Binnenlandse Francqui Leerstoel
VUB 2004-2005
2. Options and investments
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 rt  d
f 
with p 
rt
ud
e
August 23, 2004
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Mutiperiod extension: European option
•
(European and American options)
u²S
uS
S
udS
dS
d²S
Recursive method
•
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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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
100.00
4.73
100.00 2p(1-p)=
0.00
0.5000
91.70
0.00
84.10
0.00
(1-p)²=
0.2476
Risk neutral expected value = 4.77
Call value = 4.77 e-.05(.1667) = 4.73
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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
Stock price
t
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T
|5
Time
Convergence of Binomial Model
Convergence of Binomial Model
12.00
10.00
Option value
8.00
6.00
4.00
2.00
97
Number of steps
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|6
100
94
91
88
85
82
79
76
73
70
67
64
61
58
55
52
49
46
43
40
37
34
31
28
25
22
19
16
13
10
7
4
1
0.00
Understanding the PDE
• Assume we are in a risk neutral world
f
f 1  f 2 2
 rS

 S  rf
2
t
S 2 S
2
Change of the
value with
respect to time
August 23, 2004
Change of the value
with respect to the
price of the
underlying asset
OMS 2004 Greeks
Expected change
of the value of
derivative
security
Change of the
value with
respect to
volatility
|7
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) 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 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
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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 (d1 )
d1 
ln( S 0 / Ke  rT )
 T
 0.5 T
d 2  d1   T
N(x) = cumulative probability distribution function for a standardized normal variable
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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
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A closer look at d1 and d2
d1 
ln( S 0 / Ke  rT )
 T
d 2  d1   T
 0.5 T
2 elements determine d1 and d2
S0 /
Ke-rt
 T
August 23, 2004
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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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
d2 = 0.4083 – 0.15 = 0.2583
N(d2) = 0.6019
August 23, 2004
European call :
100  0.6585 - 100  0.95123  0.6019
= 8.60
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Relationship between call value and spot price
For call option,
time value > 0
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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)
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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
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Relationship between Put Value and Spot Price
For put option, time
value >0 or <0
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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)
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Black Scholes Merton with constant dividend
yield
The partial differential
equation:
(See Hull 5th ed. Appendix 13A)
f
f 1  2 f 2 2
 (r  q) S

 S  rf
2
t
S 2 S
Expected growth rate of stock
Call option
C  S 0 e  qT  N (d1 )  Ke  rT  N (d 2 )
Put option
P  Ke  rT N (d 2 )  S 0 e  qT  N (d1 )
d1 
August 23, 2004
ln( S 0 e  qT / Ke  rT )
 T
 0.5 T
d 2  d1   T
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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
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Fundamental determinants of option value
Current asset price S
Delta
Striking price K
Interest rate r
Rho
Dividend yield q
Call value
Put Value

0 < Delta < 1

- 1 < Delta < 0






Time-to-maturity T
Theta

?
Volatility
Vega


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Example
BLACK-SCHOLES OPTION PRICING FORMULA
Stock price
Dividend yield
Striking price
Maturity (days)
Interest rate
Volatility
Price
Delta
Gamma
Theta (per day)
Elasticity
Vega
Rho
August 23, 2004
100
0.00%
100
365
5.00%
20.00%
Call
10.451
0.637
0.019
-0.018
6.094
0.375
0.532
Put
5.574
-0.363
0.019
-0.005
-6.516
0.375
-0.419
A.Farber
Call
Put
Decomposition of value
Intrinsic val.
0.00
Time value
4.88
Insurance
5.57
0.00
-4.88
10.45
BS partial differential equation
Theta
-6.41
-1.66
(r-q)SDelta
3.18
-1.82
0.5²S²Gamma
3.75
3.75
rf
0.52
0.28
Put-Call Parity
Call Value
+ PV(Strike)
= S * exp(-qT)
+ Put
OMS 2004 Greeks
10.45
95.12 105.57
100.00
5.57 105.57
|21
The Greeks
•
f
S
Delta
Delta 
•
Gamma
Gamma 
•
Theta
f
Theta 
T
•
Vega (not a Greek)
Vega 
f

•
Rho
Rho 
f
r
August 23, 2004
² f
S 2
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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
Forward : Delta = + 1
Call : 0 < Delta < +1
Put : -1 < Delta < 0
August 23, 2004
Binomial model: Delta = (fu – fd) / (uS – dS)
European options:
Delta call = e-qT N(d1)
Delta put = Delta call - 1
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Calculation of delta
August 23, 2004
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Variation of delta with the stock price for a call
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Delta and maturity
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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
August 23, 2004
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Effectiveness of Delta hedging
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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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Variation of Gamma with the stock price
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Gamma and maturity
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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
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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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Variation of Theta with the stock price
August 23, 2004
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Relation between delta, gamma, theta
• Remember PDE:
f
f 1  f 2 2
 rS

