Pricing Financial Derivatives
Bruno Dupire
Bloomberg L.P/NYU
AIMS Day 1
Cape Town, February 17, 2011
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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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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OUTLINE
A) Theory
- Risk neutral pricing
- Stochastic calculus
- Pricing methods
B) Volatility
- Definition and estimation
- Volatility modeling
- Volatility arbitrage
A) THEORY
Risk neutral pricing
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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2 approaches
Naïve expectation
70  60  Buy
Replication Argument
50  60  Sell
“as if” priced with other probabilities
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instead of
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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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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  adt  b dW
drift
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a
noise
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Ito’s Dilemma
Classical calculus:
y  f (x)
dy  f ' ( x) dx
expand to the first order
Stochastic calculus:
dx  adt  b dW
should we expand further?
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dy  f ' ( x) dx  ...
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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 :
St t
S 0 exp( rT )
1
)  T)
K
2
 T
S exp( rT )
1
1
 K exp( rT ) N (
ln( 0
)  T)
K
2
 T
C BS  S 0 N (
1
ln(
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:
C BS  S 0 N (d1 )  Ke  rT N (d 2 )
ln( S 0 / 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)
CS (i, n) 
2S
C (i  1, n)  2C (i, n)  C (i  1, n)
CSS (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
B) 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
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Historical Volatility
• Measure of realized moves
• annualized SD of
 


252  n 2
2
  xti   xti 
n  1  i 1

St 1
xt  ln
St
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Estimates based on High/Low
• Commonly available information: open, close, high, low
ln S
S upper
u  ln open
S
S close
c  ln open
S
S down
d  ln open
S
• Captures valuable volatility information
n
1
2


u

d
• Parkinson estimate:  
 t t
4 n ln 2 t 1
2
P
• Garman-Klass estimate:
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
2
GK
0.5 n
0.39 n 2
2
ut  d t  

ct


n t 1
n t 1
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Move based estimation
• Leads to alternative historical vol estimation:
L( , T )
h ~ 
T
L( , T )
= number of
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 crossings of log-price over [0,T]
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Black-Scholes Model
If instantaneous volatility is constant :
dS
 dt  dW
S
Then call prices are given by :
S 0 exp( rT )
1
)  T)
K
2
 T
S exp( rT )
1
1
 K exp( rT ) N (
ln( 0
)  T)
K
2
 T
C BS  S 0 N (
1
ln(
No drift in the formula, only the interest rate r due to the
hedging argument.
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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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Skews
• Volatility Skew: slope of implied volatility as a
function of Strike
• Link with Skewness (asymmetry) of the Risk
Neutral density function  ?
Moments
1
2
3
4
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Statistics
Expectation
Variance
Skewness
Kurtosis
Finance
FWD price
Level of implied vol
Slope of implied vol
Convexity of implied vol
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Why Volatility Skews?
• Market prices governed by
– a) Anticipated dynamics (future behavior of volatility or jumps)
– b) Supply and Demand
 impl
Market Skew
Th. Skew
K
• To “ arbitrage” European options, estimate a) to capture
risk premium b)
• To “arbitrage” (or correctly price) exotics, find Risk
Neutral dynamics calibrated to the market
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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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2 mechanisms to produce Skews (1)
• To obtain downward sloping implied volatilities
 impl
K
– a) Negative link between prices and volatility
• Deterministic dependency (Local Volatility Model)
• Or negative correlation (Stochastic volatility Model)
– b) Downward jumps
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2 mechanisms to produce Skews (2)
– a) Negative link between prices and volatility
S1
S2
– b) Downward jumps
S1
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S2
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Strike dependency
• Fair or Break-Even volatility is an average of squared
returns, weighted by the Gammas, which depend on the
strike
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Strike dependency for multiple paths
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A Brief History of Volatility
Evolution Theory
constant deterministic stochastic
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nD
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A Brief History of Volatility (1)
–
dSt   dWt Q
: Bachelier 1900
–
dSt
 r dt   dWt Q
St
: Black-Scholes 1973
–
dSt
 r (t ) dt   (t ) dWt Q
St
–
dSt
 (r  k ) dt   dWt Q  dq : Merton 1976
St
–
 dSt
Q

