Binomial Flexible Option Pricing Theory - Essay Example

? Option Pricing Theory Abdullahi Warsame al affiliation: London Metropolitan Luiz Vtiello Option Pricing Theory 2.1 Background to the problem Option Pricing Theory is any model or theory used for calculating the value of an option. The Black-Scholes model and the binomial model are the most commonly used option pricing models. These theories have errors because their options are derived from assets. For instance in a company’s stock, time does affect the theories because the process of calculating option pricing takes a long time or is done after several years (Coval, 2001). An option provides the buyer the right to buy or sale the quantity of goods he or she wants at a fixed price known as the strike price. Since the process of buying an option is optional, the holder can choose not to buy or sale the assets. There are two options these are; right to buy and right to sale. Options can come in several varieties like; a put option, gives the seller an underlying price to sale an option (Bostock, 2004). A call option gives its holder the right to buy an option on its set price; these options depend on when the option is offered. Therefore, the paper aims at giving a theoretical analysis of option pricing theory. 2.2 Research questions The paper focuses on two main research questions; to determine the effects of option pricing theory and to explore ways of improving option pricing theory. 2.3 Significance of the research The research targets businessmen who take part in buying and selling of options using the option pricing theories. The research findings will provide them with the basis of calculating option prices. The study mainly delimits itself to the two option pricing theories (Black-Scholes model and binomial pricing option models). 2.4 Literature review The Binomial and Black-Scholes models do shed some light on the derivatives that can be given prices based on risk approach, and this Risk neutrality is the direction to a modern portfolio theory, and it involves the understanding that investors do not mind about risks involved, but focuses on the option prices.  Even, though several, researchers have argued that, risk neutrality is not assumed by theorists, and they accept the prices of underlying assets as determined in a fair market (Chance, 1999). Broadie and Detemple (2000) in their research provided a suggestion that binomial models are modified by replacing the binomial prices with the tree diagram analysis using the the Black-Scholes values, or by making it easy to payoff stocks at maturity, and the other option prices as usual. The major disadvantages of this model is that the option price converges; a result of changes that may take place in the prices. In order to obtain solutions that are exact, the Standard Richardson extrapolation may be applied . Burn (2003) states that, although the option pricing models were used. their patterns of convergent and rate of convergence for calculating the option ratios are not well described. Hull method and extended model can be used to come up with monotonic convergence using as the formula for coming up with deltas and gammas and deltas in this model.He adds that the models can be improved by introducing a more advanced formula, to improve the computation of the hedge ratios while calculation option pricing. The Central Limit Theorem, states that, the actual distribution of prices under the Black-Scholes model converges to its continuous-time limit. For instance, the price distribution of the model converges to a lognormal distribution. Similarly, binomial option prices calculated also converge to the Black-Scholes price. . N-Cumulative Standard normal distribution function r- rate of return (risk free) T-time (up to expiry in years) S-current stock price o- volatility of stock q- strike price Broadie and Detemple (2000) in their evaluation suggested for a binomial model called Binomial Black and Scholes model to price options. This model is identical to the Cox, Ross and Rubinsten (CRR) model apart from one thing the time before option maturity, and it replaces the usual valuation process. According to Tian (2001) a flexible binomial model with a parameter that changes shapes and alters the binomial tree. The parameter is presented as follows. . (U, d, p are represented as unknowns in the equation so that after calculations, the remaining conditions have to remain, (U=1/d).This leads to the lattice nodes of the binomial tree remain symmetrical). With positive movement of the parameter, the up movement is lager than the correspondent up movements of CRR model. The central nodes also move up words the resulting tree in the shifted up wards. 2.4.1. Black-Scholes model Black-Scholes model was derived in 1973 and it accounts for dividends; these dividends are incorporated in a reduction in stock price, and input equal to the current value of the expected dividend. The model takes this inputs; strike option term riskless rate stock expected volatility and expected dividend yield. This model is used to calculate a theoretical call price using five key elements which are; strike price, time to expiration, stock price, volatility, and short-term interest rate. Modified model of Black-Scholes and binomial pricing option models can be used to establish the impact on option prices of the distribution of non-lognormal price as measured by coefficients of variations, and to calculate the volatility as displayed by price distributions. The theories show the distribution of assets as they diverge from the normal curve. They plot the distributions of asset coefficients kurtosis and skewness. Black-Scholes model has been