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Pricing Financial Derivatives
Bruno DupireBloomberg L.P/NYU
AIMS Day 1
Cape Town, February 17, 2011
Bruno Dupire 2
Addressing Financial Risks
•volume•underlyings•products•models•users•regions
Over the past 20 years, intense development of Derivatives
in terms of:
Bruno Dupire 3
Vanilla Options
European Call:Gives the right to buy the underlying at a fixed price (the strike) at some future time (the maturity)
TSKPayoffPut
European Put:Gives the right to sell the underlying at a fixed strike at some maturity
0,max)(Payoff Call KSKS TT
Bruno Dupire 4
Option prices for one maturity
Bruno Dupire 5
Risk Management
Client has risk exposure
Buys a product from a bank to limit its risk
Risk
Not Enough Too Costly Perfect Hedge
Vanilla Hedges Exotic Hedge
Client transfers risk to the bank which has the technology to handle it
Product fits the risk
OUTLINE
A) Theory- Risk neutral pricing- Stochastic calculus- Pricing methods
B) Volatility- Definition and estimation- Volatility modeling- Volatility arbitrage
A) THEORY
Risk neutral pricing
Bruno Dupire 9
Warm-up
%30][
%70][
Black
Red
P
P
Black if$0
Red if100$
Roulette:
A lottery ticket gives:
You can buy it or sell it for $60Is it cheap or expensive?
Bruno Dupire 10
2 approaches
Buy6070
Sell6050
Naïve expectation
Replication Argument
“as if” priced with other probabilities instead of
Bruno Dupire 11
Price as discounted expectation
!?
Option gives uncertain payoff in the futurePremium: known price today
Resolve the uncertainty by computing expectation:
Transfer future into present by discounting
?!
Bruno Dupire 12
Application to option pricing
Risk Neutral ProbabilityPhysical Probability
o TTT
rT dSKSSe ))((Price
Stochastic Calculus
Bruno Dupire 14
Modeling Uncertainty
Main ingredients for spot modeling• Many small shocks: Brownian Motion
(continuous prices)
• A few big shocks: Poisson process (jumps)
t
S
t
S
Bruno Dupire 15
Brownian Motion
10
100
1000
• From discrete to continuous
Bruno Dupire 16
Stochastic Differential Equations
tW
),0(~ stNWW st
dWdtdx ba
At the limit:
Continuous with independent Gaussian increments
SDE:
drift noise
a
Bruno Dupire 17
Ito’s Dilemma
)(xfy
dWdtdx ba
Classical calculus:
expand to the first order
Stochastic calculus:
should we expand further?
dxxfdy )('
...)(' dxxfdy
Bruno Dupire 18
Ito’s Lemma
dtdW 2)(
dWbdtadx
dtbxfdxxf
dxxfdxxf
xfdxxfdf
2
2
)(''2
1)('
)()(''2
1)('
)()(
At the limit
for f(x),
If
Bruno Dupire 19
Black-Scholes PDE
• Black-Scholes assumption
• Apply Ito’s formula to Call price C(S,t)
• Hedged position is riskless, earns interest rate r
• Black-Scholes PDE
• No drift!
dWdtS
dS
dtCS
CdSCdC SStS )2
(22
dtSCCrdSCdCdtCS
C SSSSt )()2
(22
)(2
22
SCCrCS
C SSSt
SCC S
Bruno Dupire 20
P&L
St t
St
Break-even points
t
t
Option Value
St
CtCt t
S
Delta hedge
P&L of a delta hedged option
Bruno Dupire 21
Black-Scholes Model
If instantaneous volatility is constant :
dWdtS
dS
Then call prices are given by :
)2
1)
)exp(ln(
1()exp(
)2
1)
)exp(ln(
1(
0
00
TK
rTS
TNrTK
TK
rTS
TNSCBS
No drift in the formula, only the interest rate r due to the hedging argument.
