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Published on April 17, 2008

Author: Sharck

Source: authorstream.com

An Idiot’s Guide to Option Pricing:  An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP bdupire@bloomberg.net CRFMS, UCSB April 26, 2007 Warm-up:  Warm-up Roulette: A lottery ticket gives: You can buy it or sell it for $60 Is it cheap or expensive? Naïve expectation:  Naïve expectation Replication argument:  Replication argument “as if” priced with other probabilities instead of OUTLINE:  OUTLINE Risk neutral pricing Stochastic calculus Pricing methods Hedging Volatility Volatility modeling Addressing Financial Risks:  Addressing Financial Risks volume underlyings products models users regions Over the past 20 years, intense development of Derivatives in terms of: To buy or not to buy?:  $ K To buy or not to buy? Call Option: Right to buy stock at T for K $ K $ K TO BUY NOT TO BUY CALL Vanilla Options:  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 Option prices for one maturity:  Option prices for one maturity Risk Management:  Risk Management Client has risk exposure Buys a product from a bank to limit its risk Risk Not Enough Too Costly Perfect Hedge Vanilla Hedges Exotic Hedge Client transfers risk to the bank which has the technology to handle it Product fits the risk Risk Neutral Pricing:  Risk Neutral Pricing Price as discounted expectation:  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 Application to option pricing:  Application to option pricing Risk Neutral Probability Physical Probability Basic Properties:  Basic Properties Price as a function of payoff is: - Positive: - Linear: Price = discounted expectation of payoff Toy Model:  gives 1 in state Option A gives Toy Model 1 period, n possible states in state If , 0 in all other states, where is a discount factor is a probability: FTAP:  FTAP Fundamental Theorem of Asset Pricing 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 Risk Neutrality Paradox:  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 Rain: 1 Banana = 2 Apples 50% 50% Stochastic Calculus:  Stochastic Calculus Modeling Uncertainty:  Modeling Uncertainty Main ingredients for spot modeling Many small shocks: Brownian Motion (continuous prices) A few big shocks: Poisson process (jumps) Brownian Motion:  Brownian Motion 10 100 1000 From discrete to continuous Stochastic Differential Equations:  Stochastic Differential Equations At the limit: continuous with independent Gaussian increments SDE: drift noise Ito’s Dilemma:  Ito’s Dilemma Classical calculus: expand to the first order Stochastic calculus: should we expand further? Ito’s Lemma:  Ito’s Lemma At the limit for f(x), If Black-Scholes PDE:  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! P&L of a delta hedged option:  P&L of a delta hedged option Black-Scholes Model:  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. Pricing methods:  Pricing methods Pricing methods:  Pricing methods Analytical formulas Trees/PDE finite difference Monte Carlo simulations Formula via PDE:  Formula via PDE The Black-Scholes PDE is Reduces to the Heat Equation With Fourier methods, Black-Scholes equation: Formula via discounted expectation:  Formula via discounted expectation Risk neutral dynamics Ito to ln S: Integrating: Same formula Finite difference discretization of PDE:  Finite difference discretization of PDE Black-Scholes PDE Partial derivatives discretized as Option pricing with Monte Carlo methods:  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:  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:  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:  Expectation = Integral Unit hypercube Gaussian coordinates trajectory Gaussian transform techniques discretisation schemes A point in the hypercube maps to a spot trajectory therefore Generating Scenarios:  Generating Scenarios Low Discrepancy Sequences:  Low Discrepancy Sequences Hedging:  Hedging To Hedge or Not To Hedge :  To Hedge or Not To Hedge Daily Position Daily P&L Full P&L Big directional risk Small daily amplitude risk The Geometry of Hedging:  The Geometry of Hedging Risk measured as Target X, hedge H Risk is an L2 norm, with general properties of orthogonal projections Optimal Hedge: The Geometry of Hedging:  The Geometry of Hedging Slide42:  Super-replication Property: Let us call: Which implies: Slide43:  A sight of Cauchy-Schwarz Volatility:  Volatility Volatility : some definitions:  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 Historical Volatility:  Historical Volatility Measure of realized moves annualized SD of logreturns Historical volatility:  Historical volatility Implied volatility:  Implied volatility Input of the Black-Scholes formula which makes it fit the market price : Market Skews:  Market Skews Dominating fact since 1987 crash: strong negative skew on Equity Markets Not a general phenomenon Gold: FX: We focus on Equity Markets A Brief History of Volatility:  A Brief History of Volatility Evolution theory of modeling:  Evolution theory of modeling constant deterministic stochastic nD A Brief History of Volatility:  A Brief History of Volatility : Bachelier 1900 : Black-Scholes 1973 : Merton 1973 : Merton 1976 Local Volatility Model:  Local Volatility Model Dupire 1993, minimal model to fit current volatility surface The Risk-Neutral Solution:  The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution From simple to complex:  European prices Local volatilities Exotic prices From simple to complex Stochastic Volatility Models:  Stochastic Volatility Models Heston 1993, semi-analytical formulae. The End:  The End

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