Option Tutorial Answers超级完整版

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School of Commerce

Optionsr, Futures and Risk Management-

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Derivatives

Answers to Class Activities

Notes to Tutors:

A lot of work has gone into these questions.

1)

Under NO conditions are provided to the students. Exceptions to this photocopies of the answers to be case include material taken from websites. Answers are to be discussed in the tutorials only. Students who do not attend the tutorials do not deserve to receive the answers. If you think you are not going to be able to go through all the exercises, then you can make your own notes and distribute that in class.

2)

Some questions can

be

attempted using

both

compounding/discounting. Either method is acceptable students will be told the method to apply for each question.

discrete/continuous although for exams

you run out of questions to discuss in the tutorials, you should refer to the supplementary questions & answers that students have been provided with in their readers. Students may wish to go through in class some of these questions as well.

3)

If

4)

If you are unsure of how to answer a question,

please

try and

see me at least a

few days before you tutor so you are prepared. s)

If

you notice any mistakes in the class activities (including typos, etc.) please highlight them and let me know. This will be much appreciated.

6) Have fun!

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Page2

Derivatives

Topic l Activities

1.

Check out the ASX website for up-to-date details on the options market in Australia. Acquire information on the following:

i)

What is the contract size for its share options derivatives? 1,000 shares

ii)

What is the contract expiry date for index options? 3'd Thursday of the month

iii)

What is the contract expiry date for share options? Thursday of the last business Friday of the month

iv) What is the longest expiry length for an option? 3 years

v)

What is an index multiplier? What is the index multiplier for the ASX200 index option? $10 is the index multiplier. The multiplier indicates the $ value of each index point

Provide copies of the ASX website copies included in this solutions set U students have not bothered to check the website, give them the copies and allow them to read itfor 10 minutes. Then get them to answer the questions.

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Page 3

Derivatives

From the ASX website

Index options lndex options give you exposure to the securities comprising a sharemarket index. They offer you similar flexibility to that provided by options over individual or on the stocks, while allowing you to trade a view on the market as a whole,

market sector covered by the particular index' the whereas the value of a share option varies according to movements in value of the underlying shares, an index option varies according to

movements in the underlying index.

Underlying asset

ASX approved indices (currently the S&P/ASX 200 lndex, S&P/ASX 50 Index, S&P ASX 200 Property Trusts Index)

Exercise style

European cash settled with reference to the OPIC (see below)

Expiry day

third Thursday of the month, unless otherwise specified by ASX.

Last trading day

Trading wili cease at 12 noon on expiry Thursday. This means trading will continue after the settlement price has been determined.

e

expressed in points

Strike pnce

expressed in points

Index multiplier

a specified number of dollars per point e'g'

Contract value

ihe exercise price of the option multiplied by the index multiplier

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AUD

Page 4

Derivatives

Some of the differences between index options and options over securities are:

. . .

index options are cash settled. index options are European in exercise style. This means the holder can only exercise an index option on the expiry day. the strike price and premium of an index option are expressed in points. A multiplier is then applied to give a dollar figure.

lndex options are cash settled. The settlement amount is based on the opening prices of the stocks in the

underlying index on the morning of the maturity date. As the stocks in the relevant index open, the first traded price of each stock is recorded. Once all stocks in the index have opened, an index calculation (the opening Price Index Calculation (OPIC)) is made using these opening prices.

Options are currently available over the following ASX indices:

. . .

S&PTM/ASX 200TM Index - code XJO S&PTM/ASX 200TM Property Trusts Index - code XPJ S&PTM/ASX 50TM Index - code XFL

Benefits of index options:

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Page 5

Derivatives

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lndex options

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lndex(

The ability to trade all the stocks in an index with

just one trade - investing in index options approximates trading a share poftolio that tracks thai pariicular index. By using options over an index, you can trade a view on the general direction of the market with just one trade. For example, if you are bullish on the market, you could buy a call option over an index. This gives you exposur€ to the broader market which the index represents, withoui having to choose a particular stock.

Leverage - index options, like ordinary options, provide leveraged profit opportunities. When the market rises (or falls), percentage gains {or losses) are greater than rises (or falls) in the underlying index.

Protection for a share portfolio - when you buy shares you afe exposed to two types of risk: company risk - the ri$k that the specific companies you have bought into will underperform. market risk - the risk that the whole market underperforms, including your shares. You can protect your shares against market risk by buying an index put option. lf your bearish market view proves correct, the proflts on your put option will at least partly compensate you for the loss of value in the stocks in your portfolio.

Low trading 6osts - since the amount of capital outlaid in an option trade is usually much lower than that involved in a share transaction providing similar market exposure, brokerage costs are often lower in option trades. For more information on index options, please refer to the ASX explanatory booklet Understandinq Options Tfadinq. Find out more about underlyinq indices. Find out more about index strateqies. Download the lndex Options brochure 157K

'pdf documents require Adobe Acrobat Reader ASX200TM is a trade mark of ASX Operations Pty Ltd, ABN 42 004 523 782, a member of the Australian Stock Exchange Group of companies. S&PTM is a trade mark of Slandard & Poor's, a division of The h4cGraw-Hill Comoanies, Inc.

Terms of use I Pnvacy Statement I Last reviewed: 31107142 @ Austraiian Stock Exchange Limited ABN 98 008 624 691

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Page 6

Derivatives

)ption feaiures

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In order to make it easier for investors to trade into and out of options positions, three features of option contracts have been standardised. They are:

the underlying securities the expiry date lhe exercise price (or strike price) These components are found in all options traded on ASX. They may change if, during the life of the option, an adjustment is required to be made in accordance with ASX Business Rules (for example as a result of a new issue or a reorganisation of capital in the underlying share).

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The only other variable is the premium, which is the

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market price of the option.

Underlying securities Options traded on ASX's derivatives market are available for certain securities. These securities mav be ihe shares of approved ASX listed companies. or a' share price index.

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The companies over which options are listed are selected by OCH in accordance with ASX Business Rules.

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An option contract size is standardised at 1,000 underlying securities. This means that one option contract represents 1,000 underlying securities, unless an adjustment has taken place.

