Unbiased Expectations Theory and Spot Rate Curve

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, S(n) = f (0, 1) + f (1, 2) + · · · + f (n − 1, n)

n .

Spot and Forward Rates under Continuous Compounding (concluded)

• 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) .

•

f (T ) ≡ lim

∆T →0f (T, T + ∆T ) = S(T ) + T ∂S

∂T .

• f (T ) > S(T ) if and only if ∂S/∂T > 0.

Unbiased Expectations Theory

• Forward rate equals the average future spot rate,

f (a, b) = E[ S(a, b) ]. (14)

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

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

Unbiased Expectations Theory and Spot Rate Curve

• 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. (12) on p. 116.

– 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.

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.

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) ] (15) 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.

A “Bad” Expectations Theory (continued)

• Now consider two one-period strategies.

– Strategy one buys a two-period bond 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)

¸ .

A “Bad” Expectations Theory (concluded)

• Combine this with Eq. (15) on p. 131 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.

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.

Duration Revisited

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

perceived type of change.

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

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

– · · ·

• Let P (y) ≡ P

i C_i/(1 + S(i) + yc_i)ⁱ be the price associated with the cash flow C₁, C₂, . . . .

• Define duration as

−∂P (y)/P (0)

∂y

¯¯

¯¯

y=0

.

Fundamental Statistical Concepts

There are three kinds of lies:

lies, damn lies, and statistics.

— Benjamin Disraeli (1804–1881) One death is a tragedy, but a million deaths are a statistic.

— Josef Stalin (1879–1953)

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 − µ_X)(Y − µ_Y ) ] ,

where µ_X and µ_Y are the means of X and Y , respectively.

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

Correlation

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

σ_X ≡ p

Var[ X ] .

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

ρ_X,Y ≡ Cov[ X, Y ] σ_Xσ_Y ,

provided both have nonzero standard deviations.

Variance of Sum

• Variance of a weighted sum of random variables equals Var

" _n X

i=1

a_iX_i

#

=

Xn i=1

Xn j=1

a_ia_j Cov[ X_i, X_j ].

• It becomes

Xn i=1

a²_i Var[ X_i ] when X_i are uncorrelated.

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 ]. (16)

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√

2π 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.

Moment Generating Function

• The moment generating function of random variable X is

θ_X(t) ≡ E[ e^tX ].

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

·

µt + σ²t² 2

¸

. (17)

Distribution of Sum

• If X_i ∼ N (µ_i, σ_i²) are independent, then X

i

X_i ∼ N

ÃX

i

µ_i,X

i

σ_i²

! .

• Let X_i ∼ N (µ_i, σ_i²), which may not be independent.

• Then Xn

i=1

t_iX_i ∼ N



 Xn

i=1

t_i µ_i, Xn

i=1

Xn j=1

t_it_j Cov[ X_i, X_j ]



 .

• X_i are said to have a multivariate normal distribution.

Generation of Univariate Normal Distributions

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

c ≡ p

−2(ln ω)/ω .

A Dirty Trick and a Right Attitude

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

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

X12 i=1

ξ_i − 6.

• But “this is not a highly accurate approximation and should only be used to establish ballpark estimates.”^a

aJ¨ackel, Monte Carlo Methods in Finance (2002).

A Dirty Trick and a Right Attitude (concluded)

• Always blame your random number generator last.^a

• Instead, check your programs first.

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

Generation of Bivariate Normal Distributions

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

• X₁ and X₂ be independent standard normal variables.

• Set

U ≡ aX₁, V ≡ ρU + p

1 − ρ² aX₂.

• U and V are the desired random variables with Var[ U ] = Var[ V ] = a² and Cov[ U, V ] = ρa².

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 ≡ e^X.

• The mean and variance of Y are

µ_Y = e^µ+σ2/2 and σ_Y² = e^2µ+σ2

³

e^σ2 − 1

´ ,

(18) respectively.

– They follow from E[ Y ⁿ ] = e^nµ+n2σ2/2.

Option Basics

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

— Merton H. Miller (1923–2000)

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?

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.

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 because of the early exercise feature.

Convenient Conventions

• C: call value.

• P : put value.

• X: strike price.

• S: stock price.

• D: dividend.

Payoff

• 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.

• 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).

• At any time t before the expiration date, we call max(0, S_t − X) the intrinsic value of a call.

• At any time t before the expiration date, we call max(0, X − S_t) the intrinsic value of a put.

Payoff (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.

20 40 60 80 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

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

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.

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.

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.

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.

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

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

• 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.

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 current value of the portfolio.

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 X_L, X_M, and X_H to denote the strike prices with X_L < X_M < X_H.

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

– The initial investment is C_L − C_H.

– The maximum profit is (X_H − X_L) − (C_L − C_H).

– The maximum loss is C_L − C_H.

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

-4 -2 2 4 Profit

Covered Position: Spread (continued)

• Writing an X_H put and buying an X_L 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 X_L call, long one X_H call, and short two X_M calls.

85 90 95 100 105 110Stock price Butterfly

-1 1 2 3 Profit

Covered Position: Spread (concluded)

• A butterfly spread pays off a positive amount at

expiration only if the asset price falls between X_L and X_H.

• A butterfly spread with a small X_H − X_L approximates a state contingent claim, which pays $1 only when a

particular price results.

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

Covered Position: Combination

• A combination consists of options of different types on the same underlying asset, and they are 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.

