6cc4eeae4e4ea6f6d9223409afc1ecbd.ppt
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An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP bdupire@bloomberg. net CRFMS, UCSB April 26, 2007 Bruno Dupire
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
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 Pricing methods Hedging Volatility modeling
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 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 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
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
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
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 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 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 = 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
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
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 further? Bruno Dupire 22
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) • 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 Dupire 25
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 • 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 • With Fourier methods, Black-Scholes equation: Bruno Dupire 29
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 Dupire 31
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 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 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 point in the hypercube maps to a spot trajectory therefore
Generating Scenarios Bruno Dupire 36
Low Discrepancy Sequences Bruno Dupire 37
Hedging Bruno Dupire
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 • Risk is an L 2 norm, with general properties of orthogonal projections • Optimal Hedge: Bruno Dupire 40
The Geometry of Hedging Bruno Dupire 41
Super-replication • Property: Let us call: Which implies: Bruno Dupire 42
A sight of Cauchy-Schwarz Bruno Dupire 43
Volatility Bruno Dupire
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 46
Historical volatility Bruno Dupire 47
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 Not a general phenomenon Gold: FX: K K We focus on Equity Markets Bruno Dupire 49
A Brief History of Volatility Bruno Dupire
Evolution theory of modeling constant deterministic stochastic Bruno Dupire n. D 51
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 53
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
Stochastic Volatility Models Heston 1993, semi-analytical formulae. Bruno Dupire 56
The End Bruno Dupire
6cc4eeae4e4ea6f6d9223409afc1ecbd.ppt