1 Int. J. Pure Appl. Sci. Technol., 18(1) (213), pp International Journal o Pure and Applied Sciences and Technology ISS Available online at Research Paper Eect o Volatility on Binomial Model or the Valuation o American Options S.E. Fadugba 1, *, A.O. Aayi 2 and.r. wozo 3 1 Department o Mathematical Sciences, Ekiti State University, Ado Ekiti, igeria 2 Department o Mathematical Sciences, Ekiti State University, Ado Ekiti, igeria 3 Department o Mathematics, University o Ibadan, Oyo State, igeria * orresponding author, (Received: ; Accepted: ) Abstract: This paper presents the eect o volatility on binomial model or the valuation o American options. Volatility is a measure o the dispersion o an asset price about its mean level over a ixed time interval. Binomial model is a widespread numerical method o calculating the price o American options as it takes into consideration the possibilities o early exercise and other actors like dividends. The strength and weakness o volatility on the valuation o options were considered. Binomial model is computationally eicient, good or pricing American options but not adequate to price path dependent options. Keywords: American Option, Binomial Model, Black Scholes Model, European Option, Volatility. 1. Introduction Many inancial problems induce a deep study in mathematics, raising mathematical problems o the relevance. These procedures involve analysis, probability and stochastic processes, optimization and control theory and numerical analysis. The past and present inancial crisis points out that the classical models that have been traditionally used do not produce an accurate derivative pricing. The need or investigating new inancial models based on new analytical and more sophisticated instruments became evident. A vanilla option oers the right to buy or sell an underlying security by a certain date at a set strike price. In comparison to other option structures, vanilla options are not ancy or complicated. Such
2 Int. J. Pure Appl. Sci. Technol., 18(1) (213), options may be well-known in the markets and easy to trade. Increasingly, however, the term vanilla option is a relative measure o complexity, especially when investors are considering various options and structures. Examples o vanilla options are; an American option which allows exercise at any point during the lie o the option. This added lexibility over European options results in American options having a value o at least equal to that o an identical European option, although in many cases the values are very similar as the optimal exercise date is oten the expiry date. The early exercise eature o these options complicates the valuation process as the standard Black-Scholes continuous time model cannot be used. The most common mode or pricing American Options is the binomial model. A European option allows exercise to occur only at expiration or maturity date . Black and Scholes published their seminar work on option pricing  in which they described a mathematical rame work or inding the air price o a European option. They used a no-arbitrage argument to describe a partial dierential equation which governs the evolution o the price o options with respect to the maturity time and price o the underlying asset. The subect o numerical methods in the area o pricing options and hedging is very broad, putting more demands on computation speed and eiciency. A wide range o dierent types o contracts are available and in many cases there are several candidate models or the stochastic evolution o the underlying state variables . ow, we present an overview o binomial model in the context o Black-Scholes-Merton [1, 1] or pricing vanilla options based on a risk-neutral valuation which was irst suggested and derived by  and assumes that stock price movements are composed o a large number o small binomial movements . Other procedures are inite dierence methods or pricing derivatives governed by solving the underlying partial dierential equations was considered by  and Monte arlo method or pricing European option and path dependent options was introduced by . The comparative study o inite dierence method and Monte arlo method or pricing European option was considered by . Later, on the stability and accuracy o inite dierence method or option pricing was considered by .  onsidered Monte arlo method or pricing some path dependent options. Some numerical methods or options valuation was considered by . These procedures provide much o the inrastructures in which many contributions to the ield over the past three decades have been centered. In this paper we shall consider only the eect o volatility on binomial model or the valuation o American options. 2. Binomial Model Binomial model is one o the numerical methods or the valuation o options. This iterative model uses a discrete time model o the underlying inancial instrument in general, or some vanilla options such as American option; binomial model is the only choice since there is no known closed orm solution that predicts its price over a period o time. The ox-ross-rubinstein Binomial model  contains the Black-Scholes analytic ormula as the limiting case, as the number o steps tends to ininity. ext we shall present the derivation and the numerical implementation o binomial model or the valuation o American options below. 2.1 The ox-ross-rubinstein Binomial Model [4, 9, 12] We know that ater a period o time, the stock price can move up to S u with probability p or down to S d with probability ( 1 p ), where u > 1 and < d < 1as shown in the igure below