drift:
noise, SD:
tSt
tSt
Pricing methods
Bruno Dupire 23
Pricing methods
• Analytical formulas
• Trees/PDE finite difference
• Monte Carlo simulations
Bruno Dupire 24
Formula via PDE
• The Black-Scholes PDE is
• Reduces to the Heat Equation
• With Fourier methods, Black-Scholes equation:
)(2
22
SCCrCS
C SSSt
TddT
TrKSd
dNeKdNSC rTBS
12
20
1
210
,)2/()/ln(
)()(
xxUU2
1
Bruno Dupire 25
Formula via discounted expectation
• Risk neutral dynamics
• Ito to ln S:
• Integrating:
• Same formula
dWdtrS
dS
dWdtrSd )
2(ln
2
])[(])[()
2(
0
2
KeSEeKSEepremiumTWTrrT
TrT
TT WTrSS )
2(lnln
2
0
Bruno Dupire 26
Finite difference discretization of PDE
• Black-Scholes PDE
• Partial derivatives discretized as
)(),(
)(2
22
KSTSC
SCCrCS
C
T
SSSt
2)(
),1(),(2),1(),(
2
),1(),1(),(
)1,(),(),(
S
niCniCniCinC
S
niCniCniC
t
niCniCniC
SS
S
t
Bruno Dupire
Option pricing with Monte Carlo methods
• An option price is the discounted expectation of its payoff:
• Sometimes the expectation cannot be computed analytically:– complex product– complex dynamics
• Then the integral has to be computed numerically
P EP f x x dxT0
the option price is its discounted payoffintegrated against the risk neutral density of the spot underlying
Bruno Dupire
Computing expectationsbasic example
•You play with a biased die
•You want to compute the likelihood of getting
•Throw the die 10.000 times
•Estimate p( ) by the number of over 10.000 runs
B) VOLATILITY
Bruno Dupire 30
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
Bruno Dupire 31
Historical volatility
Bruno Dupire 32
Historical Volatility
• Measure of realized moves• annualized SD of
t
tt S
Sx 1ln
2
1
2
1
252ii t
n
it xx
n
Bruno Dupire 33
• Commonly available information: open, close, high, low
• Captures valuable volatility information
• Parkinson estimate:
• Garman-Klass estimate:
Estimates based on High/Low
n
tttP du
n 1
22
2ln4
1
n
tt
n
tttGK c
ndu
n 1
2
1
22 39.05.0
Sln
open
upper
S
Su ln
open
close
S
Sc ln
open
down
S
Sd ln
Bruno Dupire 34
Move based estimation
• Leads to alternative historical vol estimation:
= number of crossings of log-price over [0,T]),( TL T
TLh
),(~
Bruno Dupire 35
Black-Scholes Model
If instantaneous volatility is constant :
dWdtS
dS
Then call prices are given by :
)2
1)
)exp(ln(
1()exp(
)2
1)
)exp(ln(
1(
0
00
TK
rTS
TNrTK
TK
rTS
TNSCBS
No drift in the formula, only the interest rate r due to the hedging argument.
Bruno Dupire 36
Implied volatility
Input of the Black-Scholes formula which makes it fit the market price :
Bruno Dupire 37
Market Skews
Dominating fact since 1987 crash: strong negative skew on Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets
K
impl
K
impl
K
impl
Bruno Dupire 38
Skews
• Volatility Skew: slope of implied volatility as a function of Strike
• Link with Skewness (asymmetry) of the Risk Neutral density function ?
Moments Statistics Finance
1 Expectation FWD price
2 Variance Level of implied vol
3 Skewness Slope of implied vol
4 Kurtosis Convexity of implied vol
Bruno Dupire 39
Why Volatility Skews?
• Market prices governed by– a) Anticipated dynamics (future behavior of volatility or jumps)– b) Supply and Demand
• 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
K
impl Market Skew
Th. Skew
Supply and Demand
Bruno Dupire 40
Modeling Uncertainty
Main ingredients for spot modeling• Many small shocks: Brownian Motion
(continuous prices)
• A few big shocks: Poisson process (jumps)
t
S
t
S
Bruno Dupire 41
• To obtain downward sloping implied volatilities
– a) Negative link between prices and volatility• Deterministic dependency (Local Volatility Model)• Or negative correlation (Stochastic volatility Model)
– b) Downward jumps
K
impl
2 mechanisms to produce Skews (1)
Bruno Dupire 42
2 mechanisms to produce Skews (2)
– a) Negative link between prices and volatility
– b) Downward jumps
1S 2S
1S 2S
Bruno Dupire 43
Strike dependency
• Fair or Break-Even volatility is an average of squared returns, weighted by the Gammas, which depend on the strike
Bruno Dupire 44
Strike dependency for multiple paths
A Brief History of Volatility
Bruno Dupire 46
Evolution Theory
constant deterministic stochastic nD
Bruno Dupire 47
A Brief History of Volatility (1)
– : Bachelier 1900
– : Black-Scholes 1973
– : Merton 1973
– : Merton 1976
– : Hull&White 1987
Qt
t
t dWtdttrS
dS )( )(
Qtt dWdS
tLt
Qtt
t
t
dZdtVVad
dWdtrS
dS
)(
2
Qt
t
t dWdtrS
dS
dqdWdtkrS
dS Qt
t
t )(
Bruno Dupire 48
A Brief History of Volatility (2)
Dupire 1992, arbitrage model
which fits term structure of
volatility given by log contracts.
Dupire 1993, minimal model
to fit current volatility surface
Qt
Tt
Qtt
t
t
dZdtT
tLd
dWS
dS
2
2
22
2
,2
2
2 2,
),( )(
K
CK
KC
rKTC
TK
dWtSdttrS
dS
TK
Qt
t
t
Bruno Dupire 49
A Brief History of Volatility (3)
Heston 1993,
semi-analytical formulae.
