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## Pricing Financial Derivatives

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**Pricing Financial Derivatives**Bruno Dupire Bloomberg L.P/NYU AIMS Day 1 Cape Town, February 17, 2011**Addressing Financial Risks**Over the past 20 years, intense development of Derivatives in terms of: • volume • underlyings • products • models • users • regions**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**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**OUTLINE**Theory - Risk neutral pricing - Stochastic calculus - Pricing methods B) Volatility - Definition and estimation - Volatility modeling - Volatility arbitrage**Warm-up**Roulette: A lottery ticket gives: You can buy it or sell it for $60 Is it cheap or expensive?**2 approaches**Naïve expectation Replication Argument “as if” priced with other probabilities instead of**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**Risk Neutral Probability Physical Probability**S**t t S Modeling Uncertainty Main ingredients for spot modeling • Many small shocks: Brownian Motion (continuous prices) • A few big shocks: Poisson process (jumps)**Brownian Motion**• From discrete to continuous 10 100 1000**a**Stochastic Differential Equations At the limit: Continuous with independent Gaussian increments SDE: drift noise**Ito’s Dilemma**Classical calculus: expand to the first order Stochastic calculus: should we expand further?**Ito’s Lemma**At the limit If for f(x),**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!**Option Value**P&L Break-even points Delta hedge P&L of a delta hedged option**drift:**noise, SD: 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**• Analytical formulas • Trees/PDE finite difference • Monte Carlo simulations**Formula via PDE**• The Black-Scholes PDE is • Reduces to the Heat Equation • With Fourier methods, Black-Scholes equation:**Formula via discounted expectation**• Risk neutral dynamics • Ito to ln S: • Integrating: • Same formula**Finite difference discretization of PDE**• Black-Scholes PDE • Partial derivatives discretized as**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 expectationsbasic 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**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**• Measure of realized moves • annualized SD of**Estimates based on High/Low**• Commonly available information: open, close, high, low • Captures valuable volatility information • Parkinson estimate: • Garman-Klass estimate:**Move based estimation**• Leads to alternative historical vol estimation: = number of crossings of log-price over [0,T]**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.**Implied volatility**Input of the Black-Scholes formula which makes it fit the market price :**K**K K Market Skews Dominating fact since 1987 crash: strong negative skew on Equity Markets Not a general phenomenon Gold: FX: We focus on Equity Markets**Skews**• Volatility Skew: slope of implied volatility as a function of Strike • Link with Skewness (asymmetry) of the Risk Neutral density function ?**Market Skew**Supply and Demand Th. Skew K Why Volatility Skews? • Market prices governed by • a) Anticipated dynamics (future behavior of volatility or jumps) • b) Supply and Demand • To “ arbitrage” European options, estimate a) to capture risk premium b) • To “arbitrage” (or correctly price) exotics, find Risk Neutral dynamics calibrated to the market**S**t t S Modeling Uncertainty Main ingredients for spot modeling • Many small shocks: Brownian Motion (continuous prices) • A few big shocks: Poisson process (jumps)**K**2 mechanisms to produce Skews (1) • To obtain downward sloping implied volatilities • a) Negative link between prices and volatility • Deterministic dependency (Local Volatility Model) • Or negative correlation (Stochastic volatility Model) • b) Downward jumps**2 mechanisms to produce Skews (2)**• a) Negative link between prices and volatility • b) Downward jumps**Strike dependency**• Fair or Break-Even volatility is an average of squared returns, weighted by the Gammas, which depend on the strike**Evolution Theory**constant deterministic stochastic nD**A Brief History of Volatility (1)**• : Bachelier 1900 • : Black-Scholes 1973 • : Merton 1973 • : Merton 1976 • : Hull&White 1987**A Brief History of Volatility (2)**Dupire 1992, arbitrage model which fits term structure of volatility given by log contracts. Dupire 1993, minimal model to fit current volatility surface**A Brief History of Volatility (3)**Heston 1993, semi-analytical formulae. Dupire 1996 (UTV), Derman 1997, stochastic volatility model which fits current volatility surface HJM treatment.