3 Int. J. Pure Appl. Sci. Technol., 18(1) (213), Figure 1: Stock in a General One Step Tree Thereore the corresponding value o the call option at the irst time movement δt is given by  u d = max( S u K,) (1) = max( S d K,) (2) Where respectively. u and d are the values o the call option ater upward and downward movements We need to derive a ormula to calculate the air price o vanilla options. The risk neutral call option price at the present time is given by r t δ [ pu d ] (3) The risk neutral probability ( p ) is given by rδt e d p = u d σ δt Where σ, u and respectively. d σ δt are called volatility, upward and downward movements (4) ow, we extend the binomial model to two periods. Let uu denote the call value at time 2δt or two consecutive upward stock movements, ud or one downward and one upward movement and dd or two consecutive downward movements o the stock price  as shown in the igure below. Figure 2: Stock and Option Prices in a General Two Steps Tree
4 Int. J. Pure Appl. Sci. Technol., 18(1) (213), Then we have, uu ud dd = max( S uu K,) (5) = max( S ud K,) (6) = max( S dd K,) (7) The values o the call option at time δt are u d r t δ r t δ [ p uu ud ] [ p ud dd ] (8) (9) Substituting (8) and (9) into (3), we have rδt rδt rδt [ pe ( puu ud ) e ( pud dd )] 2r t δ [ p 2 uu + 2 p(1 p) ud 2 dd ] (1) 2 (1) is called the current call value, where the numbers p, 2 p(1 p) and neutral probabilities or the underlying asset prices Suu, Sud and Sdd respectively. ( 1 p) 2 are the risk We shall generalize the result in (1) to value an option at T = δt as ollows rδt = p (1 p) u d Thereore, rδt = p (1 p) max( S u d K,) (11)! Where = max( S u d K,) and u d = is the binomial coeicient. We ( )!! assume that m is the smallest integer or which the option s intrinsic value in (11) is greater than zero. m m This implies that S u d K. Then (11) can be written as = S e rδt rδt p (1 p) u d Ke = m = m p (1 p) (12) rδt (12) gives us the present value o the call option. The term e is the discounting actor that reduces to its present value. We can see rom the irst term o (12) that p (1 p) u d is the binomial probability o upward movements to occur ater the irst trading periods and Su d is the corresponding value o the asset ater upward movements o the stock price. The rδt second term o (12) is the present value o the option s strike price. LetQ, we substitute Q in the irst term o (12) to yield
5 Int. J. Pure Appl. Sci. Technol., 18(1) (213), = S Q = m p (1 p) u d Ke rδt = m p (1 p) = S 1 1 rδt [ Q pu] [ Q (1 p) d] Ke = m = m p (1 p) (13) ow, let Φ( m;, p) denotes the binomial distribution unction given by Φ ( m ;, p) = p (1 p) (14) = m (14) is the probability o at least m success in independent trials, each resulting in a success with 1 probability p and in a ailure with probability ( 1 p). Then let p = Q pu 1 and (1 p ) = Q (1 p) d. onsequently, it ollows that = S Φ( m;, p ) Ke rt Φ( m;, p) (15) T The model in (15) was developed by ox-ross-rubinstein , whereδ t = and we will reer to it as RR model. The corresponding put value o the European option can be obtained using call-put rt relationship o the orm E + Ke = PE + S as = Ke Φ( m;, p) S Φ( m;, p ) rt (16) Where the risk ree interest rate is denoted by r, E is the European call, PE is the European put and S is the initial stock price. European option can only be exercised at expiration, while or an American option, we check at each node to see whether early exercise is advisable to holding the option or a urther time periodδ t. When early exercise is taken into considerations, the air price must be compared with the option s intrinsic value . 2.2 umerical Implementation o Binomial Model or the Valuation o American Options ow, we present the implementation o binomial model or pricing vanilla options as ollows. When stock price movements are governed by a multi-step binomial tree, we can treat each binomial step separately. The multi-step binomial tree can be used or the American and European style options. Like the Black-Scholes, the RR ormula in (15) can only be used in the valuation o European style options and can easily be implemented in Matlab . To overcome this problem, we use a dierent multi-period binomial model or the American style options on both the dividend and non-dividend paying stocks. The stock price o the underlying asset or non-dividend and dividend paying stocks are given respectively by; i S u d, i, i (17)
6 Int. J. Pure Appl. Sci. Technol., 18(1) (213), i S (1 λ ) u d, i, i (18) Where the dividend paying stock is denoted by λ which reduces the price o the underlying asset. Suppose that the lie o American options on a non-dividend yield is divided into subintervals o lengthδ t. Deine the th node at the time iδt as the ( i, ) node, where i, i. Deine i, as the value o the option at the ( i, ) node. The stock price at the ( i, ) node is S u d i. Then, the respective American call and put options can be expressed as i ( S u d,) i ( K S u d,) i, = max K P i, = max (2) There is a probability o moving rom the ( i, ) node at the time iδt to the ( i + 1, + 1) node at time ( i + 1) δt and a probability o moving rom the ( i, ) node at the time iδt to the ( i + 1, ) node at time ( i + 1) δt, and then the neutral valuation is ( p + 1 p ) * rδt i, ( i+ 1, + 1) ( ) ( i+ 1, + 1) (19), i 1, i. (21) For an American put option, we check at each node to see whether early exercise is preerable to holding the option or a urther time periodδ t. When early exercise is taken into considerations, this value o must be compared with the option s intrinsic value or and we have that i, * = ( P, ) (22) i, i, i, Substituting (2) and (21) into (22), then we have i, = + i rδ t max [ K Su d, e ( p( i+ 1, + 1) ) ( i+ 1, 1) ] (23) (23) gives two possibilities as ollows; * i. I P >, then early exercise is advisable. ii. i, i, * I P <, then early exercise is not advisable. i, i, 2.3 Facts about the Binomial Tree onstruction i. There is a standard and constant relation between u and d u * d = 1 (24) ii. u and d do not depend on the expected return o the stock price or the subective probabilities. iii. Volatility and interest rate are constant during the lie o the option. 