The Binomial and Black-Scholes Option Pricing Models: A Pedagogical Review with Vba Implementation F.T. Oduro 1, V.K. Dedu 2

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1 The Binomial an Black-Scholes Option Pricing Moels: A Peagogical Review with Vba Implementation F.T. Ouro 1, V.K. Deu Department o Mathematics, Kwame Nkrumah University o Science an Technology, Kumasi, Ghana 1 touro@math-knust.eu.gh 31 Abstract In this paper, a peagogical review o two option pricing moels is presente; speciically, the Binomial an the Black-Scholes pricing moels. Theoretically these moels converge or a very large number o exercise perios within a single option contract by virtue o the central limit theorem being base on the ranom walk an the Brownian motion processes respectively. This relationship is graphically illustrate by the use o an MS VBA implementation o the moels. Keywors Binomial istribution, option pricing, Black- Scholes moel, convergence, VBA. 1. Introuction In recent years, the inancial markets have improve consierably. Thus people can invest using various strategies or instruments to either reuce the risk o traing an investment an also maximize proit. Many strategies have oun their way into the inancial market allowing iniviuals, corporate institutions an governments to reuce their risk an maximize proit on investments. Derivatives instruments are mostly use in recent times. Derivative instruments are agreements (contracts or inancial instruments) that have their value etermine by the price o another asset. The basic types o erivative instruments are the Forwars, Futures, Options an Swaps. They are trae on either Exchanges or on Over-the-counter (OTC) markets [1]. Among the various erivative instruments mentione above, option traing gives the holer the right an not the obligation to buy or sell an unerlying asset at a preetermine price at a speciie uture time. Options are mostly use to either reuce the risk o loss on a particular investment (heging) or to maximize proit on an investment. Option traing gives an investor the ability to combine various strategies (in aition to International Journal o Business & Inormation Technology IJBIT, E-ISSN: Copyright ExcelingTech, Pub, UK ( the long an short positions) oere by all the types o erivatives. This unerlines the lexibility o option traing. Stock options are trae on most exchanges aroun the worl. The value o stock options is epenent on the movement o stock prices. Thus an investor will take a position epening on whether prices o stocks move up or own. Investors will only exercise their option i they have a positive payo. Stock trae options have ixe settlement terms, an a ixe time to exercise right. Everything in the contract is negotiable: the quantity, type, elivery or settlement proceure o the unerlying asset, the expiry ate, an the strike price or the contract [2]. In this paper we will concentrate on option contracts where the holer can exercise his right on a speciie expiration time. 1.2 Types o Option Contracts The two major option contracts are the call option which gives the holer the right to buy, an the put option which gives the holer the right to sell [3]. The buyer o a call option pays a premium to the seller an, in return, has the right (but is not obligate) to buy a speciic number an type o stock at a ixe price, beore or at a given ate.. The buyer o a put option pays a premium to the seller an, in return, has the right (but is not obligate) to sell a speciic amount an type o oil at a ixe price, beore or at a given ate. The ixe pre-etermine price at which the holer o the option can buy or sell the unerlying asset is usually known as the strike price or exercise price. The ate agree in the contracts is usually known as the expiration ate, exercise ate or the maturity o the option.

2 Option Traing Positions There are our main option traing positions which give the right holer certain payos. These are eine below: Long Call A traer who believes that a stock's price will increase might buy the right to purchase a call option. This is to give him the right to buy the stock at lower price in the uture but not necessarily buying the stock toay. He woul have no obligation to buy the stock. I the stock price at expiration is above the exercise price by more than the premium (price) pai, he will proit. I the stock price at expiration is lower than the exercise price, he will let the call contract expire worthless, an only lose the amount o the premium. Thus the Payo on a purchase call is given by Max (0, S T K). Thus the proit o the right holer is given by Max (0, S T K) FV (premium). This is shown in igure 1: Figure 1. Pay-o an Proit (Long Call) Long Put A traer who believes that a stock's price will ecrease can buy the right to sell the stock at a ixe price. He will be uner no obligation to sell the stock, but has the right to o so until the expiration ate. I the stock price at expiration is below the exercise price by more than the premium pai, he will proit. I the stock price at expiration is above the exercise price, he will let the put contract expire worthless an only lose the premium pai. Thus the Payo on a purchase put is given by Max (0, K - S T ). Thus the proit o the right holer is given by Max (0, K - S T ) - FV(premium). This is shown in igure 2: Short Call A traer who believes that a stock price will ecrease, can sell the stock short or instea sell, or "write," a call. The traer selling a call has an obligation to sell the stock to the call buyer at the buyer's option. I the stock price ecreases, the short call position will make a proit in the amount o the premium. I the stock price increases over the exercise price by more than the amount o the premium, the short will lose money, with the potential loss unlimite. The payo o a written call is given by -max (0, S T K). Thus the proit is given by FV (Premium) - max (0, S T K). This is illustrate in Figure 3. Figure 3. Pay-o an Proit (Short call) Short Put A traer who believes that a stock price will increase can buy the stock or instea sell a put. The traer selling a put has an obligation to buy the stock rom the put buyer at the put buyer's option. I the stock price at expiration is above the exercise price, the short put position will make a proit in the amount o the premium. I the stock price at expiration is below the exercise price by more than the amount o the premium, the traer will lose money, with the potential loss being up to the ull value o the stock. The payo or this position is given by max [0, K - S T ] an has a proit o FV(premium) - max[0, K - S T ]. This is shown in Figure 4. Figure 4. Pay-o an Proit (Short put) Option Pricing Parameters There exist six actors that aect the price o a stock option namely:

