Transcript Diapositive 1 - UCSB Statistics

An Idiot’s Guide to Option Pricing
Bruno Dupire
Bloomberg LP
[email protected]
CRFMS, UCSB
April 26, 2007
Warm-up
Roulette:
P[ Red]  70%
P[ Black]  30%
$100 if Red
A lottery ticket gives: 
 $0 if Black
You can buy it or sell it for $60
Is it cheap or expensive?
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Naïve expectation
70  60  Buy
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Replication argument
50  60  Sell
“as if” priced with other probabilities
instead of
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OUTLINE
1.
2.
3.
4.
5.
6.
Risk neutral pricing
Stochastic calculus
Pricing methods
Hedging
Volatility
Volatility modeling
Addressing Financial Risks
Over the past 20 years, intense development of Derivatives
in terms of:
•volume
•underlyings
•products
•models
•users
•regions
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To buy or not to buy?
• Call Option: Right to buy stock at T for K
$
TO BUY
$
NOT TO BUY
K
ST
K
ST
$
CALL
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K
ST
7
Vanilla Options
European Call:
Gives the right to buy the underlying at a fixed price (the strike) at
some future time (the maturity)
Call Payoff (ST  K )  maxST  K ,0
European Put:
Gives the right to sell the underlying at a fixed strike at some maturity
Put Payoff K  ST 

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Option prices for one maturity
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Risk Management
Client has risk exposure
Buys a product from a bank to limit its risk
Not Enough
Risk
Too Costly
Vanilla Hedges
Perfect Hedge
Exotic Hedge
Client transfers risk to the bank which has the technology to handle it
Product fits the risk
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Risk Neutral Pricing
Price as discounted expectation
Option gives uncertain payoff in the future
Premium: known price today
!
?
Resolve the uncertainty by computing expectation:
?!
Transfer future into present by discounting
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Application to option pricing
Risk Neutral Probability
Price  e
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Physical Probability
 rT


o
 (ST )(ST  K )  dST
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Basic Properties
Price as a function of payoff is:
- Positive:
A  0  price( A)  0
- Linear:
price( A  B)  price( A)  price( B)
 Price = discounted expectation of payoff
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Toy Model
1 period, n possible states
s1 ,...,sn
Option A gives xi in state s i
If Ai gives 1 in state
A   xi Ai

s i , 0 in all other states,
price( A) 
i

 x price( A )
q x
i
i
i
i
i
i


where    price( Ai )  price  Ai  price(1) is a discount factor
qi 
price( Ai )
is a probability: qi  0 and
 price( A j )
q
i
1
i
j
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FTAP
Fundamental Theorem of Asset Pricing
1)
NA  There exists an equivalent martingale measure
2) NA + complete There exists a unique EMM
Claims attainable from 0
Cone of >0 claims
Separating hyperplanes
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Risk Neutrality Paradox
• Risk neutrality: carelessness about uncertainty?
50%
50%
Sun: 1 Apple = 2 Bananas
Rain: 1 Banana = 2 Apples
• 1 A gives either 2 B or .5 B1.25 B
• 1 B gives either .5 A or 2 A1.25 A
• Cannot be RN wrt 2 numeraires with the same probability
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Stochastic Calculus
Modeling Uncertainty
Main ingredients for spot modeling
• Many small shocks: Brownian Motion
(continuous prices) S
t
• A few big shocks: Poisson process (jumps)
S
t
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Brownian Motion
• From discrete to continuous
10
100
1000
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Stochastic Differential Equations
At the limit:
Wt continuous with independent Gaussian increments
Wt  Ws ~ N (0, t  s)
SDE:
dx  a dt  b dW
drift
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a
noise
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Ito’s Dilemma
f (x)
Classical calculus:
df  f ' ( x) dx
expand to the first order
Stochastic calculus:
dx  a dt  b dW
should we expand further?
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Ito’s Lemma
At the limit
(dW)2  dt
If dx  a dt  b dW
for f(x),
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df  f ( x  dx)  f ( x)
1
2
 f ' ( x) dx  f ' ' ( x) (dx)
2
1
2
 f ' ( x) dx  f ' ' ( x) b dt
2
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Black-Scholes PDE
dS
  dt   dW
S
• Black-Scholes assumption
• Apply Ito’s formula to Call price C(S,t)
dC  CS dS  (Ct 
 2S 2
2
CSS ) dt
• Hedged position C  CS S is riskless, earns interest rate r
(Ct 
 2S 2
2
CSS ) dt  dC  CS dS  r (C  CS S ) dt
• Black-Scholes PDE
Ct  
 2S 2
2
CSS  r (C  CS S )
• No drift!
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P&L of a delta hedged option
Option Value
P&L
Break-even
points
Delta
hedge
Ct t
Ct
  t
St
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 t
S

