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An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP bdupire@bloomberg. net CRFMS, UCSB An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP [email protected] net CRFMS, UCSB April 26, 2007 Bruno Dupire

Warm-up Roulette: A lottery ticket gives: You can buy it or sell it for Warm-up Roulette: A lottery ticket gives: You can buy it or sell it for $60 Is it cheap or expensive? Bruno Dupire 2

Naïve expectation Bruno Dupire 3 Naïve expectation Bruno Dupire 3

Replication argument “as if” priced with other probabilities instead of Bruno Dupire 4 Replication argument “as if” priced with other probabilities instead of Bruno Dupire 4

OUTLINE 1. 2. 3. 4. 5. 6. Bruno Dupire Risk neutral pricing Stochastic calculus OUTLINE 1. 2. 3. 4. 5. 6. Bruno Dupire Risk neutral pricing Stochastic calculus Pricing methods Hedging Volatility modeling

Addressing Financial Risks Over the past 20 years, intense development of Derivatives in terms Addressing Financial Risks Over the past 20 years, intense development of Derivatives in terms of: • volume • underlyings • products • models • users • regions Bruno Dupire 6

To buy or not to buy? • Call Option: Right to buy stock at To buy or not to buy? • Call Option: Right to buy stock at T for K $ TO BUY $ NOT TO BUY K K $ CALL Bruno Dupire K 7

Vanilla Options European Call: Gives the right to buy the underlying at a fixed 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 Bruno Dupire 8

Option prices for one maturity Bruno Dupire 9 Option prices for one maturity Bruno Dupire 9

Risk Management Client has risk exposure Buys a product from a bank to limit Risk Management Client has risk exposure Buys a product from a bank to limit its risk Not Enough Risk Too Costly Vanilla Hedges Perfect Hedge Exotic Hedge Client transfers risk to the bank which has the technology to handle it Product fits the risk Bruno Dupire 10

Risk Neutral Pricing Bruno Dupire Risk Neutral Pricing Bruno Dupire

Price as discounted expectation Option gives uncertain payoff in the future Premium: known price 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 Bruno Dupire 12

Application to option pricing Risk Neutral Probability Bruno Dupire Physical Probability 13 Application to option pricing Risk Neutral Probability Bruno Dupire Physical Probability 13

Basic Properties Price as a function of payoff is: - Positive: - Linear: Price Basic Properties Price as a function of payoff is: - Positive: - Linear: Price = discounted expectation of payoff Bruno Dupire 14

Toy Model 1 period, n possible states Option A gives If in state gives Toy Model 1 period, n possible states Option A gives If in state gives 1 in state , 0 in all other states, where is a discount factor is a probability: Bruno Dupire 15

FTAP Fundamental Theorem of Asset Pricing 1) NA There exists an equivalent martingale measure 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 claims Claims attainable from 0 Separating hyperplanes Bruno Dupire 16

Risk Neutrality Paradox • Risk neutrality: carelessness about uncertainty? 50% Sun: 1 Apple = Risk Neutrality Paradox • Risk neutrality: carelessness about uncertainty? 50% Sun: 1 Apple = 2 Bananas Rain: 1 Banana = 2 Apples • 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 Bruno Dupire 17

Stochastic Calculus Bruno Dupire Stochastic Calculus Bruno Dupire

Modeling Uncertainty Main ingredients for spot modeling • Many small shocks: Brownian Motion (continuous Modeling Uncertainty Main ingredients for spot modeling • Many small shocks: Brownian Motion (continuous prices) S t • A few big shocks: Poisson process (jumps) S t Bruno Dupire 19

Brownian Motion • From discrete to continuous 10 1000 Bruno Dupire 20 Brownian Motion • From discrete to continuous 10 1000 Bruno Dupire 20

Stochastic Differential Equations At the limit: continuous with independent Gaussian increments SDE: a drift Stochastic Differential Equations At the limit: continuous with independent Gaussian increments SDE: a drift Bruno Dupire noise 21

Ito’s Dilemma Classical calculus: expand to the first order Stochastic calculus: should we expand Ito’s Dilemma Classical calculus: expand to the first order Stochastic calculus: should we expand further? Bruno Dupire 22

Ito’s Lemma At the limit If for f(x), Bruno Dupire 23 Ito’s Lemma At the limit If for f(x), Bruno Dupire 23

Black-Scholes PDE • Black-Scholes assumption • Apply Ito’s formula to Call price C(S, t) 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! Bruno Dupire 24

P&L of a delta hedged option Option Value P&L Break-even points Delta hedge Bruno P&L of a delta hedged option Option Value P&L Break-even points Delta hedge Bruno Dupire 25

Black-Scholes Model If instantaneous volatility is constant : drift: noise, SD: Then call prices 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. Bruno Dupire 26

