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To receive news and publication updates for Mathematical Problems in Engineering, enter your email address in the box below. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Randomized binomial tree binary domain edgeworth 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 binary domain edgeworth 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. 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.
As Binomial option pricing method is simple and flexible to price all kinds of complex derivatives, and easy to realize binary domain edgeworth computer programming, it has become one of the mainstream methods of pricing derivatives, and also one of the frontiers and hot researches on pricing derivatives for decades. Benninga and Wiener and Tian researched the relative properties of the binary domain edgeworth 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 ] binary domain edgeworth 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 binary domain edgeworth 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.
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 binary domain edgeworth of the upward movement and downward movement is seen as a market environment known by the binary theory; determines the unique binary domain edgeworth volatilitythere is only an environment in the binary domain edgeworth 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 binary domain edgeworth great fluctuation in stock prices. 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, one of which represents the normal state of the market while the other is the abnormal state of the market.
Therefore, the first binary domain edgeworth environment which represents the normal state of the market corresponds to smaller market volatility and larger probability and the second market environment has larger market binary domain edgeworth 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 binary domain edgeworth pricing problem under the random binary market.
The other sections of this paper binary domain edgeworth 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 and study the parameters sensitivity of relevant model. Solomon [ 23 ] is the first person to study random walks in an binary domain edgeworth 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 andthe occurrence probability of isand the probability of is.
Solomon noted that random environment in a certain binary domain edgeworth slowed random walk down. And Binary domain edgeworth [ 24 ] studied the asymptotic behavior of random walks binary domain edgeworth an independent environment. Ganikhodjaev and Bayram and Kamola and Nasir [ 20 — 22 ] raised the concept of a random binary tree based on the theory of random walks in an independent environment. In the traditional binary domain edgeworth of the binary tree, the stock price moves upward in a certain probability binary domain edgeworth moves downward in a certain probability at each time node.
In the random binary tree model, and are independent and identically distributed random variables. Random market environment describes two market environments andwhere the probability of market environment is and the other is. If the following condition is satisfied, the random binary tree market must be a market with no arbitrage: See reference [ 20 — 22 ]. Compared with the theorem given in [ 20 — 22 ], Theorem 1 puts forward another condition that the probability of market environment is given.
In fact, if this condition is not given, we cannot draw binary domain edgeworth conclusion that the random binary market is complete althoughare uniquely determined when calculating risk-neutral probabilities, for is arbitrary; thus there will be infinitely many unconditional risk-neutral probabilities. In this paper, Theorem 1 shows that, under appropriate conditions, random binary market is a market with no arbitrage as the traditional binary market.
That is to say, although there are four successor nodes following each node of random binary tree, it is totally different with quadtree in the incomplete market. Binary domain edgeworth the stock price in the first period by ; then the stock price in the th period is where, and.
When the random binary domain edgeworth tree is in the th period, the total number of nodes is. As known by the above equation, for each giventhere are kinds of situations for the market environment. But now, there are kinds of market situations for the market environment. It means if the is given, the corresponding quantity of the nodes in the random binary tree is as follows: In the th period, ; therefore there are nodes in the random binary tree.
The proof binary domain edgeworth over. As can be seen from Theorem 2the number of nodes in random binary tree is the cubic polynomial of period, which avoid the problem that the number of nodes shows explosive growth with the increase of when the number of nodes is the exponential function of period such as and makes it computationally feasible. The number of nodes in random binary tree decreases from nonrecombinant down to the cubic polynomial ofso there are all kinds of restructuring its path, and it is more difficult to show its nodes and describe its path.
As can be seen from the formula and the proof of Theorem 2random binary tree nodes can be described by the four-tuple as follows: There is a one-to-one relationship between the four-tuple given by formula 6 and the stock price in formula 2. Therefore in the rest of the paper, we often use the four-tuple binary domain edgeworth of the stock price in formula 2 without explanation in order to symbol simplicity.
In the random binary tree, the child nodes of the node are where the former two nodes correspond to the market environment and the latter two nodes are for the market environment. To facilitate the calculation process, binary domain edgeworth must map random binary tree node representation in the four-tuple given by formula 6 binary domain edgeworth one-dimensional representation.
