An Idiot’s Guide to Option Pricing

Bruno DupireBloomberg LP

bdupire@bloomberg.net

CRFMS, UCSBApril 26, 2007

Bruno Dupire 2

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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Naïve expectation

Buy6070

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Replication argument

Sell6050

“as if” priced with other probabilities

instead of

OUTLINE

1. Risk neutral pricing

2. Stochastic calculus

3. Pricing methods

4. Hedging

5. Volatility

6. Volatility modeling

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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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$

TSK

To buy or not to buy?

• Call Option: Right to buy stock at T for K

$

TSK

$

TSK

TO BUY NOT TO BUY

CALL

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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

Risk Neutral Pricing

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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

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Basic Properties

0)(price0 AA

)(price)(price)(price BABA

Price as a function of payoff is:

- Positive:

- Linear:

Price = discounted expectation of payoff

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gives 1 in state

Option A gives

Toy Model

nss ,...,1

ix is

1 period, n possible states

in state

iA is

iii

iii

iii

xq

AxAAxA

)(price)(price

If , 0 in all other states,

where price(1)price)(price ii AA is a discount factor

jj

ii A

Aq

)(price

)(price1and0

iii qq is a probability:

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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

Cone of >0 claimsClaims attainable from 0

Separating hyperplanes

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Risk Neutrality Paradox

• Risk neutrality: carelessness about uncertainty?

• 1 A gives either 2 B or .5 B1.25 B• 1 B gives either .5 A or 2 A1.25 A• Cannot be RN wrt 2 numeraires with the same probability

Sun: 1 Apple = 2 Bananas

Rain: 1 Banana = 2 Apples

50%

50%

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

)(xf

dWdtdx b a

Classical calculus:

expand to the first order

Stochastic calculus:

should we expand further?

dxxfdf )('

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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))/(ln(

1(

)2

1))/(ln(

1(

0

00

TrTKST

NKe

TrTKST

NSC

rT

BS

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

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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

- when the dynamics are complexrunning a Monte Carlo path simulation

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Expectation = Integral

Unit hypercube Gaussian coordinates trajectory

Gaussian transform techniques discretisation schemes

A point in the hypercube maps to a spot trajectorytherefore

EP f x S S dx g y dy

Ng x

T t tdd d

ixi

d

.Pr ,...,,1

,1

10

0

1

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Generating Scenarios

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Low Discrepancy Sequences

dimensions1 & 2

Halton Faure Sobol

dimensions20 & 25

dimensions51 & 52

Hedging

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To Hedge or Not To Hedge

Daily Position

Daily P&L

Full P&L

Big directional risk HedgeDelta Small daily amplitude risk

S

P&L

Unhedged

0Hedged

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The Geometry of Hedging

• Risk measured as • Target X, hedge H

• Risk is an L2 norm, with general properties of orthogonal projections

• Optimal Hedge:

TPLSD

ttt HXPL

HXHXRisk TTvar

H

HXHXH

infˆ

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The Geometry of Hedging

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Super-replication

•Property:

Let us call:

Which implies:

E XY E X E Y 2 2

yx

yx

xy

PP

YPXPXY

YPXPYX

2

:Portfolio by the dominated is XY so

,0 , and allFor

22

2

P X

P Yx

y

:

:

price today of

price today of

2

2

price XYP P P P

P PP P

y x x y

x yx y

2

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A sight of Cauchy-Schwarz

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

• Measure of realized moves• annualized SD of logreturns

t

tt S

Sx 1ln

2

1

2

1

252ii t

n

it xx

n

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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

Not a general phenomenon

Gold: FX:

We focus on Equity Markets

K

impl

K

impl

K

impl

A Brief History of Volatility

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Evolution theory of modeling

constant deterministic stochastic nD

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A Brief History of Volatility

– : Bachelier 1900

– : Black-Scholes 1973

– : Merton 1973

– : Merton 1976

Qtt

t

t dWtdtrS

dS )(

Qtt dWdS

Qt

t

t dWdtrS

dS

dqdWdtkrS

dS Qt

t

t )(

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Local Volatility Model

Dupire 1993, minimal model to fit current volatility surface

2

,2

2

2 2,

),(

K

CK

KC

rKTC

TK

dWtSdtrS

dS

TK

Qt

t

t

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sought diffusion(obtained by integrating twice

Fokker-Planck equation)

1D Diffusion

s

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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European prices

Localvolatilities

Localvolatilities

Exotic prices

From simple to complex

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Stochastic Volatility Models

Heston 1993, semi-analytical formulae.

tttt

ttt

t

dZdtbd

dWdtrS

dS

)(

222

The End