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In the case of index options, the contract value is fixed at a certain number of dollars per index point, (for example,

i-

AUD $10 per index point). The size of the contract is

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equal to the index level multlplied by the dollar value per index point.

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Exercise Price

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This is the price at which you may buy or sell the underlying securities if you exercise your option.

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OCH lists a range of exercise prices for all options listed on ASX's derivatives market. Usually there is a range of

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exercise prices available for options with a given expiry day. New exercise prices are listed as the underlying share price moves. Typically, the range of exercise prices includes one exercise price close to the cunent market price of the underlying share with two exercise prices above and two exercise prices below the current share price. Exercise prices may also be adjusted during the life of the option in accordance with ASX Business Rules.

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Derivatives

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The Expiry Day is the day on which all.unexercised ootions in a particular seiies expire, lt is the last day of traOing for that particular sefies. For shares ihis is usuatf the Thuisday before the last business Friday in the month. For index options, the expiry date is the third Friday of the contraci month providing this is a trading day' The last trading date wilf be the trading day prior to the expiry date. OCH lists options over underlying securities in various expiry cycles that usually extend for a period of nine to tw;lv-e months. As one series expires a new, more distant one is created. ln addition to quarterly expiry cycles, a current or spot month is availible for most classes of options' These optiont that expire at the end of the current month "iu dt. used to trade short term price changes in the "nO shares. underlying

There are also longer term option contracts listed over certain classes, with terms of up to three years' For details of expiry cycles and expiry dates, refer to the Expirv Calendar.

Premium The premium is the price of the option' lt is not set by oCil, nut is determined by market forces' The premiums for share options are quoted on a cents o"idnut" basis. To calculate the full premium payable ioi a sianoiro size option contract, multiply the quoted pi"*iu* by 1,000' the number of shares per contract' To The oremiums for index options are quoted in points' the full premium payable for an index option' "Jt.ut"t* multiplv the premium by the index multipller' hor a irremium oi 30 points, with an index multiplier "*u*p1",dt0, reptes"nts a total premium cost of AUD

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$300 Per contract.

.iL, 07./08-/99 IoF oF Terms of use I Privacv Ftatement I Last reviewed: ABN 98 008 624 691 O Australian Stocx

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Page 8

Derivatives

Are index options on the ASX

2.

considered to be European or American options? What does this imply? ASX Index options are European. This indicates you can only exercise them at the expiry date.

J.

what is the benefit_ of trading index options? what benefits, if any, do index options have over share options? - Leverage, protection of a share portfolio, low trading costs, can trade all stocks of an index with one trade

4.

what is the difference between holding a short cail and a long put position? D^oes one of these positions result in anlbtgation on trr" pu.t of the holder? rf you have taken a SIToRT position, you are obligatei to exchange either goods/money if the taker exercises the option. noi a short call oi shares, this would indicate you needing to eichange shares in return for a specified amount of cash. The holder of the long put can chose to either exchange his/her shares for cash. A long poritioo may not entail an obligation but it does cost up-front money in the form of a-premiam.

5.

What is an efficient market and why is it important for derivatives markets? Without an efficient market, derivative instruments cannot be correcly priced.

6,

what types of transaction costs Broker commission rates Bid-ask spread Clearing house fee

are you faced

with when trading options?

Exchange fee 7.

If you wanted to physically trade an option here in Australia how would you go about it? Work out which brokers and what transaction costs they would charse.

A question like this may well appear in the exam and so its worth students researching this question. Usually brokers charge a fixed fee -$50 to cover all costs. 8.

what is the main taxation ruling that governs options taxation processes?

There is no specific taxation ruring that appties to options. rlowevero ITAA1997 covers general taxation matters relating to options. 9.

How is the treatment of the options premium different for a writer and a taker of an option? Is there any difference between calls and puts in this regard?

Writers receive the premium and therefore it is treated as assessable for tax purposes' For takers it is a cost and therefore deductible. There is no difference between puts and calls in this matter.

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Page 9

Derivatives

Topic 2 Activities

1.

Get hold of a recent copy of fr'rc Financial Review. Chose at random a stock that offers both a call and put option on it. a) Work out whether the lower and upper boundary copditions of the call and put options are being maintained. b) Determine whether Put-Call parity holds. If it doesn't, explain possible reasons for it not holding. Students to discuss in class. At least on student should demonstrate his/her calculations. There should be no arbitrage opportunities. If there are it could be because no account was made for: - dividend payments - incorrect risk-free rate used - bid/ask spread very wide (allow for deviation from the 'fair value' by 57 cents either way.

2.

A friend of yours tells you she holds

3. :

A friend of yours tells you he holds an American put option that is very deepin-the-money. He tells you he's going to exercise it now and cash in on his winnings. Is this a wise move to make? Possibly, as the limit a stock price can drop to is $0, if it is close to this you may as well cash in now, given that there is always a chance a white knight rescues the company.

4.

You notice two call options on the market that are costing the same. However, the expiration dates are different. If everything else is the same, how can this

an American call option that is very deep-in-the-money. She tells you she's going to exercise it now and cash in on her winnings. Is this a wise move to make? No, she should sell it. Sell now you get S-PV(X), rather than exercise now which leads to S-X.

be the case?

If

the calls are very deep out of the money, time value of the options be small and option value essentially zero.

5.

will

Your friend is about to offer an American call option on the market. However, she decides to be a little different and set no maturity date for the option. What would be the maximum and minimum value you would pay for this option?

The call becomes a stock!

6.

Will an option's time value be gteatest when the stock price is

near the

exercise price, when the option is deep-in-the-money, or deep out-of-themoney?

At-the-money. This is when there is most uncertainty about the final value of the option at expiration.

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Page 10

Derivatives

1

The following diagram shows the value of a put option at expiration:

,

Option

Exercise price of both Options

Value

-4

76

80

Stock Price ($)

Ignoring transaction costs, which of the following statements about the value of the put option at expiration is TRUE? A: The value of the short position in the put is $4 if the stock price is $76. B: The value of the long position in the put is -$4 if the stock price is $76. c: The long put has value when the stock price is below the $80 exercise prrce. D: The value of the short position in the put is zero for stock prices equalling or exceeding $76. Source: 2002CFALevel

1 samole exam Previous Exam Question

Which of the following statements about the value of a call option at expiration is FALSE? A: The short position in the same call option can result in a loss if the stock price exceeds the exercise price. B: The value of the long position equals zero or the stock price minus the exercise price, whichever is higher. C: The value of the long position equals zero or the exercise price minus the stock price, whichever is higher. D: The short position in the same call option has a zero value for all stock prices equal to or less than the exercise price.

