VIEWS: 13 PAGES: 29 CATEGORY: Jobs & Careers POSTED ON: 7/9/2010 Public Domain
Derivatives Inside Black Scholes Professor André Farber Solvay Business School Université Libre de Bruxelles Lessons from the binomial model • Need to model the stock price evolution • Binomial model: – discrete time, discrete variable – volatility captured by u and d • Markov process • Future movements in stock price depend only on where we are, not the history of how we got where we are • Consistent with weak-form market efficiency • Risk neutral valuation – The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate p f u (1 p ) f d e rt d f with p e rt ud July 9, 2010 Derivatives 08 Inside Black Scholes |2 Black Scholes differential equation: assumptions • S follows the geometric Brownian motion: dS = µS dt + S dz – Volatility constant – No dividend payment (until maturity of option) – Continuous market – Perfect capital markets – Short sales possible – No transaction costs, no taxes – Constant interest rate • Consider a derivative asset with value f(S,t) • By how much will f change if S changes by dS? • Answer: Ito’s lemna July 9, 2010 Derivatives 08 Inside Black Scholes |3 Ito’s lemna • Rule to calculate the differential of a variable that is a function of a stochastic process and of time: • Let G(x,t) be a continuous and differentiable function • where x follows a stochastic process dx =a(x,t) dt + b(x,t) dz • Ito’s lemna. G follows a stochastic process: G G 1 2 G 2 G dG ( a 2 b ) dt b dz x t 2 x x Drift Volatility July 9, 2010 Derivatives 08 Inside Black Scholes |4 Ito’s lemna: some intuition • If x is a real variable, applying Taylor: G G 1 2G 2 2G 1 2G 2 G x t x x t t .. x t 2 x 2 xt 2 t 2 An approximation G G dx², dt², dx dt negligeables • In ordinary calculus: dG dx dt x t G G 1 ²G • In stochastic calculus: dG dx dt dx² x t 2 x ² • Because, if x follows an Ito process, dx² = b² dt you have to keep it July 9, 2010 Derivatives 08 Inside Black Scholes |5 Lognormal property of stock prices • Suppose: dS= S dt + S dz • Using Ito’s lemna: d ln(S) = ( - 0.5 ²) dt + dz • Consequence: ln(ST) – ln(S0) = ln(ST/S0) ² Continuously compounded ln( S T ) ln( S 0 ) ~ N [( )T , T ] 2 return between 0 and T ² ln(ST) is normally distributed ln( S T ) ~ N [ln(S 0 ) ( )T , T ] so that ST has a lognormal 2 distribution July 9, 2010 Derivatives 08 Inside Black Scholes |6 Derivation of PDE (partial differential equation) • Back to the valuation of a derivative f(S,t): • If S changes by dS, using Ito’s lemna: f f 1 2 f f df ( S 2 2 S 2 ) dt S dz S t 2 S S • Note: same Wiener process for S and f • possibility to create an instantaneously riskless position by combining the underlying asset and the derivative • Composition of riskless portfolio • -1 sell (short) one derivative • fS = ∂f /∂S buy (long) DELTA shares • Value of portfolio: V = - f + fS S July 9, 2010 Derivatives 08 Inside Black Scholes |7 Here comes the PDE! • Using Ito’s lemna f 1 2 f 2 2 dV ( S ) dt t 2 S 2 • This is a riskless portfolio!!! • Its expected return should be equal to the risk free interest rate: dV = r V dt • This leads to: f f 1 2 f 2 2 rS S rf t S 2 S 2 July 9, 2010 Derivatives 08 Inside Black Scholes |8 Understanding the PDE • Assume we are in a risk neutral world Expected change f f 1 f 2 2 2 of the value of rS S rf t S 2 S 2 derivative security Change of the value with respect to time Change of the value Change of the with respect to the value with price of the respect to underlying asset volatility July 9, 2010 Derivatives 08 Inside Black Scholes |9 Black Scholes’ PDE and the binomial model • We have: • BS PDE : f’t + rS f’S + ½ ² f”SS = r f • Binomial model: p fu + (1-p) fd = ert • Use Taylor approximation: • fu = f + (u-1) S f’S + ½ (u–1)² S² f”SS + f’t t • fd = f + (d-1) S f’S + ½ (d–1)² S² f”SS + f’t t • u = 1 + √t + ½ ²t • d = 1 – √t + ½ ²t • ert = 1 + rt • Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes July 9, 2010 Derivatives 08 Inside Black Scholes |10 And now, the Black Scholes formulas • Closed form solutions for European options on non dividend paying stocks assuming: • Constant volatility • Constant risk-free interest rate Call option: C S 0 N (d1 ) Ke rT N (d 2 ) Put option: P Ke rT N ( d 2 ) S 0 N ( d 1 ) ln( S 0 / Ke rT ) d1 0.5 T d 2 d1 T T N(x) = cumulative probability distribution function for a standardized normal variable July 9, 2010 Derivatives 08 Inside Black Scholes |11 Understanding Black Scholes • Remember the call valuation formula derived in the binomial model: C = S0 – B • Compare with the BS formula for a call option: C S 0 N (d1 ) Ke rT N (d 2 ) • Same structure: • N(d1) is the delta of the option • # shares to buy to create a synthetic call • The rate of change of the option price with respect to the price of the underlying asset (the partial derivative CS) • K e-rT N(d2) is the amount to borrow to create a synthetic call N(d2) = risk-neutral probability that the option will be exercised at maturity July 9, 2010 Derivatives 08 Inside Black Scholes |12 A closer look at d1 and d2 ln( S 0 / Ke rT ) d 2 d1 T d1 0.5 T T 2 elements determine d1 and d2 A measure of the “moneyness” of the S0 / Ke-rt option. The distance between the exercise price and the stock price T Time adjusted volatility. The volatility of the return on the underlying asset between now and maturity. July 9, 2010 Derivatives 08 Inside Black Scholes |13 Example Stock price S0 = 100 Exercise price K = 100 (at the money option) Maturity T = 1 year Interest rate (continuous) r = 5% Volatility = 0.15 ln(S0 / K e-rT) = ln(1.0513) = 0.05 √T = 0.15 d1 = (0.05)/(0.15) + (0.5)(0.15) = 0.4083 N(d1) = 0.6585 European call : d2 = 0.4083 – 0.15 = 0.2583 100 0.6585 - 100 0.95123 0.6019 N(d2) = 0.6019 = 8.60 July 9, 2010 Derivatives 08 Inside Black Scholes |14 Relationship between call value and spot price For call option, time value > 0 July 9, 2010 Derivatives 08 Inside Black Scholes |15 European put option • European call option: C = S0 N(d1) – PV(K) N(d2) Delta of call option Risk-neutral probability of exercising the option = Proba(ST>X) • Put-Call Parity: P = C – S0 + PV(K) • European put option: P = S0 [N(d1)-1] + PV(K)[1-N(d2)] Delta of put option Risk-neutral probability of exercising the option = Proba(ST<X) • P = - S0 N(-d1) +PV(K) N(-d2) (Remember: N(x) – 1 = N(-x) July 9, 2010 Derivatives 08 Inside Black Scholes |16 Example • Stock price S0 = 100 • Exercise price K = 100 (at the money option) • Maturity T = 1 year • Interest rate (continuous) r = 5% • Volatility = 0.15 N(-d1) = 1 – N(d1) = 1 – 0.6585 = 0.3415 N(-d2) = 1 – N(d2) = 1 – 0.6019 = 0.3981 European put option - 100 x 0.3415 + 95.123 x 0.3981 = 3.72 July 9, 2010 Derivatives 08 Inside Black Scholes |17 Relationship between Put Value and Spot Price For put option, time value >0 or <0 July 9, 2010 Derivatives 08 Inside Black Scholes |18 Dividend paying stock • If the underlying asset pays a dividend, substract the present value of future dividends from the stock price before using Black Scholes. • If stock pays a continuous dividend yield q, replace stock price S0 by S0e-qT. – Three important applications: • Options on stock indices (q is the continuous dividend yield) • Currency options (q is the foreign risk-free interest rate) • Options on futures contracts (q is the risk-free interest rate) July 9, 2010 Derivatives 08 Inside Black