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Derivatives Introduction to option pricing
André Farber Solvay Brussels School Université Libre de Bruxelles
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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
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Options
• Standard options – Call, put – European, American
• Exotic options (non standard) – More complex payoff (ex: Asian) – Exercise opportunities (ex: Bermudian)
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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
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Models of the behavior of stock prices
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Time /Stock price Continuous Discrete Continuous Geometric
Brownion motion
Advanced calculus (Ito)
Partial differential
equation Discrete Binomial model
Elementary algebra
Discounted cash
flow
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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
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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)
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Geometric Brownian motion illustrated
Geometric Brownian motion
-100.00
-50.00
0.00
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400.00
0 8 16 24 32 40 48 56 64 72 80 88 96 104
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Drift Random shocks Stock price
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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
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Binomial representation of the geometric Brownian
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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:
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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
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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)
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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
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Pricing in 1-period binomial model
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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
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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
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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.
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Law of one price
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trd
tru
du
evevdSvuSvS
ΔΔ ×+×=
×+×=
1
State prices calculation:
Pricing formula:
dduu fvfvf +=
=> du vv ,
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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
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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
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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)
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Example: call option Example= call option
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Example put option
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A
B
C
A
B
C
C
B
A
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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
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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
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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 Δt→0
Stock price
Timet T
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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
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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)
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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
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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
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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
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Black-Scholes illustrated
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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