8.

S

ource

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Is it possible to have two calls (or puts) similar in all respects, except the

9.

exercise price, having the same market price?

lf

both options were deep out-of-the-money, they might have prices of zero. As in the previous question, tr,r..o options arc expectecl to expire out-of-the-moniy.

the

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Page 11

Derivatives

10.

-

having a value less than Max [0, So X(1+r)t1, identify the transactions you should execute to create an arbitrage profit.

If you observed a European call option

The call is underpriced, so buy the cal1, seli short the stock" and buy risk-free bonds with face value oi'X Tire cash received from the stock is greater than the cost of the call and bonds. Thus, there is a positivt' cash florv up front. The payoffs from the portiblio at expiration are as {bllows:

If

srX S, - X

Sr X -

(fi'oia the cail) (from the stock)

ifrom the bonds)

the total is zero.

fhe portfolio generates a positive cash flow up front and there is no cash outflow at expLatian.

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Page 12

Derivatives

Topic 3 Activities l.

How does the binomial model account of volatility in the stock? The u and d factors measure the spread of price movement from one period to the next. This is a measure of volatility.

z.

when would you account for early exercise of an option? How do you do it within a binomial framework? When: i) it is an Americ an option and X ii) it's a call paying dividends iii) its a put

1

How: At every node, the price of an option is calculated from its present value of future prices. This is then compared to the value if it were exercised today. If the latter is larger, then this is the value of the option and you would exercise it. 3.

see if you can apply the binomial model to your selected share option you used for the previous topic. use a binomial software package to check your option price after - 1 iterations - 10 iterations - 50 iterations - 100 iterations How many iterations are necessary before adding another 10 iterations does not change the option price by more than 0.01? For student discussion in class

4.

A stock has a 15 percent change of moving either up or down per period and is currently priced at $25. using a one period binomial model, and assuming

that the risk-free rate is 10 percent, complete the following. Determine the possibie stock prices at the end of the first period. ? b' Calculate the intrinsic values at expiration of a European call option with an exercise price of$25. c. Find the value of the option today. d' Construct a hedge by combining a position in stock with a position in the call' Show that the retum on the hedge is the risk-free rate regardless of the outcome, assuming that the call sells for the value you obtained in c. e. Determine the rate of return from a riskless hedge if the call is selling for $3.50 when the hedge is initiated. Source: Chance, An Introduction to Derivatives and Risk Management, 5thEd.

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7-

Derivatives

: 28,?s

25{ 1. 1s)

25{0.85} = ?l-25

i!{ax{0,28.?-1- ?5i = l"7i Max{0,21"25 -- 15):0

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p

:{1.10-0.85)/(l.ls -

0"85)

:

-8313,

1

- p :.1667

i.8il3)3.75+(.1667)0 * a e1

L_=--_!.O-

ilO

h = {3.15 *.0.CI)l(28.75

V will then be

F

- 2l.25}

= 0 50

500(25)

* 1.000(2.84):9,660

500(25i 500{28.75) vd =- 500i21.75) ==

V,

-

Rn

:

1'000(3'50) = 9'000

be 10'61: V" (and Vi wil! stili Ar expiration'

Sa buy 500 shares artci sell 1.000 calls

v

-

Rd

= (10'625/9'000)

- 1 *'18

- 1.000(3.75) - 10.625 - I .000(0.0) : 1ii.625

{10,63519,660)

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an option is lower than what the value is from using the binomial model what would You do? If it's a call, sell n shares and buy the call as it is underpriced. This leads

If the market price for

5.

to an arbitrage profit (riskless profit above the risk-free rate)

6.

stock index is currently trading at 50'00. The annual index standard deviation is 20 percent. Paul Tripp, CFA, wants to value two-year index options using the binomial model. To correctly value the options, he needs the formulas in Exhibit 1. The annual risk-free interest rate is 6 percent. Assume

A

no dividends are paid on any of the underlying securities in the index.

Exhibit [.J

=

1.

eo6

Wbere:

=l.212l4

O

o,,=ffi

=!U

rvhere e'^'

=Lo6l84

U:lp movement factor D: down movement factor rT,, : probfuility of an upward price movement

Exhibit 2. Discount Factors

5.00% 6.00% Period Period Period

A: B: C:

7.Q00/o

1 0.95123 0.94176 0.93239 2 0.90484 0.88692 0'86936 3 0.86071 0.83527 0.81058 C"^tt*"t a two-period binomial

lattice for the stock index' Calculate the value of a European index call option with an exercise price of 60.00. Calculate the value of a European index put option with an exercise price of 60.00.

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Source: 2000 CFA Level2

samPle exam

Page 14

Derivatives

A. over two periods, the stock

price must follow one of four patterns: up-up, up-down, dcwndown' or down-up. Ts construct a Z-period stock price latiice for a2-yearr oprion, each period consists of 365 days, for a total of 730 days.

u =eo& U =1.2214

D= |

D=: u

t.22t4

whereD=0.glg7

The binomial paramct€rs are: IT - + percentage increase in a period if the stock price rises = 1.2214 D= + percentage decrease in a period if tha stock price falls = 0.818? ft= + Risk-free rate = 1.06 I 84

,'rr{ro*$so.oo\ ,oo.no Period 0

Period

//$so.oo

-_-*rr.r,

1

Period 2

Period 1:

Up: Down:

$50x1.2214=$61.0? $50 x 0.8187 = $40.94

Period 2:

Up: Down: up, Down:

$61.0? x 1.2214=574.59 $61.0? x 0.818? = $50.00 $40-94 xl.2214 =$50.00 $40.94 x 0.8187 = $33.52

B. For each terminal

stock price, a call option has a specific value. Because the company does not pay any dividend, 6 = 0 . With a stock paying a dividend, the dividend yield rvould be subtracted from the risk-free rate.

