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? Bruno Dupire

Naïve expectation Bruno Dupire

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

OUTLINE Risk neutral pricing Stochastic calculus Pricing methods Hedging Volatility 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

To buy or not to buy? Call Option: Right to buy stock at T for K $ $ Bruno Dupire

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

Option prices for one maturity Bruno Dupire

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 Bruno Dupire

Risk Neutral Pricing

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

Application to option pricing Risk Neutral Probability Physical Probability Bruno Dupire

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

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: Bruno Dupire

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 Bruno Dupire

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 Bruno Dupire

Stochastic Calculus

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 Bruno Dupire

Brownian Motion From discrete to continuous 10 100 1000 Bruno Dupire

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

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

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

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

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

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

Pricing methods

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

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

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

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

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 Bruno Dupire

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 Bruno Dupire

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 Bruno Dupire

Expectation = Integral Gaussian transform techniques discretisation schemes Unit hypercube Gaussian coordinates trajectory A point in the hypercube maps to a spot trajectory therefore Bruno Dupire

Generating Scenarios Bruno Dupire

Low Discrepancy Sequences Bruno Dupire

Hedging

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 Bruno Dupire

The Geometry of Hedging Risk measured as Target X, hedge H Risk is an L2 norm, with general properties of orthogonal projections Optimal Hedge: Bruno Dupire

The Geometry of Hedging Bruno Dupire

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

A sight of Cauchy-Schwarz Bruno Dupire

Volatility

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

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

Historical volatility Bruno Dupire

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

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 Bruno Dupire

A Brief History of Volatility

Evolution theory of modeling constant deterministic stochastic nD Bruno Dupire

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

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

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

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

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

The End