3 33 1. The current stock price, : which is the prevailing market price o stock at expiration ate. 2. The strike price, : which is the preetermine price at which the holer will exercise right 3. The time to expiration, : which is the time uration the holer has to exercise right 4. The volatility o the stock price, : which measures the uncertainty o movement in the market. 5. The risk-ree interest rate, : which is the rate o investment on stock. 6. The iviens expecte uring the lie o the option. Variation in the above parameters aects the price o the option. The impacts on price o the option as a result o changes in the parameters are iscusse below. (a) Stock Price I stock prices increase the value o a call option increases but the value or put options ecreases. Thus, i stock prices ecrease the value o a call option ecreases but the value or put options increases. This is shown in Figure 5. Figure 6. Variation in Time to Expiration Price (c) Volatility Volatility creates parallel increase between call an put options. This is shown in Figure 7. Figure 7. Variation in Volatility () Risk Free Interest Rate The risk-ree interest rate aects the price o an option in a less clear-cut way. As interest rates in the economy increase, the expecte return require by investors rom the stock tens also to increase. In aition, the present value o any uture cash low receive by the holer o the option ecreases. The combine impact o these two eects is to increase the value o call options an ecrease the value o put options. This is shown in Figure 8. Figure 5. Variation in Stock Price (b) Time to Expiration As the time to expiration changes both call option an put option experience increase in value. Nevertheless the value o a call option becomes always greater than the put option. This is shown in Figure Moels Figure 8. Variation in Interest Rate 2.2 The Black-Scholes Moel Consiering earlier iiculties in calculating option prices, Fischer Black, Myron Scholes an Robert Merton erive the Black-Scholes Moel. This moel has been the breakthrough in the option market an is wiely use toay Black-Scholes Moel Assumptions There are several assumptions unerlying the Black-Scholes moel o calculating options pricing. These assumptions are combine with the principle

4 34 that options pricing shoul provie no immeiate gain to either seller or buyer [1]. The assumptions o the Black-Scholes Moel are: 1. Stock pays no iviens 2. Options can only be exercise upon expiration 3. Market irection cannot be preicte, hence "Ranom Walk" 4. No commissions are charge in the transaction 5. Interest rates remain constant 6. Stock returns are normally istribute, thus volatility is constant over time Black-Scholes Pricing Formula In inancial terms, the price o an option is simply the present value o the uture income stream that can be expecte rom holing the option contract [4]-[6]. The Black-Scholes ormulas or the prices at time 0 o a European call option on a non-ivien paying stock an a European put option on a nonivien paying stock are u : One step payo rom option i prices move up to S 0 u. : One step payo rom option i prices move own to S 0. Delta ( ): the number o shares neee to create a riskless portolio. The ormulae or calculating the price o option using the binomial tree moel is base on the riskless portolio approach at a risk ree interest rate. The riskless portolio can be generate rom the act that there are only two securities (the stock an the stock option) with only two possible outcomes. In creating a riskless portolio or both long an short positions, (enoting the number o stocks neee to create a riskless portolio) shoul be etermine. The value o the portolio at the en o the option lie is S 0 u - or an upwar movement an S 0 - or ownwar movement. For a riskless portolio; u S 0 u - u = S 0 - ; hence the require to create the riskless portolio is Where = u S u S 0 0 = With a risk ree interest rate o r, the present value o the portolio is = ( 0 S u e U ) rt The payo o the option is given as The unction is the cumulative probability istribution unction or a stanarize normal istribution. It is the probability that a variable with a stanar normal istribution will be less than. 2.3 The Binomial Moel Given that stock prices moving up or own is enote by u or respectively, the binomial tree illustrates a one-step price movement in stock an option The notations use are explaine as ollows: S 0 : current stock price S 0 u: new price o stock as a result o upwar movement rom S 0 S 0 : new price o stock as a result o ownwar movement rom S 0. rt e [ pu (1 p) ] Where, p is the probability that stock prices go up by u or go own by. This can be calculate as p rt e u ; Given that upwar an ownwar movements (u an ) are by einite amounts. The moel can be generalize as 2r t 2 2 e [ p 2p(1 p) (1 p) ] uu or a two-step binomial Generalizing the Formula The value o a stock at each (i,j) th noe is given as; u

