# Finance Theory

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1 Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lectures : : Options

2 Critical Concepts Motivation Payoff Diagrams Payoff Tables Option Strategies Other Option-Like Securities Valuation of Options A Brief History of Option-Pricing Theory Extensions and Qualifications Readings: Brealey, Myers, and Allen Chapters 27 Slide 2

3 Motivation Derivative: Claim Whose Payoff is a Function of Another's Warrants, Options Calls vs. Puts American vs. European Terms: Strike Price K, Time to Maturity τ At Maturity T, payoff Slide 3

4 Motivation Put Options As Insurance Differences Early exercise Marketability Dividends Slide 4

5 Payoff Diagrams Call Option: Payoff / profit, \$ Stock price Slide 5

6 Payoff Diagrams Stock Return vs. Call Option Return 133% 100% 67% 33% 0% 30-33% % -100% -133% Slide 6

7 Payoff Diagrams Put Option: Payoff / profit, \$ Stock price Slide 7

8 Payoff Diagrams Long Call Long Put Stock price 5 Stock price Stock price 5 Stock price Short Call Short Put Slide 8

9 Payoff Tables Stock Price = S, Strike Price = X Call option (price = C) if S < X if S = X if S > X Payoff 0 0 S X Profit C C S X C Put option (price = P) if S < X if S = X if S > X Payoff X S 0 0 Profit X S P P P Slide 9

10 Option Strategies Trading strategies Options can be combined in various ways to create an unlimited number of payoff profiles. Examples Buy a stock and a put Buy a call with one strike price and sell a call with another Buy a call and a put with the same strike price Slide 10

11 Option Strategies Stock + Put Payoff Payoff Buy stock Stock price Payoff Stock + put Stock price Buy a put Stock price Slide 11

12 Option Strategies Call1 Call Stock price Payoff Payoff 20 Buy call with X = Stock price Write call with X = Payoff 12 Call 1 Call Stock price Slide 12

13 Option Strategies Call + Put Payoff Payoff Buy call with X = Buy put with X = Stock price Stock price Payoff Call + Put Stock price -10 Slide 13

14 Other Option-Like Securities Corporate Liabilities: Equity: hold call, or protected levered put Debt: issue put and (riskless) lending Quantifies Compensation Incentive Problems: Biased towards higher risk Can overwhelm NPV considerations Example: leveraged equity Slide 14

15 Other Option-Like Securities Other Examples of Derivative Securities: Asian or Look Back Options Callables, Convertibles Futures, Forwards Swaps, Caps, Floors Real Investment Opportunities Patents Tenure etc. Slide 15

16 Valuation of Options Binomial Option-Pricing Model of Cox, Ross, and Rubinstein (1979): Consider One-Period Call Option On Stock XYZ Current Stock Price S 0 Strike Price K Option Expires Tomorrow, C 1 = Max [S 1 K, 0] What Is Today s Option Price C 0? 0 1 S 0, C 0 =??? S 1, C 1 = Max [S 1 K, 0] Slide 16

17 Valuation of Options Suppose S 0 S 1 Is A Bernoulli Trial: p us 0 S 1 1 p ds 0 u > d C 1 p 1 p Max [ us 0 K, 0 ] == C u Max [ ds 0 K, 0 ] == C d Slide 17

18 Valuation of Options Now What Should C 0 = f ( ) Depend On? Parameters: S 0, K, u, d, p, r Consider Portfolio of Stocks and Bonds Today: Shares of Stock, \$B of Bonds Total Cost Today: V 0 == S 0 + B Payoff V 1 Tomorrow: p us 0 + rb V 1 1 p ds 0 + rb Slide 18

19 Valuation of Options Now Choose and B So That: p us 0 + rb= C u V 1 1 p ds 0 + rb= C d * = (C u C d )/(u d)s 0 B* = (uc d dc u )/(u d)r Then It Must Follow That: C 0 = V 0 = S 0 * + B* = 1 r r d u d C u + u r u d C d Slide 19

20 Valuation of Options Suppose C 0 > V 0 Today: Sell C 0, Buy V 0, Receive C 0 V 0 > 0 Tomorrow: Owe C 1, But V 1 Is Equal To C 1! Suppose C 0 < V 0 Today: Buy C 0, Sell V 0, Receive V 0 C 0 > 0 Tomorrow: Owe V 1, But C 1 Is Equal To V 1! C 0 = V 0, Otherwise Arbitrage Exists C 0 = V 0 = S 0 * + B* = 1 r r d u d C u + u r u d C d Slide 20

