Part 2: Binomial Trees and The Black-Scholes-Merton

Options, Futures, and Other Derivatives ELEVENTH EDITION Part 2 Binomial Trees and The Black-Scholes-Merton Model P OPTIONS, FUTURES, AND OTHER DERIVATIVES Pearson John Hull Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Outline • Binomial trees Black - Scholes - Merton model Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Binomial Trees (cây nhị phân) Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved A Simple Binomial Model . A stock price is currently $20. . In 3 months, it will be either $22 or $18. Stock Price = $22 Stock price = $20 Stock Price = $18 Pearson Copyright 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved A Call Option (Figure 13.1) A 3-month call option on the stock has a strike price of $21. Stock Price = $22 Option Payoff = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Payoff = $0 Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Setting Up a Riskless Portfolio For a portfolio that is long short 1 call option values are shares and a 22-1 18Δ . Portfolio is riskless when 22-1=18Δ or = 0.25 Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Valuing the Portfolio (Risk-Free Rate is 4%) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is: 22×0.25-1=4.50 The value of the portfolio today is: 4.5e-0.04×0.25 = 4.455 Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Valuing the Option . The portfolio that is: long 0.25 shares short 1 option is worth $4.455 The value of the shares is: 5.000(= 0.25×20) The value of the option is, therefore, Pearson 5.000-4.455 =0.545 Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Generalization (Figure 13.2) (1 of 3) A derivative lasts for time T and is dependent on a stock. Sou fu S f Sod fa Pearson Copyright 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Generalization (Figure 13.2) (2 of 3) • Value of a portfolio that is long shares and short 1 derivative: Sou fu S f Sod fa The portfolio is riskless when SuA-f =SodA-for f,-fdu Pearson pos-uos Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Generalization (Figure 13.2) (3 of 3) . . Value of the portfolio at time Tis SouA-f -rT Value of the portfolio today is (SouA-f)e Another expression for the portfolio value today is SA-f Hence, Pearson f = SA- (SuA-f)eT Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Generalization continued (Equations 13.2 and 13.3) Công thức tổng quát để tính giá của quyền chọn Substituting for we obtain, f = [pf,+(1-p)f]e-rT where rT e d p = u-d Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved p as a Probability It is natural to interpret pand 1-p as probabilities of up and down movements. • The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate. Sou P fu So f (1-p) Sod fa Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved . Risk-Neutral Valuation When the probability of an up and down movements are pand 1-p the expected stock price at time T is SerT This shows that the stock price earns the risk-free rate. Binomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate. This is known as using risk-neutral valuation. P Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Original example revisited Sou = 22 P f=1 S=20 f (1-p) Sod = 18 fa=0 p is the probability that gives a return on the stock equal to the risk-free rate: 2000.04×0.25 = 22p+18(1-p) so that p = 0.5503 Alternatively: e-d e 0.04x0.25 -0.9 0.5503 u-d P Pearson 1.1-0.9 Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Valuing the Option Using Risk-Neutral Valuation S=20 f 0.5503 0.4497 The value of the option is: Sou = 22 f=1 Sod = 18 fa=0 e-0.04×0.25 (0.5503 × 1 + 0.4497 × 0) = 0.545 Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved P A Two-Step Example for a call (Figure 13.3) 24.2 22 20 19.8 18 16.2 이 K = $21, r = 4%, S, = $20 Each time step is 3 months, every 3 months, the stock price might rise 1.1 times or decrease 0.9 times. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Valuing a Call Option Figure 13.4 24.2 3.2 22 B 20 0.9497 1.7433 19.8 A 0.0 18 0.0 16.2 0.0 Value at node В: = e-0.04×0.25 (0.5503 x 3.2 + 0.4497 × 0) = 1.7433 Value at node А: Pearson = e-0.04*0.25 (0.5503 x 1.7433 + 0.4497 × 0) = 0.9497 Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved A Put Option Example Figure 13.7 72 이 60 50 1.4147 48 4.1923 4 40 9.4636 32 20 K = 52, time step = 1 year r = 5%, u =1.2, d = 0.8, p = 0.6282 Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved What Happens When Put Option Is American (Figure 13.8) 72 이 60 50 5.0894 1.4147 48 4 40 C 12.0 32 20 The American feature increases the value at node C from 9.4636 to 12.0000 ($). This increases the value of the option (original node) from 4.1923 to 5.0894 ($). Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Delta - delta hedging Delta (4) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock. The value of varies from node to node. In the first example (call option), = (1-0) /(22-18) = 0.25 When short 1 call option & long shares, the portfolio is stable when the stock price changes (Delta hedging). Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Delta hedging The current price of a non-dividend-paying stock is $30. Over the next six months, it is expected to rise to $36 or fall to $25. Assume the risk-free rate is zero. An investor sells a six-month call option with a strike price of $32. Which of the following positions would hedge the risk? A. Buy 0.6 shares for each of the above options. B. Buy 0.4 shares for each of the above options C. Sell 0.6 shares for each of the above options. D. Sell 0.4 shares for each of the above options. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Choosing u and d (Equations 13.15 and 13.16) One way of matching the volatility is to set (p284, 10 ed.): u = e and d=e where is the volatility and At is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Assets Other than Non-Dividend Paying Stocks • For options on stock indices, currencies, and futures, the basic procedure for constructing the tree is the same except for the calculation of p. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved The Probability of an Up Move a-d p= u-d rAt a=e for a non dividend paying stock a=e(r-q)t for a stock index where q is the dividend yield on the index a =ee(r-r)At for a currency where r, is the foreign risk-free rate a =1 for a future contract Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved 2.2 The Black-Scholes-Merton Model Pearson Copyright 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved The Volatility – Độ biến động (của giá cổ phiếu) . • The volatility is the standard deviation of the continuously compounded rate of return in 1 year. The standard deviation of the return in a short time period time At is approximately oAt If the volatility (sigma) is 25% per year what is the standard deviation of the rate of return in one day? At ( = 1/252 year)? Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Estimating Volatility From Historical Data 1. Take observations S, S,..., S at intervals of years (e.g. for weekly data г = 1/52) 2. Calculate the continuously compounded return in each interval as: S 4,=In S-1i-1 2 1 1 n-1i=l n(n -1) S 3. The standard deviation =S 4. The historical volatility estimate is: Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Example Table 15.1 Computation of volatility. Day Closing stock price (dollars), S Price relative Si/Si-1 Daily return u=In(S;/Si-1) 0 20.00 1 20.10 1.00500 0.00499 2 19.90 0.99005 -0.01000 3 20.00 1.00503 0.00501 4 20.50 1.02500 0.02469 5 20.25 0.98780 -0.01227 6 20.90 1.03210 0.03159 7 20.90 1.00000 0.00000 8 20.90 1.00000 0.00000 9 20.75 0.99282 -0.00720 10 20.75 1.00000 0.00000 11 21.00 1.01205 0.01198 12 21.10 1.00476 0.00475 13 20.90 0.99052 -0.00952 14 20.90 1.00000 0.00000 15 21.25 1.01675 0.01661 16 21.40 1.00706 0.00703 17 21.40 1.00000 0.00000 18 21.25 0.99299 -0.00703 19 21.75 1.02353 0.02326 20 22.00 1.01149 0.01143 Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Solution. Table 15.1 shows a possible sequence of stock prices during 21 consecutive trading days. In this case, n = 20, so that Σ 141 = 0.09531 and 11 = 0.00326 and the estimate of the standard deviation of the daily return is 0.00326 0.095312 =0.01216 19 20 x 19 or 1.216%. Assuming that there are 252 trading days per year, = 1/252 and the data give an estimate for the volatility per annum of 0.01216/252 = 0.193, or 19.3%. The standard error of this estimate is 0.193 = 0.031 2 x 20 or 3.1% per annum. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Nature of Volatility (Business Snapshot 15.2) Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed. For this reason time is usually measured in "trading days" not calendar days when options are valued. It is assumed that there are 252 trading days in one year for most assets. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Example Suppose it is April 1 and an option lasts to April 30 so that the number of days remaining is 30 calendar days or 22 trading days. The time to maturity would be assumed to be 22/252 = 0.0873 years. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Black - Scholes - Merton Model Note: Black - Scholes - Merton Model is mainly used for European options. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Assumptions for BSM model 1. The stock price is lognormally distributed. 2. The short selling of securities with full use of proceeds is permitted. 3. There are no transaction costs or taxes. All securities are perfectly divisible. 4. There are no dividends during the life of the derivative. 5. There are no riskless arbitrage opportunities. 6. Security trading is continuous. 7. The risk-free rate of interest, r, is constant and the same for all maturities. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Assumption for BSM model (cont.) As we discuss in later chapters, some of these assumptions can be relaxed. For example, r can be known functions of t. We can even allow interest rates to be stochastic provided that the stock price distribution at maturity of the option is still lognormal. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved BSM formula for European options (CT 15.20, 15.21) c=S, N(d)-KeN(d) -rT p=Ke N(-d₂)-S N(-d) where d =In(S,/K)+(r+&²12)T T d _In(S/K)+(r-0²12)7 T =d-o√T Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved The N(x) Function N(x) is the probability that a normally distributed variable Z with a mean of zero and a standard deviation of 1, also denoted by P(Z< x). 0 See tables at the end of the book. P Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved . Properties of Black-Scholes Formula • As S becomes very large, c tends to and p tends to zero. So-Ke-rT As S becomes very small, c tends to zero and p tends to Ke-rt-So Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Understanding Black-Scholes c=ee-fTN(d2)(SerN(d₁)/N(d₂)-K) -rT e: Present value factor N(d₂): Probability of exercise Se N(d,)/N(d2): Expected stock price in a risk-neutral world if option is exercised K: Strike price paid if option is exercised Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Implied Volatility The implied volatility of an option is the volatility for which the BlackScholes-Merton price equals the market price. There is a one-to-one correspondence between prices and implied volatilities. Traders and brokers often quote implied volatilities rather than dollar prices. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Three approaches for BSM model • Binomial trees • . Risk neutral evaluation PDE Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Proving Black-Scholes-Merton from Binomial Trees (Appendix to Chapter 13, p296) (1 of 2) -rT n! c=e Σ p(1-p) max(Su'd)- K, 0) j=0 (n-j)!j! Option is in the money when j > a where n In(S K) α 2 20T/n -rT c = e (SU,–KU) so that where U1 = Σ j>a n! 2 Σ j>a Pearson n! (n-j)!j! p'(1-p)nda- (n-j)!j! n-j p(1-p) Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Proving Black-Scholes-Merton from Binomial Trees (Appendix to Chapter 13) (2 of 2) The expression for U can be written n! n! U = [pu +(1-p)d]"Σ )d)(1-d) = e"Σ (p)(1-p) (n-j)!j! where du pu+(1-p)р • Both U, and U₂ can now be evaluated in terms of the cumulative binomial distribution. We now let the number of time steps tend to infinity and use the result that a binomial distribution tends to a normal distribution. P Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Options on dividend-paying stocks European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes. Only dividends with ex-dividend dates during life of option should be included. The "dividend" should be the expected reduction in the stock price expected. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved American Calls An American call on a non-dividend-paying stock should never be exercised early if investors want to hold the stock. An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date. Suppose dividend dates are at times t,t2,...t. Early exercise is sometimes optimal at time t if the dividend at that time is greater than K[1-e(+1-)] Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved Black's Approximation for Dealing With Dividends in American Call Options Set the American price equal to the maximum of two European prices: 1. The 1st European price is for an option maturing at the same time as the American option. 2. The 2nd European price is for an option maturing just before the final ex-dividend date. Pearson Copyright © 2022, 2018, 2012 Pearson Education, Inc. All Rights Reserved