Finance Case Study

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3/31/20

Chapter 11

Properties of Stock Options

Properties of Stock Options - Goals

• Discuss the factors affecting option prices • Include the current stock price, strike price, time to maturity,

volatility of the stock price, risk-free interest rate, and paid-out dividends

• Identify the upper and lower bounds for European- and American-style option prices

• Introduce the put-call parity • The optimal early exercise decision • Consider the effect of dividend payments on

• Upper and lower bounds of option prices, the put-call parity, and the early exercise decision

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Factors Affecting Option Prices - Notation

𝑐: European call option price 𝐶: American call option price

𝑝: European put option price 𝑃: American put option price

𝑆!: Current stock price 𝑆": Stock price at option maturity

𝐾: Strike price 𝐷: Dividends that are expected to be paid during option’s life

𝑇: Life of option 𝑟: Risk-free rate for maturity T with continuously compounding

𝜎: Volatility of the stock price

Sensitivity Analysis on Option Prices

※ Note that the European call (put) value can be derived as • 𝑐 = 𝑒!"#𝐸[max(𝑆# − 𝐾,0)] (𝑝 = 𝑒!"#𝐸[max(𝐾 − 𝑆#,0)])

※ The American call (put) value can be derived as • 𝐶 = 𝐸[𝑒!"$max(𝑆$ − 𝐾,0)] (𝑃 = 𝐸[𝑒!"$max(𝐾 − 𝑆$,0)]),

• where 𝜏 is the time point to exercise American options

Factors 𝑐 𝑝 𝐶 𝑃 𝑆! + – + – 𝐾 – + – + 𝑇 ? ? + + 𝜎 + + + + 𝑟 + – + – 𝐷 – + – +

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Effect of Factors on Option Pricing • Current stock price 𝑆4 ↑

• For both European and American calls, prob. of being ITM (in-the-money) ↑ and thus call values ↑

• For both European and American puts, prob. of being ITM ↓( or probability of being OTM out-of-money ↑) and thus put values ↓

*𝐾 = 50,𝑟 = 5%,𝜎 = 30%,𝐷 = 0,and 𝑇 = 1

Effect of Factors on Option Pricing

• Strike price 𝐾 ↑ • For both European and American calls, prob. of being ITM ↓ and thus call values ↓ • For both European and American puts, prob. of being ITM ↑ and thus put values ↑

*𝑆% = 50,𝑟 = 5%,𝜎 = 30%,𝐷 = 0,and 𝑇 = 1

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Effect of Factors on Option Pricing

• Time to maturity 𝑇 ↑ • For American options, the holder of the long-life option has all the exercise

opportunities open to the holder of the short-life option–and more Þ The long- life American option must be worth as least as the short-life American option

• European calls and puts generally (not always) become more valuable as the time to expiration increases

*𝑆% = 50,𝐾 = 50,𝑟 = 5%,𝜎 = 30%,and 𝐷 = 0

Effect of Factors on Option Pricing

• For European calls, • Suppose two European call options, 𝑐& and 𝑐', on a stock with the same 𝐾 and with

different maturity 𝑇& and 𝑇' (> 𝑇&) • If there is a cash dividends paid in [𝑇&,𝑇'], the stock price declines on the dividend

payment date so that the short-life call 𝑐& could be worth more than the long-life call 𝑐'

• For deeply ITM European put options, short-life put 𝑝! (with 𝑇! time to maturity) could be worth more than the long-life put 𝑝" (with 𝑇" time to maturity) • Note that the put value can be derived as 𝑒!"#𝐸[max(𝐾 − 𝑆#,0)] • Consider an extreme case in which the stock price is close to 0 so that 𝑆# can be

almost ignored when calculating payoffs of puts • The option values of the above two put options are 𝑝& = 𝑒!"#!𝐸 𝐾 − 0 = 𝑒!"#!𝐾

and 𝑝' = 𝑒!"#"𝐸 𝐾 − 0 = 𝑒!"#"𝐾 ⇒ 𝑝& > 𝑝' (inverse relationship between put values and 𝑇)

