The Binomial Option Pricing - Coursework Example

Binomial option pricing Many complex options pricing problems can find solution through use of binomial option pricing which can be described as being simple but a very powerful technique. When compared to other complex option pricing models like Black-Scholes model that involve the solving of stochastic differential equation , the binomial option is found to be very simple mathematically and is there is an assumption of no arbitrage when in use (Heston 2000) . The implication of lack of arbitration is that all investments that are risk free will earn returns that are risk free and that there is no any investment which requires zero dollars investment with an expectation of yielding positive returns. The activity of many individuals that are in operation of the context of financial markets contributes in upholding these conditions. Barone-Adesi (1997) notes that, there is always malignment of speculators or arbitrators by the media although their activity works in ensuring that financial markets work. This is through insuring financial assets like options pricing is within a narrow tolerance of the theoretical values (Leisen 1996; Jarrow, 1983). Assuming that a share of stock has a current price of $100/share. In the following month, there will be an up state to a price of $110 or a down stage to $90 with no possibility of other outcome in next month. If it is assumed that there is a call option of the stock with the call option having a strike price $100 that will mature at the end of the month. The call option value at the end of the month will be $10 incase of a stock price of $110 and 0 for a stock price of $90 with a payoff at maturity being : The question that requires an answer is the price of the call option today. To help in giving the answer we consider what will occur when the following investments is made in the stock and call option. It is assumes a one-half share stock is bought at $50 (0.5x$100), simultaneously a one call option having a strike price of $100 maturing at the end of the month. From this it is seen that the investment is $50 lower than the current price of the call option. The end of month payoff from this position will present in the following manner. When the stock price is $110, the worth of the stock position is $55 and there will be a lose of $10 incurred on the option. This mans that there would be a return of $45 incase the stock reached a price of $110. Alternatively suppose the stock was to go down to $90 it would result in the stock position being $45 with the value of the option position being 0 with the payoff being $45. When this particular option is taken on the stock that has this payoff structure the net effect is that there is a payoff of $45 which does not depend on stock price at the end of the month. On buying a half unit of the stock and writing a call it ensured that a risky position changed to a risk free whose payoff is $45 with no regard to price of stock at the end of the month. In the case of no arbitrage, the investor that makes this investment will be able to the risk-free rate of return. This simply means that a $50 investment minus the call option price will have to be equal to a present value of $45 which is a payoff that has been d9scounted foe a month at current risk-free rate of return. The process of finding the current price of the call option involve solving the following equation. $50 – Option price = $45  Option price = 50 - $45  In the equation RF represents risk-free rate while T is time to maturity given in years (Heston, 2000). If it is assumed that the current return rate is 6% per annum with a time to maturity being one month, T= 0.08333 using the formula results in current option price being $5.22. The process that has been engaged in pricing the option the example involve the same procedure that is used both the simple binomial option model or the Black-Scholes model which is more complicated. The underlying assumption is that a risk-free hedge is found followed by the price off of the risk-free hedge (Nelson, 1990; Cox, 1979). It is assumed that the risk free hedge pricing will be in such a way that it will be able to earn an equivalent of the risk-free rate return at which point arbitrageurs come into play. The activities of the individuals searching for opportunities to invest in a risk less asset with the intention of earning more than the risk free rate of return insures options pricing is according to no arbitrage conditions. The two approaches which are used in binomial model are the Risk-les Hedge approach and the Risk-Neutral Approach with both of the approaches yielding the same answer (He, 1990; Hsia, 1983 ). Estimation the risk less hedge Since in the example below the hedge ratio has been given, the next step is a demonstration of how to find the hedge ratio that is appropriate for a stock. If the following payoff structure over the next month is assumed. In this case, the current stock price is $75, and at the end of the month, the stock will be either $95 or $63. Suppose