### 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.**

### 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.

### Time Line

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

Time 0

-^{S(1)}

-^{S(2)}

-^{S(3)}

-^{S(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.

### 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).*

spot rate curve forward rate curve yield curve

(a)

spot rate curve forward rate curve yield curve

(b)

### 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) ].*

### 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)

### 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.

### 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}

### 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.

### 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* *.*

### 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.**

### 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.

### 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.

### 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.

### A “Bad” Expectations Theory

*• The expected returns*^{a} 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.

### 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).*

### 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.

### 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 in Practice

*• 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 ],**

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

**– . . . .**

*• At least one c** _{i}* should be 1 as the reference point.

### Duration in Practice (concluded)

*• Let*

*P (y)* *≡*

*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*

or *−* *P (Δy)* *− P (−Δy)*
*2P (0)Δy* *.*

*• 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.

— 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)

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

### 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
*E[ X* *| I*1 *] = E[ E[ X* *| I*2 ]*| I*1 *].* (23)

### 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 following 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 deﬁned as

*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

*.* (24)

### 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^{a}

*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.

### 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.*

*• 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)

*• The general formula is*
(_{n}

*i=1**ξ** _{i}*)

*− (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}

*• 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 as follows.*

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

*• Set*

*U* *≡ aX*_{1}*,*
*V* *≡ ρU +*

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}

*e*^{σ}^{2} *− 1*
*,*

(25) respectively.

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

### The Lognormal Distribution (continued)

*• Conversely, suppose Y is lognormally distributed with*
*mean μ and variance σ*^{2}.

*• 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 ] e**Cov[ ln X,ln Y ]**,*

*Cov[ X, Y ]* = *E[ X ] E[ Y ]*

*e**Cov[ ln X,ln Y ]* *− 1*
*.*

### The Lognormal Distribution (concluded)

*• Let Y be lognormally distributed such that*
*ln Y* *∼ N(μ, σ*^{2}).

*• Then*

_{∞}

*a*

*yf (y) dy = e*^{μ+σ}^{2}^{/2}*N*

*μ* *− ln a*

*σ* *+ σ*

*.*

*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 unit of the*
underlying asset by paying a strike price.

- 6

? ?

option primium

stock

strike price

### 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

### 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.*

### 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.

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

### 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, 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.*

*• Exchange-traded stock options are adjusted for stock*
dividends.

*• We assume options are 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.

*• 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.

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** _{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*).

*∗ When both are exercised at expiration.*

**– The maximum loss is C***L* *− C**H*.

*∗ When neither is exercised at expiration.*

**– If we buy (X**_{H}*− X** _{L}*)

*units of the bull call spread*

^{−1}*and X*

_{H}*− X*

_{L}*→ 0, a (Heaviside) step function*

emerges as the payoﬀ.

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 a positive amount at expiration*
*only if the asset price falls between X*_{L}*and X** _{H}*.

*• Take a position in (X*_{M}*− X** _{L}*)

*units of the butterﬂy spread.*

^{−1}*• When X*_{H}*− X*_{L}*→ 0, it approximates a state contingent*
claim,^{a} *which pays $1 only when the state S = X*_{M}

happens.^{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