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Effect of Factors on Option Pricing • Volatility 𝜎 ↑ (the chance that the stock will perform better or poorer increases)

• Recall: call options have limited downside risk (the most he can lose is the price of the option)à an increase in the volatility (𝜎 ↑) increases the probability of a price increase à option value ↑

• Recall: put options have limited downside risk risk (the most he can lose is the price of the option) à an increase in the volatility (𝜎 ↑) increases the probability of a price decrease à option value ↑

*𝑆% = 50,𝐾 = 50,𝑟 = 5%,𝐷 = 0,and 𝑇 = 1

Effect of Factors on Option Pricing • Risk-free rate 𝑟 ↑

• The expected return of the underlying asset ↑, and the discount rate ↑ such that the PV of future CFs ↓

• For calls, the option value ↑ because the higher expected 𝑆# and the higher prob. to be ITM dominate the effect of lower PVs

• For puts, option value ↓ due to the higher expected 𝑆#, the lower prob. to be ITM, and the effect of lower PVs

*𝑆% = 50,𝐾 = 50,𝜎 = 30%,𝐷 = 0,and 𝑇 = 1

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Effect of Factors on Option Pricing

• Dividend payment ↑ • Dividends have the effect of reducing the stock price on the ex-dividend date • For calls, prob. of being ITM ↓ and thus call values ↓ • For puts, prob. of being ITM ↑ and thus put values ↑

0 2 4 6 8 10

Call option price, c

Dividends, D 0

0 2 4 6 8 10

Put option price, p

Dividends, D

*𝑆% = 50,𝐾 = 50,𝑟 = 5%,𝜎 = 30%,and 𝑇 = 1

Upper and Lower Bounds for Option Prices

• Some assumptions

• There are no transactions costs

• The tax rate issue is ignored in this chapter • However, all results in this chapter hold when all trading profits (net of trading

losses) are subject to the same tax rate

• Borrowing and lending are always possible at the risk-free interest rate

• There is no dividends payment during the option life • In the last section of this chapter, this constraint will be released

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Upper and Lower Bounds for Option Prices • Upper bounds for the European and American call and put

• Since both American and European calls grant the holders the right to buy one share of a stock for a certain price, the option can never be worth more than the value of the stock share today

• An American put grants the holder the right to sell one share of a stock for 𝐾 at any time point, so the option value today can never be worth more than 𝑲

• For a European put, since its payoff at maturity cannot be worth more than 𝐾, it cannot be worth more than the PV of 𝑲 today

• An American option is worth at least as much as the corresponding European option, so 𝑐 ≤ 𝐶 and 𝑝 ≤ 𝑃

Upper bound for call Upper bound for put

American 𝐶 ≤ 𝑆% 𝑃 ≤ 𝐾 European 𝑐 ≤ 𝑆% (𝑐 ≤ 𝐶) 𝑝 ≤ 𝐾𝑒!"# (𝑝 ≤ 𝑃)

Upper and Lower Bounds for Option Prices

• Lower bounds for European calls and puts

• The lower bound for European calls • Portfolio A: one European call option plus a zero-coupon bond that provides a payoff of 𝐾 at

time 𝑇 • If 𝑆# > 𝐾 at 𝑇, the call is exercised, and one stock share is purchased with the principal of the

bond Þ Portfolio A is worth 𝑆# • If 𝑆# ≤ 𝐾 at 𝑇, the portfolio holder receives the repayment of the principal of the bond Þ

Portfolio A is worth 𝐾 ÞPortfolio A is worth max(𝑆#,𝐾) at 𝑇

• Portfolio B: one share of the stock Þ worth 𝑆# at 𝑇

※Portfolio A is worth more than Portfolio B Þ this should also be true in PV terms Þ 𝑐 + 𝐾𝑒!"# ≥ 𝑆% Þ 𝑐 ≥ 𝑆% − 𝐾𝑒!"# 𝑎𝑛𝑑 𝑐 ≥ 0 à 𝑐 ≥ max(𝑆% − 𝐾𝑒!"#, 0)

Lower bound for call Lower bound for put

European 𝑐 ≥ max(𝑆% − 𝐾𝑒!"#,0) 𝑝 ≥ max(𝐾𝑒!"# − 𝑆%,0)

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Proof that c> Max[0,S0 -PV(K)] • Obviously, c > 0 • Proof that c > S-PV(K) • What if c < S0 - PV(K)? • Then c – S0 + PV(K) < 0 • Then -c + S0 - PV(K) > 0 permits arbitrage, because cash is received today, and

there are no cash outflows at expiration.