2.4 Volatility Volatility reers to the amount o uncertainty or risk about the size o changes in security's value. This is also the key to understanding why option prices luctuate and act the way they do. Inact, volatility is the most important concept in the valuation o options and it is denoted byσ.
7 Int. J. Pure Appl. Sci. Technol., 18(1) (213), Volatility is one o the actors aecting option pricing. The greater the expected movement in the price o the underlying instruments due to high volatility, the greater the probability that the option can be exercised or a proit and hence the more valuable the option is. A higher volatility means that a security's value can potentially be spread out over a larger range o values. This means that the price o security can change dramatically over a short time period in either direction. A lower volatility means that a security's value does not luctuate dramatically but changes in value at a steady pace over a period o time. For options, volatility is good' because the price o volatility creates greater value or a given option, or the greater the volatility o the underlying, the greater the value o the option while or other inancial assets, volatility is `bad'. This is due to the act that the purchases o options enoy only the upside potential not downside risk. Other inancial assets have both risks. The eect o volatility on binomial model is that the higher the volatility, the higher the value o binomial model or the valuation o American call and put options. Vice versa. There are two types o volatility: (a) Implied Volatility This measures the price movement o the option itsel. It is also an estimated volatility o a security s price in real time, or as the option trades. Values or implied volatility come rom ormulas that measure the options market s expectations, oering a prediction o the volatility o the underlying asset over the lie o the option. (b) Historical Volatility Historical volatility is also called statistical volatility that measures the rate o movement in the price o the underlying asset. This is a measurement o how ast the prices o the underlying asset have been changing over time. Because historical volatility is always changing, it has to be calculated on a daily basis. It is stated as the percentage and summarizes the recent movements in price. In general, the higher the historical volatility, the more an option is worth. 3. umerical Experiments and Results This section presents numerical experiments, Table o results and discussion o results as ollows: 3.1 umerical Experiments Example 1: onsider an American option that expires in 6 months with an exercise (strike) price o$22. Assume that the underlying stock pays no dividend yield, trades at $2and the risk-ree rate is 1% per annum. The volatility varies rom 25% to 5%per annum. We compute the values o American options using binomial model as we increase the number o step and varies the value o volatility with the ollowing parameters below. S = 2, K = 22, T =.5, r =.1, σ =.25, σ = The Black Scholes model or call and put option price denoted by given below BScand BS p respectively are
8 Int. J. Pure Appl. Sci. Technol., 18(1) (213), i. Forσ 1 =.25, BS c = and BS p = ii. Forσ 2 =.5, BS c = and BS p = The results obtained are shown in the Table 1 below. Example 2: Let us consider the pricing o American put option with the ollowing parameters: S = 5, K = 5, T =.42, r =.1, σ =.4, δ t =.83 Thereore, u e e σ δt.2(.83) = = = d e e σ δt.2(.83) = = = 1.122,.891 The risk neutral probability o the upward movement is given by r t.1(.83) e δ d e.891 p = = =.58 u d and q = 1 p = 1.58 =.942 The results generated using binomial models are shown in the Figure 3 below: Figure 3: The Stock Price and the American Put Option prices are given in the graph o the Binomial Tree
9 Int. J. Pure Appl. Sci. Technol., 18(1) (213), Table o Results The results obtained using Matlab codes are shown below Table 1: Eect o Volatility on Binomial Model or the Valuation o American Options σ 1 =.25 σ 2 =.5 American call American put American call American put Discussion o Results We can see rom Table 1 above that the greater the expected movement in the price o the underlying instruments due to high volatility, the greater the probability that the option can be exercised or a proit and hence the more valuable is the option. We also see that the higher the volatility, the higher the value o binomial model or the valuation o American call and put options. The stock Price and the American put option prices are given in the graph o the Binomial Tree are shown in Figure 3. The rate o convergence or binomial model may be assessed by repeatedly doubling the number o time step. This method is very lexible, easy to implement and good or the valuation o American options.
