Variance of Sum

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.

Moments

• The variance of a random variable X is defined as Var[ X ] = E^Δ 

(X − E[ X ])²  .

• The covariance between random variables X and Y is Cov[ X, Y ] = E [ (X − μ^Δ _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 =^Δ 

Var[ X ] .

• The correlation (or correlation coefficient) between X and Y is

ρ_X,Y =^Δ Cov[ X, Y ] σ_Xσ_Y ,

provided both have nonzero standard deviations.^a

aWilmott (2009), “the correlations between financial quantities are notoriously unstable.” It may even break down “at high-frequency time

Variance of Sum

• Variance of a weighted sum of random variables equals Var

 _n



i=1

a_iX_i



=

n i=1

n j=1

a_ia_j Cov[ X_i, X_j ].

• It becomes

n i=1

a²_i Var[ X_i ] when X_i are uncorrelated.^a

aBienaym´e (1853).

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^a says E[ X ] = E[ E[ X | I ] ].

Conditional Expectation (concluded)

• If I₂ contains at least as much information as I₁, then E[ X | I₁ ] = E[ E[ X | I₂ ]| I₁ ]. (26) – I1 contains price information up to time t1, and I2

contains price information up to a later time t₂ > t₁.

• In general,

I₁ ⊆ I₂ ⊆ · · ·

means the players never forget past data so the information sets are increasing over time.^a

aHirsa & Neftci (2014). This idea is used in sigma fields and filtration

The Normal Distribution

• A random variable X has the normal distribution with mean μ and variance σ² if its probability density

function is

1 σ√

2π e^{−(x−μ)2/(2σ2)}.

• This is expressed by X ∼ N(μ, σ²).

• The standard normal distribution has zero mean, unit variance, and the following distribution function

Prob[ X ≤ z ] = N (z) =^Δ 1

√2π

 _z

e^−x2/2 dx.

Moment Generating Function

• The moment generating function of random variable X is defined as

θ_X(t) = E[ e^{Δ tX} ].

• The moment generating function of X ∼ N(μ, σ²) is θ_X(t) = exp

μt + σ²t² 2

. (27)

The Multivariate Normal Distribution

• If X_i ∼ N(μ_i, σ_i²) are independent, then



i

X_i ∼ N



i

μ_i,

i

σ_i² 

.

• Let X_i ∼ N(μ_i, σ_i²), which may not be independent.

• Suppose

n i=1

t_iX_i ∼ N

⎛

⎝ⁿ

i=1

t_i μ_i,

n i=1

n j=1

t_it_j Cov[ X_i, X_j ]

⎞

⎠

for every linear combination _n

t X with

The Multivariate Normal Distribution (concluded)

• Then X_i are said to have a multivariate normal distribution.^a

• With M ≡ C⁻¹ and the (i, j)th entry of the matrix M being M_i,j, the probability density function for the X_i is

 1

(2π)ⁿdet(C) exp

⎡

⎣ −1 2

n i=1

n j=1

(X_i − μ_i) M_ij (X_j − μ_j)

⎤

⎦ ,

with a positive-definite covariance matrix C = [ Cov[ X^Δ _i, X_j ] ]_1≤i,j≤n.

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 x₁ and x₂ from X until ω = (2x^Δ ₁ − 1)² + (2x₂ − 1)² < 1.

• Then c(2x₁ − 1) and c(2x₂ − 1) are independent standard normal variables where^a

c =^Δ 

−2(ln ω)/ω .

aAs they are normally distributed, to prove independence, it suffices

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.

a

A Dirty Trick and a Right Attitude (concluded)

• The general formula is (_n

i=1ξ_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

Generation of Bivariate Normal Distributions

• Pairs of normally distributed variables with correlation ρ can be generated as follows.

• Let X₁ and X₂ be independent standard normal variables.

• Set

U =^Δ aX1,

V =^Δ aρX1 + a

1 − ρ² X2.

Generation of Bivariate Normal Distributions (continued)

• U and V are the desired random variables with Var[ U ] = Var[ V ] = a²,

Cov[ U, V ] = ρa².

