![\"Screen](\"http:\/\/pages.vassar.edu\/magnes\/files\/2015\/04\/Screen-Shot-2015-04-19-at-9.14.10-PM.png\")
<\/a><\/p>\n\u00a0(Each number is the possible price of the asset).<\/p>\n
The binomial tree is more advantageous in pricing American options. This is because for an American option the owner at any point can exercise the option before the maturity date (when the option expires). The binomial tree works backwards to find the value of the option at T=0, when time is equal to zero. This helps determine how much one is willing to pay for the option. The binomial tree finds the difference in price between the original asset price, and the price at another discrete time interval, using the function max(K-P,0) , where K is the strike price, and P is the price of the asset at T=0. This function max(K-P,0) is used for a put option, and described more in our code.<\/p>\n
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It is important to discuss the variables the binomial tree is dependent on. It is dependent on $\\sigma$ which is volatility, simply it is just the standard deviation in returns of an asset or portfolio, $r$ is risk neutral interest rate, $u$ and $d$ are percentages the asset either goes up or down at each stage of the tree. The formula for backwards induction to find the original value of the asset is\u2026<\/p>\n
(1) <\/span> <\/span> where $V_u$ is the upper price at a discrete stage, and $V_d$ is the lower price at a discrete stage. A discrete stage meaning n=1,2\u2026 We know need to look at what p is, well p simulates the geometric Brownian motion of the underlying stock or asset with parameters $r$ and $\\sigma$. p is given by the equation below:<\/p>\n (2) <\/span> <\/span> We have finished the pricing of an American put option for a binomial tree when n=3. And plan on continuing to work on the code for the option pricing method to work for an arbitrary number for n, and hope to have completed by our next post.<\/p>\n The past week has been spent turning the basic ferromagnetic spin model (isingtest.m) created last week into a model of a basic financial system. In the ferromagnetic model, the spins of constituent atoms were represented by discrete spin values of 1 or -1. In this model, we will use the analogous atoms to represent agents in a market. This market will be based on any arbitrary good or stock, and like the ferromagnetic model, adjacent \u201cagents\u201d will be able to share information and influence each other. Here, the tendency for atoms to align with adjacent atoms represents a tendency of financial agents that work close to each other to use the same ideas.<\/p>\n In this model, a spin value of 1 signifies that an agent is currently buying the good or stock, whereas the value of -1 signifies that the agent is currently selling it. The market is governed by two rules:<\/p>\n In order to initialize this market, we create a grid to store whether or not each agent is a buyer or seller. We initially assigned the majority of agents to be buyers, represented as white pixels. The sellers are interspersed as blocks throughout the market, represented by the black pixels. Below is an example of what an initial market might look like:<\/p>\n <\/p>\n From here, our monte carlo method cycles through every agent in the market, using its position relative to its neighbors as well as an overall market trend to calculate the \u201cfield\u201d that pushes it toward one strategy or another. After every iteration, the display of the grid was altered accordingly. The equation for this field is below:<\/p>\n (3) <\/span> <\/span> The first term in the expression is analogous to the spin alignments, whereas the second term is representative to the individual agent\u2019s view of the whole market. The value C corresponds to whether an agent will be a \u201cfundamentalist(C=1)\u201d or a \u201cchartist(C=-1).\u201d If the agent is a fundamentalist, it means that he or she will be likely to follow the rule that it is best to try to get into the minority, making the second term negative and pushing it against the first term. If the agent is a \u201cchartist,\u201d it means that he or she is content with being in the majority, and the sign of the second term is the same as the first. Whether or not the agent is a fundamentalist or a chartist is determined by whether or not choosing this strategy will increase the energy, or risk, of the system, which would improve the likelihood of making money. This probability of an agent changing his or her identity is determined with:<\/p>\n (4) <\/span> <\/span> Then, the probability of an agent remaining\/becoming a buyer is given by:<\/p>\n (5) <\/span> <\/span> (6) <\/span> <\/span> The second term is the probability that the agent will remain\/become a seller. A random number is generated by the monte carlo method and compared to the probability, deciding which strategy the agent in question will use at the current time. Once the code has cycled through every agent in the market, it will update the grid displaying the identity of each agent\u2019s strategy at the given time. The average value of all of the strategies in the market is then taken, as it is representative of the price of the good or the stock. A positive average represents a positive trend in price, because there are more people that want to buy than sell. The opposite is true as well, as a negative average means that there is a surplus of sellers. Taking this average and subtracting from it the average of the last iteration gives use the change in price, or return, created by a shift in the market. An example is displayed below, plotting this average vs. time:<\/p>\n <\/p>\n <\/p>\n The periods of extreme return in a negative or positive direction correspond to a market configuration where there are large groups of buyers grouped together and large groups of sellers grouped together as shown below:<\/p>\n <\/p>\n Conversely, when the returns stay near 0, it is because there is no real trend in the market and it is more turbulent, as such:<\/p>\n<\/p>\n<\/p>\n\n
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