Pricing Financial Derivatives

Bruno DupireBloomberg L.P/NYU

AIMS Day 1

Cape Town, February 17, 2011

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Addressing Financial Risks

•volume•underlyings•products•models•users•regions

Over the past 20 years, intense development of Derivatives

in terms of:

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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)

TSKPayoffPut

European Put:Gives the right to sell the underlying at a fixed strike at some maturity

0,max)(Payoff Call KSKS TT

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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

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

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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?

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2 approaches

Buy6070

Sell6050

Naïve expectation

Replication Argument

“as if” priced with other probabilities instead of

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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

?!

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Application to option pricing

Risk Neutral ProbabilityPhysical Probability

o TTT

rT dSKSSe ))((Price

Stochastic Calculus

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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

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Brownian Motion

10

100

1000

• From discrete to continuous

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Stochastic Differential Equations

tW

),0(~ stNWW st

dWdtdx ba

At the limit:

Continuous with independent Gaussian increments

SDE:

drift noise

a

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Ito’s Dilemma

)(xfy

dWdtdx ba

Classical calculus:

expand to the first order

Stochastic calculus:

should we expand further?

dxxfdy )('

...)(' dxxfdy

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Ito’s Lemma

dtdW 2)(

dWbdtadx

dtbxfdxxf

dxxfdxxf

xfdxxfdf

2

2

)(''2

1)('

)()(''2

1)('

)()(

At the limit

for f(x),

If

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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

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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

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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

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Pricing methods

• Analytical formulas

• Trees/PDE finite difference

• Monte Carlo simulations

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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

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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

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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

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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

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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

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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

t

tt S

Sx 1ln

2

1

2

1

252ii t

n

it xx

n

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• 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

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Move based estimation

• Leads to alternative historical vol estimation:

= number of crossings of log-price over [0,T]),( TL T

TLh

),(~

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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.

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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

Not a general phenomenon

Gold: FX:

We focus on Equity Markets

K

impl

K

impl

K

impl

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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 Statistics Finance

1 Expectation FWD price

2 Variance Level of implied vol

3 Skewness Slope of implied vol

4 Kurtosis 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

• 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

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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

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• 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)

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2 mechanisms to produce Skews (2)

– a) Negative link between prices and volatility

– b) Downward jumps

1S 2S

1S 2S

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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

A Brief History of Volatility

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Evolution Theory

constant deterministic stochastic nD

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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 )(

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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

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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

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•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

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European prices

Localvolatilities

Localvolatilities

Exotic prices

From simple to complex

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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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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

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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

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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

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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

Volatility Arbitrages

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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

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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

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

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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 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

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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

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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

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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 Δ-hedging a fair game.

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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

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Theoretical Skewfrom historical prices (4)

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Theoretical Skewfrom historical prices (5)

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Theoretical Skewfrom historical prices (6)

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Theoretical Skewfrom 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

III. Dynamic arbitrage

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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

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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• The W portfolio: has a free

Volga and no other Greek up to the second order

StraddleStranglePFStraddle

Strangle

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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

K1 K2

S0S+S-

=(K1+K2)/2

• In a sticky-delta smiley market:

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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

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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 ,,

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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

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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

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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

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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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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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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

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Option prices for one maturity

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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

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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