5 35 Int. J Busi. In. Tech. Vol-2 No. 2, September, 2012,, The value o the option is given as; Where noe is the value o the option at the (i,j) th 3. Results 3.2 Black-Scholes Moel Implementation An EXCEL VBA implementation ([7] - [8]) o the ormulae or the computation o option values using either the Black Scholes or the Binomial Moels was carrie out. The results are provie below in terms o Input an Output Forms in a GUI context o VB. Thus in the computation o the value o an option using the Black-Scholes moel the require parameter values namely, the Strike Price, Stock Price, Interest Rate, Volatility as well as the type o option being consiere (Put or Call) must be entere as ata through an input winow as shown in Figure 9. Figure 10 Output Form or Black Scholes Moel The program was esigne only to calculate the value o non-ivien paying European options an the ormulae use in this section o the program inclue the ollowing: 3.3 Binomial Moel Implementation Again, in the computation o the value o an option using the Binomial Moel the require parameter values namely, the Strike Price, Stock Price, Interest Rate, Volatility, Number o Perios as well as the type o option being consiere (Put or Call an whether American or European style) must be entere as ata through an input winow as shown in Figure 11. Figure 9. Input Form or Black Scholes Moel The corresponing output orm in respect o the Black Scholes Moel then isplays the option value as shown in Figure 10. Figure 11 Input Form or Binomial Moel Clicking the OK button returns the value o the option in an output orm shown Figure 12.

6 4. Discussion 4.2 Convergence o the Binomial to the Black-Scholes Moel As the number o exercise perio between option contacts increases the binomial moel converges to the Black Scholes moel. It is base on the stochastic principle o ranom walk an Brownian motion. Though theoretically proven, using excel an VBA give more unerstaning to this poo. The iagram in Figure 14 isplays the convergence o the Binomial to the Black-Scholes base on the inputs mae in the program. 36 Figure 12 Output Form or Binomial Moel Its associate binomial tree showing the expecte stock prices an option values at the ierent noes is also isplaye ater the clicking the Display Tree button. Also or American options the program shows when exercising the option is proitable by returning the value o the option in re at that noe. The generalize ormulae or the moel state above were use or eveloping the ollowing program ormulas are employe: In the example given, the speciie number o perios was ive (5). The program generate the option price at each noe an the stock price at each noe. At each noe the price o the stock at that noe is above the price o the option. This is shown in Figure 13. Figure 13 Binomial Tree Output The tree represents an American put. In the tree, the price o option which appears re inicates that the holer will beneit i he exercise option right. The tree can also be isplaye or American call, European call an European put. Figure 14 Convergence o Binomial to Black Scholes Moel 5. Conclusion In conclusion, the program can be use to value European call an put options, American call an put options using either the Binomial or Black-Scholes moels. The application works well in proucing the value o a non- ivien paying stock or the two types o options. The application also is very simple an easy to use. The graphical proo o convergence gives more unerstaning to the theoretical proo o convergence. Developing application such as this make it easier or everyone whether stuent o proessional to use since it reuces time spent an errors attenant to a manual calculation. The graphical output in relation to the binomial tree inicates optimal points where exercise o option right becomes proitable. Reerences [1]. Hull, C. J., Options, utures an Other Derivatives, 7 th eition. Pearson Prentice Hall, NJ, (2009) [2]. Long, D., Oil Traing Manual, Supplement 3, Woohea Publishing Lt., Cambrige (2000). [3]. McDonal, L.R., Derivatives Markets, secon eition. Pearson Eucation, Inc.,NY,(2006)

7 [4]. Benninga, S.,.Financial Moelling, secon eition. The MIT Press, MA (2000) [5]. Cherry, H., Financial Economics, irst eition, Merrick, NY. (2007) [6]. Myers, S. C. an Brealey A. R., Principles o Corporate Finance 7 th eition McGraw Hill/Irwin, IL (2003) [7]. Birnbaum, D.,Microsot Excel VBA Programmin or the Absolute Beginner. Premier Press Inc, Oregon (2002) [8]. Green J.,Bullen S., Bovey R. an Alexaner M., Excel 2007 VBA. Wiley Publishing, Inc., NJ (2007) 37