Download DEXIA « Impact Seminar

Options and Speculative Markets
2004-2005
Inside Black Scholes
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
OMS 08 Inside Black Scholes
|2
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
August 23, 2004
OMS 08 Inside Black Scholes
|3
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:
G
G 1  2 G 2
G
dG  (
a 
  2  b )  dt 
 b  dz
x
t 2 x
x
Drift
August 23, 2004
Volatility
OMS 08 Inside Black Scholes
|4
Ito’s lemna: some intuition
• If x is a real variable, applying Taylor:
G
G
1 2G 2 2G
1 2G 2
G 
x 
t 
x 
x  t 
t ..
x
t
2 x 2
xt
2 t 2
• In ordinary calculus:
• In stochastic calculus:
dG 
G
G
dx 
dt
x
t
dG 
An approximation
dx², dt², dx dt negligeables
G
G
1  ²G
dx 
dt 
dx ²
x
t
2 x ²
• Because, if x follows an Ito process, dx² = b² dt you have to keep it
August 23, 2004
OMS 08 Inside Black Scholes
|5
Lognormal property of stock prices
• Suppose:
• Using Ito’s lemna:
dS=  S dt +  S dz
d ln(S) = ( - 0.5 ²) dt +  dz
• Consequence:
ln(ST) – ln(S0) = ln(ST/S0)
ln( S T )  ln( S 0 ) ~ N [(  
ln( S T ) ~ N [ln( S 0 )  (  
August 23, 2004
²
²
2
2
)T ,  T ]
)T ,  T ]
Continuously compounded
return between 0 and T
ln(ST) is normally distributed
so that ST has a lognormal
distribution
OMS 08 Inside Black Scholes
|6
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:
f
f 1  2 f
f
df  (    S 
  2   2  S 2 )  dt 
   S  dz
S
t 2 S
S
• 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
• -1
sell (short) one derivative
• fS = ∂f /∂S
buy (long) DELTA shares
• Value of portfolio: V = - f + fS S
August 23, 2004
OMS 08 Inside Black Scholes
|7
Here comes the PDE!
• Using Ito’s lemna
f 1  2 f 2 2
dV  ( 

 S )dt
2
t 2 S
• This is a riskless portfolio!!!
• Its expected return should be equal to the risk free interest rate:
dV = r V dt
• This leads to:
f
f 1  2 f 2 2
 rS

 S  rf
2
t
S 2 S
August 23, 2004
OMS 08 Inside Black Scholes
|8
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 08 Inside Black Scholes
Expected change
of the value of
derivative
security
Change of the
value with
respect to
volatility
|9
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 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
August 23, 2004
OMS 08 Inside Black Scholes
|10
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
August 23, 2004
OMS 08 Inside Black Scholes
|11
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
August 23, 2004
OMS 08 Inside Black Scholes
|12
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 /
 T
August 23, 2004
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.
OMS 08 Inside Black Scholes
|13
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
OMS 08 Inside Black Scholes
|14
Relationship between call value and spot price
For call option,
time value > 0
August 23, 2004
OMS 08 Inside Black Scholes
|15
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)
August 23, 2004
OMS 08 Inside Black Scholes
|16
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
August 23, 2004
OMS 08 Inside Black Scholes
|17
Relationship between Put Value and Spot Price
For put option, time
value >0 or <0
August 23, 2004
OMS 08 Inside Black Scholes
|18
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)
August 23, 2004
OMS 08 Inside Black Scholes
|19
Dividend paying stock: binomial model
t = 1 u = 1.25, d = 0.80
r = 5% q = 3%
Derivative: Call K = 100
uS0 eqt with dividends reinvested
128.81
uS0
S0
100
dS0 eqt with dividends reinvested
82.44
ex dividend
fd
0
80
Replicating portfolio:
 uS0 eqt + M ert = fu
 128.81 + M 1.0513 = 25
 dS0 eqt + M ert = fd
 82.44 + M 1.0513 = 0
August 23, 2004
fu
25
125
dS0
f =  S0 + M
ex dividend
f = [ p fu + (1-p) fd] e-rt = 11.64
p = (e(r-q)t – d) / (u – d) = 0.489
 = (fu – fd) / (u – d )S0eqt = 0.539
OMS 08 Inside Black Scholes
|20
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
OMS 08 Inside Black Scholes
|21
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
August 23, 2004
OMS 08 Inside Black Scholes
|22
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
August 23, 2004
OMS 08 Inside Black Scholes
|23
Option on futures: binomial model
uF0 → fu
Futures price F0
dF0 →fd
Replicating portfolio:  futures + cash
 
fu  fd
uF0  dF0
f 
pf u  (1  p) f d
e rt
 (uF0 – F0) + M ert = fu
 (dF0 – F0) + M
ert
= fd
f=M
August 23, 2004
p
OMS 08 Inside Black Scholes
1 d
ud
|24
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
pf u  (1  p) f d
f 
e rt
1 d
p
ud
Dividend paying stock
pf u  (1  p) f d
f 
e rt
e ( r  q ) t  d
p
ud
Futures prices behave in the same way as a stock paying a
continuous dividend yield at the risk-free interest rate r
August 23, 2004
OMS 08 Inside Black Scholes
|25
Black’s model
Assumption: futures price has lognormal distribution
Ce
 rT
[ F0 N (d1 )  KN (d 2 )]
F0
)
X  0.5 T
d1 
 T
ln(
August 23, 2004
P  e  rT [ KN (d 2 )  F0 N (d1 )]
F0
)
X
d2 
 0.5 T  d1   T
 T
ln(
OMS 08 Inside Black Scholes
|26
Implied volatility – Call option
August 23, 2004
OMS 08 Inside Black Scholes
|27
Implied volatility – Put option
August 23, 2004
OMS 08 Inside Black Scholes
|28
Smile
SPX Option on S&P 500
September 2002 Contract
July 2, 2002
Maturity
Strike
968.25
2%
1.86%
90 days
Call
Put
OpenInt
700
750
800
900
925
950
975
980
990
995
1005
1025
1040
1050
1075
1100
1125
1150
1200
August 23, 2004
Spot index
DivYield
IntRate
3599
3228
11806
5404
9232
2286
11145
8726
23170
7556
18173
7513
Price
42
40
34.5
30
21.4
15.1
13.1
7.5
4.6
2.4
1.6
0.45
ImpVol
24.89%
26.04%
24.17%
23.73%
22.47%
20.97%
21.07%
19.97%
19.82%
19.16%
19.67%
19.33%
OpenInt
Price
ImpVol
3801
1581
31675
21723
7799
17419
16603
4994
3193
23345
5209
15242
1.5
2.9
4
15.6
19
28
33
40.3
41
46
53.5
59.8
34.19%
31.59%
26.84%
22.17%
19.54%
19.16%
15.32%
17.68%
14.86%
15.84%
16.29%
9.95%
OMS 08 Inside Black Scholes
|29