### 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. 121.

**– 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 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. 136 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) ≡* ∑

*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)*

*.*

*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}*≡* √

*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}

∑

*i=1*

*a*_{i}*X** _{i}*
]

=

∑*n*
*i=1*

∑*n*
*j=1*

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

*• It becomes*

∑*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

*| I* *| I* *| I*

### 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*

*−∞*

*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

∑

*i*

*X*_{i}*∼ N*

(∑

*i*

*µ*_{i}*,*∑

*i*

*σ*_{i}^{2}
)

*.*

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

*• Suppose*

∑*n*
*i=1*

*t*_{i}*X*_{i}*∼ N*

∑^{n}

*i=1*

*t*_{i}*µ*_{i}*,*

∑*n*
*i=1*

∑*n*
*j=1*

*t*_{i}*t*_{j}*Cov[ X*_{i}*, X** _{j}* ]

for every linear combination ∑*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

*c* *≡* √

*−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^{a}

∑12
*i=1*

*ξ*_{i}*− 6.*

aJ¨ackel (2002), “this is not a highly accurate approximation and should only be used to establish ballpark estimates.”

### 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.*

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

*• Set*

*U* *≡ aX*^{1}*,*
*V* *≡ ρU +* √

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*)
*,*

(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}

*• 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.

aAlternatively, Arrow security.

### 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}.
**– In fact, the PV of ∂**^{2}*C/∂X*^{2} is the probability

density of the stock price at option’s maturity.^{a}

aBreeden and Litzenberger (1978).

### 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: 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

* Theorem 1 P =* ∑

*n*

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

*• Suppose the price P*^{∗}*< P .*

*• Short the zeros that match the security’s n cash ﬂows.*

*• The proceeds are P dollars.*

*• Then use P** ^{∗}* of the proceeds to buy the security.

*• 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.

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.

**– Suppose the opposite is true.**

**– Now, buy the option, promptly exercise it, and close**
the stock position.

**– The cost of buying the option is less than the payoﬀ,**
which is the intrinsic value.^{a}

**– So there is an arbitrage proﬁt.**

a*max(0, S*_{t}*− X) or max(0, X − S**t*).

### Two More Examples (concluded)

*• 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 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*

### Put-Call Parity

^{a}

*C = P + S* *− PV(X).* (22)

*• 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 X*
*and time to expiration, τ .*

*• The initial cash ﬂow is therefore*

*C* *− P − S + PV(X).*

### The Proof (continued)

*• At expiration, if the stock price S*^{τ}*≤ X, the put will be*
*worth X* *− S** ^{τ}* and the call will expire worthless.

*• 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* *− X) + 0 + S* *− X = 0.*

### The Proof (concluded)

*• 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.*

**– The other can be replicated from it in combination**
with 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 2 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 (p. 189).

### Intrinsic Value (concluded)

A European put on a non-dividend-paying stock may be worth less than its intrinsic value (p. 165).

**Lemma 3 For European puts, P***≥ max(PV(X) − S, 0).*

*• Prove it with the put-call parity.*

*• Can explain the right ﬁgure on p. 165 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 4 (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 5 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.

### A General Result

**Theorem 6 (Cox and Rubinstein (1985)) Any**

*piecewise linear payoﬀ function can be replicated using a*
*portfolio of calls and puts.*

### Convexity of Option Prices

**Lemma 7 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*

*3*

^{X}*P*_{X}_{2} *≤ ωP** ^{X}*1 + (1

*− ω) P*

*3*

^{X}*Here*

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

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

### The Intuition behind Lemma 7

^{a}

*• Consider ωC** ^{X}*1 + (1

*− ω) C*

*3*

^{X}*− C*

*2.*

^{X}*• This is a butterﬂy spread (p. 176).*

*• It has a nonnegative value as*

*ω max(S−X*^{1}*, 0)+(1−ω) max(S−X*^{3}*, 0)−max(S−X*^{2}*, 0)* *≥ 0.*

*• Therefore, ωC** ^{X}*1 + (1

*− ω) C*

*3*

^{X}*− C*

*2*

^{X}*≥ 0.*

*• In the limit, ∂*^{2}*C/∂X*^{2} *≥ 0, which has a ﬁnancial*
meaning.

aContributed by Mr. Cheng, Jen-Chieh (B96703032) on March 17, 2010.

### Option on a Portfolio vs. Portfolio of Options

*• 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}*.

*• All options expire on the same date.*

*• An option on a portfolio is cheaper than a portfolio of*
options.

**Theorem 8 The call on the portfolio with a strike price***X* *≡* ∑

*i* *ω*_{i}*X*_{i}*has a value at most* ∑

*i* *ω*_{i}*C*_{i}*.*