______At Expiration______ • Today: ST > K ST < K

• Buy call -c +(ST – K) 0 • Sell stock + S0 - ST - ST • Lend -PV(K) + K + K

>0 0 -ST+K>0

Upper and Lower Bounds for Option Prices

• Is there any an arbitrage opportunity if 𝑐 = 3, 𝑆! = 20, 𝐾 = 18, 𝑟 = 10%, 𝐷 = 0, and 𝑇 = 1?

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Upper and Lower Bounds for Option Prices

• Is there any an arbitrage opportunity if 𝑐 = 3, 𝑆! = 20, 𝐾 = 18, 𝑟 = 10%, 𝐷 = 0, and 𝑇 = 1?

• Since the call price violates the lower bound constraint ($20 − $18𝑒!%.&)& = $3.71) , the following strategy can arbitrage from this distortion (c is too low)

• Buy the underestimated call and short one share of stock Þ Generate a cash inflow of $20 – $3 = $17

• Deposit $17 at 𝑟 = 10% for one year Þ Generate an income of $17𝑒&%%)& = $18.79 at the end of the year

• If 𝑆# > $18, exercise the call to purchase one share of stock at $18 and close out the short position Þ The net income is $18.79 – $18 = $0.79

• If 𝑆# < $18, give up the right of the call, purchase 1 share at 𝑆# in the market, and close out the short position Þ The net income is $18.79 – 𝑆#, which must be higher than $0.79

Upper and Lower Bounds for Option Prices

• The lower bound for European puts • Portfolio C: one European put option plus one share

• If 𝑆# ≤ 𝐾 at 𝑇, the put is exercised and sell the one share of stock owned for 𝐾 Þ Portfolio C is worth 𝐾

• If 𝑆# > 𝐾 at 𝑇, the put expires worthless Þ Portfolio C is worth 𝑆# ÞPortfolio C is worth max(𝑆#,𝐾) at 𝑇

• Portfolio D: an amount of cash equal to 𝐾𝑒@A" (or equivalently a zero- coupon bond with the payoff 𝐾 at time 𝑇)

• Portfolio C is more valuable than Portfolio D Þ 𝑝 + 𝑆! ≥ 𝐾𝑒@A" Þ 𝑝 ≥ 𝐾𝑒@A" − 𝑆! à 𝑝 ≥ max(𝐾𝑒@A" − 𝑆! , 0)

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Proof of the European Put Lower Bound What if: p < Ke-rT - S0 ?

Then, p- Ke-rT + S0 < 0

Or, -p+ Ke-rT - S0 >0

At expiration: Today ST>K ST<K

Buy put -p 0 +(K-ST)

Borrow + Ke-rT -K -K

Buy stock -S +ST +ST >0 >0 0

So, if Pp< Ke-rT - S, an arbitrage is possible, because the trader can receive a cash in-flow today, and not have to pay money in the future (in fact, in some cases, the trader receives money in the future, too.

Upper and Lower Bounds for Option Prices

• Is there any arbitrage opportunity if 𝑝 = 1, 𝑆I = 37, 𝐾 = 40, 𝑟 = 5%, 𝐷 = 0, and 𝑇 = 0.5?

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Upper and Lower Bounds for Option Prices

• Is there any arbitrage opportunity if 𝑝 = 1, 𝑆I = 37, 𝐾 = 40, 𝑟 = 5%, 𝐷 = 0, and 𝑇 = 0.5?

