Binnenlandse Francqui Leerstoel VUB 2004-2005 2. Options and investments

1. Binnenlandse Francqui Leerstoel VUB 2004-20052. Options and investments Professor André Farber Solvay Business School Université Libre de Bruxelles

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 OMS 2004 Greeks

3. u²S uS S udS dS d²S Recursive method (European and American options) Value option at maturity Work backward through the tree. Apply 1-period binomial formula at each node Risk neutral discounting (European options only) Value option at maturity Discount expected future value (risk neutral) at the riskfree interest rate Mutiperiod extension: European option OMS 2004 Greeks

4. Data S = 100 Interest rate (cc) = 5% Volatility  = 30% European call option: Strike price X = 100, Maturity =2 months Binomial model: 2 steps Time step t = 0.0833 u = 1.0905 d= 0.9170 p = 0.5024 0 1 2Risk neutral probability 118.91 p²= 18.91 0.2524 109.05 9.46 100.00 100.00 2p(1-p)= 4.73 0.00 0.5000 91.70 0.00 84.10 (1-p)²= 0.00 0.2476 Risk neutral expected value = 4.77 Call value = 4.77 e-.05(.1667) = 4.73 Multiperiod valuation: Example OMS 2004 Greeks

5. Consider: European option on non dividend paying stock constant volatility constant interest rate Limiting case of binomial model as t0 From binomial to Black Scholes OMS 2004 Greeks

6. Convergence of Binomial Model OMS 2004 Greeks

7. 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 OMS 2004 Greeks

8. Black Scholes’ PDE and the binomial model • We have: • Binomial model: p fu + (1-p) fd = ert • Use Taylor approximation: • fu = f + (u-1) Sf’S + ½ (u–1)² S² f”SS + f’tt • fd = f + (d-1) Sf’S + ½ (d–1)² S² f”SS + f’tt • u = 1 + √t + ½ ²t • d = 1 – √t + ½ ²t • ert = 1 + rt • Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes • BS PDE : f’t + rS f’S + ½² f”SS = r f OMS 2004 Greeks

9. 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 OMS 2004 Greeks

10. 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: • 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 OMS 2004 Greeks

11. A closer look at d1 and d2 2 elements determine d1 and d2 A measure of the “moneyness” of the option.The distance between the exercise price and the stock price S0 / Ke-rt Time adjusted volatility.The volatility of the return on the underlying asset between now and maturity. OMS 2004 Greeks

12. 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 : 100  0.6585 - 100  0.95123  0.6019 = 8.60 d2 = 0.4083 – 0.15 = 0.2583 N(d2) = 0.6019 OMS 2004 Greeks

13. Relationship between call value and spot price For call option, time value > 0 OMS 2004 Greeks

14. European put option • European call option: C = S0 N(d1) – PV(K) N(d2) • Put-Call Parity: P = C – S0 + PV(K) • European put option: P = S0[N(d1)-1] + PV(K)[1-N(d2)] • P = - S0 N(-d1) +PV(K) N(-d2) Risk-neutral probability of exercising the option = Proba(ST>X) Delta of call option Risk-neutral probability of exercising the option = Proba(ST<X) Delta of put option (Remember: N(x) – 1 = N(-x) OMS 2004 Greeks

15. 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 OMS 2004 Greeks

16. Relationship between Put Value and Spot Price For put option, time value >0 or <0 OMS 2004 Greeks

17. 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 S0by 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) OMS 2004 Greeks

18. 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 OMS 2004 Greeks

19. 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 - 15.601,005 30 53.501,025 21.40 59.80 • Source: Wall Street Journal OMS 2004 Greeks

20. Fundamental determinants of option value OMS 2004 Greeks

21. Example OMS 2004 Greeks

22. The Greeks • Delta • Gamma • Theta • Vega (not a Greek) • Rho OMS 2004 Greeks

23. Delta • Sensitivity of derivative value to changes in price of underlying asset Delta = ∂f / ∂S • As a first approximation : f ~ Delta x S • In example, for call option : f = 10.451 Delta = 0.637 • If S = +1: f = 0.637→ f ~ 11.088 • If S = 101: f = 11.097 error because of convexity Binomial model: Delta = (fu – fd) / (uS – dS) European options:Delta call = e-qT N(d1)Delta put = Delta call - 1 Forward : Delta = + 1 Call : 0 < Delta < +1 Put : -1 < Delta < 0 OMS 2004 Greeks

24. Calculation of delta OMS 2004 Greeks

25. Variation of delta with the stock price for a call OMS 2004 Greeks

26. Delta and maturity OMS 2004 Greeks

27. Delta hedging • Suppose that you have sold 1 call option (you are short 1 call) • How many shares should you buy to hedge you position? • The value of your portfolio is: V = nS – C • If the stock price changes, the value of your portfolio will also change. V = n S - C • You want to compensate any change in the value of the shorted option by a equal change in the value of your stocks. • For “small” S : C = Delta S • V = 0 ↔ n = Delta OMS 2004 Greeks