indicated by some research as the standard model for valuing options in finance. This model assumes that a given asset price should be calculate using geometric Brownian procedure with constant volatility. Some of the assumptions of a based on the principle that pricing should not provide immediate gain to the buyer or seller are; one, Stock pays no dividends, two, Market direction cannot be predicted, and three, "Random Walk, Option can only be exercised upon expiration. The model does not require commissions in the transaction. It assumes that stock returns are normally distributed, and the interest rates of an option remain constant, thus volatility is constant over time. It can be seen that the assumptions of this model are invalid leading theoretical aspects that are not true. Therefore, these values are obtained from the model are important for the purpose of comparison and not indication of the price of an option. The model if used by businessmen blindly will put the at a risk of loses. For instance, the model has several limitation that should be observed by many business men when calculation option prices. These limitations include; the model assumes that stock prices are always constant yet stock price is dynamic, and changes depending with the product session. The model assumes that the stock’s volatility and risk-free rate are always constant. According to model, the company stock pays no dividend until the stock expires. The model proposes that business analyst can only estimate stock volatility instead of directly observing its they observe other inputs. Lastly the model tends to put low prices on options that have a high rate of dividends. In order to tackle these model limitations, Autoregressive Conditional Heteroskedasticity, was developed. This is a variant of Black-Scholes, and it replaces the notion that volatility is constant and states that volatility is random and dynamic, it keeps changing from time to time. Despite the six limitations, Black-Scholes model remains the most popular option pricing model in finance. The Black-Scholes model is usually calculated based on some historical data and probabilities of future stock prices obtained. It uses this formulae; option price is equal to expected future stock Price minus expected cost of exercising Option. The model adds some adjustments these are; the range of future stock, the present value of existing cost, the net price of existing cost and the exercise price may be higher than the underlying stock price (Camara 2005). According to Black-Scholes model stock is volatility is important in option pricing, in that, when stock has low volatility then it will stay within a narrow range unlike when it has high volatility. So with high volatility it tends to spread further in different directions from its current price making it have different options. The model uses pay off table to estimate expected values of a situation that has several outcomes. Lastly, the model assumes that stock does not have dividends, but some stocks do pay dividends and the dividends do affect the prices of an option. The Black-Scholes model does not ape reality this is because of its simplicity based on the assumptions. It is used widely as a useful approximation, but it does avoid risk requirements for a proper application. Summary of the call option formulae for the Black Scholes model C = S N (d1) - X e-rT N (d2)   C  =  price (call option)   S  =  price (underlying stock)   X  =  option exercise price   r  =  Interest rate(risk free)   T  =  time (until expiration)   N()  =  area under normal curve   d1  =  [ ln(S/X) + (r + ?2/2) T ] / ? T1/2   d2  =  d1 - ? T1/2 The Put-call parity requires: P = C - S + Xe-rT The price then becomes: P = Xe-rT N (-d2) - S N (-d1) 2.4.2. Binomial option model Binomial option model, on the other hand, is a solution that models the price changes over the whole option in a given known period. In some situation, there are no known options to calculate option prices over a given time, therefore, the use of the binomial model is compulsory (Bakshi 2003). In this model, the price evolution is represented as a binomial tree of all possible prices at equal time under the assumption, that at a given time the price can move up, and down in respect of pseudo-probabilities. Implying that the root price is the price at the present moment, and each column of the tree represents the every possible price of an option as the price changes over time. Each node of the tree has S child price represented by u. Where, S the initial price has u and d which are the factors that make the price move up and down (Jones 2008). The option price is then derived from the binomial tree. The model has three assumptions these are; no price changes, no arbitrage, efficient market, short duration of options. With the three assumptions, the model can calculate the option through a binomial lattice (tree). The process of valuation starts at each final node, and then is calculated backwards through the tree to the first node, and the calculated value is the price of an option. The option are described in three stapes these are; tree generation of the price, the calculation of option value at each final node and the first value of the node are the value of the option (Bunn 2003). The model has a simple structure and these presents the following advantages to the seller or buyer of an option. The model provides a line of valuations for the derivation for each node in a given time. The American option can easily be derived which allows the practice an option at any time until the option expires unlike, in Black-Scholes model when the process of option pricing occurs when stock expires. The model is simple mathematics when compared to the Black-Scholes model, and hence very easy to put in practice using computer spread sheets. This model is quite slow but, when compared to the Black-Scholes model it is more accurate in calculating the option pricing especially for options that have taken long in the market and options that have security dividends. According to (Bakshi 2003), the binomial model mainly solves the pricing equation, using a calculable procedure that is solved analytically by Black-Scholes, and this provides opportunities for checking early exercise for American options.  