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An Idiot’s Guide to Option Pricing. Bruno Dupire Bloomberg LP bdupire@bloomberg.net CRFMS, UCSB April 26, 2007. Warm-up. Roulette:. A lottery ticket gives: . You can buy it or sell it for $60 Is it cheap or expensive?. Naïve expectation. Replication argument.

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an idiot s guide to option pricing

An Idiot’s Guide to Option Pricing

Bruno Dupire

Bloomberg LP

bdupire@bloomberg.net

CRFMS, UCSB

April 26, 2007

warm up
Warm-up

Roulette:

A lottery ticket gives:

You can buy it or sell it for $60

Is it cheap or expensive?

replication argument
Replication argument

“as if” priced with other probabilities

instead of

outline

OUTLINE

Risk neutral pricing

Stochastic calculus

Pricing methods

Hedging

Volatility

Volatility modeling

addressing financial risks
Addressing Financial Risks

Over the past 20 years, intense development of Derivatives

in terms of:

  • volume
  • underlyings
  • products
  • models
  • users
  • regions
to buy or not to buy
To buy or not to buy?
  • Call Option:Rightto buy stock at T for K

$

$

TO BUY

NOT TO BUY

K

K

$

CALL

K

vanilla options
Vanilla Options

European Call:

Gives the right to buy the underlying at a fixed price (the strike) at some future time (the maturity)

European Put:

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

risk management
Risk Management

Client has risk exposure

Buys a product from a bank to limit its risk

Not Enough

Too Costly

Perfect Hedge

Risk

Exotic Hedge

Vanilla Hedges

Client transfers risk to the bank which has the technology to handle it

Product fits the risk

price as discounted expectation
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

application to option pricing
Application to option pricing

Risk Neutral Probability

Physical Probability

basic properties
Basic Properties

Price as a function of payoff is:

- Positive:

- Linear:

Price = discounted expectation of payoff

toy model
Toy Model

1 period, n possible states

Option A gives

in state

gives 1 in state

, 0 in all other states,

If

where

is a discount factor

is a probability:

slide16
FTAP

Fundamental Theorem of Asset Pricing

  • 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

risk neutrality paradox
Risk Neutrality Paradox
  • Risk neutrality: carelessness about uncertainty?
  • 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

Sun: 1 Apple = 2 Bananas

50%

Rain: 1 Banana = 2 Apples

50%

modeling uncertainty

S

t

t

S

Modeling Uncertainty

Main ingredients for spot modeling

  • Many small shocks: Brownian Motion (continuous prices)
  • A few big shocks: Poisson process (jumps)
brownian motion
Brownian Motion
  • From discrete to continuous

10

100

1000

stochastic differential equations

a

Stochastic Differential Equations

At the limit:

continuous with independent Gaussian increments

SDE:

drift noise

ito s dilemma
Ito’s Dilemma

Classical calculus:

expand to the first order

Stochastic calculus:

should we expand further?

ito s lemma
Ito’s Lemma

At the limit

If

for f(x),

black scholes pde
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!
p l of a delta hedged option

Option Value

P&L

Break-even

points

Delta hedge

P&L of a delta hedged option
black scholes model

drift:

noise, SD:

Black-Scholes Model

If instantaneous volatility is constant :

Then call prices are given by :

No drift in the formula, only the interest rate r due to the hedging argument.

pricing methods28
Pricing methods
  • Analytical formulas
  • Trees/PDE finite difference
  • Monte Carlo simulations
formula via pde
Formula via PDE
  • The Black-Scholes PDE is
  • Reduces to the Heat Equation
  • With Fourier methods, Black-Scholes equation:
formula via discounted expectation
Formula via discounted expectation
  • Risk neutral dynamics
  • Ito to ln S:
  • Integrating:
  • Same formula
finite difference discretization of pde
Finite difference discretization of PDE
  • Black-Scholes PDE
  • Partial derivatives discretized as
option pricing with monte carlo methods
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
computing expectations basic example
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
option pricing superdie
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 complex

running a Monte Carlo path simulation

expectation integral
Expectation = Integral

Gaussian transform techniques

discretisation schemes

Unit hypercube

Gaussian coordinates

trajectory

A point in the hypercube maps to a spot trajectory

therefore

to hedge or not to hedge

P&L

Unhedged

Hedged

0

To Hedge or Not To Hedge

Daily P&L

Daily Position

Full P&L

Big directional risk

Small daily amplitude risk

the geometry of hedging
The Geometry of Hedging
  • Risk measured as
  • Target X, hedge H
  • Risk is an L2 norm, with general properties of orthogonal projections
  • Optimal Hedge:
slide42

Super-replication

  • Property:
  • Let us call:
  • Which implies:
volatility some definitions
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

historical volatility
Historical Volatility
  • Measure of realized moves
  • annualized SD of logreturns
implied volatility
Implied volatility

Input of the Black-Scholes formula which makes it fit the market price :

market skews

K

K

K

Market Skews

Dominating fact since 1987 crash: strong negative skew on

Equity Markets

Not a general phenomenon

Gold: FX:

We focus on Equity Markets

evolution theory of modeling
Evolution theory of modeling

constant deterministic stochastic nD

a brief history of volatility52
A Brief History of Volatility
  • : Bachelier 1900
  • : Black-Scholes 1973
  • : Merton 1973
  • : Merton 1976
local volatility model
Local Volatility Model

Dupire 1993, minimal model to fit current volatility surface

the risk neutral solution

Risk NeutralProcesses

1D Diffusions

Compatible with Smile

The Risk-Neutral Solution

But if drift imposed (by risk-neutrality), uniqueness of the solution

from simple to complex
From simple to complex

European prices

Localvolatilities

Exotic prices

stochastic volatility models
Stochastic Volatility Models

Heston 1993, semi-analytical formulae.