An Idiot’s Guide to Option Pricing

1. An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP bdupire@bloomberg.net CRFMS, UCSB April 26, 2007

2. Warm-up Roulette: A lottery ticket gives: You can buy it or sell it for $60 Is it cheap or expensive?

3. Naïve expectation

4. Replication argument “as if” priced with other probabilities instead of

5. OUTLINE Risk neutral pricing Stochastic calculus Pricing methods Hedging Volatility Volatility modeling

6. Addressing Financial Risks Over the past 20 years, intense development of Derivatives in terms of: • volume • underlyings • products • models • users • regions

7. To buy or not to buy? • Call Option:Rightto buy stock at T for K $ $ TO BUY NOT TO BUY K K $ CALL K

8. 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

9. Option prices for one maturity

10. 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

11. Risk Neutral Pricing

12. 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

13. Application to option pricing Risk Neutral Probability Physical Probability

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

15. Toy Model 1 period, n possible states Option A gives in state gives 1 in state , 0 in all other states, If where is a discount factor is a probability:

16. 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

17. 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% Rain: 1 Banana = 2 Apples 50%

18. Stochastic Calculus

19. 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)

20. Brownian Motion • From discrete to continuous 10 100 1000

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

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

23. Ito’s Lemma At the limit If for f(x),

24. 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!

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

26. 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.

27. Pricing methods

28. Pricing methods • Analytical formulas • Trees/PDE finite difference • Monte Carlo simulations

29. Formula via PDE • The Black-Scholes PDE is • Reduces to the Heat Equation • With Fourier methods, Black-Scholes equation:

30. Formula via discounted expectation • Risk neutral dynamics • Ito to ln S: • Integrating: • Same formula

31. Finite difference discretization of PDE • Black-Scholes PDE • Partial derivatives discretized as

32. 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

33. 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

34. 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

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

36. Generating Scenarios

37. Low Discrepancy Sequences

38. Hedging

39. P&L Unhedged Hedged 0 To Hedge or Not To Hedge Daily P&L Daily Position Full P&L Big directional risk Small daily amplitude risk

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

41. The Geometry of Hedging

42. Super-replication • Property: • Let us call: • Which implies:

43. A sight of Cauchy-Schwarz

44. Volatility

45. 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

46. Historical Volatility • Measure of realized moves • annualized SD of logreturns

47. Historical volatility

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

49. 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

50. A Brief History of Volatility