• Since the put price violates the lower bound constraint ($40𝑒@!.!CD!.C − $37 = $2.01) , the following strategy can arbitrage from this distortion (p too low)

• Borrow $38 at 𝑟 = 5% for 6 months Þ Need to pay off $38𝑒+%)%.+ = $38.96 after half a year

• Use the borrowing fund to buy the underestimated put and one share of stock • If 𝑆# > $40, discard the put, sell the stock for 𝑆#, and repay the loan Þ The net income is 𝑆# – $38.96 > 0

• If 𝑆# < $40, exercise the right of the put to sell the share of stock at $40 and repay the loan Þ The net income is $40 – $38.96 = $1.04

Summary

At expiration: Today ST>K ST<K Buy put -1 0 +(40-ST) Borrow $39.01 = 40𝑒MI.INOI.N -40 -40 Buy stock -37 +ST +ST

>0 >0 0

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Upper and Lower Bounds for Option Prices

• Lower bounds for American calls and puts

• The lower bounds for American calls and puts are their exercise value because the holders of them always can exercise them to obtain the current exercise value

• The American option is worth at least as much as zero because the option holder has only the right but no obligation to exercise the option

Lower bound for call Lower bound for put

American 𝐶 ≥ max(𝑆% − 𝐾,0) 𝑃 ≥ max(𝐾 − 𝑆%,0)

Put-Call Parity

• Consider Portfolios A and : • Portfolio A: 1 European call option plus a zero-coupon bond that provides

a payoff of 𝐾 at time 𝑇 • Portfolio C: 1 European put plus 1 share of the stock

Portfolio A 𝑺𝑻 > 𝑲 𝑺𝑻 ≤ 𝑲 Call option 𝑆$ − 𝐾 0 Zero-coupon bond 𝐾 𝐾 Total 𝑆$ 𝐾

Portfolio C 𝑺𝑻 > 𝑲 𝑺𝑻 ≤ 𝑲 Put option 0 𝐾 − 𝑆$ 1 share of stock 𝑆$ 𝑆$ Total 𝑆$ 𝐾

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Put-Call Parity

• Due to the law of one price, Portfolios A and C must therefore be worth the same today

𝑐 + 𝐾𝑒@A" = 𝑝 + 𝑆!

• The above equation is known as the put-call parity • The put-call parity defines a relationship between the prices of a European call and put

option, both of which are with the identical 𝐾 and 𝑇

• Is there any arbitrage opportunity if 𝑝 = 1 or 𝑝 = 2.25 given 𝑐 = 3, 𝑆! = 31, 𝐾 = 30, 𝑟 = 10%, 𝐷 = 0, and 𝑇 = 0.25?

• Write down the strategies.

Put-Call Parity

• Is there any arbitrage opportunity if 𝑝 = 1 or 𝑝 = 2.25 given 𝑐 = 3, 𝑆! = 31, 𝐾 = 30, 𝑟 = 10%, 𝐷 = 0, and 𝑇 = 0.25? • The theoretical price of the put option is 1.26 by solving 3 + 30𝑒!%.&)%.'+ = 𝑝 + 31 • The arbitrage strategies for 𝑝 = 2.25 and 𝑝 = 1 are shown in the following table

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Put-Call Parity

• Rewrite the put-call parity: 𝑐 + 𝐾𝑒MPQ = 𝑝 + 𝑆I ⇒ 𝑐 + 𝐾𝑒MPQ − 𝑆I = 𝑝, based on which it is simpler to identify the arbitrage opportunity

Three-month put price = $2.25 (p overvalued) (Long 𝐜 + 𝐊𝐞!𝐫𝐓 − 𝐒𝟎 and short 𝐩)

Three-month put price = $1 (Short 𝐜 + 𝐊𝐞!𝐫𝐓 − 𝐒𝟎 and long 𝐩)

Buy the call at $3, short the stock to realize $31, and short the put to realize $2.25 Þ Deposit the net cash flow $30.25 at 10% for 3 months

