An Idiot

1
An Idiots Guide to Option Pricing

• Bruno Dupire
• Bloomberg LP
• bdupire_at_bloomberg.net
• CRFMS, UCSB
• April 26, 2007

2
Warm-up
Roulette
A lottery ticket gives
You can buy it or sell it for 60 Is it cheap or
expensive?
3
Naïve expectation
4
Replication argument
as if priced with other probabilities
instead of
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OUTLINE

• Risk neutral pricing
• Stochastic calculus
• Pricing methods
• Hedging
• Volatility
• Volatility modeling

6
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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To buy or not to buy?

• Call Option Right to buy stock at T for K

TO BUY
NOT TO BUY
K
K

CALL
K
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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)
European Put Gives the right to sell the
underlying at a fixed strike at some maturity
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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
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
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Risk Neutral Pricing
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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
Physical Probability
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Basic Properties
Price as a function of payoff is -
Positive
- Linear
Price discounted expectation of payoff
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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
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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 gt0 claims
Separating hyperplanes
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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
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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)

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Brownian Motion

• From discrete to continuous

10
100
1000
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Stochastic Differential Equations
At the limit
continuous with independent Gaussian increments
SDE
drift noise
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Itos Dilemma
Classical calculus expand to the first
order Stochastic calculus should we expand
further?
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Itos Lemma
At the limit
If
for f(x),
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Black-Scholes PDE

• Black-Scholes assumption
• Apply Itos formula to Call price C(S,t)
• Hedged position is riskless, earns
  interest rate r
• Black-Scholes PDE
• No drift!

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PL of a delta hedged option
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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.
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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

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Formula via discounted expectation

• Risk neutral dynamics
• Ito to ln S
• Integrating
• Same formula

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Finite difference discretization of PDE

• Black-Scholes PDE
• Partial derivatives discretized as

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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

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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 complex

running a Monte Carlo path simulation
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Expectation Integral
Gaussian transform techniques
discretisation schemes
Unit hypercube
Gaussian coordinates
trajectory
A point in the hypercube maps to a spot
trajectory therefore
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Generating Scenarios
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Low Discrepancy Sequences
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Hedging
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To Hedge or Not To Hedge
Daily PL
Daily Position
Full PL
Big directional risk
Small daily amplitude risk
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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

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The Geometry of Hedging
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Super-replication

• Property
• Let us call
• Which implies

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A sight of Cauchy-Schwarz
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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

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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

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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

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Local Volatility Model

• Dupire 1993, minimal model to fit current
  volatility surface

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The Risk-Neutral Solution
But if drift imposed (by risk-neutrality),
uniqueness of the solution
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From simple to complex
European prices
Localvolatilities
Exotic prices
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Stochastic Volatility Models

• Heston 1993, semi-analytical formulae.

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The End