The probability of an upward stock price movement in any period is:

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0.6038 = 1.06184-0.8187

eU-6rL!

. =- U-D-

t.2214 -0.8181

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D

The probability of a downward stock price movement in any period is:

I -0.6038 =0.3962

t-

The value of a call option at expiration must be:

t

where:

t

L-

max (0,S

S=

- X)

curent price of stock index

X = exercise price of

the call option

l* Ir-

L:

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Page 15

Derivatives

$14.59 $8.30

a- J -t(ue- 've-rc c.rS V"=l'q.fit^fu -J

increases. before the ex-dividend Stocks that pay large dividends should be sold short large price decline in a date and Uo"gitt a{erward to take advantage of the short time period' Source: 2002 cFA Level 1 sample exam

at-the-money American The current price of an asset is 75. A three-month, what value of the asset call option on the asset has a current value of 5. At

willacoveredcallwriterbreakevenatexpiration?

A: 70. B: 75. C: 80. D: 85.

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Source: 2002 CFALevel

1 samPle exam

Page22

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Derivatives

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5.

Below is a trickv

mink before you attempt it! ,60. Call options on the stock

LI

and put options have an

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will

expire in three months. Th" There are no transaction costs using the proceeds from the short sale of any

These options

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security.

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A: A synthetic

Treasury bill can be constructed by investing in a combination of the securities identified above. i. Identify the three transactions needed to construct a synthetic Treasury

I I

bi11.

,-

ii.

LI

bill's annualised yield {basic but remember that the yield given is for 3 months)

calculate the synthetic Treasury percentage return

-

B: An arbitrage strategy

can be constructed with 75 actual and 100 synthetic Treasury bills, producing a face amount of $750,000. i. State the arbitrage strategy. ii. Calculate the immediate incoming net cash flow.

(Notes: contract multiplier : 100 for put options, call options and stocks; face value for treasury bills is $10,000)

c:

Determine the net cash flow of the arbitrage strategy at the six-month expiration date if the stock price at expiration is $80. (Ignore any cash flows stemming from the original arbitrage profit.) Source: CFA Level 3 sample exam

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Derivatives

Topicr

Portfolio Management

Minutes:

t2

Reading References: "Option Payoffs and Option Strategies," Ch.l1, Futures, Options & Swaps,2nd edition, Robert W. Kolb (Blackwell, 1997) Purpose: To test the candidate's understanding of when and how to exploit mispricing of puts and calls relative to each other.

LOS: The candidate should be able to "Option Payoffs and Option Strategiesl'tsession 13) determine whether an arbitrage opportunity exists, given a put price, a call price, the stock price, and the price of a bond; . construct and evaluate the appropriate trade when an arbitrage opportunity exists given a put price, a caI price, the underlying asset price, and the price of a bond; . create a synthetic securiry from three of the following four instruments: a calL, a put, ths underlying asset, aird a bond

'.

Guideline Answer: A.

i.

The transactions needed to construct the synthetic T-bill would be to long the stock, long the put, and short the call.

ii.

Assuming the T-bill yield was quot€d on a bond equivalent basis, bill's annualized yield can be calculated on the same basis: $?7.50 + $4.00

t($75.00

B.

-

-

$7.75

the synthetic Treasury

= $?3.?5 initial investment and $75 ending value

$73.7s) / $73.751 x 4 = 6.78%

i.

The strategy would be to shod ?5 actual T-bilis and to long 100 synthetic T-bills.

ii.

Assuming the actual T-bill was quoted on a l.25Vo quarterly return.

a

bond equivalent basis, the actual T-bill gives

Immediately, the short actual T-bill position pays: $750,000

| l.Ol25=

$?40,741

III Guideline Answers Afternoon Section - Page l0

2000 Level

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Prge24

Derivatives

At the time of creation, the long posirion in the synthetic T-Bill would be: Long stock -$??5,0ffi Long put -$ +oooo Short call $ 77.500 -$737,500 Therefore the net cash flow is: $740'741

C.

- $737,500 = $3,241

The approach to calculating net cash flow gives the same result whether the calculation is done for three months or six months.

At the thee-month expiration, the value of the long slnrhetic position is:

E=P+S-C where E = exercise price, p = put price, S = stock price, C = call price.

Atexpiration, E =

F+S*C

= $0+$80-$5 = $75 per share or $?,50O per conlract

Total cash flow of the long synthetic position = 100 conrracts x $7,500 = $750,000 Total cash flow of thE short ?reasury bili position = ?5 actual Treasury bills x $10,000 = $?50,O00 Net cash flow = $750,000

-

$750,000

=

$0

$0 cash flow at three months would be worth $0 at six months.

Alternatively, if the stock price at expiration is $80: Long stock position = $ S0 Short call position = $?5 - $80 = -$ 5 Long put position = $0 Short Treasury bill position = *$ 75 Net position $0

2000 Level Itr Cuideline Answers Afternoon Section - Page I I

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Page 25

Derivatives

6.

Ken Webster manages a $100 million equity portfolio benchmarked to the s&P 500 index. over the past two years, the S&p 500 index has appreciated 60 percent. webster believes the market is overvalued when measured by several traditional fundamental/economic indicators. He is concerned about maintaining the excellent gains the portfolio has experienced in the past two years but recognizes that the S&P index could still move above its current 668 level. webster is considering the following option collar strategy:

o

Protection for the portfolio can be attained by purchasing an s&p 500 index put with a strike price of 665 fiust out of the money). The put can be financed by selling two 675 calls (farther out of the money, for every put purchased). Because the combined delta of the two calls is less 1 (that is, 2 x 0.36 : 0.72), the options will not lose more than thethan underlying portfolio advances.

The information in the following table describes the two options used to create the collar.

