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Options and Speculative Markets2004-2005Introduction to option pricing
André Farber
Solvay Business School
University of Brussels
OMS 06 Pricing options |2August 23, 2004
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
OMS 06 Pricing options |3August 23, 2004
Options
• Standard options
– Call, put
– European, American
• Exotic options (non standard)
– More complex payoff (ex: Asian)
– Exercise opportunities (ex: Bermudian)
OMS 06 Pricing options |4August 23, 2004
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 (Black Merton Scholes)
– numerical technique
OMS 06 Pricing options |5August 23, 2004
Model of the behavior of spot price
• Geometric Brownian motion
– continuous time, continuous stock prices
• Binomial
– discrete time, discrete stock prices
– approximation of geometric Brownian motion
OMS 06 Pricing options |6August 23, 2004
Creation of synthetic option
• Geometric Brownian motion
– requires advanced calculus (Ito’s lemna)
• Binomial
– based on elementary algebra
OMS 06 Pricing options |7August 23, 2004
Options: the family tree
Black Merton Scholes (1973)
Analyticalmodels
Numericalmodels
Analyticalapproximation
models
Term structuremodels
B & SMerton
BinomialTrinomial
Finite differenceMonte Carlo
EuropeanOption
EuropeanAmerican
Option
AmericanOption
Options onBonds &
Interest Rates
AnalyticalNumerical
OMS 06 Pricing options |8August 23, 2004
Modelling stock price behaviour
• Consider a small time interval t: S = St+t - St
• 2 components of S:– drift : E(S) = S t [ = expected return (per year)]
– volatility:S/S = E(S/S) + random variable (rv)
• Expected value E(rv) = 0
• Variance proportional to t
– Var(rv) = ² t Standard deviation = t– rv = Normal (0, t)– = Normal (0,t)– = z z :
Normal (0,t)– = t : Normal(0,1)
z independent of past values (Markov process)
OMS 06 Pricing options |9August 23, 2004
Geometric Brownian motion illustrated
Geometric Brownian motion
-100.00
-50.00
0.00
50.00
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250.00
300.00
350.00
400.00
0 8 16
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Drift Random shocks Stock price
OMS 06 Pricing options |10August 23, 2004
Geometric Brownian motion model
S/S = t + z S = S t + S z
• = S t + S t
• If t "small" (continuous model)
• dS = S dt + S dz
OMS 06 Pricing options |11August 23, 2004
Binomial representation of the geometric Brownian
• u, d and q are choosen to reproduce the drift and the volatility of the underlying process:
• Drift:
• Volatility:
• Cox, Ross, Rubinstein’s solution:
•
S
uS
dS
q
1-q
teu u
d1
du
deq
t
tSeSdqqSu )1(
tSSedSquqS t 2222222 )()1(
OMS 06 Pricing options |12August 23, 2004
Binomial process: Example
• dS = 0.15 S dt + 0.30 S dz ( = 15%, = 30%)
• Consider a binomial representation with t = 0.5
u = 1.2363, d = 0.8089, q = 0.6293
• Time 0 0.5 1 1.5 2 2.5• 28,883• 23,362• 18,897 18,897• 15,285 15,285• 12,363 12,363 12,363• 10,000 10,000 10,000• 8,089 8,089 8,089• 6,543 6,543• 5,292 5,292• 4,280• 3,462
OMS 06 Pricing options |13August 23, 2004
Call Option Valuation:Single period model, no payout
• Time step = t• Riskless interest rate = r • Stock price evolution
• uS
• S
• dS
• No arbitrage: d<er t <u
• 1-period call option
• Cu = Max(0,uS-X)
• Cu =?
• Cd = Max(0,dS-X)
q
1-q
q
1-q
OMS 06 Pricing options |14August 23, 2004
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
OMS 06 Pricing options |15August 23, 2004
Synthetic call option
• Buy shares
• Borrow B at the interest rate r per period
• Choose and B to reproduce payoff of call option
u S - B ert = Cu
d S - B ert = Cd
Solution:
Call value C = S - B
dSuS
CC du
trdu
edu
uCdCB
)(
OMS 06 Pricing options |16August 23, 2004
Call value: Another interpretation
Call value C = S - B
• In this formula:
+ : long position (buy, invest)
- : short position (sell borrow)
B = S - C
Interpretation:
Buying shares and selling one call is equivalent to a riskless investment.
OMS 06 Pricing options |17August 23, 2004
Binomial valuation: Example
• Data
• S = 100
• Interest rate (cc) = 5%
• Volatility = 30%
• Strike price X = 100, • Maturity =1 month (t = 0.0833)
• u = 1.0905 d = 0.9170
• uS = 109.05 Cu = 9.05
• dS = 91.70 Cd = 0
= 0.5216
• B = 47.64
• Call value= 0.5216x100 - 47.64
• =4.53
OMS 06 Pricing options |18August 23, 2004
1-period binomial formula
• Cash value = S - B
• Substitue values for and B and simplify:
• C = [ pCu + (1-p)Cd ]/ ert where p = (ert - 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
OMS 06 Pricing options |19August 23, 2004
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 : Expected return = risk free interest rate What are the probabilities of u and d in such a world ?
p u + (1 - p) d = ert
Solving for p:p = (ert - d)/(u-d)• Conclusion : in binomial pricing formula, p = probability of an upward
movement in a risk neutral world
OMS 06 Pricing options |20August 23, 2004
Mutiperiod extension: European option
u²SuS
S udS
dS
d²S
• Recursive method (European and American options)
Value option at maturityWork backward through the tree.
Apply 1-period binomial formula at each node
• Risk neutral discounting(European options only)
Value option at maturityDiscount expected future value
(risk neutral) at the riskfree interest rate
OMS 06 Pricing options |21August 23, 2004
Multiperiod valuation: Example
• Data
• S = 100
• Interest rate (cc) = 5%
• Volatility = 30%
• European call option:
• Strike price X = 100,
• Maturity =2 months
• Binomial model: 2 steps
• Time step t = 0.0833
• u = 1.0905 d = 0.9170
• p = 0.5024
0 1 2 Risk neutral probability118.91 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 = 4.77Call value = 4.77 e-.05(.1667) = 4.73
OMS 06 Pricing options |22August 23, 2004
From binomial to Black Scholes
• Consider:
• European option
• on non dividend paying stock
• constant volatility
• constant interest rate
• Limiting case of binomial model as t0
Stock price
Timet T
OMS 06 Pricing options |23August 23, 2004
Convergence of Binomial Model
Convergence of Binomial Model
0.00
2.00
4.00
6.00
8.00
10.00
12.00
1 4 7 10
13
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100
Number of steps
Op
tio
n v
alu
e
OMS 06 Pricing options |24August 23, 2004
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
5.0)ln( )(
1
tTdd 12
OMS 06 Pricing options |25August 23, 2004
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 - 0.151= 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
)100
100ln( 05.0
1 ed
OMS 06 Pricing options |26August 23, 2004
Black Scholes differential equation: Assumptions
• S follows a 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
OMS 06 Pricing options |27August 23, 2004
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