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# Unbiased Expectations Theory and Spot Rate Curve

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### Spot and Forward Rates under Continuous Compounding

• The pricing formula:

P =

Xn i=1

Ce−iS(i) + F e−nS(n).

• The market discount function:

d(n) = e−nS(n).

• The spot rate is an arithmetic average of forward rates,a S(n) = f (0, 1) + f (1, 2) + · · · + f (n − 1, n)

n .

(2)

### Spot and Forward Rates under Continuous Compounding (continued)

• The formula for the forward rate:

f (i, j) = jS(j) − iS(i) j − i .

• The one-period forward rate:

f (j, j + 1) = − ln d(j + 1) d(j) .

(3)

### Spot and Forward Rates under Continuous Compounding (concluded)

• Now,

f (T ) ≡ lim

∆T →0 f (T, T + ∆T )

= S(T ) + T ∂S

∂T .

• So f (T ) > S(T ) if and only if ∂S/∂T > 0 (i.e., a normal spot rate curve).

(4)

### Unbiased Expectations Theory

• Forward rate equals the average future spot rate,

f (a, b) = E[ S(a, b) ]. (17)

• It does not imply that the forward rate is an accurate predictor for the future spot rate.

• It implies the maturity strategy and the rollover strategy produce the same result at the horizon on the average.

(5)

### Unbiased Expectations Theory and Spot Rate Curve

• It implies that a normal spot rate curve is due to the fact that the market expects the future spot rate to rise.

– f (j, j + 1) > S(j + 1) if and only if S(j + 1) > S(j) from Eq. (15) on p. 123.

– So E[ S(j, j + 1) ] > S(j + 1) > · · · > S(1) if and only if S(j + 1) > · · · > S(1).

• Conversely, the spot rate is expected to fall if and only if the spot rate curve is inverted.

(6)

### More Implications

• The theory has been rejected by most empirical studies with the possible exception of the period prior to 1915.

• Since the term structure has been upward sloping about 80% of the time, the theory would imply that investors have expected interest rates to rise 80% of the time.

• Riskless bonds, regardless of their different maturities, are expected to earn the same return on the average.

• That would mean investors are indifferent to risk.

(7)

• The expected returns on all possible riskless bond strategies are equal for all holding periods.

• So

(1 + S(2))2 = (1 + S(1)) E[ 1 + S(1, 2) ] (18) because of the equivalency between buying a two-period bond and rolling over one-period bonds.

• After rearrangement, 1

E[ 1 + S(1, 2) ] = 1 + S(1) (1 + S(2))2.

(8)

### A “Bad” Expectations Theory (continued)

• Now consider two one-period strategies.

– Strategy one buys a two-period bond for (1 + S(2))−2 dollars and sells it after one period.

– The expected return is

E[ (1 + S(1, 2))−1 ]/(1 + S(2))−2.

– Strategy two buys a one-period bond with a return of 1 + S(1).

• The theory says the returns are equal:

1 + S(1)

(1 + S(2))2 = E

· 1

1 + S(1, 2)

¸ .

(9)

### A “Bad” Expectations Theory (concluded)

• Combine this with Eq. (18) on p. 139 to obtain E

· 1

1 + S(1, 2)

¸

= 1

E[ 1 + S(1, 2) ].

• But this is impossible save for a certain economy.

– Jensen’s inequality states that E[ g(X) ] > g(E[ X ]) for any nondegenerate random variable X and

strictly convex function g (i.e., g00(x) > 0).

– Use g(x) ≡ (1 + x)−1 to prove our point.

(10)

### Local Expectations Theory

• The expected rate of return of any bond over a single period equals the prevailing one-period spot rate:

E £

(1 + S(1, n))−(n−1) ¤

(1 + S(n))−n = 1 + S(1) for all n > 1.

• This theory is the basis of many interest rate models.

(11)

### Duration Revisited

• To handle more general types of spot rate curve changes, define a vector [ c1, c2, . . . , cn ] that characterizes the

perceived type of change.

– Parallel shift: [ 1, 1, . . . , 1 ].

– Twist: [ 1, 1, . . . , 1, −1, . . . , −1 ],

[ 0.8%, 0.6%, 0.4%, 0.2%, 0%, −0.2%, −0.4% . . . ], etc.

– · · ·

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### Duration Revisited (concluded)

• Let

P (y) ≡ X

i

Ci/(1 + S(i) + yci)i

be the price associated with the cash flow C1, C2, . . . .

