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Derivatives Introduction to option pricing

André Farber Solvay Brussels School Université Libre de Bruxelles

March 14, 2011 Derivatives 07 Pricing options |2

Forward/Futures: Review

•  Forward contract = portfolio –  asset (stock, bond, index) –  borrowing

•  Value f = value of portfolio f = S - PV(K)

Based on absence of arbitrage opportunities •  4 inputs:

•  Spot price (adjusted for “dividends” ) •  Delivery price •  Maturity •  Interest rate

•  Expected future price not required


Options

•  Standard options –  Call, put –  European, American

•  Exotic options (non standard) –  More complex payoff (ex: Asian) –  Exercise opportunities (ex: Bermudian)


Option Valuation Models: Key ingredients

•  Model of the behavior of spot price ⇒ new variable: volatility

•  Technique: create a synthetic option •  No arbitrage •  Value determination

–  closed form solution (ex: Black Merton Scholes) –  numerical technique

March 14, 2011

Models of the behavior of stock prices

Derivatives 07 Pricing options |5

Time /Stock price Continuous Discrete Continuous Geometric

Brownion motion

Advanced calculus (Ito)

Partial differential

equation Discrete Binomial model

Elementary algebra

Discounted cash

flow


Options: the family tree

Black Merton Scholes (1973)

Analytical models

Numerical models

Analytical approximation

models Term structure

models

B & S Merton

Binomial Trinomial

Finite difference Monte Carlo

European Option

European American

Option American

Option Options on Bonds &

Interest Rates

Analytical Numerical


Modelling stock price behaviour

•  Consider a small time interval Δt: ΔS = St+Δt - St •  2 components of return ΔS/S:

–  drift : E(ΔS/S) = µ Δt [µ = expected return (per year)] –  volatility: realized return ΔS/S = E(ΔS/S) + random variable (ε)

•  Expected value E(ε) = 0 •  Variance proportional to Δt: Var(ε) = σ² Δt

–  ⇔ Standard deviation = σ √Δt where σ is the standard deviation of annual returns

‒  ε = N(0, σ √Δt) = σ × N(0,√Δt)

–  = σ × Δz Δz : Normal (0,√Δt)

•  Δz independent of past values (Markov process)


Geometric Brownian motion illustrated

Geometric Brownian motion

-100.00

-50.00

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0 8 16 24 32 40 48 56 64 72 80 88 96 104

112

120

128

136

144

152

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168

176

184

192

200

208

216

224

232

240

248

256

Drift Random shocks Stock price


Geometric Brownian motion model

• ΔS/S = µ Δt + σ Δz • ΔS = µ S Δt + σ S Δz

•  If Δt "small" (continuous model)

•  dS = µ S dt + σ S dz

March 14, 2011

Binomial representation of the geometric Brownian


S

uS

dS

q

1-qteu Δ= σ

ud 1=

dudeq

t

−

−=

Δµ

tSeSdqqSu Δ=−+ µ)1(

tSSedSquqS t Δ=−−+ Δ 2222222 )()1( σµ

u, d and q are choosen to reproduce the drift and the volatility of the underlying process: Drift:

Volatility:

Cox-Ross-Rubinstein solution:


Binomial process: Example

•  dS = 0.15 S dt + 0.30 S dz (⇔ µ = 15%, σ = 30%) •  Consider a binomial representation with Δt = 0.5

u = 1.2363, d = 0.8089, q = 0.6293 Binomial representation of Brownian geometric motion dS= Sdt+ SdzStock price 100 Per periodExpected return Mu 15% Exp return 7.50% 7.79%Volatility 30% Volatility 21.21% 20.64%Time step dt 0.5u 1.24d 0.81Proba up q 0.63

Time (yr) 0.00 0.50 1.00 1.50 2.00 2.50 3.00100.00 123.63 152.85 188.97 233.62 288.83 357.08

80.89 100.00 123.63 152.85 188.97 233.6265.43 80.89 100.00 123.63 152.85

52.92 65.43 80.89 100.0042.80 52.92 65.43

34.62 42.8028.00


Derivative Valuation:one period model, no payout

•  Time step = Δt •  Riskless interest rate = r •  Stock price evolution

•  uS

•  S

•  dS

•  No arbitrage: d<er Δt <u

•  1-period derivative

•  fu

f =?

•  fd

q

1-q

q

1-q

Call option:

Put option:

f1 =Max(0,S1 !K )

f1 =Max(0,K ! S1)


Option valuation: Basic idea

•  Basic idea underlying the analysis of derivative securities •  Can be decomposed into basic components •  ⇒ possibility of creating a synthetic identical security •  by combining: •  - Underlying asset •  - Borrowing / lending

•  ⇒ Value of derivative = value of components

March 14, 2011

Pricing in 1-period binomial model


Law of one price: value using state prices

Synthetic derivative valuation: create and value replicating portfolio

Risk neutral valuation: discount risk neutral expected value

CAPM discount risk adjusted expected value


Synthetic valuation

•  Long (+)/ short(-) δ shares •  Invest (+), borrow(-) B at the interest rate r per period •  Choose δ and B to reproduce payoff of derivatives

δ u S + B erΔt = fu δ d S + B erΔt = fd

Solution:

Derivative value f = δ S + B

! =fu ! fduS ! dS

B = ! dfu !ufd(u! d)er"t


Derivative value: Another interpretation

Derivative value f = δ S + B •  In this formula:

+ : long position (buy, invest) - : short position (sell borrow)

B = - δ S + f Interpretation for call option: Buying δ shares and selling one call is equivalent to a riskless

investment.

