### 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,*^{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* *.*

*• The one-period forward rate:*

*f (j, j + 1) = − ln* *d(j + 1)*
*d(j)* *.*

### Spot and Forward Rates under Continuous Compounding (concluded)

*• Now,*

*f (T ) ≡* lim

*∆T →0* *f (T, T + ∆T )*

*= S(T ) + T* *∂S*

*∂T* *.*

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

### Unbiased Expectations Theory

*• Forward rate equals the average future spot rate,*

*f (a, b) = E[ S(a, b) ].* (17)

*• 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. (15) on p. 123.

*– 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) ]* (18)
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 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).*

*• 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. (18) on p. 139 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 ],*

*[ 0.8%, 0.6%, 0.4%, 0.2%, 0%, −0.2%, −0.4% . . . ], etc.*

*– · · ·*

### Duration Revisited (concluded)

*• Let*

*P (y) ≡* X

*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*

*.*

*• Modified duration equals the above when*
*[ c*_{1}*, c*_{2}*, . . . , 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.

— 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}*≡* 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.^{a}

aPaul Wilmott (2009), “the correlations between financial quantities are notoriously unstable.”

### 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} *].* (19)

### 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

¸

*.* (20)

### The Multivariate Normal Distribution

*• 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.

*• Suppose*
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}* ]

for every linear combination P_{n}

*i=1* *t*_{i}*X** _{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 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^{a}

*c ≡* p

*−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

### 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}

Ã _{12}
X

*i=1*

*ξ*_{i}

!

*− 6.*

*• But why use 12?*

*• Recall the mean and variance of ξ*_{i}*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)

*• So the general formula is*
(P_{n}

*i=1* *ξ*_{i}*) − (n/2)*

p*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.*

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

*• Set*

*U* *≡ aX*_{1}*,*
*V* *≡ ρU +* p

*1 − ρ*^{2} *aX*_{2}*.*

### Generation of Bivariate Normal Distributions (concluded)

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

*Cov[ U, V ] = ρa*^{2}*.*

*• Note that the mapping between (X*_{1}*, X*_{2}*) 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, 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*

´
*,*

(21) 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?*^{a}

a*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.

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 − S** _{t}*)

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, 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}
– It is “covered” because the stock can be delivered to

the buyer of the call if the call is exercised.

*• 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 X*_{L}*, 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 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 (continued)

*• 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,

^{a}which pays $1 only when a particular price results.

^{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*

*∂*^{2}*C*

*∂X*^{2}*.*

*– Recall that C is the call’s price.*^{a}

*• In fact, the PV of ∂*^{2}*C/∂X*^{2} 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

*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).

### Portfolio Dominance Principle

*• Consider two portfolios A and B.*

*• A should be more valuable than B if A’s payoff is at*
least as good as B’s 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.