Forward Rates

Forward Rates

• The yield curve contains information regarding future interest rates currently “expected” by the market.

• Invest $1 for j periods to end up with [ 1 + S(j) ]^j dollars at time j.

– The maturity strategy.

• Invest $1 in bonds for i periods and at time i invest the proceeds in bonds for another j − i periods where j > i.

• Will have [ 1 + S(i) ]ⁱ[ 1 + S(i, j) ]^j−i dollars at time j.

– S(i, j): (j − i)-period spot rate i periods from now.

– The rollover strategy.

Forward Rates (concluded)

• When S(i, j) equals

f (i, j) ≡

 (1 + S(j))^j (1 + S(i))ⁱ

1/(j−i)

− 1, (17)

we will end up with [ 1 + S(j) ]^j dollars again.

• By definition, f(0, j) = S(j).

• f(i, j) is called the (implied) forward rates.

– More precisely, the (j − i)-period forward rate i periods from now.

Time Line

f(0, 1) f(1, 2) f(2, 3) f(3, 4) -

Time 0

-^S(1)

-^S(2)

-^S(3)

-^S(4)

Forward Rates and Future Spot Rates

• We did not assume any a priori relation between f(i, j) and future spot rate S(i, j).

– This is the subject of the term structure theories.

• We merely looked for the future spot rate that, if realized, will equate the two investment strategies.

• f(i, i + 1) are called the instantaneous forward rates or one-period forward rates.

Spot Rates and Forward Rates

• When the spot rate curve is normal, the forward rate dominates the spot rates,

f (i, j) > S(j) > · · · > S(i).

• When the spot rate curve is inverted, the forward rate is dominated by the spot rates,

f (i, j) < S(j) < · · · < S(i).

spot rate curve forward rate curve yield curve

(a)

spot rate curve forward rate curve yield curve

(b)

Forward Rates ≡ Spot Rates ≡ Yield Curve

• The FV of $1 at time n can be derived in two ways.

• Buy n-period zero-coupon bonds and receive [ 1 + S(n) ]ⁿ.

• Buy one-period zero-coupon bonds today and a series of such bonds at the forward rates as they mature.

• The FV is

[ 1 + S(1) ][ 1 + f (1, 2) ]· · · [ 1 + f(n − 1, n) ].

Forward Rates ≡ Spot Rates ≡ Yield Curves (concluded)

• Since they are identical,

S(n) = {[ 1 + S(1) ][ 1 + f(1, 2) ]

· · · [ 1 + f(n − 1, n) ]}^1/n − 1. (18)

• Hence, the forward rates (specifically the one-period forward rates) determine the spot rate curve.

• Other equivalencies can be derived similarly, such as f (T, T + 1) = d(T )

d(T + 1) − 1. (19)

Locking in the Forward Rate f(n, m)

• Buy one n-period zero-coupon bond for 1/(1 + S(n))ⁿ dollars.

• Sell (1 + S(m))^m/(1 + S(n))ⁿ m-period zero-coupon bonds.

• No net initial investment because the cash inflow equals the cash outflow: 1/(1 + S(n))ⁿ.

• At time n there will be a cash inflow of $1.

• At time m there will be a cash outflow of (1 + S(m))^m/(1 + S(n))ⁿ dollars.

Locking in the Forward Rate f(n, m) (concluded)

• This implies the rate f(n, m) between times n and m.

6 -

?

n m

1

(1 + S(m))^m/(1 + S(n))ⁿ

Forward Contracts

• We had generated the cash flow of a financial instrument called forward contract.

• Agreed upon today, it enables one to

– Borrow money at time n in the future, and

– Repay the loan at time m > n with an interest rate equal to the forward rate

f (n, m).

• Can the spot rate curve be an arbitrary curve?^a

aContributed by Mr. Dai, Tian-Shyr (B82506025, R86526008, D88526006) in 1998.

Spot and Forward Rates under Continuous Compounding

• The pricing formula:

P =

n 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 .

Spot and Forward Rates under Continuous Compounding (continued)

• The formula for the forward rate:

f (i, j) = jS(j) − iS(i)

j − i . (20)

– Compare the above formula with Eq. (17) on p. 126.

• The one-period forward rate:

f (j, j + 1) = − ln d(j + 1) d(j) .

– Compare the above formula with Eq. (19) on p. 132.

Spot and Forward Rates under Continuous Compounding (concluded)

• Now,

f (T ) ≡ lim

ΔT →0 f (T, T + ΔT )

= S(T ) + T ∂S

∂T .

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

• If S(T ) < −T (∂S/∂T ), then f(T ) < 0.^a

aContributed by Mr. Huang, Hsien-Chun (R03922103) on March 11, 2015.

Unbiased Expectations Theory

• Forward rate equals the average future spot rate,

f (a, b) = E[ S(a, b) ]. (21)

• 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.

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. (17) on p. 126.

– 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.

A “Bad” Expectations Theory

• The expected returns^a 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) ] (22) 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.

aMore precisely, the one-plus returns.

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).

A “Bad” Expectations Theory (concluded)

• The theory says the returns are equal:

1 + S(1)

(1 + S(2))² = E

 1

1 + S(1, 2)

 .

