Advanced Finance 2005-2006 Black Scholes

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Advanced Finance2005-2006Black Scholes

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

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Binomial option pricing model

• Used to value derivative securities PVf(S)
• Evolution of underlying asset binomial model
• u and d capture the volatility of the underlying
  asset
• Replicating portfolio Delta S M
• Law of one price f Delta S M

uS
fu
S
dS
fd
?t
M is the cash positionMgt0 for investmentMlt0 for
borrowing
r is the risk-free interest rate with continuous
compounding
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Risk neutral pricing

• The value of a derivative security is equal to
  risk-neutral expected value discounted at the
  risk-free interest rate
• p is the risk-neutral probability of an up
  movement

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Road map to valuation
Binomial model uSS
dS discrete time, discrete stock prices
Model of stock price behavior
Geometric Brownian Motion dS µSdtsSdz continuou
s timecontinuous stock prices
Create synthetic option
Based on Itos lemna to calculate df
Based on elementary algebra
p fu (1-p) fd f er?t
PDE
Pricing equation
Black Scholes formula
Numerical methods
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Modelling stock price behaviour

• Consider a small time interval ?t ?S St?t -
  St
• ?S E(?S) u
• 2 components of ?S
• Expected change (drift) E(?S) ? S ?t ?
  expected return (per year)
• Unexpected change u a normal random variable
• Expected value E(u) 0
• Variance E(u²) ?² S² ?t (Variance
  proportional to ?t)
• u ? S ?z with
• ?z independent of past values (Markov process)
• ?S ? S ?t ? S ?z
• If ?t "small" (continuous model)
• dS ? S dt ? S dz

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Multiperiod binomial valuation
Risk neutral probability
u4S
p4
Risk neutral discounting (European option) (1)
At maturity, calculate - firm values - equity
and debt values - risk neutral probabilities (2)
Calculate the expected values in a neutral
world (3) Discount at the risk free rate
u3S
u²S
4p3(1 p)
u3dS
uS
u2dS
6p²(1 p)²
S
udS
u2d²S
ud²S
dS
4p (1 p)3
ud3S
?t
d²S
Recursive method (European and American
options) ?Value option at maturity ?Work backward
through the tree. Apply 1-period binomial
formula at each node
d3S
(1 p)4
d4S
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Multiperiod binomial valuation example
2-year European optionS 100Strike price
105Int.Rate 5 (annually compounded)Volatility
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4-step binomial tree ?t 0.50u 1.332, d
0.751rf 2.47 per period (1.05)1/2-1p
0.471
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Multiperiod valuation details
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From binomial to Black Scholes

• Consider
• European option
• on non dividend paying stock
• constant volatility
• constant interest rate
• Limiting case of binomial model as ?t?0

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Convergence of Binomial Model
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Toward Black Scholes formulas
Value
Increase the number to time steps for a fixed
maturity
The probability distribution of the firm value at
maturity is lognormal
Call in the money
Put in the money
Maturity
Today
Time
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Lognormal property of stock prices

• If dS ? S dt ? S dz

ln(ST) ln(S0) ln(ST/S0) Continuously
compounded return between 0 and T
ln(ST) is normally distributed. Hence the
distribution of ST is lognormal
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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
• 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
• BS PDE ft rS fS ½ ?² fSS r f

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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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Standardized normal cumulative probability
distribution

• If XN(0,1) N(x) Probability(X x)

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Black-Scholes using Excel
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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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Black-Scholes European put option

• European call option C S N(d1) PV(X) N(d2)
• Put-Call Parity P C S PV(X)
• European put option P - S N(d1)-1
  PV(X)1-N(d2)
• P - S
  N(-d1) PV(X) N(-d2)

Risk-neutral probability of exercising the option
Proba(STgtX)
Delta of call option
Delta of put option
Risk-neutral probability of exercising the option
Proba(STltX)
(Remember 1-N(x) N(-x))
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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)

Delta of call option
Risk-neutral probability of exercising the option
Proba(STgtX)
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