15.401
15.401 Finance Theory
MIT Sloan MBA Program
Andrew W. Lo
Harris & Harris Group Professor, MIT Sloan School
Lectures 10–11: Options
© 2007–2008 by Andrew W. Lo
Critical Concepts
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15.401
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:
ƒ Brealey, Myers, and Allen Chapters 27
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 2
Motivation
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Derivative: Claim Whose Payoff is a Function of Another's
ƒ Warrants, Options
ƒ Calls vs. Puts
ƒ American vs. European
ƒ Terms: Strike Price K, Time to Maturity τ
ƒ At Maturity T, payoff
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 3
Motivation
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Put Options As Insurance
ƒ Differences
– Early exercise
– Marketability
– Dividends
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 4
Payoff Diagrams
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Call Option:
12
10
Payoff / profit, $
8
6
4
2
Stock price
0
10
15
20
25
30
-2
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 5
Payoff Diagrams
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Stock Return vs. Call Option Return
133%
100%
67%
33%
0%
30
34
38
42
46
50
54
58
62
66
70
-33%
-67%
-100%
-133%
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 6
Payoff Diagrams
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Put Option:
12
10
Payoff / profit, $
8
6
4
2
Stock price
0
10
15
20
25
30
-2
-4
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 7
Payoff Diagrams
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25
25
20
20
Long Call
15
10
10
5
5
0
0
30
40
-5
Long Put
15
50
60
70
-5
Stock price
5
30
40
70
60
70
Stock price
Stock price
0
30
40
-5
-10
60
Stock price
5
0
50
50
60
70
30
40
50
-5
Short Call
-10
-15
-15
-20
-20
-25
-25
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Short Put
Slide 8
Payoff Tables
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Stock Price = S, Strike Price = X
Call option (price = C)
Payoff
Profit
if S < X
if S = X
if S > X
0
–C
0
–C
S–X
S–X–C
if S < X
if S = X
if S > X
X–S
X–S–P
0
–P
0
–P
Put option (price = P)
Payoff
Profit
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 9
Option Strategies
15.401
Trading strategies
Options can be combined in various ways to create an unlimited
number of payoff profiles.
Examples
ƒ Buy a stock and a put
ƒ Buy a call with one strike price and sell a call with another
ƒ Buy a call and a put with the same strike price
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 10
Option Strategies
15.401
Stock + Put
70
65
25
Payoff
Payoff
20
60
Buy stock
55
Buy a put
15
50
10
45
40
5
35
0
30
30
40
50
Stock price
60
70
30
40
50
Stock price
60
70
70
65
Payoff
60
55
50
45
Stock + put
40
35
30
30
Lecture 10–11: Options
40
50
Stock price
60
© 2007–2008 by Andrew W. Lo
70
Slide 11
Option Strategies
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Call1  Call2
25
20
25
Payoff
20
15
15
10
5
Buy call with
X = 50
10
5
Write call with
X = 60
0
0
-5 30
Payoff
40
-10
50
60
70
Stock price
-5 30
-15
-20
-20
50
60
70
Stock price
-10
-15
20
40
Payoff
16
Call1 – Call2
12
8
4
0
30
Lecture 10–11: Options
40
50
Stock price
60
© 2007–2008 by Andrew W. Lo
70
Slide 12
Option Strategies
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Call + Put
25
25
Payoff
20
15
15
Buy call with
X = 50
10
5
0
0
30
40
Buy put with
X = 50
10
5
-5
Payoff
20
50
60
70
-5
Stock price
-10
30
40
50
60
70
Stock price
-10
25
Payoff
20
15
Call + Put
10
5
0
-5
30
40
50
60
70
Stock price
-10
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 13
Other Option-Like Securities
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Corporate Liabilities:
ƒ Equity: hold call, or protected levered put
ƒ Debt: issue put and (riskless) lending
ƒ Quantifies Compensation Incentive Problems:
– Biased towards higher risk
– Can overwhelm NPV considerations
– Example: leveraged equity
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 14
Other Option-Like Securities
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Other Examples of Derivative Securities:
ƒ Asian or “Look Back” Options
ƒ Callables, Convertibles
ƒ Futures, Forwards
ƒ Swaps, Caps, Floors
ƒ Real Investment Opportunities
ƒ Patents
ƒ Tenure
ƒ etc.
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 15
Valuation of Options
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Binomial Option-Pricing Model of Cox, Ross, and Rubinstein (1979):
ƒ Consider One-Period Call Option On Stock XYZ
ƒ Current Stock Price S0
ƒ Strike Price K
ƒ Option Expires Tomorrow, C1= Max [S1−K , 0]
ƒ What Is Today’s Option Price C0?
0
S0 , C0 = ???
Lecture 10–11: Options
1
S1 , C1= Max [S1−K , 0]
© 2007–2008 by Andrew W. Lo
Slide 16
Valuation of Options
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Suppose S0 → S1 Is A Bernoulli Trial:
p
uS0
u>d
S1
1− p
dS0
p
Max [ uS0 − K , 0 ] == Cu
1− p
Max [ dS0 − K , 0 ] == Cd
C1
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 17
Valuation of Options
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Now What Should C0 = f (⋅) Depend On?