21 Valuation of Options Note That p Does Not Appear Anywhere! Can disagree on p, but must agree on option price If price violates this relation, arbitrage! A multi-period generalization exists: Continuous-time/continuous-state version (Black-Scholes/Merton): Slide 21

22 A Brief History of Option-Pricing Theory Earliest Model of Asset Prices: Notion of fair game The martingale Cardano s (c.1565) Liber De Ludo Aleae: The most fundamental principle of all in gambling is simply equal conditions, e.g. of opponents, of bystanders, of money, of situation, of the dice box, and of the die itself. To the extent to which you depart from that equality, if it is in your opponent's favor, you are a fool, and if in your own, you are unjust. Precursor of the Random Walk Hypothesis Slide 22

23 A Brief History of Option-Pricing Theory Louis Bachelier ( ): Théorie de la Spéculation (1900) First mathematical model of Brownian motion Modelled warrant prices on Paris Bourse Poincaré s evaluation of Bachelier: The manner in which the candidate obtains the law of Gauss is most original, and all the more interesting as the same reasoning might, with a few changes,be extended to the theory of errors. He develops this in a chapter which might at first seem strange, for he titles it ``Radiation of Probability''. In effect, the author resorts to a comparison with the analytical theory of the propagation of heat. A little reflection shows that the analogy is real and the comparison legitimate. Fourier's reasoning is applicable almost without change to this problem, which is so different from that for which it had been created. It is regrettable that [the author] did not develop this part of his thesis further. Slide 23

24 A Brief History of Option-Pricing Theory Kruizenga (1956): MIT Ph.D. Student (Samuelson) Put and Call Options: A Theoretical and Market Analysis Sprenkle (1961): Yale Ph.D. student (Okun, Tobin) Warrant Prices As Indicators of Expectations and Preferences What Is The Value of Max [ S T K, 0 ]? Assume probability law for S T Calculate E t [ Max [ S T K, 0 ] ] But what is its price? Do risk preferences matter? Slide 24

25 A Brief History of Option-Pricing Theory Samuelson (1965): Rational Theory of Warrant Pricing Appendix: A Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics, H.P. McKean Samuelson and Merton (1969): A Complete Model of Warrant Pricing that Maximizes Utility Uses preferences to value expected payoff Black and Scholes (1973): Construct portfolio of options and stock Eliminate market risk (appeal to CAPM) Remaining risk is inconsequential (idiosyncratic) Expected return given by CAPM (PDE) Slide 25

26 A Brief History of Option-Pricing Theory Merton (1973): Use continuous-time framework Construct portfolio of options and stock Eliminate market risk (dynamically) Eliminate idiosyncratic risk (dynamically) Zero payoff at maturity No risk, zero payoff zero cost (otherwise arbitrage) Same PDE More importantly, a production process for derivatives! Formed the basis of financial engineering and three distinct industries: Listed options OTC structured products Credit derivatives Slide 26

27 A Brief History of Option-Pricing Theory The Rest Is Financial History: Photographs removed due to copyright restrictions. Slide 27

28 Extensions and Qualifications The derivatives pricing literature is HUGE The mathematical tools involved can be quite challenging Recent areas of innovation include: Empirical methods in estimating option pricings Non-parametric option-pricing models Models of credit derivatives Structured products with hedge funds as the underlying New methods for managing the risks of derivatives Quasi-Monte-Carlo methods for simulation estimators Computational demands can be quite significant This is considered rocket science Slide 28

29 Key Points Options have nonlinear payoffs, as diagrams show Some options can be viewed as insurance contracts Option strategies allow investors to take more sophisticated bets Valuation is typically derived via arbitrage arguments (e.g., binomial) Option-pricing models have a long and illustrious history Slide 29

30 Additional References Bachelier, L, 1900, Théorie de la Spéculation, Annales de l'ecole Normale Supérieure 3 (Paris, Gauthier-Villars). Black, F., 1989, How We Came Up with the Option Formula, Journal of Portfolio Management 15, 4 8. Bernstein, P., 1993, Capital Ideas. New York: Free Press. Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, Kruizenga, R., 1956, Put and Call Options: A Theoretical and Market Analysis, unpublished Ph.D. dissertation, MIT. Merton, R., 1973, Rational Theory of Option Pricing, Bell Journal of Economics and Management Science 4, Samuelson, P., 1965, Rational Theory of Warrant Pricing, Industrial Management Review 6, Samuelson, P. and R. Merton, 1969, A Complete Model of Warrant Pricing That Maximizes Utility, Industrial Management Review 10, Sprenkle, C., 1961, Warrant Prices As Indicators of Expectations and Preferences, Yale Economic Essays 1, Slide 30

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