OPTIONS TO CREATE THE COLLAR Characteristics

Option price Option implied volatility Option's delta Contracts needed for collar Notes: Ignore transaction costs S&P 500 historical 30-day volatility Time to option expiration: 30 days

675 Call

665 Put

$4.30

$8.05 14.00% 0.44

rr.00% 0.36 602

301

= 12.00%

Describe the potential returns of the combined portfolio (the underlying portfolio plus the option collar) if after 30 days the S&p 500 index rras trj risen approximately 5 percent to 701.00, (2) remained at 66g (no change), and (3) declined by approximately 5 percent to 635. b. Discuss the effect on the hedge ratio (delta) of each option as the s&p 500 approaches the level for each of the potential outcomes listed in part a. Evaluate the pricing of each of the following in relation to the volatility data provided: (1) the put, (2) the call, and (3) the collar. CFA Examination Level II a.

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CFA Examination ill (1997') The suggested strategy is essentially a zero-premium collar, which is constructed by selling a call option with enough premiums to fund the purchase of the put option. In this case, at-the-money protection (provided by the put) is more expensive than the out-of-money call. Therefore, the strategy requires the sale of two calls to finance one put. The call is not struck (sold) at the money because the fund manager, Ken Webstero does not want to incur the opportunity cost for a 1 .05 percent rise above the S &P 500 Index's current 668 level. $4-ll-60 ffi-manager would expose the portfolio to iqcurring losses (from the calls) ## twlcc fhe rate of the underlying portfdtiois appreciation.rThe portfolio would experience full market participation within the 005-ezs range (plus the premium received from selling the collar). The strategy will h*s ry,eeiatedfbbyolrd 675 dt expiration date a if the' ,ndex has declined below 655 at expiration.l Potential Returns Index rises to 701. This rise would cause a substantial loss to the combined portfolio. An increase above the 675 call's strike price would require an increase above 1.05 percent (668 to 675) in the S&P 500, at which time the call would be in the money. Because two calls were sold for every put bought in the collar, the collar would lose at twice the rate of the underlying portion of the portfolio against which it was written for any appreciation of the S&P 500 above 675. (The portfolio would realize a slight increment in return because of the premium [per collar tradedl generated by the collar [2 x 4.30 - 8.05 = 0.55 per collarl.)

a.i.

a.ii. No

change. The combined portfolio would experience only a slight gain as a result of the premium (per collar traded) generated by the collar (2 x 4,30 - 8.05 = 0.55 per contract). Both the put and the call would expire worthless. Therefore, the change in the portfolio would mirror the S&P 500 at 668.

@;e

a.iii. rndex declines to 635. This decline would cause a slight loss tot he combined portfolio because of the unhedged portion (668 to 675); then, the put hedge would take effect. Below.f!ftf5' fhe put option guards againsi losses on a oo"-for-orr€ffiri*a{irre portfolio would realue a slight increment in return because of the premium [per collar traded] generated by the collar - 8.05:0.55 per collarl.)

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b.

Effect of Scenarios on Delta b.i. Index rises to 701. The initial delta of the out-of-the-money calls was 0.36. As the call option got into the money and time expired, the delta of each call would approach 1.0. As the put option went out of the money and time expired, the delta of each put would approach zero.

b.ii. No change.

The delta of both the put and call would approach zero as they both went out of the money and time was

expiring.

b.iii. Index declines to 635. The initial delta of the out-of-themoney put was -0.44, As the put option got into the money and time expired, the delta of each put would approach 1.0. As the call option went out of the money and time expired, the delta of each call would approach zero.

c.

Pricing Compared with the historical volatility of the market (12 percent), the calls (Part ii) are priced relatively cheaper (11 percent) and the puts (part l) are priced relatively more expensive (14 percent). The suggestion is that the collar (Part iii) is relatively expensive.

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I J

J

l l -l

does an

option Pricing Model relt us About Option Prices?

/ .lzirs-f I i \^\K r. /I v\lt

Read the accompanying article (next page) and answer the questions:

i i

following

Identify factors that are not part of the B-S pricing model but still would affect option prices.

-

!

-l

what

Transaction costs, supply and demand for options, Taxes, Margin requirementso Differences in contract specifications between options

Outline reasons why the B-S model may be wrong. Cannot calculate with certainty the correct volatility rate for stocks The assumptions of the B-S model not realistic. Eg, - Investors are not homogeneous - Doesn't account for the above listed factors - Doesn't explain why options with different exercise prices have different implied volatility rates - Doesn't explain why puts and calls on the same stock have different

-

implied volatility consistently underestimates the value of deep out-of-the-money options Prices may not behaviour as Brownian motion (as outlined by the general weiner process and Ito process which underlies the B-s pricing model). Stock price distributions seems to have fat-tails (excess kurtosis) probably due ,.,tg market volatility not remaining constanf for the life,,of. theoption. fiefit**i>hc + f&,tr.lan efirf,,rtJ ,-){ lr,f,l,kt* pt,sj

,

' Lil* ou, cl4rhifr\r, t

Explain how the B-S model may calculate the correct value for an option but still be different from the market price. - two cases in point: Deep out-of-the-money options tend to affract investors for the potential high gains that can be made with relative low costs involved (the option premiums will be very low) - just as with purchasing lottery tickets. This is a human perception factor relating to how people view risk and return which is not always as rationale as the equations make out that we are. - Sometimes the markets find it too difficult to properly calculate true values for certain options, such as American and. Embedded options due to the computing time and effort involved.

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@tls$

Cunrent lssue$: Opti0n$

= =

Pricing Model Tell Us What Does an Option ' Option Prices? by Stephen Figlewski, prcfessor of Fi- "That depends. Are you buying iorrr,' Stern Schoal of Business,'New or selling them?" The story is told of a Seeker of

Knowl-

the

them

u.rJ*",. to a question that has him for a long time. In his travels he hears of tr.rro wise men lvho are said by many to be very knowledgeable and experienced in such matters. The first, a famous guru, lives at the top of a mountain, high above the hustle and bustle of everyday life. After a strenuous clirnb, the Seeker is able to pose his question: "What is a call option worth?" The guru ans"ers immediately, "It is not hard to prove that

=

=

price above the theoretical value an: will have infinite demand at any pncr

=

=

,IlT:ilii::xr.llT,'?i,ll';,iH:::"il*:',:1i:l';T::ii:lT":1i:,==

the words, and equations, of the

first

guru, but he is quickly intermpted troubled with: "I don't care about all of that

edge who sets off in search of

About

is the reasoning behind the first guru

answer'

