options.docx

8/10/2019 Options.docx

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Options:

The Binomial Model

Consider the following security:

Note that this security has no risk. Assuming an equilibrium rate of return of 8% on other assets withzero risk, the aroriate discount rate in this case should yield a rice!resent "alue of #$$!#.$8 &'.(&.

Now, let)s look at another case.


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*hat is the resent "alue in this case+ et us assume that the -/ ($$ 0nde1 reresents the marketortfolio. 2hen, since the asset co"aries erfectly with the market inde1, the discount rate is simlythe equilibrium rate of return on the market ortfolio. et)s assume that this is #'%. 0n this case, we

take the e1ected "alue of the ayoff on the security, and discount it back to eriod tat #'%, which inthis case, works out to 38(.4#.

Now what about the following case:

2he e1ected cash flow in eriod t+1is 3&8. 0f we discount this at 8%, the resent "alue is 3&$.456 ifwe discount it at #'%, the answer is 384.($. 7ut which is the correct answer+

Using the Arbitrage Principle for valuation

-ecurity )s cashflows are a simle a"erage of the cash flows of securities # and ' in the ' scenarios9states of the world. ;ence, the "alue of security is also a simle a"erage of the "alues of securities# and '. 2he resent "alue of security is $.(9&'.(& < $.(98(.4# 8&.#(. 0f we work out thee1ected rate of return that is consistent with this rice, we see that it is =$.89#$$


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Note that we did not need to know the robabilities of the two states, once we had the rices of theriskfree asset and the -/ ortfolio. 2his is because we are using the rincile of no riskless arbitrage.@or the alication of this rincile, we do not need to know robabilities. 2his rincile states:

0f the cashflows on an asset can be e1ressed as a weighted a"erage of the cashflows on a gi"en setof traded assets, the market rice of the first asset must be equal to the same weighted a"erage ofthe market rices of the gi"en set of traded assets. 0f this is not so, then there will be oortunities forriskless arbitrage rofits.

7ut once we ha"e this idea of a security being reckoned in terms of an a"erage of other securities, wecan take this further. 2ake security 5.

Can security 5 be considered a weighted a"erage of securities # and '+ et)s conecture that this isindeed ossible. 2hen, let the weights be land s. 0n this case, it must be true that

l9Cashflow in state # for sec. # < s9Cashflow in state # for sec. ' #$6

l9Cashflow in state ' for sec. # < s9Cashflow in state ' for sec. ' $, which gi"es us the twosimultaneous equations:

#$$l< #$$s #$

#$$l< 8$s $

-ol"ing these equations simultaneously, we get s $.(, and l ?$.5. Beconstituting the cashflows, wesee that #$$ 9?$.5 < #$$ 9$.( #$ in state #, and #$$ 9?$.5 < 8$ 9$.( $ in state ', as required.7ut this imlies that the "alue of security 5 is simly 9?$.5 9&'.(& < 9$.( 98(.4# 3(.8'.

et)s go back to -ecurity , now. 2he corresonding equations in this case are:

#$$l< #$$s #$$

#$$l< 8$s &$

-ol"ing these equations simultaneously, we get s $.(, and l $.'. Beconstituting the cashflows, wesee that #$$ 9$.( < #$$ 9$.( #$$ in state #, and #$$ 9$.( < 8$ 9$.( &$ in state ', as


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required. 7ut this imlies that the "alue of security 5 is simly 9$.( 9&'.(& < 9$.( 98(.4# 38&.#(, which we ha"e already seen.

2he interesting thing that we ha"e deri"ed in addition to what we had before is that security 5 isequi"alent to a ortfolio consisting of a ($% in"estment in the market ortfolio and a ($% in"estmentin the risk?free asset. ;ence its beta is $.(9#


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Assume that only the "alues 35$$ and 3($$ are ossible for the inde1 ortfolio at time t+1. 2hen, ifwe draw the binomial tree for this call otion, we see that the cashflows are indeed the same as forsecurity 5, and hence its "alue is the same.

2he relicating ortfolio of the inde1 and the riskfree asset can be comuted by sol"ing the equations:

#$$l< ($$s #$

#$$l< 5$$s $

2he solution is l ?$.5, and s $.#. 2he otion "alue, as we remarked abo"e is 3(.8'.