St t
St
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Black-Scholes Model
If instantaneous volatility is constant :
drift:
Stt
dS
 dt  dW
S
noise, SD:
Then call prices are given by :
1
 T)
2
 T
1
1
 Ke  rT N (
(ln(S 0 / K )  rT )   T )
2
 T
C BS  S 0 N (
1
St t
(ln(S 0 / K )  rT ) 
No drift in the formula, only the interest rate r due to the
hedging argument.
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Pricing methods
Pricing methods
• Analytical formulas
• Trees/PDE finite difference
• Monte Carlo simulations
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Formula via PDE
• The Black-Scholes PDE is
Ct  
 2S 2
2
CSS  r (C  CS S )
• Reduces to the Heat Equation
1
U   U xx
2
• With Fourier methods, Black-Scholes equation:
CBS  S0 N (d1 )  Ke  rT N (d 2 )
ln(S0 / K )  (r   2 / 2)T
d1 
, d 2  d1   T
 T
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Formula via discounted expectation
dS
 r dt   dW
S
• Risk neutral dynamics
• Ito to ln S:
d ln S  (r 
2
2
) dt   dW
2
• Integrating: ln ST ln S 0  ( r  2 ) T  WT
premium e
rT

E[(ST  K ) ]  e
rT
E[(S0e
(r 
2
2
)T  WT
 K ) ]
• Same formula
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Finite difference discretization of PDE
• Black-Scholes PDE
Ct  
 2S 2
CSS  r (C  CS S )
2
C ( S , T )  ( ST  K ) 
• Partial derivatives discretized as
C (i, n)  C (i, n  1)
t
C (i  1, n)  C (i  1, n)
C S (i, n) 
2S
C (i  1, n)  2C (i, n)  C (i  1, n)
C SS (n, i ) 
(S ) 2
Ct (i, n) 
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Option pricing with Monte Carlo methods
• An option price is the
discounted expectation of its
payoff:
P0  EPT    f  x   x dx
• Sometimes the expectation
cannot be computed
analytically:
– complex product
– complex dynamics
• Then the integral has to be
computed numerically
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the option price is its discounted payoff
integrated against the risk neutral density of the spot underlying
Computing expectations
basic example
•You play with a biased die
•You want to compute the likelihood of getting
•Throw the die 10.000 times
•Estimate p(
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) by the number of
over 10.000 runs
Option pricing = superdie
Each side of the superdie represents a possible state of the
financial market
• N final values
in a multi-underlying model
• One path
in a path dependent model
• Why generating whole paths?
- when the payoff is path dependent
running a Monte Carlo path simulation
- when the dynamics are complex
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Expectation = Integral
Gaussian transform techniques
Unit hypercube
discretisation schemes
Gaussian coordinates
trajectory
A point in the hypercube maps to a spot trajectory
therefore
EPT 

d


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
f  x .Pr  St1 ,..., Std   dx 
1
 g x i 
N xi  0 ,1d

 0 ,1 d
g ydy
Generating Scenarios
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Low Discrepancy Sequences
Halton
Faure
Sobol
dimensions
1&2
dimensions
20 & 25
dimensions
51 & 52
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Hedging
To Hedge or Not To Hedge
Daily P&L
Daily Position
P&L
Unhedged
Full P&L
0
Hedged
S
Delta Hedge
Big directional risk  Small daily amplitude risk
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The Geometry of Hedging
• Risk measured as SDPLT 
• Target X, hedge H
PLt  X t  H t
Risk  varX T  H T   X  H

• Risk is an L2 norm, with general properties of orthogonal
projections
• Optimal Hedge:
Hˆ
X  Hˆ  inf X  H
H 
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The Geometry of Hedging
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Super-replication
E XY   E X 2  EY 2 
•Property:
Let us call:
Px : price today of X 2
Py : price today of Y 2
For all X and Y ,


2
Py X  Px Y  0,
so XY is dominated by thePortfolio:
 XY 
Px X 2  PyY 2
2 Px Py
Which implies:
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price XY  
Py Px  Px Py
2 Px Py
 Px Py
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A sight of Cauchy-Schwarz
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Volatility
Volatility : some definitions
Historical volatility :
annualized standard deviation of the logreturns; measure of
uncertainty/activity
Implied volatility :
measure of the option price given by the market
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Historical Volatility
• Measure of realized moves
• annualized SD of logreturns
 


252  n 2
2
  xti   xti 
n  1  i 1

St 1
xt  ln
St
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Historical volatility
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Implied volatility
Input of the Black-Scholes formula which makes it fit the
market price :
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Market Skews
Dominating fact since 1987 crash: strong negative skew on
Equity Markets
 impl
K
Not a general phenomenon
FX:  impl
Gold:  impl
K
K
We focus on Equity Markets
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A Brief History of Volatility
Evolution theory of modeling
constant deterministic stochastic
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A Brief History of Volatility
–
dSt   dWt Q
: Bachelier 1900
–
dSt
 r dt   dWt Q
St
: Black-Scholes 1973
–
dSt
 rt dt   (t ) dWt Q
St
: Merton 1973
–
dSt
 (r  k ) dt   dWt Q  dq
St
: Merton 1976
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Local Volatility Model
dSt
 r dt   ( S , t ) dWt Q
St
C
C
 rK
 2 K , T   2 T 2 K
2  C K ,T
K
K 2
Dupire 1993, minimal model to fit current
volatility surface
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The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Risk
Neutral
Processes
1D
Diffusions
Compatible
with Smile
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sought diffusion
(obtained by integrating twice
Fokker-Planck equation)
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From simple to complex
European
prices
Local
volatilities
Exotic prices
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Stochastic Volatility Models
 dSt
 S  r dt   t dWt
 t
d 2  b( 2   2 )dt   dZ

t
t
t
 t
Heston 1993, semi-analytical formulae.
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The End