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

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

• 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.

• 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 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 (R86526008, D8852600) in 1998.

### Spot and Forward Rates under Continuous Compounding

• The pricing formula:

P =

n

X

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, f (0, 1) + f (1, 2) + · · · + f(n − 1, 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) ]. (13)

• 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. (11) on p. 115.

– 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) ] (14)
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

### A “Bad” Expectations Theory (concluded)

• Combine this with Eq. (14) on p. 130 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 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})^{i} be the price
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

### Variance of Sum

• Variance of a weighted sum of random variables equals Var

" _{n}
X

i=1

a_{i}X_{i}

#

=

n

X

i=1 n

X

j=1

a_{i}a_{j} Cov[ X_{i}, X_{j} ].

• It becomes

n

X

i=1

a^{2}_{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 I^{2} contains at least as much information as I_{1}, then

### 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

. (16)

### Distribution of Sum

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

i

X_{i} ∼ N X

i

µ_{i},X

i

σ_{i}^{2}

! .

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

• Then

n

X

i=1

t_{i}X_{i} ∼ N

n

X

i=1

t_{i} µ_{i},

n

X

i=1 n

X

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

12

X

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}

e^{σ}^{2} − 1
,

(17) respectively.

n 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.

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

• 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.

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.

### 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

### 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}).

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.

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

### 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 = Pn

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

6 6 6 6 -

C^{1} C^{2} C^{3}

· · ·

C^{n}

? ? ? ?

C^{1} C^{2} C^{3}

· · ·

C^{n}

6

P

?

P^{∗}

_{security}

_{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). (18)

• 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

### The Proof (concluded)

• 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 zero in either case.

• The no-arbitrage principle implies that the initial

investment to set up the portfolio must be nil as well.

### 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

### 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 (see p. 157), but:

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. 157 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.

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 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.

## Option Pricing Models

If the world of sense does not fit mathematics, so much the worse for the world of sense.

— Bertrand Russell (1872–1970) Black insisted that anything one could do with a mouse could be done better with macro redefinitions of particular keys on the keyboard.

— Emanuel Derman, My Life as a Quant (2004)

### The Setting

• The no-arbitrage principle is insufficient to pin down the exact option value.

• Need a model of probabilistic behavior of stock prices.

• One major obstacle is that it seems a risk-adjusted interest rate is needed to discount the option’s payoff.

• Breakthrough came in 1973 when Black (1938–1995) and Scholes with help from Merton published their celebrated option pricing model.

### Terms and Approach

• C: call value.

• P : put value.

• X: strike price

• S: stock price

• ˆr > 0: the continuously compounded riskless rate per period.

• R ≡ e^{r}^{ˆ}: gross return.

• Start from the discrete-time binomial model.

### Binomial Option Pricing Model (BOPM)

• Time is discrete and measured in periods.

• If the current stock price is S, it can go to Su with probability q and Sd with probability 1 − q, where 0 < q < 1 and d < u.

– In fact, d < R < u must hold to rule out arbitrage.

• Six pieces of information suffice to determine the option value based on arbitrage considerations: S, u, d, X, ˆr, and the number of periods to expiration.

### S

### Su q

### 1 q

### Sd

### Call on a Non-Dividend-Paying Stock: Single Period

• The expiration date is only one period from now.

• C^{u} is the call price at time one if the stock price moves
to Su.

• C^{d} is the call price at time one if the stock price moves
to Sd.

• Clearly,

C_{u} = max(0, Su − X),
C = max(0, Sd − X).

C

Cu= max( 0, Su X ) q

1 q

Cd = max( 0, Sd X )

### Call on a Non-Dividend-Paying Stock: Single Period (continued)

• Set up a portfolio of h shares of stock and B dollars in riskless bonds.

– This costs hS + B.

– We call h the hedge ratio or delta.

• The value of this portfolio at time one is either hSu + RB or hSd + RB.

• Choose h and B such that the portfolio replicates the payoff of the call,

hSu + RB = C ,

### Call on a Non-Dividend-Paying Stock: Single Period (concluded)

• Solve the above equations to obtain
h = C_{u} − C^{d}

Su − Sd ≥ 0, (19)

B = uC_{d} − dC^{u}

(u − d) R . (20)

• By the no-arbitrage principle, the European call should cost the same as the equivalent portfolio, C = hS + B.

• As uC^{d} − dC^{u} < 0, the equivalent portfolio is a levered
long position in stocks.