2he relicating ortfolio aroach has an alternati"e interretation. Note that the "alue of the otionis l=/E9#$$> < s9the current rice of the underlying asset. 0n other words, the rice of the otion isequal to the current market "alue of a ortfolio consisting of sunits of the underlying asset and lunitsof a bond aying 3#$$ at time t+1. *e see, in this case, that the otion is equi"alent to a ortfoliolong $.# units of the underlying asset and short $.5 units of the riskfree asset. Feeing in mind that


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being short the riskfree asset is equi"alent to borrowing, we note that a call otion can be relicatedby simultaneously going long the underlying asset and borrowing.

Gtions that are written on an underlying inde1, such as the otion that we ust "alued, are calledinde1 otions.

Determinants of Option Prices

2he "alue of an otion deends on se"eral arameters and we need to know how changing thesearameters, one at a time, affects the otion "alue:

the current asset rice

the e1ercise rice

the asset return "olatility

the rate of interest

the time to maturity

*e will use the re"ious e1amle as our base e1amle: a call otion on the ortfolio of -/ ($$ inde1stocks with an e1ercise rice of 35&$.

ffect of changes in the asset return volatilit!"

2he "olatility in the return on an asset is usually measured by the standard de"iation of the return onthe asset. et us increase the "olatility of the return on the -/ ($$ 0nde1, keeing the mean returnconstant. @or e1amle, let us assume that the "alues of the inde1 ortfolio at time t+1can be eitherH$ or (#$. 9Note that the e1ected "alue of the -/ ($$ 0nde1 at time t


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#$$l< (#$s '$

#$$l< H$s $

-ol"ing, we find l ?#'!'(, and s '!#(. 2his gi"es us an otion "alue of #'.4$, which is greaterthan (.8', as conectured.

ffect of changes in the ris#$free rate"

*e ha"e assumed a riskfree rate of 8%. 0f we increase the riskfree rate to #$%, the otion 9being

equi"alent to a ortfolio long the stock and short the riskless asset will increase in "alue, since theresent "alue of the amount required at t+1has droed. 2he equations to be sol"ed are the same asbefore, and hence the relicating ortfolio stays the same.

#$$l< ($$s #$#$$l< 5$$s $

;owe"er, the "alue of the otion, now, is ?$.5 9#$$!#.# < $.# 95'8.(4 H.5& I (.8'.

ffect of changes in the current asset price"

et us now change the current rice of the asset. -ince we need to kee the return "olatitliy constant,this can be achie"ed by changing the discount rate for the asset. -uose the discount rate is #5%9instead of #'%. 2hen the stock rice will dro, and the "alue of the otion, which consists of aortfolio long in stocks will dro as well. 2he equations to be sol"ed are the same as before, andhence the relicating ortfolio stays the same.

#$$l< ($$s #$#$$l< 5$$s $

;owe"er, the "alue of the otion, now, is ?$.5 9#$$!#.$8 < $.# 958$!#.#5 (.$4 J (.8'.

ffect of changes in the e%ercise price"

0f we change the e1ercise rice to 58$ instead of 5&$, the equations to be sol"ed change to:

#$$l< ($$s '$#$$l< 5$$s $

2he solution to these equations is: ?$.86 $.'6 hence the otion "alue is ?$.8 9#$$!#.$8 < $.'958$!#.#' ##.H5 I (.8'.

0ntuiti"ely, this is clear, because the higher the e1ercise rice, the lower the uside otential of theotion.

Multi$period e%tensions of the binomial model

et us assume that there is yet another eriod before the otion matures. Now, from date tto t+1,the stock rice has been assumed to either increase by #H.H4% 9from 5'8.(4 to ($$ or decrease by

H.H4% 9from 5'8.(4 to 5$$. et us assume that during the ne1t eriod too, only the same kind ofincrease or decrease is ossible. ;ence the scenario can be shown as follows:


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0n this instance, we need to comute the "alue of the call otion at time tin two stages. *e firstcomute the "alue of the otion at time t+1conditional on the inde1 "alue. 0n other words, wecomute what the call otion "alue would be at t+1if the inde1 "alue were ($$ or 5$$. 2hen wecomute the otion "alue at t, gi"en the otion "alues at t+1. 2his can be done in e1actly the sameway as we did it before.

Ksing this technique, we sol"e the equations:

#$$l< (8.s &. and #$$l< 5HH.HHs $

to obtain the "alue of the call otion at t+1if the inde1 "alue is ($$.

Ans: l ?.46 s $.86 call "alue ?.49#$$!#.$8 < 9$.89($$ (5.H.

0f the "alue of the inde1 "alue at t+1is 5$$, the equations to sol"e are:

#$$l< 5HH.HHs $ and #$$l< 4.s $

2his gi"es us l $ and s $, or a call otion "alue of $.


