VIEWS: 216 PAGES: 15 CATEGORY: Jobs & Careers POSTED ON: 7/9/2010
Finance with Dr. John Elder 1 1 Black Scholes Option Pricing • Read Hull chapters 12 and 13. • Assumptions • Model and Derivation • Estimating Volatility • Implied Volatility • Dividends (see section 13.1 and 13.2) • Currency Options (see section 13.5) • Black-Scholes Formula c0 = So e–qT N(d1) – K e–rT N(d2) d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2) • Put-Call parity – Put-Call parity with no dividends c0 – p0 = S0 – Ke –rfT – Put-Call parity with known dollar div c0 – p0 = S0 – Ke –rT – PV(Div) – Put-Call parity with constant div yield c0 – p0 = S0e –qT – Ke –rT Finance with Dr. John Elder 2 2 Binomial Model Review • Step 1: Consider a stock (S0=??) and call (K=??) – compute payoff for call at each node. • Step 2: Construct portfolio replicating risk-free investment. – Hedge ratio “D” for riskless hedge portfolio D = (fu–fd) / (Su–Sd). • Step 3: Price call by setting cost (PV) of RHP equal to discounted payoff. – Find fair price of call C t at each node: PV of riskless hedge portfolio = FVe –rdt DS0 – f = (DS0u – fu)*e –r(δ t) OR (DS0d – fd)*e –r(δ t) • If we choose up/down moves carefully, as we decrease time between nodes, possible future stock prices are distributed log-normal! (+/– tree moves not absolute!) Finance with Dr. John Elder 3 3 Assumptions for Black-Scholes • Black-Scholes (B-S) model – option pricing formula derived under no-arbitrage conditions with set of formal assumptions, e.g., no taxes, transaction costs, margin: – Underlying is risky; no div; trades continuously in fractional units. – Risk-free rate, r, constant for all maturities. Volatility (σ) constant over time. – Option is euro; exercise price is K; expiration is T; ∆ denotes small change. • Distributional assumptions – C.c. return normally distributed R0,t = ln(St/S0) ~ N([µ–½σ2]∆t , σ√(∆t)). – Then St is distributed log-normal, since St = S0 exp(R0,t ). – Note µ∆t is analogous to arithmetic avg, while [µ–½σ2]∆t is geometric avg. Normal Distribution Log-Normal Distribution 0.45 0.35 0.40 s=1 0.30 0.35 s=2 0.25 0.30 0.25 0.20 0.20 0.15 0.15 0.10 0.10 s=1 0.05 s=1.5 0.05 0.00 0.00 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Finance with Dr. John Elder 4 4 Assumptions – Are they Reasonable? • Actual Stock Returns – histogram of index daily returns, with normal overlay. – Normality means 95% of daily returns between 0.04%–2% and 0.04%+2% – Stock returns tend to have big negatives, higher peak and fatter tails. – Technically, neg skew (–0.3), excess kurtosis (4.1); and heteroskedastic. – – – Stock Index (daily for 9 yrs) and Random Walk Histogram of Stock Index Daily Returns(cc) 0.16 3500 Which is which? Mean=0.04% Std=1.00% 0.14 3000 Annualized Mean =8.90% Std=15.85% 0.12 2500 0.1 2000 0.08 1500 0.06 1000 ??? 0.04 500 ??? 0.02 0 0 1001 1101 1201 1301 1401 1501 1601 1701 1801 1901 2001 2101 -32201 101 201 301 401 501 601 701 801 901 1 0% 0% 0% 0% % % % % % 0% 0% 0% % % 0% .50 .00 00 .50 50 50 00 0.5 2.5 3.5 1.5 0.0 2.0 3.0 1.0 -2. -0. -2. -1. -3 -1 Finance with Dr. John Elder 5 5 Black-Scholes Derivation • Derivation - The objective is to derive an equation that expresses value of call option as function of its strike price and other variables. • Step 1. Set-up riskless hedge portfolio long ∆ shares and short one call option. – Denoting Π as current market value of portfolio, Π = ∆S – C. – Change in value of portfolio is w.r.t to change in S is dΠ = ∆dS – dC. – To ensure riskless hedge portfolio, we find value of ∆ such that dΠ = 0. – ∆ = dC/dS. • Step 2. Construct portfolio that replicates a risk-free investment. – • Step 3: Price call option by setting cost (PV) of RHP equal to discounted payoff. – Price of call must satisfy second order partial differential equation. – Seldom easy to solve, but B-S-M redefined variables to give familiar form. – Finance with Dr. John Elder 6 6 Black-Scholes for Euro Options (with dividends) c= S0 e–qT N(d1) – K e–rT N(d2) p= –S0e–qT N(– d1) + K e–rT N(– d2) S0 σ2 ln K + r −q+ T d = d − σ T 2 2 1 d1 = σ T where c0 = current value of European call. p0 = current value of European put. S0 = current stock price N(d) = probability that random draw from normal distribn is less than d. K = Exercise price. q = annualized div yield of underlying stock (text ignores div yld for now). e = 2.71828, base of natural log. r = Risk-free interest rate matching maturity of option, c.c. T = time to maturity of option in years. σ= Std deviation of return on stock, c.c. Note: Black-Scholes is derived for European calls! No value for early exercise! Finance with Dr. John Elder 7 7 Black-Scholes Characteristics 1 • What is Black-Scholes? c = S0 e–qT N(d1) – K e–rT N(d2) – N(di) interpreted as risk-adjusted probability option will expire in-the-money – Recall that value of call is discounted expected payoff, Max(0,ST - K). – Suppose for below