Short the call to realize $3, buy the stock at $31, buy put at $1, and borrow $29 at 10% for 3 months Þ The net cash flow is 0

If 𝑆# > 30 after 3 months: Receive $31.02 from the deposit, exercise the call to buy the stock at $30 Þ Net profit = $1.02

If 𝑆# > 30 after 3 months: The call is exercised and thus need to sell the stock for $30, and use $29.73 to repay loan Þ Net profit = $0.27

If 𝑆# < 30 after 3 months: Receive $31.02 from the deposit, the put is exercised and thus need to buy the stock at $30 Þ Net profit = $1.02

If 𝑆# < 30 after 3 months: Exercise the put to sell the stock for $30, and use $29.73 to repay loan Þ Net profit = $0.27

Put-Call Parity

• Extension of the put-call parity for the American call and put (exercise 18)

𝑆% − 𝐾 ≤ 𝐶 − 𝑃 ≤ 𝑆% − 𝐾𝑒&'(

• Identify the upper and lower bounds of 𝑃 given 𝐶 = 1.5, 𝑆! = 19, 𝐾 = 20, 𝑟 = 10%, 𝐷 = 0, and 𝑇 = 5/12

19 − 20 ≤ 1.5 − 𝑃 ≤ 19 − 20𝑒@!.LDC/LN ⇒ 1.68 ≤ 𝑃 ≤ 2.50

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Optimal Early Exercise Decision

• Usually there is some chance that an American option will be exercised early • The early exercise occurs when 𝐶 < exercise value, where 𝐶 reflects the PV of holding

all future exercise opportunities

• An exception is an American call on a non-dividend paying stock, which should never be exercised early

∵ 𝑐 ≥ 𝑆, − 𝐾𝑒-.# and 𝐶 ≥ 𝑐 (bounds) ∴ 𝐶 ≥ 𝑐 ≥ 𝑆% − 𝐾𝑒!"# > 𝑆% − 𝐾 if r>0 𝐶 > 𝑆% − 𝐾 (𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑣𝑎𝑙𝑢𝑒)

• This means that C is always greater than the option’s intrinsic value prior to maturity. If it were optimal to exercise at a particular time prior to maturity, C would equal the option’s intrinsic value at that time.

Þ It is not optimal to exercise American call option if there is no dividend payments

Early Exercise

• For a deeply ITM American call option: 𝐶 = 42, 𝑆I = 100, 𝐾 = 60, 𝑇 = 0.25, and 𝐷 = 0

• Should you exercise the call immediately if

1. You intend to hold the stock (after exercising the option) for the next 3 months? • No, it is better to delay paying the strike price 3 months later

2. You still want to hold the stock, but you do not feel that the stock is worth holding for the next 3 months? • No, it is possible to purchase the stock at a price lower than 𝐾 = 60 after 3 months

3. You decide to sell the stock share immediately after the exercise? • No, selling the American call for $42 is better than undertaking the above strategy,

which is with the payoff of $100 – $60 = $40

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Early Exercise

• A summary of reasons for not exercising an American call early if there are no dividends

• Due to no dividends, no income is sacrificed if you hold the American call instead of holding the underlying stock shares

• Payment of the strike price can be delayed (Q1 on previous slide)

• Holding the call provides the possibility that the purchasing price could be lower than but never higher than the strike price (Q2 on previous slide)

• The payoff from exercising the American call is lower than the payoff from selling the American call directly (Q3 on previous slide)

Early Exercise – American call options

• For an American option only the dividend value can negatively affect the value of the call option. • Call Value = Intrinsic Value + Interest Rate Value + Volatility Value - Dividend Value • If the underlying stock pays no dividend (or no dividend is to be paid prior to expiration of the option), a

call option can never be less than parity (intrinsic value). • However, if the negative effects of the dividend are greater than the positive effects of the other

components, it might be possible for the call, if it is European, to be less than parity (intrinsic value).