28. Effectiveness of Delta hedging OMS 2004 Greeks

29. Gamma • A measure of convexity Gamma = ∂Delta / ∂S = ∂²f / ∂S² • Taylor: df = f’S dS + ½ f”SS dS² • Translated into derivative language: • f = Delta S + ½ Gamma S² • In example, for call : f = 10.451 Delta = 0.637 Gamma = 0.019 • If S = +1: f = 0.637 + ½ 0.019→ f ~ 11.097 • If S = 101: f = 11.097 OMS 2004 Greeks

30. Variation of Gamma with the stock price OMS 2004 Greeks

31. Gamma and maturity OMS 2004 Greeks

32. Gamma hedging • Back to previous example. • We have a delta neutral portfolio: • Short 1 call option • Long Delta = 0.637 shares • The Gamma of this portfolio is equal to the gamma of the call option: • V = nS – C →∂V²/∂S² = - Gammacall • To make the position gamma neutral we have to include a traded option with a positive gamma. To keep delta neutrality we have to solve simultaneously 2 equations: • Delta neutrality • Gamma neutrality OMS 2004 Greeks

33. Theta • Measure time evolution of asset Theta = - ∂f / ∂T • (the minus sign means maturity decreases with the passage of time) • In example, Theta of call option = - 6.41 • Expressed per day: Theta = - 6.41 / 365 = -0.018 (in example) • Theta = -6.41 / 252 = - 0.025 (as in Hull) OMS 2004 Greeks

34. Variation of Theta with the stock price OMS 2004 Greeks

35. Relation between delta, gamma, theta • Remember PDE: Gamma Theta Delta OMS 2004 Greeks

36. Trading strategies • A single option and a stock: covered call, protective put • * Covered call: S-C • * Protective put: S+P • Spreads: bull, bear, butterfly, calendar • Bull: +C(X1) – C(X2) X1<X2 • Bear: +C(X1) – C(X2) X1>X2 • Butterfly: +C(X1) + C(X3) – 2C(X2) X1<X2<X3 • Calendar: +C(T1)-C(T2) T1>T2 • Combinations: straddle, strips and straps, strangle • Straddle: +C+P • Strip: +C + 2P • Strap: +2C+P • Strangle: +C(X2)+P(X1) X1<X2 OMS 2004 Greeks

37. Protective Put OMS 2004 Greeks

38. (See Lehman Brother – Equity Linked Note: An Introduction) Equity Linked Note Capital garantee Equity Linked Note Bond = = + + Equity Participation Call option OMS 2004 Greeks

39. Equity Linked Note: Example • 5-year 100% principal protected ELN with 100% participation in the upside of the S&P 500 index. • See Excel file. OMS 2004 Greeks

40. Covered Call Profit At maturity Immediate Stock Price OMS 2004 Greeks

41. Reverse Convertible • Robeco: Eerste Reverse Convertible op beleggingsfonds • Van 17 februari tot 6 maart 2003 uur is het mogelijk in te schrijven op de Robeco Reverse Convertible op Robeco N.V. mrt 03/04 (Robeco Reverse Convertible), uitgebracht door Rabo Securities in samenwerking met Robeco. • De Robeco Reverse Convertible is een obligatielening met een looptijd van één jaar waarop een couponrentevan 9% wordt gegeven, hoger dan een gewone éénjaarslening. De uitgevende instelling, Rabo Securities N.V., heeft aan het einde van de looptijd de keuze om de obligatie af te lossen in contanten of af te lossen in een van tevoren vastgesteld aantal aandelen in het beleggingsfonds Robeco. Dit is afhankelijk van de koers van het aandeel Robeco N.V. Bijzondere omstandigheden daargelaten, zal Rabo Securities kiezen voor een aflossing in aandelen als de koers aan het einde van de looptijd lager is dan die op 7 maart 2003. Het aantal aandelen is gelijk aan de nominale inleg gedeeld door de openingskoers van Robeco op 7 maart 2003. Hierdoor bestaat het risico voor de belegger aan het einde van de looptijd aandelen Robeco te ontvangen, die een lagere waarde vertegenwoordigen dan de nominale inleg. Is de koers per saldo gelijk gebleven of gestegen, dan wordt de nominale inleg in contanten teruggegeven. • . OMS 2004 Greeks

42. Portfolio insurance • Use synthetic put option with dynamic hedging • V = S + P same value as with put • ΔV = ΔS + ΔP same sensitivity to underlying asset • = (1 + δPut) ΔS • V = n S + B n shares + bond • 1 + δPut = n • Dynamic hedging • LOR and the crash of October 19, 1987: see Rubinstein 1999 • Illustration: Excell worksheet PorfolioInsurance OMS 2004 Greeks

43. Bull Call Spread OMS 2004 Greeks

44. Bear Call Spread OMS 2004 Greeks

45. Butterfly OMS 2004 Greeks

46. Straddle OMS 2004 Greeks

47. Strip OMS 2004 Greeks

48. Strap OMS 2004 Greeks

49. Strangle OMS 2004 Greeks