I

.=

But in trying to apply a theoretic- =,E

stuff. Tell him to make me a bid, Then valuation model to the real world, i!,, E we can talk about what a call option is immediately clear that none of tf - E really worth." odel assumptions actually .hold. Somewhat confused and not at all The arbitrage strategy, lvhich is risi ,f = sure the wise men's answers have less and costless in theory, is neithe: E brought htT lty closer to enlighten- in practice. There is risk because ti'.: E ment, the Seeker goes away to medi- poritiot can't be rebalanced contin- ff tate further on his question' ously when markets are closed, an: ff

and Pricing

there are costs because even less ff

than-continuous rebalancing can lea: I to large transaction costs. Even ti. I Models theoretical optionvalue,itself is unce: I two different very offers This parable tain, because it depends-on the voi: question. basic same the C = SN tdl - Xe-,rN Id - o{t1, uns*Lrc to tility of the stock, which cannot !- |I reflects them between distinction The where S is the stock price, X the strike the not often recognized difference be- known exactly' Unlimited arbitra; I price, T time to expiration, r the risk- twe€n theories of option valuation and does not dominate the market' il ln actualmarkets, oPtion prices, li' I iree interest rate, rvolatility, N[.] de- option pricirzg. notes the cumulative normal distribu- The Btack-Scholes model and others prices for everything else, are det' I tion and like it are theories that try to derive the mined by supply and demand. T: I aalue of an option so that it is consis- lncludes supply t"o,"l;TflJ:.--f, d.: (log(srX) + (r +

Valuation Models

ttz)T)tl.Vi.

::l:Jfi"f:Jff":t#-:i*lInq

ffi#3:ntiXll;,i"1"" oprionsothis has to be modiried in practice to take into ac- ment in which a dynamic riskless ar- pariicipation in stock price moveme: = E " count dividends, the value of early bitrage strategy with the stock and the on the upside, limit risk on the do.' = exercise and a few other technical de- option is possible, and find the value side, and allow inveslors l: contrt a. si: tails (see the appendix)." oi the opiior, as a component of the large amount of stock with investment. Call writers supply - ,E This answer seerns pretfy exact, if a arbitrage portfolio. = In this ideal market, if the option's options to the market because it ' bit complicated. The Seeker thanks the = price should differ from the model way to generate in,come when t-'= guru warmly and goes on his way. sha:: rise E not will The second wise man lives in the value, an arbitrageur can trade it exPect stock price middle of a city, surrounded by a againstthecorrectnumberof sharesof in the near future. Option demand arrd supply are ' € continuous swirl of noise and activity. siock to produce a position that is = Once the Seeker is able to get his riskless over the next instant of time. influenced by the market envi: attention, he poses the question again: Continuous rebalancing keeps the ment, rt'hich encompasses taxes, tl' = "What is a call option worth?" hedged position riskless until the op- acfion costs, margin treatment ot : Again the answer is immediate: tion"expiration date. But its return will ferent securities, delivery featurt, f= be higher than the risk-free interest option contracts, constraints on -

or course, somewhat

:ffiiJ*:TlHl:Tli:

Lt*'l:

$::r,T:l::n#f.i**'

$u'

=

contracts, and r Brcwn, loel Hasbrouck, Mark Rubinisteh there are no corrstraints on the size of related futures that affec:' E and Wiltiam Silber for comments on an their positions, arbitrageurs will offer other things. Anything but is : = decisions hading vestors' at any of options number ,lnii*itud ur, earlier draft of this p,aper.

=

FINANCIAL ANALYSTS TOURNAL / SEPTEMBER-OCrOBER 1e8e n

12

=

Deriyatives

i

tends to push the market

lyay from the model value. As

nd wise man indicated, a

ca-ll

is wcrth exactly the price

at

it can be kaded in the market, does not depend on just the might ask: l{ithout unlimarbitrage betr+'een the option and

risk. Moreover, ii is dear that, given the cost and uncertainty of the trade, arbitrageurs wi-ll not take unlimited positions, e1'€n at prices r:utside these

wide bounds. This makes for something less than impenetrable barriers" . We are left with distressingly little in the lvay of a response to the skeptic's question.

ying stock as a foundation,

can a theoretical valuation model

us anything about actual market possible answ€r is that, under conditions, the equilibriun oF price is the Black-Scholes value

when there is no arbitrage, ber,r'hen investors evaluate the opas if it were any other asset, that is t its pa-Voff pattern is worth.l Unnately, one of the necessary conions for this result is that all invesbe identical, which is no rnsre true real markets than the continuous itrage assumptions it replaces An argument that is on stronger nd, but has weaker implications, 'that as long as the arbitrage is posble, it will be done in spite of transn costs and risk whenever the profit is large enough to csmnsate for thern. This leads to arbibounds around the model value. lf ithi.n these bounds, there is no arbitrage and market price can move feely, but if the price skays too far irom the model value, arbitrage hecomes profitable and rcill tend to push :::ce back into the bounded range. Horv much information the riiodel a::ually gives us about what the mar!,:i price u'ill be depends on ho.w wide ;,: arbitrage bounds are. This is not :;s',' to determine, because the trans:-on costs and risk for the arbitrage r-: a function of r,.'hich random path :: = stock price follor+.s. ln a recent :::er, I simulated a large number of ::,:e paths and discovered that the .::i;rage bounds are disturbingly :''.ie. even for routine cases.z For ex.::.rle, the price of a one-month, at:.i-inoney call rvith a Black-scholes ' : . re of 52.05 could be anywhere frorn S- :1 to 52.35 without giving an arbi::i:eur el1en a 50/50 chance of cover:.: ;csfs. or any compensation for : :::r-.otes appear at end of

arhicie.

