1. An Idiot’s Guide to Option Pricing
Bruno Dupire
Bloomberg LP
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?
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3. Naïve expectation
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4. Replication argument “as if” priced with other probabilities
instead of
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5. 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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7. To buy or not to buy? Call Option: Right to buy stock at T for K $ $
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8. 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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9. Option prices for one maturity
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10. 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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11. Risk Neutral Pricing

12. 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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13. Application to option pricing
Risk Neutral Probability
Physical Probability
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14. Basic Properties Price as a function of payoff is: - Positive:
- Linear:
Price = discounted expectation of payoff
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15. Toy Model 1 period, n possible states Option A gives in state If
, 0 in all other states,
where
is a discount factor
is a probability:
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16. 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
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17. 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%
50%
Rain: 1 Banana = 2 Apples
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18. Stochastic Calculus

19. 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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20. Brownian Motion
From discrete to continuous 10
100
1000
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21. Stochastic Differential Equations
At the limit:
continuous with independent Gaussian increments a
SDE:
drift noise
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22. Ito’s Dilemma Classical calculus: expand to the first order
Stochastic calculus:
should we expand further?
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23. Ito’s Lemma
At the limit If
for f(x),
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24. 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!
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25. P&L of a delta hedged option
Break-even
points
Option Value
Delta hedge
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26. Black-Scholes Model If instantaneous volatility is constant :
drift:
noise, SD:
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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27. Pricing methods

28. Pricing methods Analytical formulas Trees/PDE finite difference
Monte Carlo simulations
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29. Formula via PDE The Black-Scholes PDE is Reduces to the Heat Equation
With Fourier methods, Black-Scholes equation:
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30. Formula via discounted expectation
Risk neutral dynamics
Ito to ln S:
Integrating:
Same formula
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31. Finite difference discretization of PDE
Black-Scholes PDE
Partial derivatives discretized as
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32. 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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33. Computing expectations basic example
You play with a biased die
You want to compute the likelihood of getting
Throw the die times
Estimate p( ) by the number of over runs
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34. 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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35. 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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36. Generating Scenarios
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37. Low Discrepancy Sequences
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38. Hedging

39. To Hedge or Not To Hedge Daily P&L Daily Position Full P&L
Unhedged
Hedged
Full P&L
Big directional risk
Small daily amplitude risk
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40. 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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41. The Geometry of Hedging
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42. Super-replication
Property:
Let us call:
Which implies:
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43. A sight of Cauchy-Schwarz
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44. Volatility

45. 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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46. Historical Volatility
Measure of realized moves
annualized SD of logreturns
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47. Historical volatility
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48. Implied volatility
Input of the Black-Scholes formula which makes it fit the market price :
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49. 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 K K
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50. A Brief History of Volatility

51. Evolution theory of modeling
constant deterministic stochastic nD
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52. A Brief History of Volatility
: Bachelier 1900
: Black-Scholes 1973
: Merton 1973
: Merton 1976
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53. Local Volatility Model
Dupire 1993, minimal model to fit current volatility surface
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54. The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
1D Diffusions
Risk Neutral Processes
Compatible with Smile
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55. From simple to complex European prices Local volatilities
Exotic prices
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56. Stochastic Volatility Models
Heston 1993, semi-analytical formulae.
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57. The End