Pricing methods Bruno Dupire Pricing methods Bruno Dupire

Pricing methods • Analytical formulas • Trees/PDE finite difference • Monte Carlo simulations Bruno Pricing methods • Analytical formulas • Trees/PDE finite difference • Monte Carlo simulations Bruno Dupire 28

Formula via PDE • The Black-Scholes PDE is • Reduces to the Heat Equation Formula via PDE • The Black-Scholes PDE is • Reduces to the Heat Equation • With Fourier methods, Black-Scholes equation: Bruno Dupire 29

Formula via discounted expectation • Risk neutral dynamics • Ito to ln S: • Formula via discounted expectation • Risk neutral dynamics • Ito to ln S: • Integrating: • Same formula Bruno Dupire 30

Finite difference discretization of PDE • Black-Scholes PDE • Partial derivatives discretized as Bruno Finite difference discretization of PDE • Black-Scholes PDE • Partial derivatives discretized as Bruno Dupire 31

Option pricing with Monte Carlo methods • An option price is the discounted expectation 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 • You play with a biased die • You want Computing expectations basic 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 Each side of the superdie represents a possible state of 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 Gaussian transform techniques Unit hypercube discretisation schemes Gaussian coordinates trajectory A Expectation = Integral Gaussian transform techniques Unit hypercube discretisation schemes Gaussian coordinates trajectory A point in the hypercube maps to a spot trajectory therefore

Generating Scenarios Bruno Dupire 36 Generating Scenarios Bruno Dupire 36

Low Discrepancy Sequences Bruno Dupire 37 Low Discrepancy Sequences Bruno Dupire 37

Hedging Bruno Dupire Hedging Bruno Dupire

To Hedge or Not To Hedge Daily P&L Daily Position P&L Unhedged Full P&L To Hedge or Not To Hedge Daily P&L Daily Position P&L Unhedged Full P&L 0 Big directional risk Bruno Dupire Hedged Small daily amplitude risk 39

The Geometry of Hedging • Risk measured as • Target X, hedge H • The Geometry of Hedging • Risk measured as • Target X, hedge H • Risk is an L 2 norm, with general properties of orthogonal projections • Optimal Hedge: Bruno Dupire 40

The Geometry of Hedging Bruno Dupire 41 The Geometry of Hedging Bruno Dupire 41

Super-replication • Property: Let us call: Which implies: Bruno Dupire 42 Super-replication • Property: Let us call: Which implies: Bruno Dupire 42

A sight of Cauchy-Schwarz Bruno Dupire 43 A sight of Cauchy-Schwarz Bruno Dupire 43

Volatility Bruno Dupire Volatility Bruno Dupire

Volatility : some definitions Historical volatility : annualized standard deviation of the logreturns; measure 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 Bruno Dupire 45

Historical Volatility • Measure of realized moves • annualized SD of logreturns Bruno Dupire Historical Volatility • Measure of realized moves • annualized SD of logreturns Bruno Dupire 46

Historical volatility Bruno Dupire 47 Historical volatility Bruno Dupire 47

Implied volatility Input of the Black-Scholes formula which makes it fit the market price Implied volatility Input of the Black-Scholes formula which makes it fit the market price : Bruno Dupire 48

Market Skews Dominating fact since 1987 crash: strong negative skew on Equity Markets K Market Skews Dominating fact since 1987 crash: strong negative skew on Equity Markets K Not a general phenomenon Gold: FX: K K We focus on Equity Markets Bruno Dupire 49

A Brief History of Volatility Bruno Dupire A Brief History of Volatility Bruno Dupire

Evolution theory of modeling constant deterministic stochastic Bruno Dupire n. D 51 Evolution theory of modeling constant deterministic stochastic Bruno Dupire n. D 51

A Brief History of Volatility – : Bachelier 1900 – : Black-Scholes 1973 – A Brief History of Volatility – : Bachelier 1900 – : Black-Scholes 1973 – : Merton 1976 Bruno Dupire 52

Local Volatility Model Dupire 1993, minimal model to fit current volatility surface Bruno Dupire Local Volatility Model Dupire 1993, minimal model to fit current volatility surface Bruno Dupire 53

The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution Risk The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution Risk Neutral Processes 1 D Diffusions Compatible with Smile Bruno Dupire 54

From simple to complex European prices Local volatilities Exotic prices Bruno Dupire 55 From simple to complex European prices Local volatilities Exotic prices Bruno Dupire 55

Stochastic Volatility Models Heston 1993, semi-analytical formulae. Bruno Dupire 56 Stochastic Volatility Models Heston 1993, semi-analytical formulae. Bruno Dupire 56

The End Bruno Dupire The End Bruno Dupire