we also had a call option with a strike price of $65. The payoff for this call option at the end of the month is it can be observed from the figure that the current stock price is $75 with the stock price being either $95 or $65 at the end of the month. In case there was also a call option that had a strike price of $65, its payoff at the end of the month would be as illustrated: If H represents the number of units of stock being held, then the investment would be HX$75 which is the call option price. Suppose at the end of the month the stock price was be $95 then the value of the call option would have been $30 with the payoff on total investment being HX$95-$30. On the other hand if the stock price was $63 at the end of the month, the call option value would be $0 while payoff resulting from the total position on the stock and the call option being HX$63-$0 The appropriate risk-free hedge H can be found ensuring that setting of payoffs in both up state and down state are equal. We find the appropriate risk-free hedge, H, by setting the payoff in the up state equal to the payoff in the down state, H x $95 - $30 = H x $63 - 0. H = = 0.9375 If we solve for H1 it will be found that payoffs will be equal in each state, incase H has a value of .9375. For up state payoff is .9375x$95-30 = $59.0625 while for down state payoff is .9375x$63-0 = $59.0625. This is an illustration that if .9375 shares is bought and writing a call option a riskless hedge would have been created whose payoff is $59.0625 regardless of the end month stock price. If it is assumed that there will be no arbitrage opportunities the .9375x$75-C investment (C being the call price) will be equal to the present value payoff of $59.0625 which receives a discount at risk-free rate of return. Suppose 6% per annum risk-free return is assumed with a time to maturity of one month, T=.08333 and the call price will found to be $11.54 as shown in the following calculations. .9375 x $75 - C = $59.0625 x e -RF ×T $70.3125 - C = $59.0625 x e -.06×.08333 C = $70.3125 - $59.0625 x e -.06×.08333 C= $ 11.54 Binomial option-pricing model for more than one period This can well be illustrated by having two price changes in a month instead of one (Johnson 1979). It can be seen that when the month is divided in 2 periods the end result 3 possible outcome can be realized at the end of the month. We would like to establish what the price of an option today that has a strike price of $100 if it matures at the end of the month. The solution for the problem is breaking it up into two period models and then work backwards. If the branch where the stock reaches $104.88 is put into consideration, in 2-week period the stock will increase to $110 or reduce to $100. The question to ask at this point is that if it is assumed that the stock price is $104.88 what will the option be worth within two weeks? Lets it be assumed that H units of stock is bought when the remaining time is two weeks and then write one call option. The risk-free hedge is the number of units which need to be bought and then the payoff structure is determined as follows. H should be chosen so that HX$110-10 = Hx$100-0. From the equation the value of H is 1.0 and when H =1.0 the payoff will always be $100. This implies that for a stock price of 104.88 and a unit of the stock being bought and then writing one call option, there would be a payoff of $100 for any of that will be recorded at the end of the 2-week period. If a risk-free rate of 6% per annum is assumed with a time to maturity of 2 weeks (T=0.04167) then the call option value for a stock price $104.88 will be $5.13 1.000 x $104.88 - C = $100.000 x e -RF ×T $104.88 - C = $100.00 x e -.06×.04167 C = $104.88 - $100.00 x e -.06×.04167 C= $ 5.13 Suppose at the end of the first 2-week the stock price value is $94.86, and then in the following 2-week period the stock price will rise to $100 or will fall to $90. In both cases the call options are worthless at the end of the period as no one can choose to exercise a call option that has a strike price of $100. If the stock price is 94.86 at the end of first two weeks then the option price would be $0. Moving back to first 2-week period the payoff structure as a result of buying the stock and then writing H call options will be as follows. The value of the H is found to be 0.51198 as shown in this calculation H x $104.88 - $ 5.13 = H x $94.86 - $ 0. H = .51198 In this case if 0.51198 units of stock is bought and one call option written there will be a payoff of $48.566 not withstanding whatever happens to the stock price. The call options value need to satisfy the following equation: .51198 × $100 -Option price=$48.566 × e- RF ×T Option price solution would be Option price = $51.198 - $48.56 × e-.06×.04167 = $2.753 Risk Neutral Approach The risk neutral approach is an alternative that can be used instead of risk less hedge approach when valuation of options is being done by binomial model (Cutland, 1993). The risk neutral approach has its base on the argument that due to the fact the valuation of options