### Spot and Forward Rates under Continuous Compounding

*• The pricing formula:*

*P =*

X*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,*
*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 →0**f (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., g*^{00}*(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 [ c*_{1}*, c*_{2}*, . . . , 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}*)

*be the price*

^{i}*associated with the cash flow C*

_{1}

*, C*

_{2}

*, . . . .*

*• 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*_{X}*σ*_{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*_{i}*X*_{i}

#

=

X*n*
*i=1*

X*n*
*j=1*

*a*_{i}*a*_{j}*Cov[ X*_{i}*, X*_{j}*].*

*• It becomes*

X*n*
*i=1*

*a*^{2}_{i}*Var[ X** _{i}* ]

*when X*

*are uncorrelated.*

_{i}### 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 I*_{2} *contains at least as much information as I*_{1}, then
*E[ X | I*_{1} *] = E[ E[ X | I*_{2} *] | I*_{1} *].* (16)

### The Normal Distribution

*• A random variable X has the normal distribution with*
*mean µ and variance σ*^{2} if its probability density

function is

1
*σ√*

*2π* *e*^{−(x−µ)}^{2}^{/(2σ}^{2}^{)}*.*

*• This is expressed by X ∼ N (µ, σ*^{2}).

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

*Prob[ X ≤ z ] = N (z) ≡* 1

*√2π*

Z _{z}

*−∞*

*e*^{−x}^{2}^{/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, σ*^{2}) is
*θ*_{X}*(t) = exp*

·

*µt +* *σ*^{2}*t*^{2}
2

¸

*.* (17)

### Distribution of Sum

*• If X*_{i}*∼ N (µ*_{i}*, σ*_{i}^{2}) are independent, then
X

*i*

*X*_{i}*∼ N*

ÃX

*i*

*µ*_{i}*,*X

*i*

*σ*_{i}^{2}

!
*.*

*• Let X*_{i}*∼ N (µ*_{i}*, σ*_{i}^{2}), which may not be independent.

*• Then*
X*n*

*i=1*

*t*_{i}*X*_{i}*∼ N*

X*n*

*i=1*

*t*_{i}*µ*_{i}*,*
X*n*

*i=1*

X*n*
*j=1*

*t*_{i}*t*_{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 x*_{1} *and x*_{2} *from X until*
*ω ≡ (2x*_{1} *− 1)*^{2} *+ (2x*_{2} *− 1)*^{2} *< 1.*

*• Then c(2x*_{1} *− 1) and c(2x*_{2} *− 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.*

*• X*_{1} *and X*_{2} be independent standard normal variables.

*• Set*

*U* *≡ aX*_{1}*,*
*V* *≡ ρU +* p

*1 − ρ*^{2} *aX*_{2}*.*

*• U and V are the desired random variables with*
*Var[ U ] = Var[ V ] = a*^{2} *and Cov[ U, V ] = ρa*^{2}.

### 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, σ*^{2}*) and Y ≡ e** ^{X}*.

*• The mean and variance of Y are*

*µ*_{Y}*= e*^{µ+σ}^{2}^{/2}*and σ*_{Y}^{2} *= e*^{2µ+σ}^{2}

³

*e*^{σ}^{2} *− 1*

´
*,*

(18) respectively.

*– They follow from E[ Y* ^{n}*] = e*^{nµ+n}^{2}^{σ}^{2}* ^{/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*

*call with the same expiration date.*

_{H}*– 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*

*calls.*

_{M}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*_{i}*d(i) for a certain cash flow C*_{1}*, C*_{2}*, . . . , 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 -

*C*_{1} *C*_{2} *C*_{3}

*· · ·* *C*_{n}

? ? ? ?

*C*_{1} *C*_{2} *C*_{3}

*· · ·*

*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 X*_{1} *< X*_{2} *< X*_{3}*,*

*C*_{X}_{2} *≤ ωC*_{X}_{1} *+ (1 − ω) C*_{X}_{3}
*P*_{X}_{2} *≤ ωP*_{X}_{1} *+ (1 − ω) P*_{X}_{3}
*Here*

*ω ≡ (X*_{3} *− X*_{2}*)/(X*_{3} *− X*_{1}*).*

*(Equivalently, X*_{2} *= ωX*_{1} *+ (1 − ω) X*_{3}*.)*

### 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*_{i}*X*_{i}*has a value*
*at most* P

*i* *ω*_{i}*C*_{i}*. All options expire on the same date.*

The same result holds for European puts.