that q=0 (no dividends). • Suppose there is low probability call option will be exercised (S0 << K). – • Suppose there is high probability call option will be exercised (S0 >> K). – Then N(d1) and N(d2) are close to one, and value of call is c0 = S0 – Ke –rT – • Using Black-Scholes – derived for European options – American calls on non-div stocks not exercised prior to expiration. – Finance with Dr. John Elder 8 12 8 Black-Scholes Prices (Euro, no div) Call (X=100, T=1-mo, r=5%, Volatility=15%) 10 Parity for Euro: c–p = S0 – K e–rT 8 Option Value 6 Black-Scholes: 4 c=S0N(d1)–K e–rT N(d2) 2 N(di): risk-adj prob 0 option expires 90 92 94 96 98 100 102 104 106 108 110 -2 in-the-money Stock Price 12 Put (X=100, T=1-mo, r=5%, Volatility=15%) Euro Put: possible 10 early exercise Option Value 8 exceeds value if deep 6 in-the-money. 4 2 0 90 92 94 96 98 100 102 104 106 108 110 -2 Stock Price Finance with Dr. John Elder 9 9 Black-Scholes Example • Suppose S0 = $50, K= $45, T = 6 months, r = 10% c.c., and s=28%. Calculate the value of a call and a put option. • Black-Scholes c= S0 e–qT N(d1) – K e–rT N(d2) d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2) ( ) ln 50 45 + 0.10 − 0 + 0.282 2 0.50 d1 = = 0.884 0.28 0.50 d 2 = 0.884 − 0.28 0.50 = 0.686 c0 = S0 e–qT N(d1) – K e–rT N(d2) c0 = = = p0 = = Note: Excel worksheet fn =NORMSDIST(x), but need to load Analysis Toolpack. Finance with Dr. John Elder 10 10 Black-Scholes – Calculating interest rate • Compute interest rate from T-bills, based on avg of bid/ask Maturity Days Maturity Bid Ask • Compute average of bid and ask – • Compute price of 100 par value bill based on bank discount rate. • Compute the EAR – EAR = • Compute the continuously compounded, annual rate of return. – (1+EAR) = – • Finance with Dr. John Elder 11 11 Black-Scholes – Estimating Volatility • c= S0 e–qT N(d1) – K e–rT N(d2); d1=[ln(S0/K)+(r–q+σ2/2)T]/(σ T1/2); d2=d1–σ T1/2 • Estimating volatility - Let St be daily price and let ut=ln(St/St–1) be daily return. – Use 90-180 trading days, or maturity. Discard data around ex-div date (tax). – Find daily variance of ut; multiply by 252 days; take sqrt(σ2) for std dev. ∑i =1 (ut − E (ut )) 1 N σ 2 annual = 252 *σ 2 daily = 252 * 2 N −1 • Implied Volatility – level of volatility implied by B-S or binomial model. – May avg from several liquid options on same asset; used to price less liquid. – Issues: non-simultaneity; bid-ask; model mispecification. – VIX – – Volatility skew - Volatility Skews for IBM Options 60 Implied Standard Deviation (%) 58 56 54 52 Put ISDs 50 48 46 Call ISDs 44 42 40 110 115 120 125 130 135 140 145 Strikes ($) Finance with Dr. John Elder 12 12 Black-Scholes with Dividends • Black-Scholes – c= S0 e–qT N(d1) – K e–rT N(d2) – d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2) • Dividend Yield – basic Black-Scholes formula does not include dividends. – Accounting for div yield – discount current stock price by div yield. – • Dividends paid as known dollar values – – Recall that stock price (and call value) drops on ex-div date. – – • Black adjustment for dividends – devised by Fischer Black before PC era. – Price option as greater of (1) Euro option (2) Euro maturing just prior to latest ex-div date. Finance with Dr. John Elder 13 13 Binomial Model vs Black Scholes • Binomial Model vs. Black Scholes – Euro call no div – – American calls on div paying stocks – – Euro and American puts – • In practice – computing power is now cheap. – Analysts use binomial model to price American and other options. – Consider American call/put with K=100; S0=100; T=6-mo; r=5%; σ=25%. Call Prices Put Prices Time Binomial Black- Binomial Black Steps Scholes Scholes 5 8.601 8.26 6.334 5.791 10 8.087 8.26 5.932 5.791 15 8.373 8.26 6.128 5.791 20 8.173 8.26 5.977 5.791 30 8.202 8.26 5.992 5.791 50 8.225 8.26 6.004 5.791 99 8.277 8.26 6.039 5.791 Finance with Dr. John Elder 14 14 Tips on Pricing Options for the Savvy Investor • Black-Scholes is powerful formula for pricing Euro options – Derived before binomial model. – Some assumptions are strong (continuous trading, normally distributed cc returns), but model works well. – No arbitrage assumption cornerstone in finance and revolutionized field. – B-S can be viewed as special binomial model with many steps. – B-S less useful for pricing Amer puts and calls on div paying stocks. • There are many adjustments to Black-Scholes, but still very useful benchmark. – Options on stock indexes used with dividend yield – Options on indiv stocks often adjusted by subtracting PV of div for S0. • Black-Scholes Formula: c0 = So e–qT N(d1) – K e–rT N(d2) d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2) • Put-Call parity – Put-Call parity with known dollar div c0 – p0 = S0 – Ke –rT – PV(Div) – Put-Call parity with constant div yield c0 – p0 = S0e –qT – Ke –rT Finance with Dr. John Elder 15 15 Standard Normal Probabilities