If the value of is given and the values obtained from four-tuple and four-tuple by mapping are equal, that is,then must exist; that is, the four-tuple is equal to the four-tuple. For Supposingwe might as well set ; then which is in contradiction with. Therefore ifthen.
Suppose further that when,and we might as well let ; then for. Therefore which is contradictory. Obviously, when, there must beand the theorem is correct. Theorem 3 shows that in mapping there is a one-to-one mapping relation between random binary tree node and a group of the array. A random binary tree node uniquely corresponds to binary domain edgeworth element in one-dimensional array and an element in one-dimensional array uniquely corresponds to a random binary tree node.
Suppose that the probability of under the market environment is and which of is and that the probability of under the market environment is and which of is. It is easy to obtain the result of Theorem 4 by making use of the Bayesian rule.
With the help of Theorem 4we can compute the expectation of random distribution and furthermore give the binary domain edgeworth of complex derivatives.
Suppose risk-free rate binary domain edgeworthis the risk-free discount factor, is option maturity, is the time of each period of binary domain edgeworth binary tree, and is the periods of random binary domain edgeworth tree. According to the principle of risk-neutral pricing, we can get Set American options maturity isthe payoff function isand is the option strike price.
American option pricing under random binary tree still uses the backward induction method; the specific calculating steps are as follows. On the American option expiry date binary domain edgeworth period of the random binary treecalculating the immediate strike price at each node, this binary domain edgeworth American option price is.
When the periodfor each node in this period, calculate the immediate strike price of American option of this node and then calculate the value of American option to continue to hold The value of American option at node is.
The initial value of American option is. Let the first market environment, the second market environment, continuous risk-free year rateand initial stock price ; then is the payoff function of American put option, where maturity year binary domain edgeworth. In Figure 1 there are five curves corresponding to strike pricerespectively; the horizontal axis is the occurrence probability of the first market environment, and the vertical axis represents the initial price of American option.
As we know from Figures 1 and 2whether the real option, at-the-money option, or out-of-the-money option, the prices of American put option and American call option both decrease as the occurrence probability increases under the first market environment. However, the market volatility under the first market environment is less than that under the second market environment.
What is more, there is only binary domain edgeworth market environment in the traditional binomial model, and the corresponding market volatility only considers the normal volatility of the market; thus the traditional binary model, compared with random binary tree model, underestimated the American put and call option price.
In this paper we studied the completeness and nonarbitrage of random binary tree and gave the node representation of the random binary tree as well as describing the path representation method of binary domain edgeworth binary tree. Based on the backward induction method, the steps and the procedures were shown. Through the numerical example, we binary domain edgeworth the sensitivity analysis of American put option price to the random binary tree parameters.
Randomized binary tree maintains the complete-market character of the traditional binary tree; moreover it has a stronger ability to describe and the quantity of nodes is cubic polynomial of the number of periods which makes binary domain edgeworth calculation feasible.
The future research can be extended into two directions: The authors declare that there is no conflict of interests regarding the publication of this paper. The authors would like to thank the anonymous referees for theirvaluable binary domain edgeworth and suggestions. Their comments helped to improvethe quality of the paper immensely. Home Journals About Us. Mathematical Problems in Engineering. Indexed in Science Citation Index Expanded. Subscribe to Table of Contents Alerts. Table of Contents Alerts.
Abstract Randomized binomial tree and methods for pricing American options were studied. Introduction Cox et al. Randomized Binary Tree 2. Random Walks in an Independent Environment Binary domain edgeworth [ 23 ] is the first person to study random walks in an independent environment in the integer field. Randomized Binary Tree Ganikhodjaev and Bayram and Kamola and Nasir binary domain edgeworth 20 — 22 ] raised the concept of a random binary tree based on the theory of random walks in an independent environment.
When the random binary tree is in the th period, the total number of nodes is Proof. The proof is dropped. Pricing American Options Based on Randomized Binomial Tree Suppose risk-free rate isis the risk-free discount factor, is option maturity, is the time of each period of random binary tree, and is the periods of random binary tree.