• Define duration as

−∂P (y)/P (0)

∂y

¯¯

¯¯

y=0

.

• Modified duration equals the above when [ c1, c2, . . . , cn ] = [ 1, 1, . . . , 1 ],

S(1) = S(2) = · · · = S(n).

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### Some Loose Ends on Dates

• Holidays.

• Weekends.

• Business days (T + 2, etc.).

• Shall we treat a year as 1 year whether it has 365 or 366 days?

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### Fundamental Statistical Concepts

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There are three kinds of lies:

lies, damn lies, and statistics.

— Benjamin Disraeli (1804–1881) If 50 million people believe a foolish thing, it’s still a foolish thing.

— George Bernard Shaw (1856–1950) One death is a tragedy, but a million deaths are a statistic.

— Josef Stalin (1879–1953)

(16)

### Moments

• The variance of a random variable X is defined as Var[ X ] ≡ E £

(X − E[ X ])2 ¤ .

• The covariance between random variables X and Y is Cov[ X, Y ] ≡ E [ (X − µX)(Y − µY ) ] ,

where µX and µY are the means of X and Y , respectively.

• Random variables X and Y are uncorrelated if Cov[ X, Y ] = 0.

(17)

### Correlation

• The standard deviation of X is the square root of the variance,

σX p

Var[ X ] .

• The correlation (or correlation coefficient) between X and Y is

ρX,Y Cov[ X, Y ] σXσY ,

provided both have nonzero standard deviations.a

aPaul Wilmott (2009), “the correlations between financial quantities are notoriously unstable.”

(18)

### Variance of Sum

• Variance of a weighted sum of random variables equals Var

" n X

i=1

aiXi

#

=

Xn i=1

Xn j=1

aiaj Cov[ Xi, Xj ].

• It becomes

Xn i=1

a2i Var[ Xi ] when Xi are uncorrelated.

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### Conditional Expectation

• “X | I” denotes X conditional on the information set I.

• The information set can be another random variable’s value or the past values of X, say.

• The conditional expectation E[ X | I ] is the expected value of X conditional on I; it is a random variable.

• The law of iterated conditional expectations:

E[ X ] = E[ E[ X | I ] ].

• If I2 contains at least as much information as I1, then E[ X | I1 ] = E[ E[ X | I2 ] | I1 ]. (19)

(20)

### The Normal Distribution

• A random variable X has the normal distribution with mean µ and variance σ2 if its probability density

function is

1 σ√

e−(x−µ)2/(2σ2).

• This is expressed by X ∼ N (µ, σ2).

• The standard normal distribution has zero mean, unit variance, and the distribution function

Prob[ X ≤ z ] = N (z) ≡ 1

√2π

Z z

−∞

e−x2/2 dx.

(21)

### Moment Generating Function

• The moment generating function of random variable X is

θX(t) ≡ E[ etX ].

• The moment generating function of X ∼ N (µ, σ2) is θX(t) = exp

·

µt + σ2t2 2

¸

. (20)

(22)

### The Multivariate Normal Distribution

• If Xi ∼ N (µi, σi2) are independent, then X

i

Xi ∼ N

ÃX

i

µi,X

i

σi2

! .

• Let Xi ∼ N (µi, σi2), which may not be independent.

• Suppose Xn

i=1

tiXi ∼ N

 Xn

i=1

ti µi, Xn

i=1

Xn j=1

titj Cov[ Xi, Xj ]

for every linear combination Pn

i=1 tiXi.a

• Xi are said to have a multivariate normal distribution.

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### Generation of Univariate Normal Distributions

• Let X be uniformly distributed over (0, 1 ] so that Prob[ X ≤ x ] = x, 0 < x ≤ 1.

• Repeatedly draw two samples x1 and x2 from X until ω ≡ (2x1 − 1)2 + (2x2 − 1)2 < 1.

• Then c(2x1 − 1) and c(2x2 − 1) are independent standard normal variables wherea

c ≡ p

−2(ln ω)/ω .

aAs they are normally distributed, to prove independence, it suffices to prove that they are uncorrelated, which is easy. Thanks to a lively

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### A Dirty Trick and a Right Attitude

• Let ξi are independent and uniformly distributed over (0, 1).

• A simple method to generate the standard normal variable is to calculatea

Ã 12 X

i=1

ξi

!

− 6.

• But why use 12?

• Recall the mean and variance of ξi are 1/2 and 1/12, respectively.

aackel (2002), “this is not a highly accurate approximation and

(25)

### A Dirty Trick and a Right Attitude (concluded)

• So the general formula is (Pn

i=1 ξi) − (n/2)

pn/12 .