Scholes |19 Dividend paying stock: binomial model t = 1 u = 1.25, d = 0.80 r = 5% q = 3% uS0 eqt with dividends reinvested Derivative: Call K = 100 128.81 uS0 ex dividend fu 125 25 S0 dS0 eqt with dividends reinvested 100 82.44 fd dS0 ex dividend 0 f = S0 + M 80 f = [ p fu + (1-p) fd] e-rt = 11.64 Replicating portfolio: uS0 eqt + M ert = fu 128.81 + M 1.0513 = 25 p = (e(r-q)t – d) / (u – d) = 0.489 dS0 eqt + M ert = fd 82.44 + M 1.0513 = 0 = (fu – fd) / (u – d )S0eqt = 0.539 July 9, 2010 Derivatives 08 Inside Black Scholes |20 Black Scholes Merton with constant dividend yield The partial differential f f 1 2 f 2 2 equation: (r q) S S rf (See Hull 5th ed. Appendix 13A) t S 2 S 2 Expected growth rate of stock Call option C S 0 e qT N (d 1 ) Ke rT N (d 2 ) Put option P Ke rT N ( d 2 ) S 0 e qT N (d1 ) ln( S 0 e qT / Ke rT ) d 2 d1 T d1 0.5 T T July 9, 2010 Derivatives 08 Inside Black Scholes |21 Options on stock indices • Option contracts are on a multiple times the index ($100 in US) • The most popular underlying US indices are – the Dow Jones Industrial (European) DJX – the S&P 100 (American) OEX – the S&P 500 (European) SPX • Contracts are settled in cash • Example: July 2, 2002 S&P 500 = 968.65 • SPX September • Strike Call Put • 900 - 15.60 1,005 30 53.50 1,025 21.40 59.80 • Source: Wall Street Journal July 9, 2010 Derivatives 08 Inside Black Scholes |22 Options on futures • A call option on a futures contract. • Payoff at maturity: • A long position on the underlying futures contract • A cash amount = Futures price – Strike price • Example: a 1-month call option on a 3-month gold futures contract • Strike price = $310 / troy ounce • Size of contract = 100 troy ounces • Suppose futures price = $320 at options maturity • Exercise call option » Long one futures » + 100 (320 – 310) = $1,000 in cash July 9, 2010 Derivatives 08 Inside Black Scholes |23 Option on futures: binomial model uF0 → fu Futures price F0 dF0 →fd Replicating portfolio: futures + cash fu fd uF0 dF0 (uF0 – F0) + M ert = fu pf u (1 p) f d (dF0 – F0) + M ert = fd f e rt f=M 1 d p ud July 9, 2010 Derivatives 08 Inside Black Scholes |24 Options on futures versus options on dividend paying stock Compare now the formulas obtained for the option on futures and for an option on a dividend paying stock: Futures Dividend paying stock pf u (1 p) f d pf u (1 p) f d f f e rt e rt 1 d e ( r q ) t d p p ud ud Futures prices behave in the same way as a stock paying a continuous dividend yield at the risk-free interest rate r July 9, 2010 Derivatives 08 Inside Black Scholes |25 Black’s model Assumption: futures price has lognormal distribution Ce rT [ F0 N (d1 ) KN (d 2 )] P e rT [ KN (d 2 ) F0 N (d1 )] F0 F0 ) ln( ) ln( d1 X 0.5 T d2 X 0.5 T d T T 1 T July 9, 2010 Derivatives 08 Inside Black Scholes |26 Implied volatility – Call option July 9, 2010 Derivatives 08 Inside Black Scholes |27 Implied volatility – Put option July 9, 2010 Derivatives 08 Inside Black Scholes |28 Smile SPX Option on S&P 500 Spot index 968.25 September 2002 Contract DivYield 2% IntRate 1.86% July 2, 2002 Maturity 90 days Strike Call Put OpenInt Price ImpVol OpenInt Price ImpVol 700 3801 1.5 34.19% 750 1581 2.9 31.59% 800 31675 4 26.84% 900 21723 15.6 22.17% 925 7799 19 19.54% 950 17419 28 19.16% 975 16603 33 15.32% 980 3599 42 24.89% 4994 40.3 17.68% 990 3228 40 26.04% 3193 41 14.86% 995 11806 34.5 24.17% 23345 46 15.84% 1005 5404 30 23.73% 5209 53.5 16.29% 1025 9232 21.4 22.47% 15242 59.8 9.95% 1040 2286 15.1 20.97% 1050 11145 13.1 21.07% 1075 8726 7.5 19.97% 1100 23170 4.6 19.82% 1125 7556 2.4 19.16% 1150 18173 1.6 19.67% 1200 7513 0.45 19.33% July 9, 2010 Derivatives 08 Inside Black Scholes |29