• Note that the mapping from (X₁, X₂) to (U, V ) is a one-to-one correspondence for a = 0.

Generation of Bivariate Normal Distributions (concluded)

• The mapping in matrix form is

⎡

⎣ U V

⎤

⎦ = a

⎡

⎣ 1 0

ρ 

1 − ρ²

⎤

⎦

⎡

⎣ X¹ X2

⎤

⎦ . (28)

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, σ²) and Y = e^{Δ X}.

• The mean and variance of Y are μ_Y = e^μ+σ2/2, σ_Y² = e^2μ+σ2



e^σ2 − 1

, (29)

respectively.^a

a n nμ+n²σ²/2

The Lognormal Distribution (continued)

• Conversely, suppose Y is lognormally distributed with mean μ and variance σ².

• Then ln Y has a normal distribution with 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 ] eCov[ ln X,ln Y ],

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



eCov[ ln X,ln Y ] − 1 .

The Lognormal Distribution (concluded)

• Let Y be lognormally distributed such that ln Y ∼ N (μ, σ²).

• Then

 _∞

a

yf (y) dy = e^μ+σ2/2 N

μ − ln a

σ + σ



. (30)

Option Basics

The shift toward options as the center of gravity of finance [ . . . ]

— Merton H. Miller (1923–2000) Too many potential physicists and engineers spend their careers shifting money around in the financial sector, instead of applying their talents to innovating in the real economy.

— Barack Obama (2016)

Calls and Puts

• A call gives its holder the right to buy a unit of the underlying asset by paying a strike price.^a

- 6

? ?

option premium

stock

strike price

aThe cash flow at expiration is contingent.

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 premium

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.

• Some options can be exercised prior to the expiration date.

– This is called 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.^a

• D: dividend.

aAssume S ≥ 0. Contributed by Mr. Tang, Bert (B08902102) on March 10, 2021.

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.

Long a put

10 20 30 40 50 Payoff

20 40 60 80 Price

Short a put

-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.

aAbout 11% of option holders let in-the-money options expire worth-

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 option 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^a 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.

aOr shorting.

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

Short Selling (concluded)

• Not all assets can be shorted.

• In reality, short selling is not simply the opposite of going long.^a

aKosowski & Neftci (2015). See

https://tw.news.appledaily.com/headline/daily/20180307/37950481/

for an example in Taiwan on February 6, 2018.

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 above a certain level, so they are bullish.

aA short position has a payoff opposite in sign to that of a long

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 payoff is nonnegative.

– The maximum payoff is X_H − X_L.

∗ When both are exercised at expiration.

– The maximum profit is (X_H − X_L) − (C_L − C_H).

– The maximum loss is C_L − CH.

∗ When neither is exercised at expiration.

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

-4 -2 2 4 Profit

Covered Position: Spread (continued)

• If we buy (X_H − X_L)⁻¹ units of the bull call spread and X_H − X_L → 0, a (Heaviside) step function emerges as the payoff.

• This payoff defines the binary (or digital) call.

• The binary call thus costs

− ∂C

∂X today.

– Recall that C is the (standard) call’s price.

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

• A bear spread amounts to selling a bull spread.

• It profits from declining stock prices.

aSee https://www.businesstoday.com.tw/article/category/80392/post/201803070 for a sad example in Taiwan on February 6, 2018.

Covered Position: Spread (continued)

• 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_M calls.

• Same as long a bull call spread with strike prices X_L and X_M and short a bull call spread with strike prices X_M and X_H.

• A butterfly spread has a positive payoff at expiration only if the asset price falls between X_L and X_H.

85 90 95 100 105 110Stock price Butterfly

-1 1 2 3 Profit

Covered Position: Spread (continued)

• Assume X_M = (X_H + X_L)/2.

• Take a position in (X_M − X_L)⁻¹ units of the butterfly spread.

• When X_H − X_L → 0, it approximates a state contingent claim,^a which pays $1 only in the state S = X_M.^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 state price equals^a

∂²C

∂X² .

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

• You can buy a butterfly spread if you believe the

probability of S_T ≈ X is higher than this probability.