has its base on arbitrage making it independent of risk preferences, it is possible to value options when a set of risk preferences is assumed and this will result is the same answer (Tian,1999 Heston, 2000). This therefore makes the risk neutral model to be the easiest. While in the risk less hedge approach the probabilities of price increasing (Pu) or decreasing (Pd =1-Pu) does not feature in the analysis, in the neutral approach for a particular stock price process an attempt is made to get an approximation of the probabilities for the risk neutral individual and then using the risk neutral probabilities a call option is priced. To illustrate this price process in the risk less hedge example is used. For individuals who are risk neutral there is indifference to risk and for them there is an expectation of payoff that is discounted at the riskfree rate interest for the current stock. If the riskfree rate is 6% the following assessment would be made by a risk neutral individual. $75 =  Incase RF is 6% and T is .08333, then Pu will be .38675 which is the risk neutral probability for the stock to increase to $95the end of the month. There is a (1-.38675) = 0.61325 of the stock price reducing to $63. If the stock price go up to $95 then a call option having an exercise price of $65 has a payoff of $30 and $0 incase the stock price will rise to $63. Thus a risk neutral individual would have 0.38657 probability of getting $30 while the probability of receiving $0 by owning the call option would be 0.61325. The risk neutral value will end up as : Call Option Value = [Pu .$30 + (1- Pu ).$0 ]. e-.06×.08333 Call Option Value = [.38657 × $30] × e-.06×.08333 Call Option Value = $11.54 This happens to be the same value that is obtained when risk less hedge approach is used. Black-Scholes call option pricing The Black-Scholes formula is derived as the limit of the limit of the binomial pricing formula when there is shrinkage in time between trades or can be done directly by the continuous time model by use of an arbitrage argument (Amin, K1991). The option value has the stock price and time as its function and the computation of local movement in the stock price is through an extension of chain rule called It^o's lemma. It is not possible to use the standard version of chain due to the fact that it is not possible to differentiate the stock price in the lognormal model. Even in situation where a function representing a stock has a zero derivative at some point, the expected rate of increase may turn to be positive owing to the volatility in local price movement (Broadie, 1996). When using in the calculation of local change in the option value in term of derivatives of a function of both stock price and time lack of arbitrage has implication of restricting the derivatives in the function that is similar to the per period hedge seen in the binomial model. When there is no arbitrage it implies a differential equation solution with boundary condition being the known option value at the end. The Greeks To find their option positions it is a common practice for option traders to make references to delta, gamma, vega and theta which have a collective term of Greeks (Ho, 1997). The Greeks provide a way in which measurement to sensitivity in option’s prices when some factors are put into consideration. To new option traders the terms often tend to be confusing but when broken down the Greeks are realized to be simple concepts which bring a better understanding or risk and potential reward in an option position. The number given for the Greeks are usually theoretical as they are projected on a basis of mathematical models. Delta gives the sensitivity of an option’s theoretical value when the price of underlying asset is put under consideration. This is often represented by a number ranging from -1 to +1 and this act as an indicator of the anticipated change in the value of an option when there is a rise of a dollar in the price of underlying stock (Heston, 2000). Gamma on the other hand gives a measure of the rate at which delta changes when thee is a one-point increase in underlying asset and this makes it a valuable tool used in forecasting of changes in the delta of an option. The time decay for an option is measured by theta which can simply be stated as the amount in dollars lost by an option daily as a result of passage of time. For the case of at-the-money option there will be increase in theta when the option will be approaching the date of expiration (Duffie, 1992).There will be a decrease in theta for the case of in-and out-of-the-money option as expiration is approached. The vega gives a measure on the sensitivity of price of an option when there is a change in volatility. REFERENCES Amin, K. I. A(1991) On the Computations of Continuous Time Option Prices Using Discrete Approxmations.@ Journal of Financial and Quantitative Analysis 26, 477-496. Barone-Adesi, G., E. 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(1999), “A Flexible Binomial Option Pricing Model,” Journal of Futures Markets, 817-843. Read More