### 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.

*• 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*

*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}*∗ 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

*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.

### Two Simple Corollaries

*• 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.**

*• Two portfolios that yield the same return in every*
possible scenario must have the same price.^{a}

aAristotle, “those who are equal should have everything alike.”

### The PV Formula (p. 32) Justiﬁed

**Theorem 1 For a certain cash ﬂow C**_{1}*, C*_{2}*, . . . , C*_{n}*,*
*P =*

*n*
*i=1*

*C*_{i}*d(i).*

*• Suppose the price P*^{∗}*< P .*

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

*• The proceeds are P dollars.*

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}

### The Proof (concluded)

*• 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.

*• If P*^{∗}*> P , just reverse the trades.*

### Two More Examples

*• An American option cannot be worth less than the*
intrinsic value.^{a}

**– Suppose the opposite is true.**

**– So the American option is cheaper than its intrinsic**
value.

**– For the call: Short the stock and lend X dollars.**

**– For the put: Borrow X dollars and buy the stock.**

**– In either case, the payoﬀ is the intrinsic value.**

amax(0*, S**t* *− X) for the call and max(0, X − S**t*) for the put.

### Two More Examples (continued)

*• (continued)*

**– At the same time, buy the option, promptly exercise**
it, and close the stock position.

*∗ For the call, call the lent money to exercise it.*

*∗ For the put, deliver the stock and use the received*
strike price to settle the debt.

**– The cost of buying the option is less than the**
intrinsic value.

**– So there is an immediate arbitrage proﬁt.**

### Two More Examples (concluded)

*• A put or a call must have a nonnegative value.*

**– Suppose otherwise and the option has a negative**
price.

**– Buy the option for a positive cash ﬂow now.**

**– It will end up with a nonnegative amount at**
expiration.

**– So an arbitrage proﬁt is realized now.**

### 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;**

**– 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*

### 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.

*• The loan is now X.*

*• 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.*

*• The net future cash ﬂow is again zero:*

*−(S*_{τ}*− X) + 0 + S*_{τ}*− X = 0.*

### The Proof (concluded)

*• The net future cash ﬂow is zero in either case.*

*• The no-arbitrage principle (p. 196) 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): A stock is equivalent to a portfolio*
*containing a long call, a short put, and lending PV(X).*

*• C − P = S − PV(X): 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 (p. 202).*

*• It follows that C ≥ max(S − X, 0), the intrinsic value.*

*• An American call also cannot be worth less than its*

### Intrinsic Value (concluded)

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

**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. 173 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 S − X.*

*• But*

*max(S − PV(X), 0) ≥ 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, however.

**– 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.^{a}

**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.

aSee Theorem 8.4.2 of the textbook.

### A General Result

^{a}

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

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

**Corollary 7 Any suﬃciently well-behaved payoﬀ function***can be approximated by a portfolio of calls and puts.*

aSee Exercise 8.3.6 of the textbook.

### Convexity of Option Prices

^{a}

**Lemma 8 For three otherwise identical calls or puts with***strike prices X*1 *< X*2 *< X*3*,*

*C** _{X}*2

*≤ ωC*

*1 + (1*

_{X}*− ω) C*

*3*

_{X}*P** _{X}*2

*≤ ωP*

*1 + (1*

_{X}*− ω) P*

*3*

_{X}*Here*

*ω ≡ (X*3 *− X*2*)/(X*3 *− X*1*).*

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

aSee Lemma 8.5.1 of the textbook.

### The Intuition behind Lemma 8

^{a}

*• Set up the following portfolio:*

*ωC** _{X}*1

*− C*

*2 + (1*

_{X}*− ω) C*

*3*

_{X}*.*

*• This is a butterﬂy spread (p. 184).*

*• It has a nonnegative value as, for any S at maturity,*

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

*• Therefore,*

*ωC** _{X}*1

*− C*

*X*2 + (1

*− ω) C*

*X*3

*≥ 0.*

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

### 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.*

### Option on a Portfolio vs. Portfolio of Options (concluded)

An option on a portfolio is cheaper than a portfolio of
options.^{a}

**Theorem 9 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}*.*
*The same 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.^{a}

**– Known as the Black-Scholes 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** ^{ˆr}*: 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 will 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 1 if the stock price moves to

*Su.*

*• C** _{d}* is the call price at time 1 if the stock price moves to

*Sd.*

*• Clearly,*

*C** _{u}* =

*max(0, Su − X),*

*C*

*=*

_{d}*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*
*hSu + RB* *(up move),*

*hSd + RB* *(down move).*

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

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

*hSu + RB* = *C*_{u}*,*
*hSd + RB* = *C*_{d}*.*

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

*• Solve the above equations to obtain*
*h =* *C*_{u}*− C*_{d}

*Su − Sd* *≥ 0,* (23)

*B* = *uC*_{d}*− dC*_{u}

*(u − d) R* *.* (24)

*• By the no-arbitrage principle, the European call should*
cost the same as the equivalent portfolio,^{a}

*C = hS + B.*

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

### American Call Pricing in One Period

*• Have to consider immediate exercise.*

*• C = max(hS + B, S − X).*

* – When hS + B ≥ S − X, the call should not be*
exercised immediately.

* – When hS + B < S − X, the option should be*
exercised immediately.

*• For non-dividend-paying stocks, early exercise is not*
optimal by Theorem 4 (p. 210).

*• So*

*C = hS + B.*

### Put Pricing in One Period

*• Puts can be similarly priced.*

*• The delta for the put is (P*_{u}*− P*_{d}*)/(Su − Sd) ≤ 0, where*
*P** _{u}* =

*max(0, X − Su),*

*P** _{d}* =

*max(0, X − Sd).*

*• Let B =* ^{uP}_{(u−d) R}^{d}^{−dP}* ^{u}*.

*• The European put is worth hS + B.*

*• The American put is worth max(hS + B, X − S).*

**– Early exercise is always possible with American puts.**

### Risk

*• Surprisingly, the option value is independent of q.*^{a}

*• Hence it is independent of the expected gross return of*
*the stock, qSu + (1 − q) Sd.*

*• It therefore does not directly depend on investors’ risk*
preferences.

*• The option value depends on the sizes of price changes,*
*u and d, which the investors must agree upon.*

*• Then the set of possible stock prices is the same*
*whatever q is.*

aMore precisely, not directly dependent on *q. Thanks to a lively class*

### Pseudo Probability

*• After substitution and rearrangement,*

*hS + B =*

*R−d*
*u−d*

*C** _{u}* +

*u−R*
*u−d*

*C*_{d}

*R* *.*

*• Rewrite it as*

*hS + B =* *pC** _{u}* + (1

*− p) C*

_{d}*R* *,*

where

*p ≡* *R − d*

*u − d* *.* (25)

### Risk-Neutral Probability

*• The expected rate of return for the stock is equal to the*
riskless rate ˆ*r under p as*

*pSu + (1 − p) Sd = RS.*

*• The expected rates of return of all securities must be the*
riskless rate when investors are risk-neutral.

*• For this reason, p is called the risk-neutral probability.*

*• The value of an option is the expectation of its*

discounted future payoﬀ in a risk-neutral economy.

*• So the rate used for discounting the FV is the riskless*