ƒ Parameters: S0 , K , u , d , p , r
Consider Portfolio of Stocks and Bonds Today:
ƒ ∆ Shares of Stock, $B of Bonds
ƒ Total Cost Today: V0 == S0∆ + B
ƒ Payoff V1 Tomorrow:
p
uS0∆ + rB
1− p
dS0 ∆ + rB
V1
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 18
Valuation of Options
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Now Choose ∆ and B So That:
p
uS0∆ + rB = Cu
⇒
V1
1− p
dS0 ∆ + rB = Cd
∆* = (Cu− Cd)/(u − d)S0
B* = (uCd− dCu)/(u − d)r
Then It Must Follow That:
C0 = V0 = S0 ∆* + B* =
Lecture 10–11: Options
1
r
r −d
u −r
Cu +
C
u−d
u−d d
© 2007–2008 by Andrew W. Lo
Slide 19
Valuation of Options
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Suppose C0 > V0
ƒ Today: Sell C0 , Buy V0 , Receive C0 − V0 > 0
ƒ Tomorrow: Owe C1, But V1 Is Equal To C1!
Suppose C0 < V0
ƒ Today: Buy C0 , Sell V0 , Receive V0 − C0 > 0
ƒ Tomorrow: Owe V1, But C1 Is Equal To V1!
C0 = V0 , Otherwise Arbitrage Exists
C0 = V0 = S0 ∆* + B* =
Lecture 10–11: Options
1
r
r −d
u −r
Cu +
C
u−d
u−d d
© 2007–2008 by Andrew W. Lo
Slide 20
Valuation of Options
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Note That p Does Not Appear Anywhere!
ƒ Can disagree on p, but must agree on option price
ƒ If price violates this relation, arbitrage!
ƒ A multi-period generalization exists:
ƒ Continuous-time/continuous-state version (Black-Scholes/Merton):
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 21
A Brief History of Option-Pricing Theory
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Earliest Model of Asset Prices:
ƒ Notion of “fair” game
ƒ The martingale
ƒ 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.
ƒ Precursor of the Random Walk Hypothesis
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 22
A Brief History of Option-Pricing Theory
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Louis Bachelier (1870–1946):
ƒ Théorie de la Spéculation (1900)
ƒ First mathematical model of Brownian motion
ƒ Modelled warrant prices on Paris Bourse
ƒ 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.
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 23
A Brief History of Option-Pricing Theory
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Kruizenga (1956):
ƒ MIT Ph.D. Student (Samuelson)
ƒ “Put and Call Options: A Theoretical and Market Analysis”
Sprenkle (1961):
ƒ Yale Ph.D. student (Okun, Tobin)
ƒ “Warrant Prices As Indicators of Expectations and Preferences”
What Is The “Value” of Max [ ST − K , 0 ]?
ƒ Assume probability law for ST
ƒ Calculate Et [ Max [ ST − K , 0 ] ]
ƒ But what is its price?
ƒ Do risk preferences matter?
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 24
A Brief History of Option-Pricing Theory
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Samuelson (1965):
ƒ “Rational Theory of Warrant Pricing”
ƒ “Appendix: A Free Boundary Problem for the Heat Equation Arising
from a Problem in Mathematical Economics”, H.P. McKean
Samuelson and Merton (1969):
ƒ “A Complete Model of Warrant Pricing that Maximizes Utility”
ƒ Uses preferences to value expected payoff
Black and Scholes (1973):
ƒ Construct portfolio of options and stock
ƒ Eliminate market risk (appeal to CAPM)
ƒ Remaining risk is inconsequential (idiosyncratic)
ƒ Expected return given by CAPM (PDE)
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 25
A Brief History of Option-Pricing Theory
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Merton (1973):
ƒ 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
ƒ More importantly, a “production process” for derivatives!
ƒ Formed the basis of financial engineering and three distinct industries:
– Listed options
– OTC structured products
– Credit derivatives
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 26
A Brief History of Option-Pricing Theory
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The Rest Is Financial History:
Photographs removed due to copyright restrictions.
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 27
Extensions and Qualifications
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ƒ The derivatives pricing literature is HUGE
ƒ The mathematical tools involved can be quite challenging
ƒ 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
ƒ Computational demands can be quite significant
ƒ This is considered “rocket science”
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 28
Key Points
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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
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 29
Additional References
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ƒ Bachelier, L, 1900, “Théorie de la Spéculation”, Annales de l'Ecole Normale Supérieure 3 (Paris,
Gauthier-Villars).
ƒ Black, F., 1989, “How We Came Up with the Option Formula”, Journal of Portfolio Management 15, 4–8.
ƒ Bernstein, P., 1993, Capital Ideas. New York: Free Press.
ƒ Black, F. and M. Scholes, 1973, “The Pricing of Options and Corporate Liabilities”, Journal of Political
Economy 81, 637–659.
ƒ Kruizenga, R., 1956, Put and Call Options: A Theoretical and Market Analysis, unpublished Ph.D.
dissertation, MIT.
ƒ Merton, R., 1973, “Rational Theory of Option Pricing”, Bell Journal of Economics and Management
Science 4, 141–183.
ƒ Samuelson, P., 1965, “Rational Theory of Warrant Pricing”, Industrial Management Review 6, 13–31.
ƒ Samuelson, P. and R. Merton, 1969, “A Complete Model of Warrant Pricing That Maximizes Utility”,
Industrial Management Review 10, 17–46.
ƒ Sprenkle, C., 1961, “Warrant Prices As Indicators of Expectations and Preferences”, Yale Economic
Essays 1, 178–231.
Lecture 10–11: Options
© 2007–2008 by Andrew W. Lo
Slide 30
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15.401 Finance Theory I
Fall 2008
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