Dupire 1996 (UTV), Derman 1997,
stochastic volatility model
which fits current volatility
surface HJM treatment.
tttt
ttt
t
dZdtbd
dWdtrS
dS
)(
222
K
V
dZbdtdV
TK
QtTKTKTK
T
,
,,,
S
tolconditiona
varianceforward ousinstantane :
Local Volatility Model
Bruno Dupire 51
•Black-Scholes:
•Merton:
• Simplest extension consistent with smile:
(S,t) is called “local volatility”
dS
St dWt
dS S t dWt ,
dS
SdWt
The smile model
Bruno Dupire 52
European prices
Localvolatilities
Localvolatilities
Exotic prices
From simple to complex
Bruno Dupire 53
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.
Bruno Dupire 54
sought diffusion(obtained by integrating twice
Fokker-Planck equation)
Diffusions
Risk Neutral
Processes
Compatible with Smile
The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Bruno Dupire 55
Forward Equation
• BWD Equation: price of one option for different
• FWD Equation: price of all options for current
• 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
22
2
),(
S
CtSb
t
C
tS ,
00 , tS TKC ,
00 ,TKC
2
22
2
),(
K
CTKb
T
C
2
2
/2),(K
C
T
CTKb
Bruno Dupire 56
Summary of LVM Properties
is the initial volatility surface
• compatible with local vol
• compatible with
(calibrated SVM are noisy versions of LVM)
• 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)
tS, 0
Tk,̂
),(220 TKKSE locTT
0
Extracting information
Bruno Dupire 58
MEXBOL: Option Prices
Bruno Dupire 59
Non parametric fit of implied vols
Bruno Dupire 60
Risk Neutral density
Bruno Dupire 61
S&P 500: Option Prices
Bruno Dupire 62
Non parametric fit of implied vols
Bruno Dupire 63
Risk Neutral densities
Bruno Dupire 64
Implied Volatilities
Bruno Dupire 65
Local Volatilities
Bruno Dupire 66
Interest rates evolution
Bruno Dupire 67
Fed Funds evolution
Volatility Arbitrages
Bruno Dupire 69
Volatility as an Asset Class:A Rich Playfield
S
C(S) VIX
RVC(RV) VS
Index
Vanillas
Variance Swap
Options on Realized Variance
Realized VarianceFutures
Implied VolatilityFutures
VIX options
Outline
I. Frequency arbitrageII. Fair SkewIII. Dynamic arbitrageIV. Volatility derivatives
I. Frequency arbitrage
Bruno Dupire 72
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?
Bruno Dupire 73
Historical Vol / Historical Vol Arbitrage
weeklyTQV
dailyTQV
If weekly historical vol < daily historical vol :
• buy strip of T options, Δ-hedge daily
• sell strip of T options, Δ-hedge weekly
Adding up :
• do not buy nor sell any option;
• play intra-week mean reversion until T;
• final P&L : weeklyT
dailyT QVQV
Bruno Dupire 74
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 is the j-th day of the i-th week
-It amounts to follow an intra-week mean reversion strategy
jit ,
)11
.(1,, iji tt SS
II. Fair Skew
Bruno Dupire 76
Market Skews
Dominating fact since 1987 crash: strong negative skew on Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets
K
impl
K
impl
K
impl
Bruno Dupire 77
Why Volatility Skews?
• Market prices governed by– a) Anticipated dynamics (future behavior of volatility or jumps)– b) Supply and Demand
• 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
K
impl Market Skew
Th. Skew
Supply and Demand
Bruno Dupire 78
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.
?
=>
Bruno Dupire 79
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 Δ-hedging a fair game.
Bruno Dupire 80
Theoretical Skewfrom historical prices (3)
How to get a theoretical Skew just from spot price history?
Example:
3 month daily data
1 strike – a) price and delta hedge for a given within Black-Scholes
model– b) compute the associated final Profit & Loss: – c) solve for– d) repeat a) b) c) for general time period and average– e) repeat a) b) c) and d) to get the “theoretical Skew”
1TSkK
PL 0/ kPLk
t
S
1T 2T
K1T
S
Bruno Dupire 81
Theoretical Skewfrom historical prices (4)
Bruno Dupire 82
Theoretical Skewfrom historical prices (5)
Bruno Dupire 83
Theoretical Skewfrom historical prices (6)
Bruno Dupire 84
Theoretical Skewfrom historical prices (7)
Bruno Dupire 85
EUR/$, 2005
Bruno Dupire 86
Gold, 2005
Bruno Dupire 87
Intel, 2005
Bruno Dupire 88
S&P500, 2005
Bruno Dupire 89
JPY/$, 2005
Bruno Dupire 90
S&P500, 2002
Bruno Dupire 91
S&P500, 2003
Bruno Dupire 92
MexBol 2007
III. Dynamic arbitrage
Bruno Dupire 94
Arbitraging parallel shifts
• Assume every day normal implied vols are flat• Level vary from day to day
• Does it lead to arbitrage?