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 “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) The problem with QE is it works in practice, but it doesn’t work in theory.

— Ben Bernanke (2014)

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).^a

aForbes (2013), “In the real world of investments, however, there are obvious arguments against the EMH [efficient market hypothesis]. There are investors who have beaten the market—Warren Buffett.”

Portfolio Dominance Principle

• Consider two portfolios A and B.

• Suppose A’s payoff is at least as good as B’s under all circumstances and better under some.

• Then A should be more valuable than B.

Two Simple Corollaries

• A portfolio yielding a zero return in every possible scenario must have a zero PV.^a

– 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.^b

a“We have incurred net losses each year since our inception and we may not be able to achieve or maintain profitability in the future.” (Lyft, Inc, 2019).

The PV Formula (p. 41) Justified

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

n i=1

C_id(i).

• Suppose the price P^∗ < P .

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

• The proceeds are P dollars.

6 6 6 6 -

C1 C2 C3

· · · Cn

? ? ? ?

C1 C2 C3

· · ·

Cn

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.

One More Example

Theorem 2 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^a or margin requirements, borrowing and lending are available at the riskless interest rate,

interest rates are nonnegative, and there are no arbitrage opportunities.

aSchwab cut the fees of online trades of stocks and ETFs to zero on October 7, 2019.

Relative Option Prices (concluded)

• 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 expiration.

Put-Call Parity

C = P + S − PV(X). (31)

• 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:

The Proof (concluded)

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

• The no-arbitrage principle^a implies that the initial investment to set up the portfolio must be nil as well.

aRecall p. 221.

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.^a

• 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 3 An American call or a European call on a non-dividend-paying stock is never worth less than its intrinsic value.

• An American call cannot be worth less than its intrinsic value.^a

• For European options, the put-call parity implies C = (S − X) + (X − PV(X)) + P ≥ S − X.

• Recall C ≥ 0 (p. 225).

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

Intrinsic Value (concluded)

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

Lemma 4 For European puts, P ≥ max(PV(X) − S, 0).

• Prove it with the put-call parity.^a

• Can explain the right figure on p. 195 why P < X − S when S is small.

aSee Lemma 8.3.2 of the textbook.

Early Exercise of American Calls

European calls and American calls are identical when the underlying stock pays no dividends!

Theorem 5 (Merton, 1973) An American call on a non-dividend-paying stock should not be exercised before expiration.

• By Exercise 8.3.2 of the text, C ≥ max(S − PV(X), 0).

• If the call is exercised, the value is S − X.

• But

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.

– Options are assumed to be unprotected.

– 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 6 (Merton, 1973) 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

Theorem 7 (Cox & Rubinstein, 1985) Any piecewise linear payoff function can be replicated using a portfolio of calls and puts.

Corollary 8 Any sufficiently well-behaved payoff function can be approximated by a portfolio of calls and puts.

Theorem 9 (Bakshi & Madan, 2000) Any payoff function with bounded expectation can be replicated by a continuum of out-of-the-money European calls and puts.

Option Pricing Models

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 (2004), My Life as a Quant So we would bring in smart folks.

They didn’t know anything about finance.^a James Simons (2015, May 13, 33:27)

ahttps://www.youtube.com/watch?v=QNznD9hMEh0

The Setting

• The no-arbitrage principle is insufficient to pin down the exact option value.

• Need a model of probabilistic behavior of stock prices.

• An obstacle is that it seems a risk-adjusted interest rate is needed to discount the option’s expected payoff.^a

• Breakthrough came in 1973 when Black (1938–1995) and Scholes with help from Merton published their celebrated option pricing model.^b

– Known as the Black-Scholes option pricing model.

aLike Eq. (30) on p. 182.

Fischer Black (1938–1995)

Myron Scholes (1941–)

Robert C. Merton (1944–)

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^{Δ 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.^a

• 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.

aSee Exercise 9.2.1 of the textbook. The sufficient condition is d <

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, (32)

B = uC_d − dC_u

(u − d) R . (33)

• 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 5 (p. 234).

• So

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−dPu.

• The European put is worth hS + B.

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

– Early exercise is possible with American puts.