portfoli ment Analysis. and Po! inte of Rs 60 and if the es : ava 448 - the call option has @ i i . ce rise, ‘ a Je stock price Mya of RS 0- hoe ah expected future value ¢ i jon has 4 h We know that Oe ET asa eal call opti stock price falls the call ON gra, the cal Mr Rs 0) ence, investors are 9 Probability of . Rs 24 esent value of th (Probability of TS ae : = 0.40 x Rs 60 40.60 x RS inthe pr ie of the call option os Rs 24 peg 21.82 e current V = Expected future vate en (0. Risk-free rate : sed answer we got by using the oF ii i ial world. he value of an option in the binomial wor and loan that imitates the eee | payoffs in the future, they mus! e expected future value: tion equivalent Not surprisingly, this is exactly the method fer Thus, we have two ways of calculating folio of shares talent Method Find a port f share atives have identical Option Equiv in its payoff. Since the two altern command the same price today. i ected return .stors are risk-neutral, so that the exp* sta .d future value of the Risk Neutral Method Assume that im i asthe stock is the same as the interest rate. Calculate the expect option and discount it at the risk-free interest rate. iz *|14.6 BLACK AND SCHOLES MODEL The above analysis was based on the assumption that there for the stock price at the end of one year. If we assume that there are two possible stock prices at the end of each 6-month period, the number of possible end-of-year prices increases. As the period is further shortened (from 6-months to 3-months or 1 month), we get more frequent changes in stock price and a wider range of possible end-of-year prices. Eventually, we would reach a situation where prices change more or less continuously, leading to a continuum of possible prices at the end of the year. Theoretically, even for this situation we could set up a portfolio which has a payoff identical to that of a call option. However, the composition of this portfolio will have to be changed continuously as the year progresses. _ Calculating the value of such a portfolio and through that the value of the call option in such a situation appears to be an unwieldy task, but Black and Scholes developed a formula that does precisely that. Their formula is: C= Sy NUdy)- FN (a) were two possible values (14.13) Cy = equilibrium value of a call option now So = price of the stock now where © scanned with OKEN Scannerb Gluk : [{ I i Io i ; dyrbons 449 k 7 ' cise price = base of natural logarithm r= annualised continuously compounded risk-free interest rate {= length of time in years to the expiration date tion N(@) = value of the cumulative normal density fur h (Belz) (14.14) ot (14.15) where 1, = natural logarithm = standard deviation of the annualised continuously compounded of return on the stock Though one of the most complicated formulae in finance, it is one of the most practical. The formula has great appeal because four of the parameters, namely, Sp E,T, and t are observable. Only one of the parameters, namely 7, has to be estimated. Note that the value of a call option is affected by neither the risk aversion of the investor nor the expected return on the stock. - ' } | © Assumptions ‘You may have guessed by now that the Black and Scholes model, like other important models in economics and finance, is based on a set of simplifying assumption. Yes, you are right. The assumptions underlying the Black and Scholes model are as follows: ® The call option is the European option = The stock price is continuous and is distributed lognormally ® There are no transaction costs and taxes « There are no restrictions on or penalties for short selling © The stock pays no dividend = The risk-free interest rate is known and constant. These assumptions may appear very severe. However when some of them do not hold, a variant of the Black and Scholes model applies. Further, empirical studies indicate that the Black and Scholes model applies to American options as well. Applying the Black-Scholes Formula Though the Black-Scholes formula appears difficult it is fairly easy to apply. This may be illustrated with an example. © scanned with OKEN Scanner- aN 450 Investment Analysis eal ain stock Consic following data for a cet Price of stock now = Sy = Rs 60. © Exercise price E= Rs 56 turns = 0203 Standard deviation of continuously compounded annual ret © Years to maturity & Interest rate pe Applying the Blac nnum = 0. choles formula involves four steps: Step 1: Calculate d; and dy 068993 + 0.0925 _ 0.161493. _ gy ne14 0.2121 0.2121 dy =d,-o Vt 7614 ~ 0.2121 = 0.5493 Step 2: Find N(d,) and N(d,). N(d,) and N(d;) represent the probabilities that a random variable that has a standardised