Strikes
Implied vol
Bruno Dupire 95
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• The W portfolio: has a free
Volga and no other Greek up to the second order
StraddleStranglePFStraddle
Strangle
Bruno Dupire 96
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
Bruno Dupire 97
M arbitrage
K1 K2
S0S+S-
=(K1+K2)/2
• In a sticky-delta smiley market:
Bruno Dupire 98
S0
K
T1 T2t0
Deterministic future smiles
It is not possible to prescribe just any future smile
If deterministic, one must have
Not satisfied in general
dSTSCTStStSC TKTK 1,10000, ,,,,,22
Bruno Dupire 99
Det. Fut. smiles & no jumps => = FWD smile
If
stripped from implied vols at (S,t)
Then, there exists a 2 step arbitrage:Define
At t0 : Sell
At t:
gives a premium = PLt at t, no loss at T
Conclusion: independent of from initial smile
TTKKTKTKtSVTKtS imp
TK
TK
,,,lim,,/,,, 2
00
2,
TKtSK
CtSVTKPL TKt ,,,,,
2
2
,2
tStSt DigDigPL ,, S
t0 t T
S0
K
TKt TKK
SSS ,2
TK,2, sell , CS
2buy , if
TKtSVtS TK ,,, 200, tSV TK ,,
Bruno Dupire 100
Consequence of det. future smiles
• Sticky Strike assumption: Each (K,T) has a fixed independent of (S,t)
• Sticky Delta assumption: depends only on moneyness and residual maturity
• In the absence of jumps,– Sticky Strike is arbitrageable– Sticky Δ is (even more) arbitrageable
),( TKimpl
),( TKimpl
Bruno Dupire 101
Each CK,T lives in its Black-Scholes ( ) world
P&L of Delta hedge position over dt:
If no jump
Example of arbitrage with Sticky Strike
21,2,1 assume2211
TKTK CCCC
!
free , no
02
22
21
221
1
22
22
21
tSCCPL
tSSCPL iii
),( TKimpl
1 21C
2C 11
2 C
ttS
tS
11
2 C
2C
11
22 CC
tS ttS
Bruno Dupire 102
Arbitrage with Sticky Delta
1K
2K
tS
• 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:
12 KK PCPF
21,S
Vega > Vega2K 1K
PF
21,
Vega < Vega2K 1K
S PF
Δ-hedged PF gains from S induced volatility moves.
IV. Volatility derivatives
VIX Future Pricing
Bruno Dupire 105
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 :
Bruno Dupire 106
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)
Bruno Dupire 107
Log Profile
TS
SE T
20
2
ln
dKK
KSdK
K
SK
S
SS
S
S
S
S
0
0
20
20
0
0
)()()ln(
dKK
CdK
K
PFutures
S
TKS
TKS
0
0
0 2,
02,1
Under BS: dS=σS dW, price ofFor all S,The log profile is decomposed as:
In practice, finite number of strikes CBOE definition:2
02
112 )1(1
),(2
2
K
F
TTKXe
K
KK
TVIX i
rT
i
iit
Put if Ki<F,
Call otherwise
FWD adjustment
Bruno Dupire 108
Option prices for one maturity
Bruno Dupire 109
Perfect Replication of 2
1TVIX
1
1
1ln
22
T
TTtT S
Sprice
TVIX
00
11 ln2
ln2
S
S
TS
S
Tprice TTT
t
We can buy today a PF which gives VIX2T1 at T1:
buy T2 options and sell T1 options.
PFpricet
Bruno Dupire 110
Theoretical Pricing of VIX Futures FVIX before launch
• FVIXt: price at t of receiving at T1
.
VIXTTT FVIXPF
111
)(][][ UBBoundUpperPFPFEPFEF tTtTtVIXt
•The difference between both sides depends on the variance of PF (vol vol).
Bruno Dupire 111
RV/VarS
• The pay-off of an OTC Variance Swap can be replicated by a string of Realized Variance Futures:
• 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
t T
T1 T2 T3T0 T4
Bruno Dupire 112
RV/VIX
• Assume that RV and VIX, with prices RV and F are defined on the same future period [T1 ,T2]
• If at T0 , then buy 1 RV Futures and sell 2 F0 VIX Futures
• at T1
• If sell the PF of options for and Delta hedge in S until maturity to replicate RV.
• In practice, maturity differ: conduct the same approach with a string of VIX Futures
200 FRV
201
21
0102
01
010011
)(
)(2
)(2
1FFFRV
FFFFRV
FFFRVRVPL
211 FRV
21F
Bruno Dupire 113
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
The End