Derivatives Inside Black Scholes

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DerivativesInside Black Scholes

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

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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

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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 Itos lemna

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Itos 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
• Itos lemna. G follows a stochastic process

Drift
Volatility
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Itos 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

An approximation dx², dt², dx dt negligeables
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Lognormal property of stock prices

• Suppose dS ? S dt ? S dz
• Using Itos lemna d ln(S) (? - 0.5 ?²) dt
  ? dz
• Consequence

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
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Derivation of PDE (partial differential equation)

• Back to the valuation of a derivative f(S,t)
• If S changes by dS, using Itos 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

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Here comes the PDE!

• Using Itos 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

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Understanding the PDE

• Assume we are in a risk neutral world

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
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Black Scholes PDE and the binomial model

• We have
• BS PDE ft rS fS ½ ?² fSS r f
• Binomial model p fu (1-p) fd er?t
• Use Taylor approximation
• fu f (u-1) S fS ½ (u1)² S² fSS ft ?t
• fd f (d-1) S fS ½ (d1)² S² fSS ft ?t
• u 1 ?v?t ½ ?²?t
• d 1 ?v?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

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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
Put option
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
• 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
2 elements determine d1 and d2
A measure of the moneyness of the option.The
distance between the exercise price and the stock
price
S0 / Ke-rt
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
?vT 0.15
d1 (0.05)/(0.15) (0.5)(0.15) 0.4083
N(d1) 0.6585
European call 100 ? 0.6585 - 100 ? 0.95123 ?
0.6019 8.60
d2 0.4083 0.15 0.2583
N(d2) 0.6019
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Relationship between call value and spot price
For call option, time value gt 0
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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)

Risk-neutral probability of exercising the option
Proba(STgtX)
Delta of call option
Risk-neutral probability of exercising the option
Proba(STltX)
Delta of put option
(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 gt0 or lt0
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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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Dividend paying stock binomial model
?t 1 u 1.25, d 0.80r 5 q
3Derivative Call K 100
uS0 eq?t with dividends reinvested128.81
fu25
uS0 ex dividend125
S0100
dS0 eq?t with dividends reinvested82.44
fd0
dS0 ex dividend80
f ? S0 M
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
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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
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Black Scholes Merton with constant dividend yield
The partial differential equation(See Hull 5th
ed. Appendix 13A)
Expected growth rate of stock
Call option
Put option
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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 SP 100 (American) OEX
• the SP 500 (European) SPX
• Contracts are settled in cash
• Example July 2, 2002 SP 500 968.65
• SPX September
• Strike Call Put
• 900 - 15.601,005 30 53.501,025 21.40 59.80
• Source Wall Street Journal

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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

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Option on futures binomial model
uF0 ? fu
Futures price F0
dF0 ?fd
Replicating portfolio ? futures cash
? (uF0 F0) M er?t fu
? (dF0 F0) M er?t fd
f M
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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
Futures prices behave in the same way as a stock
paying a continuous dividend yield at the
risk-free interest rate r
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Blacks model
Assumption futures price has lognormal
distribution
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Implied volatility Call option
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Implied volatility Put option
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