normal distribution will assume values less than d, and d,. The simplest way to find N(d,) and N(d;) is to use the Excel function NORMSDIST. Alternatively, you can use Table A.5 in Appendix A at the end of the book N(@d,) = N (0.7614) = 0.7768 N(&) = N (0.5493) = 0.7086 Step 3: Estimate the present value of the exercise price, using continuous discounting principle Step 4: Plug the numbers obtained in the previous steps in the Black-Scholes formula Co = Rs 60 x 0.7768 - Rs 52.21 x 0.7086 "= Rs 46.61 - Rs 37.00 = Rs 9.61 46) — Go+ 32) 61-92 Ge e1s%2 Note that the principle of replicating portfolio used in the binomial model also undergirds the Black-Scholes model. Exhibit 14.14 shows the replicating portfolios for calls and puts, in the binomial and the Black-Scholes models. Replicating Portfolio f, : © scanned with OKEN Scanner\Y yement Investment Analysis: and Portfolio Managemen mnvesiment Analysis and Po! 452 ce (14.18) C= Se MN (dy) = BE" N Oh) A 5 ° els yt a where hey adic the stock to This adjustment essentially does two things: () It discounts the oe oe the present at the dividend yield to reflect the expected drop in 197000 race the dividend payments . (ii) It offsets the interest rate by the dividene y lower cost of carrying the stock (in the replicating portfolio). * Revisiting the Put-Call Parity We learnt earlier that just before the expiration the following holds: C,=S)+P,-E (14.3) If there is some time before the expiration date, then Eq. (14.3) has to be restated as follows: Cy = Sy + Py- E/e™ (14.19) where — Cy = value of the call option now Sp = current stock price P E = exercise price alue of the put option now t = time left for the expiration of option r= interest rate Note that E/e'*is the present value of the exercise price. Inter alia, Eq. (14.19) can be used to establish the price of a put option and determine whether the put-call parity is working. Example Calculate the value of a call option given the following information, using the Black-Scholes formula: Stock price = Rs 60 Exercise price = Rs 50 Risk free rate = 8% Time of expiration = 3 months (t = 0.25) Standard deviation = 0 = 0.4 © scanned with OKEN Scanner453 What is the value of the put using the The value of the call option is calculated Step 1: Calculate dy and dy d,-ovt 1.1115 - 0.20 = 0.9115 Step 2: Find N(d,) and N(@,). N(d,) and N(d,) represent the probabilities that a random variable that has a standardised normal distribution will assume values less than 4d; and d,. N(d,) = N(L.1115) = 0.8668 N(d,) = N(0.9115) = 0.8190 Step 3: Estimate the present value of the exercise price, using the continuous discounting principle. Rs50___Rs50 SoS = pos = RS 49.01 eee ee 10202 : Step 4: Plug the numbers obtained above in the Black-Scholes formula. - FF E Co= So N(di) - oe N(da) = 60 x 0.8668 — 49.01 x 0.8190 = Rs 11.87 The price of the put option is: E Py = Cy So+ 0 0 ot en = 1187-60 + 49.01 = Rs 0.88 © scanned with OKEN Scanner; @ i SOLVED PROBLEMS 1 the stock exercisable a 9 " "The risk-free interest rate ¥y W per cent, By what ll A. A stock is eu how at an exer 12 per cent. Th ently selling for Rs 60. The call option or 1 price of Rs 55 is currently selling for Rs 15. tock can either rise or fall after a year, It can fall by per cent can it rise? 1 Solution 47" According to the binomial model on ash asap as Cu-Gu ae sqe= dy Cy dey (u-aR C, = Max (uS ~E, 0) Cy = Max (45 ~E, 0) eit P yer In the problem the following are given a $= Rs 60, E=Rs55,C= R515, R= 112, d= 07 Cy = Max (0.7 x 60-55, 0) = 0 Since C = Rs 15, uS ~ E has to be positive. So AS 4 9=0.7(60u~55) T1Q\u-07) Multiplying both the sides by (w - 0.7) we get gif? 2 2 a. 0.7(60u ~55) 15 (u—0.7) = 60u ~55 - °7EPEE SD gy 185 xo [Solving this for w we get u = 1.35 s=8l Sojthe a 2. Consider the following data: S=60,u=14,d=08 E=50,r=0.12,R=1.12 What is the value of the call option? © scanned with OKEN Scanner460 Investment Analysis and Portfolio Management Solution : The values of A (hedge ratio) and B (amount borrowed) can be obtained as follows: = Gh (=ds wy Be ( C, = Max (60 x 1.4 - 50,0) = 34 Cy = Max (60 x 0.8 — 50,0) = 0 34-0 4 G4-0.860 ~ 36-04 _ 14x0-08%34 (i4—08)112 C= AS +B= 944 x 60-4048 = 16.16 3. The following information is available for the equity stock of Prakash Limited So= Rs 120, E = Rs 110, r = 0.12, = 0.40 Calculate the price of a 6 month call option as per the Black-Scholes model. Solution A= ~ 40.48 d,=d,- ovt 120 |, h(a) +(oa2+3 xo16}05 0. 0870 +0.10 0.2828 4, = 0.6612 - 0.2828 = 0.3784 N(q,) = N(0.6612) = 0.7457 N(d,) = N(0.3784) = 0.6474 Eo 10 10 an eoPxOS ~ 10618 Loi Rs 120 x 0.7457 — Rs 103.60 x 0.6474 = Rs 22.41 .6612 © scanned with OKEN Scanner