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DerivativesInside Black Scholes
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
Derivatives 08 Inside Black Scholes |2April 18, 2023
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(
Derivatives 08 Inside Black Scholes |3April 18, 2023
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
Derivatives 08 Inside Black Scholes |4April 18, 2023
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:
dGG
xa
G
t
G
xb dt
G
xb dz ( )
1
2
2
22
Drift Volatility
Derivatives 08 Inside Black Scholes |5April 18, 2023
Ito’s lemna: some intuition
• If x is a real variable, applying Taylor:
• In ordinary calculus:
• In stochastic calculus:
• Because, if x follows an Ito process, dx² = b² dt you have to keep it
GG
xx
G
tt
G
xx
G
x tx t
G
tt
1
2
1
2
2
22
2 2
22 ..
dGG
xdx
G
tdt
An approximationdx², dt², dx dt negligeables
²²
²
2
1dx
x
Gdt
t
Gdx
x
GdG
Derivatives 08 Inside Black Scholes |6April 18, 2023
Lognormal property of stock prices
• Suppose: dS= S dt + S dz
• Using Ito’s lemna: d ln(S) = ( - 0.5 ²) dt + dz
• Consequence:
],)2
²[(~)ln()ln( 0 TTNSST
],)2
²()[ln(~)ln( 0 TTSNST
ln(ST) – ln(S0) = ln(ST/S0)
Continuously compounded return between 0 and T
ln(ST) is normally distributed so that ST has a lognormal distribution
Derivatives 08 Inside Black Scholes |7April 18, 2023
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:
• 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
dff
SS
f
t
f
SS dt
f
SS dz ( )
1
2
2
22 2
Derivatives 08 Inside Black Scholes |8April 18, 2023
Here comes the PDE!
• Using Ito’s lemna
• This is a riskless portfolio!!!
• Its expected return should be equal to the risk free interest rate:
dV = r V dt
• This leads to:
dVf
t
f
SS dt ( )
1
2
2
22 2
f
trS
f
S
f
SS rf
1
2
2
22 2
Derivatives 08 Inside Black Scholes |9April 18, 2023
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
Derivatives 08 Inside Black Scholes |10April 18, 2023
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 = 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
Derivatives 08 Inside Black Scholes |11April 18, 2023
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
Derivatives 08 Inside Black Scholes |12April 18, 2023
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
Derivatives 08 Inside Black Scholes |13April 18, 2023
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
T Time adjusted volatility.The volatility of the return on the underlying asset between now and maturity.
Derivatives 08 Inside Black Scholes |14April 18, 2023
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
Derivatives 08 Inside Black Scholes |15April 18, 2023
Relationship between call value and spot price
0.00
10.00
20.00
30.00
40.00
50.00
60.00
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
Derivatives 08 Inside Black Scholes |16April 18, 2023
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)
Derivatives 08 Inside Black Scholes |17April 18, 2023
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
Derivatives 08 Inside Black Scholes |18April 18, 2023
Relationship between Put Value and Spot Price
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
50.00
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
Derivatives 08 Inside Black Scholes |19April 18, 2023
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)
Derivatives 08 Inside Black Scholes |20April 18, 2023
Dividend paying stock: binomial model
S0
100
uS0 eqt with dividends reinvested
128.81
dS0 eqt with dividends reinvested
82.44
uS0 ex dividend
125
dS0 ex dividend
80
Replicating portfolio:
uS0 eqt + M ert = fu
128.81 + M 1.0513 = 25
dS0 eqt + M ert = fd
82.44 + M 1.0513 = 0
f = S0 + M
= (fu – fd) / (u – d )S0eqt = 0.539
f = [ p fu + (1-p) fd] e-rt = 11.64
p = (e(r-q)t – d) / (u – d) = 0.489
t = 1 u = 1.25, d = 0.80r = 5% q = 3%Derivative: Call K = 100
fu
25
fd
0
Derivatives 08 Inside Black Scholes |21April 18, 2023
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
Derivatives 08 Inside Black Scholes |22April 18, 2023
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
Derivatives 08 Inside Black Scholes |23April 18, 2023
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
Derivatives 08 Inside Black Scholes |24April 18, 2023
Option on futures: binomial model
00 dFuF
ff du
trdu
e
fppff
)1(
Futures price F0
uF0 → fu
dF0 →fd
Replicating portfolio: futures + cash
(uF0 – F0) + M ert = fu
(dF0 – F0) + M ert = fd
f = Mdu
dp
1
Derivatives 08 Inside Black Scholes |25April 18, 2023
Options on futures versus options on dividend paying stock
trdu
e
fppff
)1(
du
dp
1
trdu
e
fppff
)1(
Compare now the formulas obtained for the option on futures and for an option on a dividend paying stock:
du
dep
tqr
)(
Futures prices behave in the same way as a stock paying a continuous dividend yield at the risk-free interest rate r
Futures Dividend paying stock
Derivatives 08 Inside Black Scholes |26April 18, 2023
Black’s model
)]()([ 210 dKNdNFeC rT )]()([ 102 dNFdKNeP rT
TTX
F
d
5.0)ln( 0
1
Assumption: futures price has lognormal distribution
TdTTX
F
d
1
0
2 5.0)ln(
Derivatives 08 Inside Black Scholes |27April 18, 2023
Implied volatility – Call option
0.00
10.00
20.00
30.00
40.00
50.00
60.00
15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30% 31% 32% 33% 34% 35%
Volatility
Implied volatility
Market price
Derivatives 08 Inside Black Scholes |28April 18, 2023
Implied volatility – Put option
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%
Volatility
Market price
Implied volatility
Derivatives 08 Inside Black Scholes |29April 18, 2023
Smile
SPX Option on S&P 500 Spot index 968.25September 2002 Contract DivYield 2%
IntRate 1.86%
July 2, 2002Maturity 90 days
Strike Call PutOpenInt 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%
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