1. Options and Speculative Markets 2004-2005 Inside Black Scholes Professor André Farber Solvay Business School Université Libre de Bruxelles

2. August 23, 2004 OMS 08 Inside Black Scholes |2 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

3. August 23, 2004 OMS 08 Inside Black Scholes |3 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

4. August 23, 2004 OMS 08 Inside Black Scholes |4 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: DriftVolatility

5. August 23, 2004 OMS 08 Inside Black Scholes |5 Ito’s lemna: some intuition If x is a real variable, applying Taylor: In ordinary calculus: In stochastic calculus: Because, if x follows an Ito process, dx² = b² dt you have to keep it An approximation dx², dt², dx dt negligeables

6. August 23, 2004 OMS 08 Inside Black Scholes |6 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(S T ) – ln(S 0 ) = ln(S T /S 0 ) Continuously compounded return between 0 and T ln(S T ) is normally distributed so that S T has a lognormal distribution

7. August 23, 2004 OMS 08 Inside Black Scholes |7 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: 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 -1sell (short) one derivative f S = ∂f /∂S buy (long) DELTA shares Value of portfolio: V = - f + f S S

8. August 23, 2004 OMS 08 Inside Black Scholes |8 Here comes the PDE! Using Ito’s lemna This is a riskless portfolio!!! Its expected return should be equal to the risk free interest rate: dV = r V dt This leads to:

9. August 23, 2004 OMS 08 Inside Black Scholes |9 Understanding the PDE Assume we are in a risk neutral world Expected change of the value of derivative security Change of the value with respect to time Change of the value with respect to the price of the underlying asset Change of the value with respect to volatility

10. August 23, 2004 OMS 08 Inside Black Scholes |10 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 f u + (1-p) f d = e r  t Use Taylor approximation: f u = f + (u-1) S f’ S + ½ (u–1)² S² f” SS + f’ t  t f d = f + (d-1) S f’ S + ½ (d–1)² S² f” SS + f’ t  t u = 1 +  √  t + ½  ²  t d = 1 –  √  t + ½  ²  t e r  t = 1 + r  t Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes

11. August 23, 2004 OMS 08 Inside Black Scholes |11 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: Put option: N(x) = cumulative probability distribution function for a standardized normal variable

12. August 23, 2004 OMS 08 Inside Black Scholes |12 Understanding Black Scholes Remember the call valuation formula derived in the binomial model: C =  S 0 – B Compare with the BS formula for a call option: Same structure: N(d 1 ) 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 C S ) K e -rT N(d 2 ) is the amount to borrow to create a synthetic call N(d 2 ) = risk-neutral probability that the option will be exercised at maturity

13. August 23, 2004 OMS 08 Inside Black Scholes |13 A closer look at d 1 and d 2 2 elements determine d 1 and d 2 S 0 / Ke -rt A measure of the “moneyness” of the option. The distance between the exercise price and the stock price Time adjusted volatility. The volatility of the return on the underlying asset between now and maturity.

14. August 23, 2004 OMS 08 Inside Black Scholes |14 Example Stock price S 0 = 100 Exercise price K = 100 (at the money option) Maturity T = 1 year Interest rate (continuous) r = 5% Volatility  = 0.15 ln(S 0 / K e -rT ) = ln(1.0513) = 0.05  √T = 0.15 d 1 = (0.05)/(0.15) + (0.5)(0.15) = 0.4083 N(d 1 ) = 0.6585 d 2 = 0.4083 – 0.15 = 0.2583 N(d 2 ) = 0.6019 European call : 100  0.6585 - 100  0.95123  0.6019 = 8.60

15. August 23, 2004 OMS 08 Inside Black Scholes |15 Relationship between call value and spot price For call option, time value > 0

16. August 23, 2004 OMS 08 Inside Black Scholes |16 European put option European call option: C = S 0 N(d 1 ) – PV(K) N(d 2 ) Put-Call Parity: P = C – S 0 + PV(K) European put option: P = S 0 [N(d 1 )-1] + PV(K)[1-N(d 2 )] P = - S 0 N(-d 1 ) +PV(K) N(-d 2 ) Delta of call option Risk-neutral probability of exercising the option = Proba(S T >X) Delta of put option Risk-neutral probability of exercising the option = Proba(S T <X) (Remember: N(x) – 1 = N(-x)

17. August 23, 2004 OMS 08 Inside Black Scholes |17 Example Stock price S 0 = 100 Exercise price K = 100 (at the money option) Maturity T = 1 year Interest rate (continuous) r = 5% Volatility  = 0.15 N(-d 1 ) = 1 – N(d 1 ) = 1 – 0.6585 = 0.3415 N(-d 2 ) = 1 – N(d 2 ) = 1 – 0.6019 = 0.3981 European put option - 100 x 0.3415 + 95.123 x 0.3981 = 3.72

18. August 23, 2004 OMS 08 Inside Black Scholes |18 Relationship between Put Value and Spot Price For put option, time value >0 or <0

19. August 23, 2004 OMS 08 Inside Black Scholes |19 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 S 0 by S 0 e -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)

20. August 23, 2004 OMS 08 Inside Black Scholes |20 Dividend paying stock: binomial model S 0 100 uS 0 e q  t with dividends reinvested 128.81 dS 0 e q  t with dividends reinvested 82.44 uS 0 ex dividend 125 dS 0 ex dividend 80 Replicating portfolio:  uS 0 e q  t + M e r  t = f u  128.81 + M 1.0513 = 25  dS 0 e q  t + M e r  t = f d  82.44 + M 1.0513 = 0 f =  S 0 + M  = (f u – f d ) / (u – d )S 0 e q  t = 0.539 f = [ p f u + (1-p) f d ] e -r  t = 11.64 p = (e (r-q)  t – d) / (u – d) = 0.489  t = 1 u = 1.25, d = 0.80 r = 5% q = 3% Derivative: Call K = 100 f u 25 fd0fd0

21. August 23, 2004 OMS 08 Inside Black Scholes |21 Black Scholes Merton with constant dividend yield The partial differential equation: (See Hull 5th ed. Appendix 13A) Expected growth rate of stock Call option Put option

22. August 23, 2004 OMS 08 Inside Black Scholes |22 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 StrikeCallPut 900 - 15.60 1,0053053.50 1,02521.4059.80 Source: Wall Street Journal

23. August 23, 2004 OMS 08 Inside Black Scholes |23 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

24. August 23, 2004 OMS 08 Inside Black Scholes |24 Option on futures: binomial model Futures price F 0 uF 0 → f u dF 0 →f d Replicating portfolio:  futures + cash  (uF 0 – F 0 ) + M e r  t = f u  (dF 0 – F 0 ) + M e r  t = f d f = M

25. August 23, 2004 OMS 08 Inside Black Scholes |25 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 prices behave in the same way as a stock paying a continuous dividend yield at the risk-free interest rate r Futures Dividend paying stock

26. August 23, 2004 OMS 08 Inside Black Scholes |26 Black’s model Assumption: futures price has lognormal distribution

27. August 23, 2004 OMS 08 Inside Black Scholes |27 Implied volatility – Call option

28. August 23, 2004 OMS 08 Inside Black Scholes |28 Implied volatility – Put option

29. August 23, 2004 OMS 08 Inside Black Scholes |29 Smile