The relationships between the two models are the assumptions in relation to stock prices affect both the binomial and Black-scholes models. In that the prices must follow a given process describes by geometric Brownian motion. For European options, the binomial model replaces the Black-scholes model as the number of binomial calculations steps increases. This model is important where the binomial number is infinite, and the binomial model provides discrete prices of options to the continuous process based on the Black-Scholes model (Bakshi 2003). The binomial model summarizes the option based on complex equation derived using different equations. It can be noted, the valuation of options using five factors known to the buyer or seller. The model can be said to be a complex model unlike in Black-Scholes model where the valuation are clear to understand. The model assumes the idealistic assumption of constant volatility that seems to be not true. Bunn (2003) in his research stated that, the Black-Scholes model is being view by many businessmen as the model of choice, though is being criticized if its advantages should be considered. Outside, the finance specialization the model’s usage is not understood, and some of its limitations remain unsolved. The various versions of the model allow the seller or the buyer to the assumptions that are not strict in valuing stock options. The model assumptions does affects the option prices even, though many businessmen, do like making use of the modified versions of the model that allows loosening the assumptions, and the resulting values do not differ from the original model adjusted for dividends (Burn 2003). In order to calculate binomial values, the following is the summary: Binomial Value = [p ? Option up + (1-p) ? Option down] ? exp (- r ? ?t), Or Ct- ? t,i = e -r ?t (pC t,i+1(1-p)Ct t,i -1) Where Ct, i Is the option's value for the ith node at time t, p=e(r-q) ?t-d/ u-d r and ?, - refers to the underlying stock q- Dividends If p lies between intervals of (0, 1) the following formulae has be applied on ?t ?t< ?2/ (r-q) 2 2.5. Conclusion Binomial theory model builds up a tree or a lattice. The approach is open and numerical. The model induces the price by constructing a map that can be used in future. The Black-Scholes model is a partial solution for difference equation. This model is at times known as closed or analytical approach to option pricing. Black-Scholes model mainly solves possible options under presumed condition. Therefore, binomial induces the price options where as black-scholes deduces the price options. Same theoretical foundations and assumptions form the basis of these models like the risk-neutral valuation theory and geometric Brownian motion theory of stock price behavior. The Black-Scholes model is mainly used to search for prices that are too low to buy, or too high to sell, and the option does not have a reduction risk. Options can be sold on losses in stagnant stocks. This mode does have limitations, but the model uses normal standard deviations and normal standardized, curves for future stocks. The model discounts prices with low probabilities but the lower price might turn out to be the future. The Binomial Option Pricing Model or the Binomial Model which was invented in 1979 is a varient of Black-Scholes model. The Black-Scholes model was established for pricing European style options while the Binomial Option Pricing Model was the acceptable for calculation option prices in American Style, as it allows for the possibility of early exercise in option pricing. This model was invented by Cox-Rubinstein, and he used it as a tool to explain the Black-Scholes Model to his finance students. However, it is evident that the model was more accurate than the Black-Scholes model. The review has presented a theoretical analysis of the option pricing models giving their delimitations and limitations, from the review there is an indication that more research needs to be conducted on the area of option returns especially in the European markets. Businessmen should always use the most appropriate model when calculating option prices to avoid undercharging of overcharging stock prices. 2.6. Bibliography Bakshi, G, 2003, Delta-hedged gains and the negative market volatility risk premium, The Review of Financial Studies, Vol. 16, pp. 527–566.  Broadie, J, 2000, Approximation, and a comparison of pricing Option methods. Review of Financial Studies Vol 9, 1211-1250. Bostock, P, 2004, The equity premium: Journal of Portfolio Management, vol. 30 pp 2 Camara, A, 2005, Option prices sustained by risk-preferences. The Journal of Business, Vol 78 pp 45 Coval, J, 2001, Expected option returns, Journal of Finance, Vol. 56, pp. 983–1009. Chance, D, 1999, Derivatives and Risk Management Since Black- Scholes Journal of Portfolio Management, Vol 7 pp 78-99. Jones, C, 2006, 'A nonlinear factor analysis of option Pricing, The Journal of Finance, Vol. 66, pp. 2325–2363. Tian, Y, 1999, A binomial flexible option pricing model. The Journal of Futures Markets Vol 19, 817-843. Read More