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

85 90 95 100 105 110Stock price Straddle

-5 -2.5 2.5 5 7.5 10 Profit

85 90 95 100 105 110Stock price Strangle

-2 2 4 6 8 10 Profit

Arbitrage in Option Pricing

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

— David Hume (1711–1776)

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).

• The portfolio dominance principle says portfolio A

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

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.

The PV Formula Justified

P = P_n

i=1 C_id(i) for a certain cash flow C₁, C₂, . . . , C_n.

• If the price P^∗ < P , short the zeros that match the security’s n cash flows and use P^∗ of the proceeds P to buy the security.

• Since the cash inflows of the security will offset exactly the obligations of the zeros, a riskless profit of P − P^∗ dollars has been realized now.

• If the price P^∗ > P , a riskless profit can be realized by reversing the trades.

6 6 6 6 -

C₁ C₂ C₃

· · · C_n

? ? ? ?

C₁ C₂ C₃

· · ·

C_n

6

P

?

P^∗

¾ _security

¾ _zeros

Two More Examples

• An American option cannot be worth less than the intrinsic value.

– Otherwise, one can buy the option, promptly exercise it and sell the stock with a profit.

• A put or a call must have a nonnegative value.

– Otherwise, one can buy it for a positive cash flow now and end up with a nonnegative amount at expiration.

Relative Option Prices

• These relations hold regardless of the probabilistic model for stock prices.

• Assume, among other things, that there are no

transactions costs or margin requirements, borrowing and lending are available at the riskless interest rate, interest rates are nonnegative, and there are no

arbitrage opportunities.

• Let the current time be time zero.

• PV(x) stands for the PV of x dollars at expiration.

• Hence PV(x) = xd(τ ) where τ is the time to expiration.

Put-Call Parity (Castelli, 1877)

C = P + S − PV(X). (19)

• Consider the portfolio of one short European call, one long European put, one share of stock, and a loan of PV(X).

• All options are assumed to carry the same strike price and time to expiration, τ .

• The initial cash flow is therefore C − P − S + PV(X).

• At expiration, if the stock price S_τ ≤ X, the put will be worth X − S_τ and the call will expire worthless.

The Proof (concluded)

• After the loan, now X, is repaid, the net future cash flow is zero:

0 + (X − S_τ) + S_τ − X = 0.

• On the other hand, if S_τ > X, the call will be worth S_τ − X and the put will expire worthless.

• After the loan, now X, is repaid, the net future cash flow is again zero:

−(S_τ − X) + 0 + S_τ − X = 0.

• The net future cash flow is zero in either case.

• The no-arbitrage principle implies that the initial

Consequences of Put-Call Parity

• There is only one kind of European option because the other can be replicated from it in combination with the underlying stock and riskless lending or borrowing.

– Combinations such as this create synthetic securities.

• S = C − P + PV(X) says a stock is equivalent to a

portfolio containing a long call, a short put, and lending PV(X).

• C − P = S − PV(X) implies a long call and a short put amount to a long position in stock and borrowing the PV of the strike price (buying stock on margin).

Intrinsic Value

Lemma 1 An American call or a European call on a non-dividend-paying stock is never worth less than its intrinsic value.

• The put-call parity implies

C = (S − X) + (X − PV(X)) + P ≥ S − X.

• Recall C ≥ 0.

• It follows that C ≥ max(S − X, 0), the intrinsic value.

• An American call also cannot be worth less than its intrinsic value.

Intrinsic Value (concluded)

A European put on a non-dividend-paying stock may be worth less than its intrinsic value (p. 159).

Lemma 2 For European puts, P ≥ max(PV(X) − S, 0).

• Prove it with the put-call parity.

• Can explain the right figure on p. 159 why P < X − S when S is small.

Early Exercise of American Calls

European calls and American calls are identical when the underlying stock pays no dividends.

Theorem 3 (Merton (1973)) An American call on a non-dividend-paying stock should not be exercised before expiration.

• By an exercise in text, C ≥ max(S − PV(X), 0).

• If the call is exercised, the value is the smaller S − X.

Remarks

• The above theorem does not mean American calls should be kept until maturity.

• What it does imply is that when early exercise is being considered, a better alternative is to sell it.

• Early exercise may become optimal for American calls on a dividend-paying stock.

– Stock price declines as the stock goes ex-dividend.

Early Exercise of American Calls: Dividend Case

Surprisingly, an American call should be exercised only at a few dates.

Theorem 4 An American call will only be exercised at expiration or just before an ex-dividend date.

In contrast, it might be optimal to exercise an American put even if the underlying stock does not pay dividends.

Convexity of Option Prices

Lemma 5 For three otherwise identical calls or puts with strike prices X₁ < X₂ < X₃,

C_X2 ≤ ωC_X1 + (1 − ω) C_X3 P_X2 ≤ ωP_X1 + (1 − ω) P_X3 Here

ω ≡ (X₃ − X₂)/(X₃ − X₁).

(Equivalently, X₂ = ωX₁ + (1 − ω) X₃.)

Option on Portfolio vs. Portfolio of Options

An option on a portfolio of stocks is cheaper than a portfolio of options.

Theorem 6 Consider a portfolio of non-dividend-paying assets with weights ω_i. Let C_i denote the price of a

European call on asset i with strike price X_i. Then the call on the portfolio with a strike price X ≡ P

i ω_iX_i has a value at most P

i ω_iC_i. All options expire on the same date.

The same result holds for European puts.