An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP bdupire@bloomberg.net CRFMS, UCSB April 26, 2007 Warm-up Roulette: P[ Red ] 70% P[ Black ] 30% $100 if Red A lottery ticket gives: $0 if Black You can buy it or sell it for $60 Is it cheap or expensive? Bruno Dupire 2 Naïve expectation 70 60 Buy Bruno Dupire 3 Replication argument 50 60 Sell “as if” priced with other probabilities instead of Bruno Dupire 4 OUTLINE 1. 2. 3. 4. 5. 6. 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 6 To buy or not to buy? • Call Option: Right to buy stock at T for K $ TO BUY $ NOT TO BUY K ST K ST $ CALL Bruno Dupire K ST 7 Vanilla Options European Call: Gives the right to buy the underlying at a fixed price (the strike) at some future time (the maturity) Call Payoff ( ST K ) max ST K ,0 European Put: Gives the right to sell the underlying at a fixed strike at some maturity Put Payoff K ST 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 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 Price e Bruno Dupire Physical Probability rT o (ST )( ST K ) dST 13 Basic Properties Price as a function of payoff is: - Positive: A 0 price ( A) 0 - Linear: price ( A B) price ( A) price ( B) Price = discounted expectation of payoff Bruno Dupire 14 Toy Model 1 period, n possible states s1 ,..., s n Option A gives xi in state s i If Ai gives 1 in state A xi Ai s i , 0 in all other states, price ( A) i x price ( A ) q x i i i i i i where price ( Ai ) price qi A price(1) i price ( Ai ) is a probability: qi 0 and price ( A j ) is a discount factor q i 1 i j 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 Claims attainable from 0 Cone of >0 claims Separating hyperplanes Bruno Dupire 16 Risk Neutrality Paradox • Risk neutrality: carelessness about uncertainty? 50% 50% Sun: 1 Apple = 2 Bananas Rain: 1 Banana = 2 Apples • 1 A gives either 2 B or .5 B1.25 B • 1 B gives either .5 A or 2 A1.25 A • Cannot be RN wrt 2 numeraires with the same probability Bruno Dupire 17 Stochastic Calculus 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 10 100 1000 Bruno Dupire 20 Stochastic Differential Equations At the limit: Wt continuous with independent Gaussian increments Wt Ws ~ N (0, t s) SDE: dx a dt b dW drift Bruno Dupire a noise 21 Ito’s Dilemma f (x ) Classical calculus: df f ' ( x) dx expand to the first order Stochastic calculus: dx a dt b dW should we expand further? Bruno Dupire 22 Ito’s Lemma At the limit (dW ) 2 dt If dx a dt b dW for f(x), Bruno Dupire df f ( x dx) f ( x) 1 2 f ' ( x) dx f ' ' ( x) (dx) 2 1 2 f ' ( x) dx f ' ' ( x) b dt 2 23 Black-Scholes PDE dS dt dW S • Black-Scholes assumption • Apply Ito’s formula to Call price C(S,t) dC CS dS (Ct 2S 2 2 CSS ) dt • Hedged position C CS S is riskless, earns interest rate r (Ct 2S 2 2 CSS ) dt dC CS dS r (C CS S ) dt • Black-Scholes PDE Ct 2S 2 2 CSS r (C CS S ) • No drift! Bruno Dupire 24 P&L of a delta hedged option Option Value P&L Break-even points Delta hedge Ct t Ct t St Bruno Dupire t S St t St 25 Black-Scholes Model If instantaneous volatility is constant : drift: Stt dS dt dW S noise, SD: Then call prices are given by : 1 T) 2 T 1 1 Ke rT N ( (ln( S 0 / K ) rT ) T ) 2 T C BS S 0 N ( 1 St t (ln( S 0 / K ) rT ) No drift in the formula, only the interest rate r due to the hedging argument. Bruno Dupire 26 Pricing methods Pricing methods • Analytical formulas • Trees/PDE finite difference • Monte Carlo simulations Bruno Dupire 28 Formula via PDE • The Black-Scholes PDE is Ct 2S 2 2 CSS r (C CS S ) • Reduces to the Heat Equation 1 U U