The Binomial Options Pricing Model - Statistics Project Example

Binomial Options Pricing Model Financial trading presents itself as both lucrative and risky at the same time. There is dire need to look into financial trading on strong scientific grounds in order to reduce the amount of risk incurred. One of the more speculative means of financial trading involves dealing with stocks. Stock trading involves the purchase and holding of stocks until their prices rise enough to release an acceptable level of profit. However, the involvement of risks means that making a profit on every transaction may not be a possibility. Stock trading may also end up in loss especially if the market is too volatile or if the risk tends to rise too steeply. In order to tabulate the various possibilities that may confront an investor, various kinds of mathematical models have been derived. The results produced using these models are typically known as derivatives and tend to highlight various strategies and their corresponding investment results. The various mathematical models used for tabulating financial possibilities in general and stock based possibilities in particular are the Black Scholes model, the Monte Carlo model and the binomial options pricing model. Each different model has its own peculiar merits and demerits making each model suitable for tabulating different kinds of market conditions and investment strategies. The Black Scholes model is employed in order to tabulate continuous data and time investment possibilities. On the other hand, the Monte Carlo model is used to deal with complex data relationships where several variables are often interlinked to affect the final outcomes. The Monte Carlo model is seen as being too excessive in terms of computational usage and time usage for use in regular financial modelling. In contrast to the Black Scholes and the Monte Carlo models, the binomial options pricing model is used in situations where the underlying asset is not suspected to vary by a large degree. In addition, another major difference between the binomial options pricing model and the Black Scholes and the Monte Carlo model is the use of discrete data points. The binomial options pricing model is used in situations where discrete information relating to the financial model is available. Both the Black Scholes model and the Monte Carlo model use continuous data instead of discrete data to predict financial behaviour. This has significant impacts on the formulation and the outcome of the overall financial problem. The binomial options pricing model is generally formulated in the form of a discrete numerical problem. The solution cannot be derived using closed form solutions in the cases encountered with the binomial options pricing form so the solutions are generally derived numerically. In contrast, both the Black Scholes model and the Monte Carlo model allow the formulation of problems and finding their solutions in exact forms. This precludes the use of any numerical methods in order to derive the solutions for problems formulated using the Black Scholes model and the Monte Carlo model. These issues also have a special implication for computational models. The binomial options pricing model is implemented in its original form in computational solutions while the Black Scholes model and the Monte Carlo model need to be taken into numerical forms for computational solutions. This ensures that the binomial options pricing model performs the fastest calculations and produces the most reliable results when compared to the other models. Explanation of the Binomial Options Pricing Model Like other financial models, the binomial options pricing model is based on a number of assumptions. The primary assumption in the binomial options pricing model is that any kind of price change will only take place after a specific time. This is essentially another measure of discreteness that ensures that the collected data comes after a specific time. The specific time after which each piece of data is taken is better known as the step width and is taken to be constant for the entire model. The binomial options pricing model uses the price of the underlying asset in order to construct a complete binomial tree that can be used to discern the price of the asset in various situations. The overall number of steps allowed determines the overall size of the binomial tree. In each step, the price of the underlying asset can either move up or down or could be one of its combinations. For the very first step, it is assumed that the price of the underlying asset will either decrease or increase by a fixed amount. This will tend to produce two nodes for the first step of the binomial tree. For the next step, the price is assumed to decrease or increase in amount from each node produced in the previous step. This leads to a total of four nodes for the second step. This method of developing the binomial tree continues until the last step is reached. The value of the underlying asset is assumed to go up or down depending on market volatility and the amount of time involved. The up factor, can be expressed as: While the down factor, can be expressed as: Where: is the volatility is the time interval As each step of the binomial tree proceeds, there is the addition of new possibilities. When a binomial options pricing model problem is being solved, the typical way forward is to calculate the last step of the binomial tree and to work back to the very first step. There is typically little need to produce the binomial tree in order to solve the possibilities. Another notable thing is that any step of the binomial tree can be calculated independent of the other steps using prescribed formulas. For the binomial tree the nodal value at any given node can be expressed as: Where: is the price at node is the price at the first node or the underlying price is the number of up ticks is the number of down ticks Another thing to keep in mind is that each node allows the tracking down of both the call option and the put option. For any given node , the call price and the put price can be expressed as: for a call option for a put option Where: is the strike price is the price at node Option Evaluation Methods Option evaluation can be done in various ways. The most popular are the American options and the European options. Under the American options method, the investor can decide to hold or exercise his call and put options as he desires. In order to evaluate the American options methods, the maximum of the binomial value and the exercise value is used. In contrast, the European option evaluation method strictly implies that the binomial value at each node applies regardless to any other thing. Bibliography Cox, J.C., Ross, S.A. & Rubinstein, M., 1979. Option Pricing: A Simplified Approach. Journal of Financial Economics, 7, pp.229-63. Georgiadis, E., 2011. Binomial Options Pricing Has No Closed-Form Solution. Algorithmic Finance, 1(1). Rendleman, R.J. & Bartter, B.J., 1979. Two-State Option Pricing. The Journal of Finance, 35(5), pp.1093-110. Read More