• When a stock pays a dividend, the value of the stock is diminished by the amount of that dividend. • Since the stockholder receives the value of the dividend, the two changes offset, such that there is no

net change of value for the stockholder. • On the other hand, when a stock pays a dividend, the option holder owns no right to the paid dividend. • The option value will decrease to represent the new intrinsic value as a result of the stock value decrease,

and the option holder will lose value on the option with no offsetting gain from the paid dividend.

• Then the only reason a trader would ever consider to exercise a call stock option early is to receive the dividend. • If the stock pays a dividend, the time a trader should consider early exercise is the day before the stock

goes ex-dividend.

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Early Exercise

• For a European put option the upper and lower bounds are: • max(𝑆, − 𝐾𝑒-.#) ≤ 𝑝 ≤ 𝐾𝑒-.#

• The lower bounds for American puts are their exercise value P ≥ max(𝐾 − 𝑆!) and P ≤ 𝐾

max(𝐾 − 𝑆!) ≤ 𝑃 ≤ 𝐾

• It can be optimal to exercise an American put option on a non-dividend-paying stock early

∵ 𝑝 ≥ 𝐾𝑒@A" − 𝑆! and 𝑃 ≥ 𝑝 ∴ 𝑃 ≥ 𝑝 ≥ 𝐾𝑒@A" − 𝑆!,

Þ For American puts, as long as their values are lower than max(𝐾 − 𝑆,,0), they are early exercised and the option value rises to become max(𝐾 − 𝑆,,0)

Early Exercise – American put options

• For a put option the only component that can negatively affect its price is the interest rate value. • Put Value = Intrinsic Value - Interest Rate Value + Volatility Value + Dividend Value

• Unlike the call option, the time a put option is a candidate for early exercise is anytime the interest which can be earned through the sale of the stock at the exercise price is considerably large.

• Determining the exact time at which this occurs is quite difficult. If the underlying stock pays a significant dividend it is most likely to occur on the day after the stock goes ex-dividend.

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Effects of Dividend Payments

• The no dividends assumption is unrealistic

• The underlying stocks of most exchange-traded stock options are issued by large firms

• Large firms usually pay dividends periodically (quarterly or annually)

• Denote 𝐷 to be the amount of dividend payment at time 𝑡 (𝑡 < 𝑇) and 𝐷! = 𝐷𝑒@AO to be the PV of the dividend payment • If there are multiple dividend payments during the life of the option, 𝐷% is the sum

of the PV of these dividend payments

Effects of Dividend Payments

• Similar to determining the forward (or future) price, 𝐷I should be deducted from the current stock price to derive the lower bounds and the put-call parity of options • The lower bounds for European calls and puts

𝑐 ≥ 𝑆! − 𝐷! − 𝐾𝑒@A" = 𝑆! − 𝐷! − 𝐾𝑒@A" 𝑝 ≥ 𝐾𝑒@A" − 𝑆! − 𝐷! = 𝐷! + 𝐾𝑒@A" − 𝑆!

• The put-call parity for European options 𝑐 + 𝐾𝑒@A" = 𝑝 + 𝑆! − 𝐷! ⇒ 𝑐 + 𝐷! + 𝐾𝑒@A" = 𝑝 + 𝑆!

• The put-call parity for American options (𝑆! − 𝐷!) − 𝐾 ≤ 𝐶 − 𝑃 ≤ 𝑆! − 𝐾𝑒@A"

(The only exception for the rule of replacing 𝑆, with 𝑆, −𝐷, is the upper bounds of the put-call parity for American options)

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Effects of Dividend Payments

• When dividends are expected, we can no longer assert that an American call option will not be exercised early

∵ 𝑐 ≥ 𝑆! − 𝐷! − 𝐾𝑒@A" and 𝐶 ≥ 𝑐 ∴ 𝐶 ≥ 𝑐 ≥ 𝑆! − 𝐷! − 𝐾𝑒@A", which is not necessarily larger than the exercise value, 𝑆! − 𝐾

• It is inclined to exercise an American call immediately prior to an ex-dividend date • In fact, it is never optimal to exercise a call at any other time points

(discussed in Appendix of Ch. 13)