• 沒有找到結果。

# Forward Rates

N/A
N/A
Protected

Share "Forward Rates"

Copied!
71
0
0

(1)

### 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.

(2)

### 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.

(3)

### Time Line

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

Time 0

-S(1)

-S(2)

-S(3)

-S(4)

(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.

(5)

### 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).

(6)

spot rate curve forward rate curve yield curve

(a)

spot rate curve forward rate curve yield curve

(b)

(7)

### 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) ].

(8)

### 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)

(9)

### 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.

(10)

### 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

(11)

### 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.

(12)

### 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 .

(13)

### 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.

(14)

### 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.

(15)

### 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.

(16)

### 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.

(17)

• The expected returnsa 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.

(18)

### 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).

(19)

### 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.

(20)

### 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.

(21)

### Duration in Practice

• 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 ],

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

– . . . .

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

(22)

### Duration in Practice (concluded)

• Let

P (y) 

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

or P (Δy) − P (−Δy) 2P (0)Δy .

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

S(1) = S(2) = · · · = S(n).

(23)

### 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?

(24)

## Fundamental Statistical Concepts

(25)

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)

(26)

### 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.

(27)

### Correlation

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

σX 

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.” It may even break down “at high-frequency time intervals” (Budish, Cramton, and Shim, 2015).

(28)

### Variance of Sum

• Variance of a weighted sum of random variables equals Var

n



i=1

aiXi

=

n i=1

n j=1

aiaj Cov[ Xi, Xj ].

• It becomes

n i=1

a2i Var[ Xi ] when Xi are uncorrelated.

(29)

### 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)

(30)

### 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 following distribution function

Prob[ X ≤ z ] = N(z) ≡ 1

√2π

z

−∞

e−x2/2 dx.

(31)

### Moment Generating Function

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

θX(t) ≡ E[ etX ].

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



μt + σ2t2 2



. (24)

(32)

### The Multivariate Normal Distribution

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



i

Xi ∼ N



i

μi,

i

σi2

.

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

• Suppose

n i=1

tiXi ∼ N

⎝n

i=1

ti μi,

n i=1

n j=1

titj Cov[ Xi, Xj ]

for every linear combination n

i=1 tiXi.a

• Xi are said to have a multivariate normal distribution.

(33)

### 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 

−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.

(34)

### 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

12



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

(35)

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

• The general formula is (n

i=1ξi) − (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

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.

(36)

### 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.

(37)

### 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.

(38)

### 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



eσ2 − 1 ,

(25) respectively.

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

(39)

### 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 .

(40)

### The Lognormal Distribution (concluded)

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

• Then

a

yf (y) dy = eμ+σ2/2 N

μ − ln a

σ + σ

 .

(41)

## Option Basics

(42)

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

— Merton H. Miller (1923–2000)

(43)

### 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

(44)

### 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

(45)

### 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.

(46)

### 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.

(47)

### 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.

(48)

### Convenient Conventions

• C: call value.

• P : put value.

• X: strike price.

• S: stock price.

• D: dividend.

(49)

### 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.

(50)

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

(51)

### 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.

(52)

### 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.

(53)

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

(54)

### 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.

(55)

### 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.

• We assume options are unprotected.

(56)

### 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.

(57)

### 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.

(58)

### 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

(59)

### 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.

(60)

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.

(61)

• 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.

(62)

• 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).

∗ When both are exercised at expiration.

– The maximum loss is CL − CH.

∗ When neither is exercised at expiration.

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

emerges as the payoﬀ.

(63)

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

-4 -2 2 4 Profit

(64)

• 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.

(65)

85 90 95 100 105 110Stock price Butterfly

-1 1 2 3 Profit

(66)

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

• Take a position in (XM − XL)−1 units of the butterﬂy spread.

• When XH − XL → 0, it approximates a state contingent claim,a which pays \$1 only when the state S = XM

happens.b

aAlternatively, Arrow security.

bSee Exercise 7.4.5 of the textbook.

(67)

• 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).

(68)

### 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.

(69)

85 90 95 100 105 110Stock price Straddle

-5 -2.5 2.5 5 7.5 10 Profit

(70)

### Covered Position: Combination (concluded)

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

(71)

85 90 95 100 105 110Stock price Strangle

-2 2 4 6 8 10 Profit

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30.. – By the

• Similar to futures options except that what is delivered is a forward contract with a delivery price equal to the option’s strike price. – Exercising a call forward option results

• Similar to futures options except that what is delivered is a forward contract with a delivery price equal to the option’s strike price.. – Exercising a call forward option results

• 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

• A put gives its holder the right to sell a number of the underlying asset for the strike price.. • An embedded option has to be traded along with the

• Similar to futures options except that what is delivered is a forward contract with a delivery price equal to the option’s strike price.. – Exercising a call forward option results

• Similar to futures options except that what is delivered is a forward contract with a delivery price equal to the option’s strike price.. – Exercising a call forward option results

• It works as if the call writer delivered a futures contract to the option holder and paid the holder the prevailing futures price minus the strike price.. • It works as if the