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Unbiased Expectations Theory and Spot Rate Curve

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(1)

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)

A “Bad” Expectations Theory

• 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.

– · · ·

(12)

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).

(13)

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?

(14)

Fundamental Statistical Concepts

(15)

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.

(19)

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.

(23)

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

(24)

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

à 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

• Instead, check your programs first.

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.

(28)

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.

(36)

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.

• Exchange-traded stock options are adjusted for stock dividends.

• 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.

(46)

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)

Covered Position: Spread

• 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)

Covered Position: Spread (continued)

• 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)

Covered Position: Spread (continued)

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

• A bear spread amounts to selling a bull 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)

Covered Position: Spread (continued)

• 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)

Covered Position: Spread (concluded)

• 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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