Real Options Using Tf$lg, Model Prices The probiems associated with applving a theoretical option valuation model to the real tvorld have obr.! ously not prevented virtuall-v every serious option trader from doing just that. But they use the model in ways that are more consistent rt'ith the second rvise man's approach than with the model's theoretical underpinnings. Fe*" investors use the model mainiv

for finding mispriced ophons that can be arbitraged against the underlying stock, A major reason for this is that no one feels confident they know ihe true volatility. Esrimaring volatility from a sample of past prices is a routine exercise, but you can't be sure the future will be exactly like the past. Based on historical volatilitr, for example, the price change that occurred on October 19, 1987 rvas essentially impossible.3

Option traders normally pay more attention to the implied volatility that sets the racdel option value equal to the market price. This is easv to compute and, in principle, gles a direct reading of the market's volatilitl' estimate. But ii has the disadvantage that you ha1'e to _ag-sur!e_--thq.-,g1arkg1[. is. pricing the option according to the model, rvhich rules out using implied volafilih, to detect mispricing. Another problem is that, because volatility is the one inptrt to the model thai can't be directly* observed. implied volatili$ actually sen'es as a free parameter. It impounds expected volatiliW nnd nerything e/se that affects option supply and demand but is nat in the rnodel" If a change in the tax law makes writing options less attractive, for instance, the price effect will shorv up in implied volatility. Any time the market prices options differently from a given model, for any reason, the

FL\ANCIAL A\'.41\5TS

implied volatility derived from that model is not going t0 be the market,s frue volatility esfimate. Generally, implied voiatilities d.ifer across slrike prices in a regtrlar way, even though it is logically inconsistent for a stock tryt have more than one v alatrlity Tput-o f-t he-mo ney op tio ns hpically have higher implied volatilities than at and in-the-money options, puts are often priced on different valatilities than calls, and ttge pattems vary from time to nmer/ffis is evidence that the market difes not price options strictly according to the rnodel, or at least not according to the particular model that produces the di{fering implied volatilities. Traders don't care much abcut these problems. In fact. they seldom think about ihe technical details of the computerized models they use, or even about rvhether they are pncing Euro, pean or American options. Instead, they tend to take whatever theoretical model happens to be available on the computer and (eat the implied volatili$ it produces lbi-a,qarticular option a$,q kind of index of h$.v the option is currently beisg;.p*c€d in the market relative to other options and relative to how it was priced at other times. It is perfecily normal, for example, for an ophion trader to reason, ',yesterday the ccmputer said these XyZ options were trading on a volatilify of 20 per cent, but this morning the options market is a little soft, so l,ll use 19.5 tod,ay and 21.5 {or the outof-the-money puts that always are a iittle richer." These volatilirl* "estimates" are then plugged into the computer's model, and bids and offers are based on the theoretical values that it produces. The idea behind this procedure is not that the trader thinks 19.5 is the best available estimate of XYZ's volatility until option expiration, but ihat it is a reasonable way to summarize the current state of supply and demand for XYZ options and that conditions will probablv remain fairly stable for a whileuntil thev change. For many option traders, having exactly the right model and volatilify may not be of such great importance, because r.r'hat they really care about is the delta, which tells how to hedse an .

.

/ SEn-EMBER-OCTOBER 1989

'OURNAL

-i j.3

Derivatives

option, and delta is much less sensitive to these things than theoretical value. Also, markeFmakers and active traders frequently hedge options against each other, rather than against the stock, so the effects of changing volatility and other model inaccuracies on the different options partly offset each other. In ofher words, *F-.,nut tto, , rnatter so much if your model misprices an option as long as the option. vnll conti,nr.re to be mispriced in ttre same wqy when lhe stock price, changes, or as long as it is hedged by other options that are similar$ mirs-

in$y

complex alternative rnodels tb pirical examination of those explain the phenomenon, : has not led to widespread reje Tinkering with a model to try to them, suggests that, at least fc: make it fit market prices better con- maturity, exchange-traded oph :r: fuses valuation and pricing. What afi tracts, the models work acc: arbitrage-based option model says is t well. that buy,ing the option at its fair value " But some situations wa:: and following the arbitrage strategy I greater degree of skepticism i:; until expiration will r€tum the risk- trs. There are problems of tr^, free rate of interest., If empirical tests The model can be wrong, sc show that this would happen in prac-. theoretical value is not the tru: trce, qss.uqtiyg you could buy the op- value, or the model may give tion for its model value and had no " rect value, but the market p::l transaction costs, then the valuatbn option differently. formula is correct.*How those optiory In the first category we inc might be priced in the market h6s uations in which the model ar priced. Still, when a stock has a large price nothing to do with whether the model tions are violated (to a large: | than normal). This is true fc: {nove/ implied volatilities also typi- aatrues them correctly! What model-testers have in mind, of maturity options, where par: cally change, meaning that the actual change in the ophion's price is not course, is that the theoretical valuation such as volatility and intere, what was predicted by the original model should also be the market,s (which one can treat as beini delta. The skeptic might wonder if we pricing equation. As we have seen, iin , constant over a month) ca. can even be sure that the apparent the real world an arbitrage-based vaL: widely and unpredictably over stabilify of implied volatility under uation model produces a band around periods. We can have confider: normal circumstances isn't partly due the' theorbfiCal'Option value,i withinry valuation model only to the exte to a kind of self-fulfilling prophecy, which the excess profit that could be" \A'e are confident of our forecas= what with so many option traders us- made is smaller than the cost to set up' parameters out to option exp: ing the model to change their bids and tLre arbitrage tradd. In this context, the which may be many years in :r model only says that the price should ture. otfers as the stock price moves. be within the bounds.t It is perfectly We can also expect inaccuraci* l4lhile it was conceived as an arbiconsistent with the model for the maring from errors in modeling st trage-based valuation theory, rthe ket price to be always at the lower prices as geometric brownian r:, Black-Scholes rnodel is actually used bound or the upper bound, or followThere is considerable evidenc: by option traders as a pricing equaing any kind of pattern in between. actual price changes have "fat t= tion, to predict how the option price The finding of ',bias,, a consistent fn that is, there is a greater probat: w|!l,,Sl-1enge ;when the underlying stock market prices does not indicate a rejbca large change in a short inten'i rnoves. But baders treat the model almost as if it lvere a rule of thumb, tion of the modbl, unless the bias is the model assumes, leading tc rather than a formula that gives the large enough that pricqs lie well out- ing problems and undervalua:r side the arbitrage bounfus. And those short-maturity options. There . true option value with confidence. bounds may be very wide. growing evidence that over boi; and short horizons, there is somr Testing Models with Real Is the Skeptic Right? randomness in stock price move Option Prices A true skeptic might argue that this There is certainly reason to dc In a sense, the mirror image of an discussion has shown fwo things. arbitrage-based model's valuat: option trader evaluating market prices First, option valuation models dorr'g an option on an underlying ass= using the model is the academic theo- give correct fair valges because the is not traded, even though th. rist "testing" a model on market data. continuous arbitrage can't be done in routinely apply the Black-Schoi The standard procedure is to compute practice. r5econd,",F'?,,en if modefS rdla mula to all manner of cases in . theoretical values for a set of actual give correct values, option prices in the arbitrage is irnpossible, or . options and compare them with mar- the market would not equal those val- tially so. For example, it is an ket prices. Small random differences ues because of all of the other factors tant theoretical insight that lim can be explained as "noise," but sys- affecting supply and den{and. ability in trankruptcy makes the,. tematic deviations are viewed as eviI would not go so far, although I of a firm with outstanding debt s dence of problems with the model. think some skepticism is a healthy, to a call option to buy the firm's: The common finding that deep-out- and risk-averse, attifude in this case. A from the bondholders by payrr: of-the-money options seem to be model does not have to be exactly debt. But the firm as a whole is: priced higher in the market than' right for it to be of use. The general asset that can be traded indepen: Black-Scholes would suggest has acceptance of option models in the real from its securities, so there ls r. given rise to any number of increas- world, and the fact that extensive em- investors could arbitrage the -