xx 2 • With Fourier methods, Black-Scholes equation: C BS S 0 N (d1 ) Ke rT N (d 2 ) ln( S 0 / K ) (r 2 / 2)T d1 , d 2 d1 T T Bruno Dupire 29 Formula via discounted expectation dS r dt dW S • Risk neutral dynamics • Ito to ln S: d ln S (r 2 2 ) dt dW 2 • Integrating: ln ST ln S 0 ( r 2 ) T WT premium e rT E[( ST K ) ] e rT E[( S0e (r 2 2 ) T WT K ) ] • Same formula Bruno Dupire 30 Finite difference discretization of PDE • Black-Scholes PDE Ct 2S 2 CSS r (C CS S ) 2 C ( S , T ) ( ST K ) • Partial derivatives discretized as C (i, n) C (i, n 1) t C (i 1, n) C (i 1, n) CS (i, n) 2S C (i 1, n) 2C (i, n) C (i 1, n) CSS (n, i ) (S ) 2 Ct (i, n) Bruno Dupire 31 Option pricing with Monte Carlo methods • An option price is the discounted expectation of its payoff: P0 EPT f x x dx • Sometimes the expectation cannot be computed analytically: – complex product – complex dynamics • Then the integral has to be computed numerically Bruno Dupire the option price is its discounted payoff integrated against the risk neutral density of the spot underlying 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( Bruno Dupire ) 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 running a Monte Carlo path simulation - when the dynamics are complex Bruno Dupire Expectation = Integral Gaussian transform techniques Unit hypercube discretisation schemes Gaussian coordinates trajectory A point in the hypercube maps to a spot trajectory therefore EPT d Bruno Dupire f x .Pr St1 ,..., St d dx 1 g x i N xi 0 ,1d 0 ,1 d g ydy Generating Scenarios Bruno Dupire 36 Low Discrepancy Sequences Halton Faure Sobol dimensions 1&2 dimensions 20 & 25 dimensions 51 & 52 Bruno Dupire 37 Hedging To Hedge or Not To Hedge Daily P&L Daily Position P&L Unhedged Full P&L 0 Hedged S Delta Hedge Big directional risk Small daily amplitude risk Bruno Dupire 39 The Geometry of Hedging • Risk measured as SDPLT • Target X, hedge H PLt X t H t Risk var X T H T X H • Risk is an L2 norm, with general properties of orthogonal projections • Optimal Hedge: Ĥ X Hˆ inf X H H Bruno Dupire 40 The Geometry of Hedging Bruno Dupire 41 Super-replication E XY E X 2 EY 2 •Property: Let us call: Px : price today of X 2 Py : price today of Y 2 For all X and Y , 2 Py X Px Y 0, so XY is dominated by the Portfolio : XY Px X 2 PyY 2 2 Px Py Which implies: Bruno Dupire price XY Py Px Px Py 2 Px Py Px Py 42 A sight of Cauchy-Schwarz Bruno Dupire 43 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 45 Historical Volatility • Measure of realized moves • annualized SD of logreturns 252 n 2 2 xti xti n 1 i 1 St 1 xt ln St 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 impl K Not a general phenomenon FX: impl Gold: impl K K We focus on Equity Markets Bruno Dupire 49 A Brief History of Volatility Evolution theory of modeling constant deterministic stochastic Bruno Dupire nD 51 A Brief History of Volatility – dSt dWt Q : Bachelier 1900 – dSt r dt dWt Q St : Black-Scholes 1973 – dS t rt dt (t ) dWt Q St : Merton 1973 – dSt (r k ) dt dWt Q dq St : Merton 1976 Bruno Dupire 52 Local Volatility Model dSt r dt ( S , t ) dWt Q St C C rK 2 K , T 2 T 2 K 2 C K ,T K K 2 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 1D Diffusions Compatible with Smile Bruno Dupire sought diffusion (obtained by integrating twice Fokker-Planck equation) 54 From simple to complex European prices Local volatilities Exotic prices Bruno Dupire 55 Stochastic Volatility Models dSt S r dt t dWt t d 2 b( 2 2 )dt dZ t t t t Heston 1993, semi-analytical formulae. Bruno Dupire 56 The End