Derivatives Inside Black Scholes

DerivativesInside Black Scholes

Professor André FarberSolvay Business SchoolUniversité Libre de Bruxelles

Derivatives 08 Inside Black Scholes |2April 22, 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

dudep

efpfp

ftr

trdu

with )1(


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


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:

dG Gx

a Gt

Gx

b dt Gx

b dz ( )

12

2

22

Drift Volatility


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

G Gx

x Gt

t Gx

x Gx t

x t Gt

t

12

12

2

22

2 2

22 ..

dG Gx

dx Gt

dt

An approximationdx², dt², dx dt negligeables

²²

²21 dx

xGdt

tGdx

xGdG


Lognormal property of stock prices

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

• Consequence:

],)2²[(~)ln()ln( 0 TTNSST

],)2²()[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


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

df fS

S ft

fS

S dt fS

S dz ( )

12

2

22 2


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:

dV ft

fS

S dt ( )

12

2

22 2

ft

rS fS

fS

S rf 12

2

22 2


Understanding the PDE

• Assume we are in a risk neutral world

rfSS

fSfrS

tf

222

2

21

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: • 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


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

TTKeS

drT

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

TTKeS

drT

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.


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

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


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

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


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)


Dividend paying stock: binomial model

S0

100

uS0 eqt with dividends reinvested128.81

dS0 eqt with dividends reinvested82.44

uS0 ex dividend125

dS0 ex dividend80

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


Black Scholes Merton with constant dividend yield

rfSS

fSfSqr

tf

222

2

21)(

)()( 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 - 15.60

1,005 30 53.501,025 21.40 59.80

• Source: Wall Street Journal


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


Option on futures: binomial model

00 dFuFff du

trdu

efppf

f

)1(

Futures price F0

uF0 → fu

dF0 →fd

Replicating portfolio: futures + cash

(uF0 – F0) + M ert = fu

(dF0 – F0) + M ert = fd

f = Mdudp

1


Options on futures versus options on dividend paying stock

trdu

efppf

f

)1(

dudp

1

trdu

efppf

f

)1(

Compare now the formulas obtained for the option on futures and for an option on a dividend paying stock:

dudep

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


Black’s model

)]()([ 210 dKNdNFeC rT )]()([ 102 dNFdKNeP rT

TTXF

d

5.0)ln( 0

1

Assumption: futures price has lognormal distribution

TdTTXF

d

1

0

2 5.0)ln(


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


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


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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Derivatives Inside Black Scholes

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