r
dt


dW
t
t
S
 t
d 2  a(V  V )dt     dZ
L
t
 t
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: Merton 1973
: Hull&White 1987
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A Brief History of Volatility (2)
dSt
  t dWt Q
St
 2 LT t 
Q
d  2
dt


dZ
t
2
T
2
t
dSt
 r (t ) dt   ( S , t ) dWt Q
St
C
C
 rK
 2  K , T   2 T 2  K
2  C K ,T
K
K 2
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Dupire 1992, arbitrage model
which fits term structure of
volatility given by log contracts.
Dupire 1993, minimal model
to fit current volatility surface
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A Brief History of Volatility (3)
 dSt
 S  r dt   t dWt
 t
d 2  b( 2   2 )dt   dZ

t
t
t
 t
dVK ,T   K ,T dt  bK ,T dZ tQ
VK ,T
: instantaneous forward variance
condit ional to
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ST  K
Heston 1993,
semi-analytical formulae.
Dupire 1996 (UTV),
Derman 1997,
stochastic volatility model
which fits current volatility
surface HJM treatment.
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Local Volatility Model
The smile model
•Black-Scholes:
dS
  dWt
S
•Merton:
dS
   t  dWt
S
• Simplest extension consistent with smile:
dS    S , t  dWt
(S,t) is called “local volatility”
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From simple to complex
European
prices
Local
volatilities
Exotic prices
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One Single Model
• We know that a model with dS = (S,t)dW
would generate smiles.
– Can we find (S,t) which fits market smiles?
– Are there several solutions?
ANSWER: One and only one way to do it.
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The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Risk
Neutral
Processes
Diffusions
Compatible
with Smile
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sought diffusion
(obtained by integrating twice
Fokker-Planck equation)
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Forward Equation
• BWD Equation: price of one option C K 0 ,T0  for different S, t 
C
b 2 ( S , t )  2C

t
2
S 2
• FWD Equation: price of all options C K , T  for current S0 ,t0 
C b 2 ( K , T )  2C

T
2
K 2
• Advantage of FWD equation:
– If local volatilities known, fast computation of implied volatility
surface,
– If current implied volatility surface known, extraction of local
volatilities:
2
b( K , T )  2
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C  C
/ 2
T K
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Summary of LVM Properties
0
is the initial volatility surface
•
 S,t  compatible with 0  
•
2
(K ,T )
   compatible with 0  E T2 ST  K    loc
local vol
(calibrated SVM are noisy versions of LVM)
•
̂ k,T deterministic function of (S,t) (if no jumps)
 future smile = FWD smile from local vol
• Extracts the notion of FWD vol (Conditional Instantaneous Forward
Variance)
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Extracting information
MEXBOL: Option Prices
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Non parametric fit of implied vols
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Risk Neutral density
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S&P 500: Option Prices
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Non parametric fit of implied vols
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Risk Neutral densities
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Implied Volatilities
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Local Volatilities
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Interest rates evolution
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Fed Funds evolution
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Volatility Arbitrages
Volatility as an Asset Class:
A Rich Playfield
Index
S
Vanillas
Implied
Volatility
Futures
VIX
C(S)
VIX options
C(RV)
Options on Realized
Variance
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VS
Variance
Swap
RV
Realized
Variance
Futures
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Outline
I.
II.
III.
IV.
Frequency arbitrage
Fair Skew
Dynamic arbitrage
Volatility derivatives
I. Frequency arbitrage
Frequencygram
Historical volatility tends to depend on the sampling
frequency: SPX historical vols over last 5 (left) and 2
(right) years, averaged over the starting dates
Can we take advantage of this pattern?
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Historical Vol / Historical Vol Arbitrage
If weekly historical vol < daily historical vol :
• buy strip of T options, Δ-hedge daily
• sell strip of T options, Δ-hedge weekly
 QVTdaily
 QVTweekly
Adding up :
• do not buy nor sell any option;
• play intra-week mean reversion until T;
• final P&L :
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 QVTdaily  QVTweekly
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Daily Vol / weekly Vol Arbitrage
-On each leg: always keep $ invested in the index and update every t
-Resulting spot strategy: follow each week a mean reverting strategy
-Keep each day the following exposure:
 .(
where
1
1