• Choosing n = 12 yields a formula without the need of division and square-root operations.a

• Always blame your random number generator last.b

aContributed by Mr. Chen, Shih-Hang (R02723031) on March 5, 2014.

b“The fault, dear Brutus, lies not in the stars but in ourselves that we are underlings.” William Shakespeare (1564–1616), Julius Caesar.

(26)

### Generation of Bivariate Normal Distributions

• Pairs of normally distributed variables with correlation ρ can be generated.

• Let X1 and X2 be independent standard normal variables.

• Set

U ≡ aX1, V ≡ ρU + p

1 − ρ2 aX2.

(27)

### Generation of Bivariate Normal Distributions (concluded)

• U and V are the desired random variables with Var[ U ] = Var[ V ] = a2,

Cov[ U, V ] = ρa2.

• Note that the mapping between (X1, X2) and (U, V ) is one-to-one.

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### The Lognormal Distribution

• A random variable Y is said to have a lognormal distribution if ln Y has a normal distribution.

• Let X ∼ N (µ, σ2) and Y ≡ eX.

• The mean and variance of Y are

µY = eµ+σ2/2 and σY2 = e2µ+σ2

³

eσ2 − 1

´ ,

(21) respectively.

– They follow from E[ Y n ] = enµ+n2σ2/2.

(29)

### Option Basics

(30)

The shift toward options as the center of gravity of finance [ . . . ]

— Merton H. Miller (1923–2000)

(31)

### Calls and Puts

• A call gives its holder the right to buy a number of the underlying asset by paying a strike price.

• A put gives its holder the right to sell a number of the underlying asset for the strike price.

• How to price options?a

aIt can be traced to Aristotle’s (384 B.C.–322 B.C.) Politics, if not earlier.

(32)

### Exercise

• When a call is exercised, the holder pays the strike price in exchange for the stock.

• When a put is exercised, the holder receives from the writer the strike price in exchange for the stock.

• An option can be exercised prior to the expiration date:

early exercise.

(33)

### American and European

• American options can be exercised at any time up to the expiration date.

• European options can only be exercised at expiration.

• An American option is worth at least as much as an otherwise identical European option.

(34)

### Convenient Conventions

• C: call value.

• P : put value.

• X: strike price.

• S: stock price.

• D: dividend.

(35)

### Payoff, Mathematically Speaking

• The payoff of a call at expiration is C = max(0, S − X).

• The payoff of a put at expiration is P = max(0, X − S).

• A call will be exercised only if the stock price is higher than the strike price.

• A put will be exercised only if the stock price is less than the strike price.

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Price Long a put

10 20 30 40 50 Payoff

20 40 60 80 Price

Short a put

-50 -40 -30 -20 -10

Payoff

20 40 60 80 Price

Long a call

10 20 30 40 Payoff

20 40 60 80 Price

Short a call

-40 -30 -20 -10

Payoff

(37)

### Payoff, Mathematically Speaking (continued)

• At any time t before the expiration date, we call max(0, St − X)

the intrinsic value of a call.

• At any time t before the expiration date, we call max(0, X − St)

the intrinsic value of a put.

(38)

### Payoff, Mathematically Speaking (concluded)

• A call is in the money if S > X, at the money if S = X, and out of the money if S < X.

• A put is in the money if S < X, at the money if S = X, and out of the money if S > X.

• Options that are in the money at expiration should be exercised.a

• Finding an option’s value at any time before expiration is a major intellectual breakthrough.

a11% of option holders let in-the-money options expire worthless.

(39)

80 85 90 95 100 105 110 115 Stock price

0 5 10 15 20

Call value

80 85 90 95 100 105 110 115 Stock price

0 2 4 6 8 10 12 14

Put value

(40)

### Cash Dividends

• Exchange-traded stock options are not cash dividend-protected (or simply protected).

– The option contract is not adjusted for cash dividends.

• The stock price falls by an amount roughly equal to the amount of the cash dividend as it goes ex-dividend.

• Cash dividends are detrimental for calls.

• The opposite is true for puts.

(41)

### Stock Splits and Stock Dividends

• Options are adjusted for stock splits.

• After an n-for-m stock split, the strike price is only

m/n times its previous value, and the number of shares covered by one contract becomes n/m times its

previous value.

• Options are assumed to be unprotected.

(42)

### Example

• Consider an option to buy 100 shares of a company for

\$50 per share.