10 Int. J. Pure Appl. Sci. Technol., 18(1) (213), onclusion Since volatility o the underlying asset price is a critical actor aecting option prices, the modeling o volatility and its dynamics is o vital interest to traders, investors, and risk managers. This modeling is a diicult task because the path o volatility during the lie o an option is highly unpredictable. learly, the Black- Scholes assumption o constant volatility can be improved upon by incorporating time variation in volatility. In a recent empirical test o deterministic-volatility models, including binomial tree approaches, show that the Black-Scholes model does a better ob o predicting uture option prices. The option delta, which is derived rom an option pricing model and measures the sensitivity o the option price to changes in the underlying asset price, can be used to speciy positions in options that oset underlying asset price movements in a portolio. Binomial option pricing model approach is widely used as it is able to handle a variety o conditions in which other models cannot easily be applied. This is largely because the model is based on the description o underlying instruments over a period o time rather than a single point. In general, binomial model has its strengths and weaknesses o use. This model is good or the valuation o American options that are exercisable at any time in a given interval, accurate particularly or longerdated options on securities with dividend payment, converges aster, widely used by practitioners in the inancial market and it is relatively easy to implement but path dependent option remains problematic and can be quite hard to adapt to more complex situations. Reerences  F. Black and M. Scholes, The pricing o options and corporate liabilities, Journal o Political Economy, 81(1973),  P. Boyle, Options: A Monte arlo approach, Journal o Financial Economics, 4(1977),  M. Brennan and E. Schwartz, Finite dierence methods and ump processes arising in the pricing o contingent claims, Journal o Financial and Quantitative Analysis, 5(1978),  J. ox, S. Ross and M. Rubinstein, Option pricing: A simpliied approach, Journal o Financial Economics, 7(1979),  S.E. Fadugba, F.H. Adeolau and O.H. Akindutire, On the stability and accuracy o inite dierence method or option pricing, International Institute o Science, Technology and Education, Mathematical Theory and Modeling, 2(212),  S. Fadugba,. wozo and T. Babalola, The comparative study o inite dierence method and Monte arlo method or pricing European option, International Institute o Science, Technology and Education, Mathematical Theory and Modeling, 2(212),  A. Etheridge, Financial alculus, ambridge University Press, ambridge, 22.  B. Hahn, Essential Matlab or Scientists and Engineers (Third Edition), Pearson Education Publishers, South Arica, 22.  J. Hull, Options, Futures and other Derivatives (Fith Edition), Pearson Education Inc., Prentice Hall, ew Jersey, 23.  R.. Merton, Theory o rational option pricing, Bell Journal o Economics and Management Science, 4(1973), .R. wozo and S.E. Fadugba, Monte arlo method or pricing some path dependent options, International Journal o Applied Mathematics, 25(212), .R. wozo and S.E. Fadugba, Some numerical methods or options valuation, JAMB, ommunications in Mathematical Finance, Scienpress Ltd, 1(212),  S.M. Ross, An Introduction to the Mathematical Finance: Options and other Topics, ambridge University Press, ambridge, 1999.
11 Int. J. Pure Appl. Sci. Technol., 18(1) (213),  J. Weston, T. opeland and K. Shastri, Financial Theory and orporate Policy, (Fourth Edition), ew York, Pearson Addison Wesley, 25.  P. Wilmott, S. Howison and J. Denwynne, The Mathematics o Financial Derivatives: A Student Introduction, ambridge University Press, ambridge, 1995.