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

Binnenlandse Francqui Leerstoel VUB 2004-20052. Options and investments

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

OMS 2004 Greeks |2August 23, 2004

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

du

dep

e

fpfpf

tr

trdu

with )1(


Mutiperiod extension: European option

u²SuS

S udS

dS

d²S

• Recursive method (European and American options)

Value option at maturityWork backward through the tree.

Apply 1-period binomial formula at each node

• Risk neutral discounting(European options only)

Value option at maturityDiscount expected future value

(risk neutral) at the riskfree interest rate


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 probability118.91 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 = 4.77Call value = 4.77 e-.05(.1667) = 4.73


From binomial to Black Scholes

• Consider:

• European option

• on non dividend paying stock

• constant volatility

• constant interest rate

• Limiting case of binomial model as t0

Stock price

Timet T


Convergence of Binomial Model

Convergence of Binomial Model

0.00

2.00

4.00

6.00

8.00

10.00

12.00

1 4 7 10

13

16

19

22

25

28

31

34

37

40

43

46

49

52

55

58

61

64

67

70

73

76

79

82

85

88

91

94

97

100

Number of steps

Op

tio

n v

alu

e


Understanding the PDE

• Assume we are in a risk neutral world

rfSS

f

S

frS

t

f 22

2

2

2

1

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


Black Scholes’ PDE and the binomial model

• We have:

• 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

• BS PDE : f’t + rS f’S + ½ ² f”SS = r f


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

)()( 210 dNKedNSC rT Call option:

Put option: )()( 102 dNSdNKeP rT

TT

KeSd

rT

5.0)/ln( 0

1

Tdd 12

N(x) = cumulative probability distribution function for a standardized normal variable


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

)()( 210 dNKedNSC rT

N(d2) = risk-neutral probability that the option will be exercised at maturity


A closer look at d1 and d2

TT

KeSd

rT

5.0)/ln( 0

1

Tdd 12

2 elements determine d1 and d2

S0 / Ke-rtA measure of the “moneyness” of the option.The distance between the exercise price and the stock price

TTime adjusted volatility.The volatility of the return on the underlying asset between now and maturity.


Example

Stock price S0 = 100Exercise price K = 100 (at the money option)Maturity T = 1 yearInterest 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

d2 = 0.4083 – 0.15 = 0.2583

N(d2) = 0.6019

European call :100 0.6585 - 100 0.95123 0.6019 = 8.60


Relationship between call value and spot price

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50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

Intrinsic value

Time value

Premium

For call option, time value > 0


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)

Delta of call option Risk-neutral probability of exercising the option = Proba(ST>X)

Delta of put option Risk-neutral probability of exercising the option = Proba(ST<X)

(Remember: N(x) – 1 = N(-x)


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


Relationship between Put Value and Spot Price

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50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

Intrinsic value

Time value

For put option, time value >0 or <0


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)


Black Scholes Merton with constant dividend yield

rfSS

f

S

fSqr

t

f 22

2

2

2

1)(

)()( 210 dNKedNeSC rTqT

)()( 102 dNeSdNKeP qTrT

The partial differential equation:(See Hull 5th ed. Appendix 13A)

Expected growth rate of stock

Call option

Put option

TT

KeeSd

rTqT

5.0)/ln( 0

1 Tdd 12


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


Fundamental determinants of option value

Call value Put Value

Current asset price S

Delta

0 < Delta < 1

- 1 < Delta < 0

Striking price K

Interest rate r Rho

Dividend yield q

Time-to-maturity T Theta ?

Volatility Vega


Example

BLACK-SCHOLES OPTION PRICING FORMULA A.Farber

Stock price 100 Call PutDividend yield 0.00% Decomposition of valueStriking price 100 Intrinsic val. 0.00 0.00Maturity (days) 365 Time value 4.88 -4.88Interest rate 5.00% Insurance 5.57 10.45Volatility 20.00%

BS partial differential equationTheta -6.41 -1.66

Call Put (r-q)SDelta 3.18 -1.82Price 10.451 5.574 0.5²S²Gamma 3.75 3.75Delta 0.637 -0.363 rf 0.52 0.28Gamma 0.019 0.019Theta (per day) -0.018 -0.005 Put-Call ParityElasticity 6.094 -6.516 Call Value 10.45Vega 0.375 0.375 + PV(Strike) 95.12 105.57Rho 0.532 -0.419 = S * exp(-qT) 100.00

+ Put 5.57 105.57


The Greeks

• Delta

• Gamma

• Theta

• Vega (not a Greek)

• Rho

S

fDelta

2

²

S

fGamma

T

fTheta

f

Vega

r

fRho


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 = + 1Call : 0 < Delta < +1Put : -1 < Delta < 0