• A 2-for-1 split changes the term to a strike price of \$25 per share for 200 shares.

(43)

### Short Selling

• Short selling (or simply shorting) involves selling an asset that is not owned with the intention of buying it back later.

– If you short 1,000 XYZ shares, the broker borrows them from another client to sell them in the market.

– This action generates proceeds for the investor.

– The investor can close out the short position by buying 1,000 XYZ shares.

– Clearly, the investor profits if the stock price falls.

• Not all assets can be shorted.

(44)

### Payoff of Stock

20 40 60 80 Price

Long a stock

20 40 60 80 Payoff

20 40 60 80 Price

Short a stock

-80 -60 -40 -20

Payoff

(45)

### Covered Position: Hedge

• A hedge combines an option with its underlying stock in such a way that one protects the other against loss.

• Protective put: A long position in stock with a long put.

• Covered call: A long position in stock with a short call.a – It is “covered” because the stock can be delivered to

the buyer of the call if the call is exercised.

• Both strategies break even only if the stock price rises, so they are bullish.

aA short position has a payoff opposite in sign to that of a long position.

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85 90 95 100 105 110Stock price Protective put

-2.5 2.5 5 7.5 10 12.5

Profit

85 90 95 100 105 110Stock price Covered call

-12 -10 -8 -6 -4 -2 2 Profit

Solid lines are profits of the portfolio one month before

maturity assuming the portfolio is set up when S = 95 then.

(47)

• A spread consists of options of the same type and on the same underlying asset but with different strike prices or expiration dates.

• We use XL, XM, and XH to denote the strike prices with

XL < XM < XH.

(48)

• A bull call spread consists of a long XL call and a short XH call with the same expiration date.

– The initial investment is CL − CH.

– The maximum profit is (XH − XL) − (CL − CH).

– The maximum loss is CL − CH.

(49)

85 90 95 100 105 110Stock price Bull spread (call)

-4 -2 2 4 Profit

(50)

• Writing an XH put and buying an XL put with identical expiration date creates the bull put spread.

• It profits from declining stock prices.

• Three calls or three puts with different strike prices and the same expiration date create a butterfly spread.

– The spread is long one XL call, long one XH call, and short two XM calls.

(51)

85 90 95 100 105 110Stock price Butterfly

-1 1 2 3 Profit

(52)

• A butterfly spread pays off a positive amount at

expiration only if the asset price falls between XL and XH.

• A butterfly spread with a small XH − XL approximates a state contingent claim,a which pays \$1 only when a particular price results.b

aAlternatively, Arrow security.

bSee Exercise 7.4.5 of the textbook.

(53)

• The price of a state contingent claim is called a state price.

• The (undiscounted) state price equals

2C

∂X2.

– Recall that C is the call’s price.a

• In fact, the PV of ∂2C/∂X2 is the probability density of the stock price ST = X at option’s maturity.b

aOne can also use the put (see Exercise 9.3.6 of the textbook).

bBreeden and Litzenberger (1978).

(54)

### Covered Position: Combination

• A combination consists of options of different types on the same underlying asset.

– These options must be either all bought or all written.

• Straddle: A long call and a long put with the same strike price and expiration date.

– Since it profits from high volatility, a person who buys a straddle is said to be long volatility.

– Selling a straddle benefits from low volatility.

(55)

85 90 95 100 105 110Stock price Straddle

-5 -2.5 2.5 5 7.5 10 Profit

(56)

### Covered Position: Combination (concluded)

• Strangle: Identical to a straddle except that the call’s strike price is higher than the put’s.

(57)

85 90 95 100 105 110Stock price Strangle

-2 2 4 6 8 10 Profit

(58)

### Arbitrage in Option Pricing

(59)

All general laws are attended with inconveniences, when applied to particular cases.

— David Hume (1711–1776)

(60)

### Arbitrage

• The no-arbitrage principle says there is no free lunch.

• It supplies the argument for option pricing.

• A riskless arbitrage opportunity is one that, without any initial investment, generates nonnegative returns under all circumstances and positive returns under some.

• In an efficient market, such opportunities do not exist (for long).

(61)

### Portfolio Dominance Principle

• Consider two portfolios A and B.

• A should be more valuable than B if A’s payoff is at least as good as B’s under all circumstances and better under some.

(62)

### A Corollary

• A portfolio yielding a zero return in every possible scenario must have a zero PV.

– Short the portfolio if its PV is positive.

– Buy it if its PV is negative.

– In both cases, a free lunch is created.

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