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Binnenlandse Francqui Leerstoel VUB 2004-2005 2. Options and investments Professor André Farber Solvay Business School Université Libre de Bruxelles

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August 23, 2004 OMS 2004 Greeks |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

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August 23, 2004 OMS 2004 Greeks |3 Mutiperiod extension: European option u²S uS SudS 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

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August 23, 2004 OMS 2004 Greeks |4 Multiperiod valuation: Example 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 2 Risk neutral probability 118.91 p²= 18.91 0.2524 109.05 9.46 100.00100.00 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

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August 23, 2004 OMS 2004 Greeks |5 From binomial to Black Scholes Consider: European option on non dividend paying stock constant volatility constant interest rate Limiting case of binomial model as t 0

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August 23, 2004 OMS 2004 Greeks |6 Convergence of Binomial Model

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August 23, 2004 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

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August 23, 2004 OMS 2004 Greeks |8 Black Scholes’ PDE and the binomial model We have: 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 BS PDE : f’ t + rS f’ S + ½ ² f” SS = r f

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August 23, 2004 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

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August 23, 2004 OMS 2004 Greeks |10 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

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August 23, 2004 OMS 2004 Greeks |11 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.

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August 23, 2004 OMS 2004 Greeks |12 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

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August 23, 2004 OMS 2004 Greeks |13 Relationship between call value and spot price For call option, time value > 0

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August 23, 2004 OMS 2004 Greeks |14 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

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August 23, 2004 OMS 2004 Greeks |15 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

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August 23, 2004 OMS 2004 Greeks |16 Relationship between Put Value and Spot Price For put option, time value >0 or <0

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August 23, 2004 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 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)

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August 23, 2004 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

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August 23, 2004 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 StrikeCallPut 900 - 15.60 1,0053053.50 1,02521.4059.80 Source: Wall Street Journal

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August 23, 2004 OMS 2004 Greeks |20 Fundamental determinants of option value Call valuePut Value Current asset price S Delta 0 < Delta < 1 - 1 < Delta < 0 Striking price K Interest rate rRho Dividend yield q Time-to-maturity TTheta ? VolatilityVega

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August 23, 2004 OMS 2004 Greeks |21 Example

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August 23, 2004 OMS 2004 Greeks |22 The Greeks Delta Gamma Theta Vega (not a Greek) Rho

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August 23, 2004 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 = (f u – f d ) / (uS – dS) European options: Delta call = e -qT N(d 1 ) Delta put = Delta call - 1 Forward : Delta = + 1 Call : 0 < Delta < +1 Put : -1 < Delta < 0

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August 23, 2004 OMS 2004 Greeks |24 Calculation of delta

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August 23, 2004 OMS 2004 Greeks |25 Variation of delta with the stock price for a call

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August 23, 2004 OMS 2004 Greeks |26 Delta and maturity

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August 23, 2004 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 = n S – 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

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August 23, 2004 OMS 2004 Greeks |28 Effectiveness of Delta hedging

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August 23, 2004 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

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August 23, 2004 OMS 2004 Greeks |30 Variation of Gamma with the stock price

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August 23, 2004 OMS 2004 Greeks |31 Gamma and maturity

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August 23, 2004 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 = n S – C →∂V²/∂S² = - Gamma call 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

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August 23, 2004 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)

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August 23, 2004 OMS 2004 Greeks |34 Variation of Theta with the stock price

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August 23, 2004 OMS 2004 Greeks |35 Relation between delta, gamma, theta Remember PDE: ThetaDelta Gamma

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August 23, 2004 OMS 2004 Greeks |36 Trading strategies 1.A single option and a stock: covered call, protective put * Covered call: S-C * Protective put: S+P 2.Spreads: bull, bear, butterfly, calendar Bull: +C(X 1 ) – C(X 2 ) X 1

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August 23, 2004 OMS 2004 Greeks |37 Protective Put

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August 23, 2004 OMS 2004 Greeks |38 Equity Linked Note (See Lehman Brother – Equity Linked Note: An Introduction) Bond Call option + Equity Linked Note Capital garantee Equity Participation + = =

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August 23, 2004 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.

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August 23, 2004 OMS 2004 Greeks |40 Covered Call Profit Stock Price Immediate At maturity

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August 23, 2004 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 couponrente van 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..

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August 23, 2004 OMS 2004 Greeks |42 Portfolio insurance Use synthetic put option with dynamic hedging V = S + Psame value as with put ΔV = ΔS + ΔPsame sensitivity to underlying asset = (1 + δ Put ) ΔS V = n S + Bn shares + bond 1 + δ Put = n Dynamic hedging LOR and the crash of October 19, 1987: see Rubinstein 1999 Illustration: Excell worksheet PorfolioInsurance

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August 23, 2004 OMS 2004 Greeks |43 Bull Call Spread

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August 23, 2004 OMS 2004 Greeks |44 Bear Call Spread

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August 23, 2004 OMS 2004 Greeks |45 Butterfly

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August 23, 2004 OMS 2004 Greeks |46 Straddle

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August 23, 2004 OMS 2004 Greeks |47 Strip

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August 23, 2004 OMS 2004 Greeks |48 Strap

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August 23, 2004 OMS 2004 Greeks |49 Strangle

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