Calculation of delta

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50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

Delta


Variation of delta with the stock price for a call

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0.100

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0.400

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0.700

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0.900

1.000

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Stock price


Delta and maturity

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0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price

30 days 182 days 365 days


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


Effectiveness of Delta hedging

-1.200

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Stock price


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


Variation of Gamma with the stock price

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0.005

0.010

0.015

0.020

0.025

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price


Gamma and maturity

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0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

60 70 80 90 100 110 120 130 140 150

Stock price

90 days 182 days 365 days


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 = - 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


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)


Variation of Theta with the stock price

-0.020

-0.018

-0.016

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Stock price


Relation between delta, gamma, theta

• Remember PDE:

f

trS

f

S

f

SS rf

1

2

2

22 2

Theta Delta Gamma


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(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

3. Combinations: straddle, strips and straps, strangle

• Straddle: +C+P

• Strip: +C + 2P

• Strap: +2C+P

• Strangle: +C(X2)+P(X1) X1<X2


Protective PutMaturity Prot.put Price Delta

Stock 1 1,000.00 1.00Call 950 0.25 0 91.02 0.68Call 1000 0.25 0 63.37 0.55Call 1050 0.25 0 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 1 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Prot.put 1055.90 0.55

-100.00

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800 850 900 950 1000 1050 1100 1150 1200


Equity Linked Note

• (See Lehman Brother – Equity Linked Note: An Introduction)

Bond

Call option

+Equity

Linked

Note

Capital garantee

Equity Participation

+= =


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.


Covered Call

MaturityCovered

call Price DeltaStock 1 1,000.00 1.00Call 950 0.25 0 91.02 0.68Call 1000 0.25 -1 63.37 0.55Call 1050 0.25 0 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 0 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Covered call 936.63 0.45

-200.00

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-100.00

-50.00

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50.00

100.00

800 850 900 950 1000 1050 1100 1150 1200

Profit

Stock Price

Immediate

At maturity


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.

• .


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


Bull Call Spread

MaturityBull spread

Price DeltaStock 0 1,000.00 1.00Call 950 0.25 1 91.02 0.68Call 1000 0.25 0 63.37 0.55Call 1050 0.25 -1 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 0 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Bull spread 48.76 0.26

-60.00

-40.00

-20.00

0.00

20.00

40.00

60.00

800 850 900 950 1000 1050 1100 1150 1200


Bear Call Spread

MaturityBear

spread Price DeltaStock 0 1,000.00 1.00Call 950 0.25 -1 91.02 0.68Call 1000 0.25 0 63.37 0.55Call 1050 0.25 1 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 0 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Bear spread -48.76 (0.26)

-60.00

-40.00

-20.00

0.00

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60.00

800 850 900 950 1000 1050 1100 1150 1200


Butterfly

Maturity Butterfly spreadPrice DeltaStock 0 1,000.00 1.00Call 950 0.25 1 91.02 0.68Call 1000 0.25 -2 63.37 0.55Call 1050 0.25 1 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 0 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Butterfly spread 6.54 0.00

-10.00

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50.00

800 850 900 950 1000 1050 1100 1150 1200


Straddle

Maturity Straddle Price DeltaStock 0 1,000.00 1.00Call 950 0.25 0 91.02 0.68Call 1000 0.25 1 63.37 0.55Call 1050 0.25 0 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 1 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Straddle 119.27 0.10

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

800 850 900 950 1000 1050 1100 1150 1200


Strip

Maturity Strip Price DeltaStock 0 1,000.00 1.00Call 950 0.25 0 91.02 0.68Call 1000 0.25 1 63.37 0.55Call 1050 0.25 0 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 2 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Strip 175.17 (0.35)

-200.00

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-100.00

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800 850 900 950 1000 1050 1100 1150 1200


StrapMaturity Strap Price Delta

Stock 0 1,000.00 1.00Call 950 0.25 0 91.02 0.68Call 1000 0.25 2 63.37 0.55Call 1050 0.25 0 42.26 0.42Put 950 0.25 0 33.92 -0.32Put 1000 0.25 1 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Strap 182.64 0.65

-250.00

-200.00

-150.00

-100.00

-50.00

0.00

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800 850 900 950 1000 1050 1100 1150 1200


Strangle

Maturity Strangle Price DeltaStock 0 1,000.00 1.00Call 950 0.25 0 91.02 0.68Call 1000 0.25 0 63.37 0.55Call 1050 0.25 1 42.26 0.42Put 950 0.25 1 33.92 -0.32Put 1000 0.25 0 55.90 -0.45Put 1050 0.25 0 84.42 -0.58Strangle 76.19 0.10

-100.00

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

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