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2he solutions to sol"e, in this case, are:

#$$l< ($$s (5.H and #$$l< 5$$s $

-ol"ing, we get l ?'.#8(6 s $.(5H, for an otion "alue of #.8# I (.8'.

2he reason that a greater time to maturity increases the "alue of the otion is simly that the usideotential is greater, while the downside otential is always fi1ed.

Parameter Effect

the current asset rice ositi"e

the e1ercise rice negati"e

the asset return "olatility ositi"e

the rate of interest ositi"e

the time to maturity ositi"e

Pricing Options using the Blac#$&choles method

2he 7lack?-choles formula gi"es the "alue, C, of a call otion with e1ercise rice Eand e1iring

in tyears, written on an asset with "ariance of the continuous return equal to 2er year and

currently selling at S. 92his formula assumes that there is continuous trading, and that the otion canonly be e1ercised at maturity:

C - N9d# ? D e?rtN9d'6 and d' d#?

N9. is the area under the standardized normal cur"e 9i.e. the lot of robability against "alue for anormally distributed "ariable with mean zero and s.d. #, and can be obtained by consulting normalrobability tables.


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D1amle:

Consider a call otion with e1ercise rice 5&$ and time to maturity equal to # year, on a stock sellingat 35'8.(4, with a "ariance of $.$#'5( er year. 2he riskless rate of return is 8% er year. *hat is the"alue of the otion+

d# ?$.5'446 d' ?$.(&6 N9d# $.5( and N9d' $.'&5H.

C 5'8.(4 9$.5( ? 5&$ 9.&'# 9.'&5H #$.#$

Put Options and Put$Call Parit!

A ut otion on an asset is the right 9but not the obligation to sell a gi"en quantity of that asset at afi1ed 9e1ercise rice within a fi1ed time eriod.

D1amle: -uose we are considering a ut otion on the ortfolio of -/ ($$ inde1 stocks with ane1ercise rice of 35&$. 0f the inde1 9ortfolio "alue at time t+1is 3($$, then the "alue of the utotion is worthless. 0f the inde1 "alue at time t+1is 35$$, then the ut otion is worth e1actly 3&$,since we can buy the inde1 ortfolio for 35$$ and sell it for 35&$. *e can grah this relation:


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A erson who sells a call otion 9with an e1ercise rice of 35&$ and a erson who sells a ut otionwith an e1ercise rice of 35&$ will, howe"er, ha"e the following ayoffs:


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*e can also grah the ayoff to the holder of one unit of an asset as a function of the asset rice atsome time in the future.

Now consider combining a long osition in the inde1 ortfolio along with a ut on the inde1 ortfolio.

@inally, let us combine a call with an in"estment in riskless bonds maturing at the same time as thecall.


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2he ayoff at e1iration of the two combinations is the same. ;ence, the resent "alue of the twocombinations must also be the same. 0n general:

C < D e?rt / < -, where / is the rice of the ut.

D1amle:

*e can grah the "alue of the call otion before maturity as a function of the underlying asset rice.0n this case, the grah line will no longer be linear.


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&toc#s and Bonds as Options

*e can e1lain some of the agency roblems of debt by looking at the equity in the firm as a callotion on the assets of the firm. Consider what haens to the equityholders of a firm. 0f the firm doeswell, they ay off the bondholders and they get the entire uside. 0f the firm is not doing well, theydon)t ha"e to ayoff the bondholders in full6 they can ust walk away from the firm. ;ence, what theyha"e is similar to an otion.

*e know that the "alue of an otion is increasing in the "olatility of the return on the underlyingasset. -imilarly, the management of a firm can increase the "alue of equity by increasing the riskiness9"olatility of returns on the firm)s roects.

D1amle: Consider a firm with a roect currently worth 3#$$. Ne1t eriod, if the roect succeeds,the firm will be worth 3'$$6 else the "alue of the firm will dro to 35$. -uose the firm currently hasdebt with a romised ayment of 3H$ ne1t eriod.


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Assuming an interest rate of (%, the equations to be sol"ed are:

'$$s< #.$( #5$ and 5$s< #.$( $

2he solution is s $.84(, and ?., for an otion "alue 9equity "alue of 3(5.#4.

Corresondingly, the "alue of the debt can be diagrammed as follows:


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Uses of options"

Gtions can be used as insurance. @or e1amle, ut otions guarantee a floor "alue for the

underlying asset.

0n"estors can lace bets on underlying stock return "olatility by trading in stock call otions.

0n"estors 9who are bearish on a stock can lace negati"e bets on the rosects for a stock by

buying uts or selling calls.

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