives. Introduction Cox et al.

### Mathematical Problems in Engineering

As Binomial option pricing method is simple and flexible to price all kinds of complex derivatives, and easy to realize the computer programming, it has become one of the mainstream methods of pricing derivatives, and also one of the frontiers and hot researches on binomial binary options derivatives for decades. Benninga and Wiener and Tian researched the relative properties of the binary tree and priced complex financial derivatives by binary tree to improve the computational efficiency of binary tree algorithm [ 45 ].

Rubinstein [ 6 ] expanded the built Edgeworth binary tree with random distribution of Edgeworth, which effectively involved the information of randomly distributed skewness and kurtosis, and made binary tree approach nonnormal distribution when applied to option pricing. Walsh [ 7 ] proved the effectiveness of binary tree algorithm via the study of the convergence and convergence speed problems of binary method from the theoretical point of view.

Gerbessiotis [ 8 ] gave a parallel binomial option pricing method with independent architecture, studied algorithm parameter adjustment method of achieving the optimal theory acceleration, and verified the feasibility and effectiveness of the algorithm under different parallel computing environments.

Georgiadis [ 9 ] tested that there is no so-called closed-form solution when pricing options with binary tree method.

Simonato [ 10 ] posed Johnson binary tree based on the approximation to Johnson distribution of the random distribution, overcoming some possible problems in Edgeworth binary tree that the combination of skewness and kurtosis cannot constitute qualified random distribution.

Due to the theory that Hermite orthogonal polynomials can approximate random distribution with the arbitrary precision, Leccadito et al.

Cui et al. Wen et al. Yuen et al. Aluigi et al.

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Jimmy [ 19 ] proposed robust binomial lattices for pricing derivatives where probabilities can be chosen to match local densities. However, in the traditional binary tree market, if the formed combination of the upward movement is seen as a market environment known by the binary theory; determines the unique market volatilitythere is only an environment binomial binary options the traditional binary tree market and at each node the binary tree moved upward or downward only once it means that market volatility is the same at any time.

However, this assumption is far from the reality of the financial markets, because the stock prices will respond immediately to the various information from domestic and abroad, and thus it is very sensitive. For example, a sudden change in the risk-free interest rate, the conflict with its neighbor countries, good performance of the rival company, a new CEO, and other random emergency information will lead to great fluctuation in stock prices.

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Ganikhodjaev and Bayram and Kamola and Nasir [ 20 — 22 ] put forward the random binary tree applied to European option pricing.

In this random binary tree market, there are at least two market environments, binomial binary options of which represents the normal state of the market while the other is the abnormal state of the market.

The binomial option pricing model is an options valuation method developed in The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option's expiration date. Key Takeaways The binomial option pricing model values options using an iterative approach utilizing multiple periods to value American options. With the model, there are two possible outcomes with each iteration—a move up or a move down that follow a binomial tree. The model is intuitive and is used more frequently in practice than the well-known Black-Scholes model.

Therefore, the first market environment which represents the normal state of the market corresponds to smaller market volatility and larger probability and the second market environment has larger binomial binary options volatility and smaller probability. The contribution of this paper is studying the related properties of random binary tree from the viewpoint of complete market and the number of nodes, giving the storage structure of random binary, describing the path characteristics of random binary tree, and researching the American option pricing problem under the random binary market.

### Nonlinear Problems: Mathematical Modeling, Analyzing, and Computing for Finance

The other sections of this paper are as follows. In Section 2we introduce random binary tree and its properties; the American option pricing problem under random binary environment is studied in Section 3 ; In Section 4we demonstrate the effectiveness of the algorithm through a numerical example learning binary options study the parameters sensitivity of relevant model.

Randomized Binary Tree 2.

Random Walks in an Independent Environment Solomon [ 23 ] is the first person to study random walks in an independent environment in the integer field. Letfor all be a sequence of independent and identically distributed random variables; then the random walks in an independent environment in the integer domain are a random sequencewhere.

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- Understanding the Binomial Option Pricing Model