)
S t i , j S t i ,1
ti , j is the j-th day of the i-th week
-It amounts to follow an intra-week mean reversion strategy
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II. Fair Skew
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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Why Volatility Skews?
• Market prices governed by
– a) Anticipated dynamics (future behavior of volatility or jumps)
– b) Supply and Demand
 impl
Market Skew
Th. Skew
K
• To “ arbitrage” European options, estimate a) to capture
risk premium b)
• To “arbitrage” (or correctly price) exotics, find Risk
Neutral dynamics calibrated to the market
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Theoretical Skew from Prices
?
=>
Problem : How to compute option prices on an underlying without
options?
For instance : compute 3 month 5% OTM Call from price history only.
1) Discounted average of the historical Intrinsic Values.
Bad : depends on bull/bear, no call/put parity.
2) Generate paths by sampling 1 day return recentered histogram.
Problem : CLT => converges quickly to same volatility for all
strike/maturity; breaks autocorrelation and vol/spot dependency.
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Theoretical Skew from Prices (2)
3) Discounted average of the Intrinsic Value from recentered 3 month
histogram.
4) Δ-Hedging : compute the implied volatility which makes the Δhedging a fair game.
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Theoretical Skew
from historical prices (3)
How to get a theoretical Skew just from spot price
history?
S
K
Example:
ST
3 month daily data
t
T1
T2
1 strike K  k ST1
– a) price and delta hedge for a given  within Black-Scholes
1
–
–
–
–
model
b) compute the associated final Profit & Loss: PL 
c) solve for  k / PL  k  0
d) repeat a) b) c) for general time period and average
e) repeat a) b) c) and d) to get the “theoretical Skew”
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
  
 
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Theoretical Skew
from historical prices (4)
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Theoretical Skew
from historical prices (5)
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Theoretical Skew
from historical prices (6)
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Theoretical Skew
from historical prices (7)
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EUR/$, 2005
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Gold, 2005
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Intel, 2005
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S&P500, 2005
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JPY/$, 2005
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S&P500, 2002
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S&P500, 2003
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MexBol 2007
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III. Dynamic arbitrage
Arbitraging parallel shifts
• Assume every day normal implied vols are flat
• Level vary from day to day
Implied vol
Strikes
• Does it lead to arbitrage?
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W arbitrage
• Buy the wings, sell the ATM
• Symmetric Straddle and Strangle have no Delta and no
Vanna
• If same maturity, 0 Gamma => 0 Theta, 0 Vega

PF

Strangle

Straddle has a free Volga
• The W portfolio:

and no other Greek up to the second order
Strangle
Straddle
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Cashing in on vol moves
• The W Portfolio transforms all vol moves
into profit
• Vols should be convex in strike to prevent
this kind of arbitrage
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M arbitrage
•
In a sticky-delta smiley market:
K1
K2
S+
SS0
=(K1+K2)/2
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Deterministic future smiles
It is not possible to prescribe just any future
smile
If deterministic, one must have
C K ,T S 0 , t0     S 0 , t0 , S , T1  C K ,T S , T1 dS
2
2
Not satisfied in general
K
S0

t0
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T1
T2
98
Det. Fut. smiles & no jumps
=> = FWD smile
2
2
If S , t , K , T  / VK ,T S , t    K , T   lim  imp K , T , K  K , T  T 
K  0
T  0
stripped from implied vols at (S,t)
Then, there exists a 2 step arbitrage:
Define
 2C
2
S , t , K , T 
PLt   K , T   VK ,T S , t 
2
K
S0
S
K
At t0 : Sell PLt  Dig S  ,t  Dig S  ,t 
At t: if S t  S   , S   
t0
2
buy 2 CSK,T , sell 
K
gives a premium = PLt at t, no loss at T
Conclusion: VK ,T S , t  independent of
from initial smile
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t
2
T
K , T  K ,T
S , t   VK ,T S0 , t0    2 K , T 
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Consequence of det. future smiles
• Sticky Strike assumption: Each (K,T) has a fixed  impl ( K , T )
independent of (S,t)
• Sticky Delta assumption:  impl ( K , T ) depends only on
moneyness and residual maturity
•
In the absence of jumps,
– Sticky Strike is arbitrageable
– Sticky Δ is (even more) arbitrageable
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Example of arbitrage with Sticky Strike
Each CK,T lives in its Black-Scholes (  impl ( K , T ) ) world
C1  C K1 ,T1
assume  1   2
C2  C K 2 ,T2
P&L of Delta hedge position over dt:


PLCi   12 S 2   i S 2t i

 

PL C2  2 C1   2 S 2  12   22 t  0
1  2

no , free 


If no jump
C2
2
C1
1
C2
C2 
C1
1
St
2
St t
Bruno Dupire
!
2
C1
1
2
C1
1
St
S t t
101
Arbitrage with Sticky Delta
• In the absence of jumps, Sticky-K is arbitrageable and Sticky-Δ even more so.
• However, it seems that quiet trending market (no jumps!) are Sticky-Δ.
In trending markets, buy Calls, sell Puts and Δ-hedge.
Example:
K1
PF  C K 2  PK1
S
K2
1 , 2
VegaK > Vega
2
S
1 , 2
VegaK < Vega
2
Bruno Dupire
St
PF
K1
PF
Δ-hedged PF gains
from S induced
volatility moves.
K1
102
IV. Volatility derivatives
VIX Future Pricing
Vanilla Options
Simple product, but complex mix of underlying and volatility:
Call option has :
 Sensitivity to S : Δ
 Sensitivity to σ : Vega
These sensitivities vary through time and spot, and vol :
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105
Volatility Games
To play pure volatility games (eg bet that S&P vol goes
up, no view on the S&P itself):
Need of constant sensitivity to vol;
Achieved by combining several strikes;
Ideally achieved by a log profile : (variance swaps)
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106
Log Profile
 ST 
2
Under BS: dS=σS dW, price of E ln    2 T
 S0 
For all S,

S  S0 0 ( K  S ) 
S
(S  K )
ln( ) 

dK  
dK
2
2
S0
S0
K
K
0
S0
S
The log profile is decomposed as:
S0
1
S0
Futures  
0
PK ,T
K
2

dK  
S0
CK ,T
K
2
dK
In practice, finite number of strikes
VIX t2 
Ki 1  Ki 1 rT
2
1 F
e
X
(
K
,
T
)

(  1) 2

i
2
T
2Ki
T K0
Put if Ki<F,
Call otherwise
Bruno Dupire
CBOE definition:
FWD adjustment
107
Option prices for one maturity
Bruno Dupire
108
Perfect Replication of
VIX T21  
VIX
2
T1
 ST T 
2
pricet ln 1 
T
ST1 

ST1 2
ST1 T 
2
 pricet  ln

ln


T
S

T
S0 
0

 pricet PF 
We can buy today a PF which gives VIX2T1 at T1:
buy T2 options and sell T1 options.
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Theoretical Pricing of VIX Futures
FVIX before launch
• FVIXt: price at t of receiving
PFT1  VIX T1  FTVIX
1
at T1 .
FtVIX  Et [ PFT ]  Et [ PFT ]  PFt  Upper Bound (UB)
•The difference between both sides depends on the
variance of PF (vol vol).
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110
RV/VarS
• The pay-off of an OTC Variance Swap can be replicated
by a string of Realized Variance Futures:
t
T0
T
T1
T2
T3
T4
• From 12/02/04 to maturity 09/17/05, bid-ask in vol:
15.03/15.33
• Spread=.30% in vol, much tighter than the typical 1%
from the OTC market
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111
RV/VIX
• Assume that RV and VIX, with prices RV and F are defined on the
same future period [T1 ,T2]
2
• If at T0 , RV0  F0 then buy 1 RV Futures and sell 2 F0 VIX Futures
PL1  RV1  RV0  2 F0 ( F1  F0 )
• at T1
 RV1  F02  2 F0 ( F1  F0 )
 RV1  F12  ( F1  F0 ) 2
2
2
• If RV1  F1 sell the PF of options for F1 and Delta hedge in S until
maturity to replicate RV.
• In practice, maturity differ: conduct the same approach with a string
of VIX Futures
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Conclusion
• Volatility is a complex and important field
• It is important to
- understand how to trade it
- see the link between products
- have the tools to read the market
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The End