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Payoﬀ, Mathematically Speaking

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(1)

Convenient Conventions

• C: call value.

• P : put value.

• X: strike price.

• S: stock price.

• D: dividend.

(2)

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.

(3)

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

(4)

Payoﬀ, Mathematically Speaking (continued)

• At any time t before the expiration date, we call max(0, St − X)

the intrinsic value of a call.

• At any time t before the expiration date, we call max(0, X − St)

the intrinsic value of a put.

(5)

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.

(6)

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

(7)

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.

(8)

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.

• Options are assumed to be unprotected.

(9)

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.

(10)

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.

(11)

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

(12)

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.

(13)

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.

(14)

• 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 XL, XM, and XH to denote the strike prices with

XL < XM < XH.

(15)

• A bull call spread consists of a long XL call and a short XH call with the same expiration date.

– The initial investment is CL − CH.

– The maximum proﬁt is (XH − XL) − (CL − CH).

∗ When both are exercised at expiration.

– The maximum loss is CL − CH.

∗ When neither is exercised at expiration.

– If we buy (XH − XL)−1 units of the bull call spread and XH − XL → 0, a (Heaviside) step function

emerges as the payoﬀ.

(16)

85 90 95 100 105 110Stock price Bull spread (call)

-4 -2 2 4 Profit

(17)

• Writing an XH put and buying an XL put with identical expiration date creates the bull put 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 XL call, long one XH call, and short two XM calls.

(18)

85 90 95 100 105 110Stock price Butterfly

-1 1 2 3 Profit

(19)

• A butterﬂy spread pays a positive amount at expiration only if the asset price falls between XL and XH.

• Take a position in (XM − XL)−1 units of the butterﬂy spread.

• When XH − XL → 0, it approximates a state contingent claim,a which pays \$1 only when the state S = XM

happens.b

aAlternatively, Arrow security.

bSee Exercise 7.4.5 of the textbook.

(20)

• The price of a state contingent claim is called a state price.

• The (undiscounted) state price equals

2C

∂X2 .

– Recall that C is the call’s price.a

• In fact, the PV of ∂2C/∂X2 is the probability density of the stock price ST = X at option’s maturity.b

aOne can also use the put (see Exercise 9.3.6 of the textbook).

bBreeden and Litzenberger (1978).

(21)

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.

(22)

85 90 95 100 105 110Stock price Straddle

-5 -2.5 2.5 5 7.5 10 Profit

(23)

Covered Position: Combination (concluded)

• Strangle: Identical to a straddle except that the call’s strike price is higher than the put’s.

(24)

85 90 95 100 105 110Stock price Strangle

-2 2 4 6 8 10 Profit

(25)

Arbitrage in Option Pricing

(26)

All general laws are attended with inconveniences, when applied to particular cases.

— David Hume (1711–1776)

(27)

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).

(28)

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.

(29)

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.”

(30)

The PV Formula (p. 32) Justiﬁed

Theorem 1 For a certain cash ﬂow C1, C2, . . . , Cn, P =

n i=1

Cid(i).

• Suppose the price P < P .

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

• The proceeds are P dollars.

(31)

6 6 6 6 -

C1 C2 C3

· · · Cn

? ? ? ?

C1 C2 C3

· · ·

Cn

6

P P?

 security

 zeros

(32)

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.

(33)

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, St − X) for the call and max(0, X − St) for the put.

(34)

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.

(35)

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.

(36)

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

(37)

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

(38)

The Proof (continued)

• At expiration, if the stock price Sτ ≤ X, the put will be worth X − Sτ and the call will expire worthless.

• The loan is now X.

• The net future cash ﬂow is zero:

0 + (X − Sτ) + Sτ − X = 0.

• On the other hand, if Sτ > X, the call will be worth Sτ − X and the put will expire worthless.

• The net future cash ﬂow is again zero:

−(Sτ − X) + 0 + Sτ − X = 0.

(39)

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.

(40)

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).

(41)

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

(42)

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.

(43)

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.

(44)

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.

(45)

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.

(46)

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.

(47)

Convexity of Option Prices

a

Lemma 8 For three otherwise identical calls or puts with strike prices X1 < X2 < X3,

CX2 ≤ ωCX1 + (1 − ω) CX3

PX2 ≤ ωPX1 + (1 − ω) PX3

Here

ω ≡ (X3 − X2)/(X3 − X1).

(Equivalently, X2 = ωX1 + (1 − ω) X3.)

aSee Lemma 8.5.1 of the textbook.

(48)

The Intuition behind Lemma 8

a

• Set up the following portfolio:

ωCX1 − CX2 + (1 − ω) CX3.

• This is a butterﬂy spread (p. 184).

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

ω max(S − X1, 0) − max(S − X2, 0) + (1 − ω) max(S − X3, 0) ≥ 0.

• Therefore,

ωCX1 − CX2 + (1 − ω) CX3 ≥ 0.

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

(49)

Option on a Portfolio vs. Portfolio of Options

• Consider a portfolio of non-dividend-paying assets with weights ωi.

• Let Ci denote the price of a European call on asset i with strike price Xi.

• All options expire on the same date.

(50)

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

ωiXi has a value at most 

i

ωiCi. The same holds for European puts.

(51)

Option Pricing Models

(52)

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)

(53)

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.

(54)

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 ≡ eˆr: gross return.

• Start from the discrete-time binomial model.

(55)

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.

(56)

(57)

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

• The expiration date is only one period from now.

• Cu is the call price at time 1 if the stock price moves to Su.

• Cd is the call price at time 1 if the stock price moves to Sd.

• Clearly,

Cu = max(0, Su − X), Cd = max(0, Sd − X).

(58)

C

Cu= max( 0, Su X ) q

1 q

Cd = max( 0, Sd X )

(59)

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).

(60)

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 = Cu, hSd + RB = Cd.

(61)

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

• Solve the above equations to obtain h = Cu − Cd

Su − Sd ≥ 0, (23)

B = uCd − dCu

(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 uCd − dCu < 0, the equivalent portfolio is a levered long position in stocks.

(62)

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.

(63)

Put Pricing in One Period

• Puts can be similarly priced.

• The delta for the put is (Pu − Pd)/(Su − Sd) ≤ 0, where Pu = max(0, X − Su),

Pd = max(0, X − Sd).

• Let B = uP(u−d) Rd−dPu.

• 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.

(64)

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

(65)

Pseudo Probability

• After substitution and rearrangement,

hS + B =

R−d u−d



Cu +

u−R u−d

 Cd

R .

• Rewrite it as

hS + B = pCu + (1 − p) Cd

R ,

where

p ≡ R − d

u − d . (25)

(66)

Risk-Neutral Probability

• The expected rate of return for the stock is equal to the riskless rate ˆr 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

• 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

• It works as if the call writer delivered a futures contract to the option holder and paid the holder the prevailing futures price minus the strike price.. • It works as if the

• Use the Black-Scholes formula with the stock price reduced by the PV of the dividends.. • This essentially decomposes the stock price into a riskless one paying known dividends and

 If I buy a call option from you, I am paying you a certain amount of money in return for the right to force you to sell me a share of the stock, if I want it, at the strike price,

• ‘ content teachers need to support support the learning of those parts of language knowledge that students are missing and that may be preventing them mastering the

 Promote project learning, mathematical modeling, and problem-based learning to strengthen the ability to integrate and apply knowledge and skills, and make. calculated

This kind of algorithm has also been a powerful tool for solving many other optimization problems, including symmetric cone complementarity problems [15, 16, 20–22], symmetric