 S  rf
2
t
S 2 S
2
Theta
August 23, 2004
Delta
Gamma
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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
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Protective Put
Maturity
Stock
Call
Call
Call
Put
Put
Put
Prot.put
950
1000
1050
950
1000
1050
0.25
0.25
0.25
0.25
0.25
0.25
Prot.put
1
0
0
0
0
1
0
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
1055.90
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.55
200.00
150.00
100.00
50.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-50.00
-100.00
August 23, 2004
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Equity Linked Note
• (See Lehman Brother – Equity Linked Note: An Introduction)
Capital
garantee
Bond
Equity
+
Call option
August 23, 2004
=
Linked
=
Note
+
Equity
Participation
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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.
August 23, 2004
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Covered Call
Covered
call
1
0
-1
0
0
0
0
Maturity
Stock
Call
950
Call
1000
Call
1050
Put
950
Put
1000
Put
1050
Covered call
0.25
0.25
0.25
0.25
0.25
0.25
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
936.63
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.45
100.00
Profit
At maturity
50.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
Immediate
-50.00
-100.00
-150.00
-200.00
August 23, 2004
Stock Price
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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.
.
August 23, 2004
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Portfolio insurance
• Use synthetic put option with dynamic hedging
• V=S+P
• ΔV = ΔS + ΔP
•
= (1 + δPut) ΔS
same value as with put
same sensitivity to underlying asset
• V=nS+B
n shares + bond
• 1 + δPut = n
• Dynamic hedging
• LOR and the crash of October 19, 1987: see Rubinstein 1999
• Illustration: Excell worksheet PorfolioInsurance
August 23, 2004
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Bull Call Spread
Bull spread
Maturity
Stock
Call
950
Call
1000
Call
1050
Put
950
Put
1000
Put
1050
Bull spread
0
1
0
-1
0
0
0
0.25
0.25
0.25
0.25
0.25
0.25
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
48.76
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.26
60.00
40.00
20.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-20.00
-40.00
-60.00
August 23, 2004
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Bear Call Spread
Maturity
Stock
Call
950
Call
1000
Call
1050
Put
950
Put
1000
Put
1050
Bear spread
0.25
0.25
0.25
0.25
0.25
0.25
Bear
spread
0
-1
0
1
0
0
0
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
-48.76
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
(0.26)
60.00
40.00
20.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-20.00
-40.00
-60.00
August 23, 2004
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Butterfly
Maturity Butterfly spreadPrice
Stock
0 1,000.00
Call
950
0.25
1
91.02
Call
1000
0.25
-2
63.37
Call
1050
0.25
1
42.26
Put
950
0.25
0
33.92
Put
1000
0.25
0
55.90
Put
1050
0.25
0
84.42
Butterfly spread
6.54
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.00
50.00
40.00
30.00
20.00
10.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-10.00
August 23, 2004
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Straddle
Maturity
Stock
Call
Call
Call
Put
Put
Put
Straddle
950
1000
1050
950
1000
1050
Straddle
0
0
1
0
0
1
0
0.25
0.25
0.25
0.25
0.25
0.25
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
119.27
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.10
150.00
100.00
50.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-50.00
-100.00
-150.00
August 23, 2004
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Strip
Maturity
Stock
Call
Call
Call
Put
Put
Put
Strip
950
1000
1050
950
1000
1050
0.25
0.25
0.25
0.25
0.25
0.25
Strip
0
0
1
0
0
2
0
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
175.17
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
(0.35)
250.00
200.00
150.00
100.00
50.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-50.00
-100.00
-150.00
-200.00
August 23, 2004
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Strap
Maturity
Stock
Call
Call
Call
Put
Put
Put
Strap
950
1000
1050
950
1000
1050
0.25
0.25
0.25
0.25
0.25
0.25
Strap
0
0
2
0
0
1
0
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
182.64
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.65
300.00
250.00
200.00
150.00
100.00
50.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-50.00
-100.00
-150.00
-200.00
-250.00
August 23, 2004
OMS 2004 Greeks
|48
Strangle
Maturity
Stock
Call
Call
Call
Put
Put
Put
Strangle
950
1000
1050
950
1000
1050
0.25
0.25
0.25
0.25
0.25
0.25
Strangle
0
0
0
1
1
0
0
Price
1,000.00
91.02
63.37
42.26
33.92
55.90
84.42
76.19
Delta
1.00
0.68
0.55
0.42
-0.32
-0.45
-0.58
0.10
120.00
100.00
80.00
60.00
40.00
20.00
0.00
800
850
900
950
1000
1050
1100
1150
1200
-20.00
-40.00
-60.00
-80.00
-100.00
August 23, 2004
OMS 2004 Greeks
|49