:_

FINANCIAL ANALYSTS JOURNAL / SEPTEMBER.OCTOBER 1989 tr 14

Derivatives

asset, such as a unique piece of propstv/ causes difficutty lvhen there is no wav to form a hedge portfi:lio that ran

jle reUalanced. There are clearly many f**r in rvhich one must be sieptical about deriving an option's value from

i ltre

seccnd categorv of problems irertains lvhen the model gives the -. r';e value of an opfion, but the market $pn."r it differentlv. The more difficult je-rd costly the arbikage trade is to do. Fre greater the scope is for factors not foi'ered in the model to move the Frarket price away from its theoretical :',:lue. Several situations harre proved difficult for arbitrage*::.rticularlv. E-:,:SeO

models.

". Cut-of-ilrc-tn{in€y aptions: people par_ :-r-rlarly like the combination of a large

;::ential payoff and limited rlsk arici s=:: rviiling to pay a premium for it. ,.iat is why they buy lottery tickets at ::.es_that embody an expected loss. f j:-ot-the-monev ;options offer a sirn; "; payoff pattern, At the same fime, -:'--. ',r.riters cf those options are ex-f :,,,-ed to substanial risk hecause it is '--::C to hedge against large price ::::nges. l^Ilry -r,-:-of-the-moneyshould we fiof expect options to seil flr a ;:.-::rtum over fair value? .:-',;ericet! optians: The possibilifv of 6 :::.'. exercise makes American option, " .-::: iL, r.alue theoretically, especiallv :*::use the early-exercise pravision is , _:'i.:.-'m exercised optimally according ,' :.-.t theorv. f his is an erlormous ::- :lem with mortgage-backed securi::: trecause of the homeclwner,s r.)p_ f

tion to prepay the mortgage lcan, but for computing option values; further_ all American opticns share it to some more, the harder ihe arbitrage is to do, extent. We should not be surprised if the less con.fidence these investors can the market pricrs American options have that the moclel is going to give differently from their model values be- either the truer4ption value or the cause of the uncertainty. market pnce. f{edging options r+,ith Emfudtled aptions: Valuation models options, rathei thaffiffiTIFiiriHerlv_ treat a security rvith embedded option iirff{dck, can provicle some defense features, such as a callable bond or a against inaccurate valatility estimates security with default risk, as if it were and mod' simply the sum of a straight security In general, investars are nOt and the opticn. But the market doei ffueg as snarket-making arbitrageurs not generally price things this way_ should be less cotcern*d with Glua_ For example, when coupon strippers tion models than with using options to unbundle government bonds. or produce overall payoff patterns that when mcrtgage pass-throughs are re- suit their market expectations and risk packaged into CMOs, the sum of the preferences. When they think the mar_ parts sells for more than the original ket might drop sharpltL it makes sense whole. Whv should we expect" ihe for them to buy put options, er.en if market to pric€ ernbedded options as if they have to pay rncrre than ,'far{' they could be traded separately rt'hen value. this is nct true of other securities? Times of crisis: The period around the Footnotes crash of October 1987 showed that in 1. See M. Rrrbinstein, ,'The Valuation times of financial crisis, arbitrase beof Uncertain Income Streams ancl comes even harder to do and Jption the Pricing of Opticns,,, BeIl fourna! prices can be subject to tremeridous of Econombs and Management Scietzrc, pressures. At such fimes, rve should Aufumn 1976, ar Nl[. J. Brennan, not experl to be able to explain market 'The Pricing of Ccntingent Claims prices n'ell with ur, utbifr"ge-based in Discrete Time Models," Th€ l$urvaluation model. nal ot' Finance, lv{arch 1979. 2. S. Figlewski, "Options Arbitrage in Whete Do We Go From Here? lmperfect Markets," The [ournal of If what is really u'anted is a model to Financt, forthcoming I 989. explain hor*- the market prices cp- 3. And including that day,s price tions, it doesn't make sense for acl_ change in volatility estimates after demics and builders of optian models the er..ent meant that ii dominateei to restrict their attention entirelv to the calculaiian. There lyas then a elaborating arbitrage-based valuaiion spurious sharp fall in estimated volmodels in an ideal market. They abilitv months later, on the day Ocshould at least examine broader tober 19 dropped out of the data classes of theories that include factors sample. such as expectations, risk aversion and 4. See, for example, E. Fama and K. market "imperfections" that do not French, "Permanent and Tempoenter arbitrage-based valuation modrary- Cornponents of Stock Prices," els but do affect option dernand and lournal of P