March 14, 2011

Law of one price


trd

tru

du

evevdSvuSvS

ΔΔ ×+×=

×+×=

1

State prices calculation:

Pricing formula:

dduu fvfvf +=

=> du vv ,


Risk neutral pricing

•  Derivative value = δ S + B •  Substitue values for δ and B and simplify:

•  f = [ pfu + (1-p)fd ]/ erΔt where p = (erΔt - d)/(u-d)

•  As 0< p<1, p can be interpreted as a probability

•  p is the “risk-neutral probability”: the probability such that the expected return on any asset is equal to the riskless interest rate


Risk neutral valuation

•  There is no risk premium in the formula ⇔ attitude toward risk of investors are irrelevant for valuing the option

•  ⇒ Valuation can be achieved by assuming a risk neutral world •  In a risk neutral world :

r  Expected return = risk free interest rate r  What are the probabilities of u and d in such a world ?

p u + (1 - p) d = erΔt

r  Solving for p:p = (erΔt - d)/(u-d)

•  Conclusion : in binomial pricing formula, p = probability of an upward movement in a risk neutral world

March 14, 2011

Risk adjusted valuation - CAPM

Market price of risk: 0)()(

λ=−

=S

fS

RVarrRE

RiskpremiumRisk

Pricing equation: f

S

rRffEf

+×−

=1

),cov()( 101 λ

Discount certaintu equivalent (risk adjusted expected CF)

March 14, 2011

Example: call option Example= call option

March 14, 2011

Example put option

March 14, 2011

A

B

C

A

B

C

C

B

A


Mutiperiod extension: European option

u²S uS

S udS dS d²S

•  Recursive method (European and American options) �Value option at maturity �Work backward through the tree.

Apply 1-period binomial formula at each node

•  Risk neutral discounting (European options only) �Value option at maturity �Discount expected future value

(risk neutral) at the riskfree interest rate


Multiperiod valuation: Example

•  Data •  S = 100 •  Interest rate (cc) = 5% •  Volatility σ = 30% •  European call option: •  Strike price K = 100, •  Maturity =2 months •  Binomial model: 2 steps •  Time step Δt = 0.0833

•  u = 1.0905 d = 0.9170 •  p = 0.5024

0 1 2 Risk neutral probability 118.91 p²=

18.91 0.2524 109.05 9.46

100.00 100.00 2p(1-p)= 4.73 0.00 0.5000

91.70 0.00 84.10 (1-p)²= 0.00 0.2476

Risk neutral expected value = 4.77 Call value = 4.77 e-.05(.1667) = 4.73


From binomial to Black Scholes

•  Consider: •  European option •  on non dividend paying stock •  constant volatility •  constant interest rate

•  Limiting case of binomial model as Δt→0

Stock price

Timet T


Convergence of Binomial Model

Convergence of Binomial Model

0.00

2.00

4.00

6.00

8.00

10.00

12.001 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

Number of steps

Opt

ion

valu

e

March 14, 2011 Derivatives 10 Options on bonds and IR |29

DerivaGem 2.01: option pricing software

•  Available on John Hull website (Free!!) •  Underlying Asset:

–  Equity / Currency /Stock Index / Futures –  Bonds –  Interest rate / Swap

•  Option types: –  Standard (European / American) –  Exotic (Asian, barrier, binary, chooser, compound, lookback)

•  Models: –  Analytic –  Numerical method (binomial, trinomial)


Black Scholes formula

•  European call option: •  C = S × N(d1) - K e-r(T-t) × N(d2)

•  N(x) = cumulative probability distribution function for a standardized normal variable

•  European put option: •  P= K e-r(T-t) × N(-d2) - S × N(-d1)

•  or use Put-Call Parity

tTtT

KeS

dtTr

−+−

=−−

σσ

5.0)ln( )(

1

tTdd −−= σ12


Black Scholes: Example

•  Stock price S = 100 •  Exercise price = 100 (at the money

option) •  Maturity = 1 year (T-t = 1) •  Interest rate (continuous) = 5% •  Volatility = 0.15

•  Reminder: N(-x) = 1 - N(x)

•  d1 = 0.4083 •  d2 = 0.4083 - 0.15√1= 0.2583 •  N(d1) = 0.6585 N(d2) = 0.6019 •  European call : •  100 × 0.6585 - 100 × 0.95123 × 0.6019 = 8.60 •  European put : •  100 × 0.95123 × (1-0.6019)

•  - 100 × (1-0.6585) = 3.72

0.115.05.00.115.0

)100100ln( 05.0

1 ×+=−ed


Black Scholes differential equation: Assumptions

•  S follows a geometric Brownian motion:dS = µS dt + σ S dz •  Volatility σ constant •  No dividend payment (until maturity of option) •  Continuous market •  Perfect capital markets •  Short sales possible •  No transaction costs, no taxes •  Constant interest rate


Black-Scholes illustrated

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Action Option Valeur intrinséque

Lower boundIntrinsic value Max(0,S-K)

Upper boundStock price

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