• Combine this with Eq. (22) on p. 141 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., g^(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 in Practice

• 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 ],

[ 1.8%, 1.6%, 1.4%, 1%, 0%, −1%, −1.4%, . . . ], etc.

– . . . .

• At least one c_i should be 1 as the reference point.

Duration in Practice (concluded)

• Let

P (y) ≡ 

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

or − P (Δy) − P (−Δy) 2P (0)Δy .

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

S(1) = S(2) = · · · = S(n).

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?

Fundamental Statistical Concepts

There are three kinds of lies:

lies, damn lies, and statistics.

— Misattributed to 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)

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 ≡ 

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.^a

aPaul Wilmott (2009), “the correlations between financial quantities are notoriously unstable.” It may even break down “at high-frequency time intervals” (Budish, Cramton, and Shim, 2015).

Variance of Sum

• Variance of a weighted sum of random variables equals Var

_n



i=1

a_iX_i

=

n i=1

n j=1

a_ia_j Cov[ X_i, X_j ].

• It becomes

n 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 ]. (23)

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 following distribution function

Prob[ X ≤ z ] = N(z) ≡ 1

√2π

 _z

−∞

e^−x2/2 dx.

Moment Generating Function

• The moment generating function of random variable X is defined as

θ_X(t) ≡ E[ e^tX ].

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



μt + σ²t² 2



. (24)

The Multivariate Normal Distribution

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



i

X_i ∼ N



i

μ_i,

i

σ_i²

.

• Let Xi ∼ N(μi, σ_i²), which may not be independent.

• Suppose

n i=1

t_iX_i ∼ N

⎛

⎝ⁿ

i=1

t_i μ_i,

n i=1

n j=1

t_it_j Cov[ X_i, X_j ]

⎞

⎠

for every linear combination _n

i=1 t_iX_i.^a

• 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, 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^a

c ≡ 

−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 class discussion on March 5, 2014.

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^a

 ₁₂



i=1

ξ_i

− 6.

• But why use 12?

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

aJ¨ackel (2002), “this is not a highly accurate approximation and

A Dirty Trick and a Right Attitude (concluded)

• The general formula is (_n

i=1ξ_i) − (n/2)

n/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.

Generation of Bivariate Normal Distributions

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

• Let X1 and X₂ be independent standard normal variables.

• Set

U ≡ aX₁, V ≡ ρU + 

1 − ρ² aX₂.

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.

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



e^σ2 − 1 ,

(25) respectively.

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

The Lognormal Distribution (continued)

• Conversely, suppose Y is lognormally distributed with mean μ and variance σ².

• Then

E[ ln Y ] = ln(μ/

1 + (σ/μ)²), Var[ ln Y ] = ln(1 + (σ/μ)²).

• If X and Y are joint-lognormally distributed, then E[ XY ] = E[ X ] E[ Y ] eCov[ ln X,ln Y ],

Cov[ X, Y ] = E[ X ] E[ Y ]



eCov[ ln X,ln Y ] − 1 .

The Lognormal Distribution (concluded)

• Let Y be lognormally distributed such that ln Y ∼ N(μ, σ²).

• Then

 _∞

a

yf (y) dy = e^μ+σ2/2 N

μ − ln a

σ + σ

 .

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 unit of the underlying asset by paying a strike price.

- 6

? ?

option primium

stock

strike price

Calls and Puts (continued)

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

- 6

? ?

option primium

strike price

stock

Calls and Puts (concluded)

• An embedded option has to be traded along with the underlying asset.

• How to price options?

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

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.

Convenient Conventions

• C: call value.

• P : put value.

• X: strike price.

• S: stock price.

• D: dividend.

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.

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

Payoff, Mathematically Speaking (continued)

• 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 − St)

the intrinsic value of a put.

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.

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, m shares become n shares.

• Accordingly, 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.

• We assume options are 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.

• 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.

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

• 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 profits of the portfolio one month before

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

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, X_M, and X_H to denote the strike prices with

X_L < X_M < X_H.

Covered Position: Spread (continued)

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

– The initial investment is C_L − C_H.

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

∗ When both are exercised at expiration.

– The maximum loss is CL − CH.

∗ When neither is exercised at expiration.

– If we buy (X_H − X_L)⁻¹ units of the bull call spread and X_H − X_L → 0, a (Heaviside) step function

emerges as the payoff.

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

-4 -2 2 4 Profit

Covered Position: Spread (continued)

• Writing an XH 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 XL 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 (continued)

• A butterfly spread pays a positive amount at expiration only if the asset price falls between X_L and X_H.

• Take a position in (X_M − X_L)⁻¹ units of the butterfly spread.

• When X_H − X_L → 0, it approximates a state contingent claim,^a which pays $1 only when the state S = X_M

happens.^b

aAlternatively, Arrow security.

bSee Exercise 7.4.5 of the textbook.

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 S_T = X at option’s maturity.^b

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

bBreeden and Litzenberger (1978).

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.

85 90 95 100 105 110Stock price Straddle

-5 -2.5 2.5 5 7.5 10 Profit

Covered Position: Combination (concluded)

• 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 Strangle

-2 2 4 6 8 10 Profit