# Derivatives Inside Black Scholes by ulf16328

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```									Derivatives
Inside Black Scholes

Professor André Farber
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 
e rt                     ud

July 9, 2010                      Derivatives 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?

July 9, 2010                 Derivatives 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                               Volatility

July 9, 2010                      Derivatives 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
An approximation
G      G                        dx², dt², dx dt negligeables
• In ordinary calculus:        dG       dx     dt
x      t

G      G      1  ²G
• In stochastic calculus:         dG          dx     dt         dx²
x      t      2 x ²

• Because, if x follows an Ito process, dx² = b² dt you have to keep it

July 9, 2010                      Derivatives 08 Inside Black Scholes                       |5
Lognormal property of stock prices

• Suppose:                             dS=  S dt +  S dz
• Using Ito’s lemna:                   d ln(S) = ( - 0.5 ²) dt +  dz

• Consequence:
ln(ST) – ln(S0) = ln(ST/S0)
²                                 Continuously compounded
ln( S T )  ln( S 0 ) ~ N [(            )T ,  T ]
2                                 return between 0 and T

²                                     ln(ST) is normally distributed
ln( S T ) ~ N [ln(S 0 )  (          )T ,  T ]                        so that ST has a lognormal
2
distribution

July 9, 2010                                    Derivatives 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

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Here comes the PDE!

• Using Ito’s lemna

f 1  2 f 2 2
dV  (             S ) dt
t 2 S  2

• This is a riskless portfolio!!!
• Its expected return should be equal to the risk free interest rate:
dV = r V dt

f      f 1  2 f 2 2
 rS            S  rf
t      S 2 S  2

July 9, 2010                     Derivatives 08 Inside Black Scholes    |8
Understanding the PDE

• Assume we are in a risk neutral world
Expected change
f      f 1  f 2 2  2
of the value of
 rS           S  rf
t      S 2 S 2                                    derivative
security
Change of the
value with
respect to time     Change of the value                          Change of the
with respect to the                          value with
price of the                                 respect to
underlying asset                             volatility

July 9, 2010                       Derivatives 08 Inside Black Scholes                   |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

July 9, 2010                    Derivatives 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 (  d 1 )

ln( S 0 / Ke  rT )
d1                           0.5 T                    d 2  d1   T
 T
N(x) = cumulative probability distribution function for a standardized normal variable

July 9, 2010                                  Derivatives 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
July 9, 2010                         Derivatives 08 Inside Black Scholes      |12
A closer look at d1 and d2

ln( S 0 / Ke  rT )                              d 2  d1   T
d1                           0.5 T
 T
2 elements determine d1 and d2

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

 T
The volatility of the return on
the underlying asset between
now and maturity.

July 9, 2010                                  Derivatives 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
European call :
d2 = 0.4083 – 0.15 = 0.2583
100  0.6585 - 100  0.95123  0.6019
N(d2) = 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)

July 9, 2010                     Derivatives 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

July 9, 2010                             Derivatives 08 Inside Black Scholes   |17
Relationship between Put Value and Spot Price

For put option, time
value >0 or <0

July 9, 2010            Derivatives 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)

July 9, 2010                    Derivatives 08 Inside Black Scholes     |19
Dividend paying stock: binomial model

t = 1 u = 1.25, d = 0.80
r = 5% q = 3%                       uS0 eqt with dividends reinvested
Derivative: Call K = 100            128.81
uS0            ex dividend                       fu
125                                              25
S0
dS0 eqt with dividends reinvested
100
82.44
fd
dS0             ex dividend                          0
f =  S0 + M                80
f = [ p fu + (1-p) fd] e-rt = 11.64
Replicating portfolio:
 uS0 eqt + M ert = fu
 128.81 + M 1.0513 = 25                        p = (e(r-q)t – d) / (u – d) = 0.489
 dS0 eqt + M ert = fd
 82.44 + M 1.0513 = 0
 = (fu – fd) / (u – d )S0eqt = 0.539

July 9, 2010                                Derivatives 08 Inside Black Scholes                |20
Black Scholes Merton with constant dividend
yield
The partial differential               f             f 1  2 f 2 2
equation:                                  (r  q) S            S  rf
(See Hull 5th ed. Appendix 13A)
t             S 2 S  2

Expected growth rate of stock

Call option
C  S 0 e  qT  N (d 1 )  Ke  rT  N (d 2 )

Put option                        P  Ke  rT N ( d 2 )  S 0 e  qT  N (d1 )

ln( S 0 e  qT / Ke  rT )                              d 2  d1   T
d1                                  0.5 T
 T

July 9, 2010                                 Derivatives 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

July 9, 2010                                    Derivatives 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

July 9, 2010                   Derivatives 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
 (uF0 – F0) + M ert = fu
pf u  (1  p) f d
 (dF0 – F0) + M   ert   = fd                                   f 
e rt

f=M                                                                1 d
p
ud

July 9, 2010                              Derivatives 08 Inside Black Scholes                      |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                                           Dividend paying stock

pf u  (1  p) f d                                 pf u  (1  p) f d
f                                                 f 
e rt                                              e rt

1 d                                                 e ( r  q ) t  d
p                                                   p
ud                                                        ud

Futures prices behave in the same way as a stock paying a
continuous dividend yield at the risk-free interest rate r

July 9, 2010                        Derivatives 08 Inside Black Scholes             |25
Black’s model

Assumption: futures price has lognormal distribution

Ce         rT
[ F0 N (d1 )  KN (d 2 )]
P  e  rT [ KN (d 2 )  F0 N (d1 )]

F0                                                 F0
)
ln(                                                )
ln(
d1    X  0.5 T                                  d2    X  0.5 T  d   T
 T
1
 T

July 9, 2010                            Derivatives 08 Inside Black Scholes        |26
Implied volatility – Call option

July 9, 2010             Derivatives 08 Inside Black Scholes   |27
Implied volatility – Put option

July 9, 2010             Derivatives 08 Inside Black Scholes   |28
Smile
SPX Option on S&P 500            Spot index            968.25
September 2002 Contract          DivYield                 2%
IntRate               1.86%

July 2, 2002
Maturity                   90 days

Strike          Call                                            Put
OpenInt          Price          ImpVol         OpenInt       Price   ImpVol

700                                                           3801         1.5    34.19%
750                                                           1581         2.9    31.59%
800                                                          31675           4    26.84%
900                                                          21723        15.6    22.17%
925                                                           7799          19    19.54%
950                                                          17419          28    19.16%
975                                                          16603          33    15.32%
980            3599              42          24.89%           4994        40.3    17.68%
990            3228              40          26.04%           3193          41    14.86%
995           11806            34.5          24.17%          23345          46    15.84%
1005            5404              30          23.73%           5209        53.5    16.29%
1025            9232            21.4          22.47%          15242        59.8     9.95%
1040            2286            15.1          20.97%
1050           11145            13.1          21.07%
1075            8726             7.5          19.97%
1100           23170             4.6          19.82%
1125            7556             2.4          19.16%
1150           18173             1.6          19.67%
1200            7513            0.45          19.33%
July 9, 2010                                                 Derivatives 08 Inside Black Scholes                    |29

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