OptionsTradingin7Days

SimonGleadall

1stEdition

Volcube

©2014VolcubeLtd.All

rightsreserved.

Tableofcontents

DisclaimerHow touse thisbookAbout theauthorAboutVolcube1.Introductiontooptions

2. Optionvaluation

3. Waysto tradeoptions

4. Pricingoptions

5.Managingoptionrisk : theGreeks

6.UnderstandingImpliedVolatility

7.Tradingoptionsandvolatility

How to

continueyouroptionseducation

SolutionstoExercises

Glossary

DisclaimerThis book does not constitute an offeror solicitation for brokerage services,investment advisory services, or otherproductsorservicesinanyjurisdiction.

The book’s content, tools andcalculations are being provided to youfor educational purposes only. Noinformation presented constitutes arecommendation by Volcube to buy,sell or hold any security, financial

productorinstrumentdiscussedthereinortoengageinanyspecificinvestmentstrategy. The content, tools andcalculations neither are, nor should beconstrued as, an offer, or a solicitationof an offer, to buy, sell, or hold anysecurities by Volcube. Volcube doesnot offer or provide any opinionregarding the nature, potential, value,suitability or profitability of anyparticular investment or investmentstrategy, and you are fully responsibleforanyinvestmentdecisionsyoumake.Such decisions should be based solelyon your evaluation of your financialcircumstances, investment objectives,risktoleranceandliquidityneeds.

Options involve risk and are not

suitable for all investors. Optionstransactions are complex and carry ahigh degree of risk. They are intendedfor sophisticated investors and are notsuitableforeveryone.

HowtousethisbookThis ebook can be used as astand-alone 7 day course inoptions. No prior knowledgeisassumed.Eachofthesevenlessons includes a set ofExerciseswithfullanswersatthe back. The lessonsintroduce options, explainhow they function and ways

they can be traded. Alsoconsidered ishowoptionsbevalued or priced and theirassociated risk. The last twolessons focus on options andvolatility.To get the most out of thecourse, it is recommendedthat you attempt theAssignments in each lesson.These are performed usingtheVolcubeoptionssimulatorandtrainingtool.Youcanget

afree7daylogintoVolcubefrom volcube.com. You donot need a credit card andyour login will just expireautomatically at the end ofyour7daytrial.Volcubeisapowerful options educationtechnologywhichallowsyouto play simulated optionstrading games and alsoincludes useful videos andarticles relating to optionstrading.

A glossary of importantoption-related terms will befound towards the endof thebook. Recommendations forhow to continue with anoptions education are alsogiven.

AbouttheauthorSimonGleadall is one of theco-founders of Volcube andhas traded options and otherderivatives since 1999. Heworks closely with theVolcube development teamonupgradestothesimulationtechnology and also co-producesmuchoftheoriginal

learningcontent.

AboutVolcubeVolcube provides a leadingoptions education technologyto firms and individualswhowant to learn aboutprofessional options andvolatility trading. TheVolcubetechnologyisaweb-based option marketsimulator with embedded,automated teaching tools anda rich learninglibrary.Volcube was founded

in2010.Please visitwww.volcube.com to learnmoreandtryouttheVolcubesimulatorforfree.About the VolcubeAdvanced Options TradingGuidesseriesClearandconciseguidesthatexplore more advanced

options trading topics.Checkout the other volumes in theseries…volumeI:OptionGammaTradingvolumeII:Option Volatility Trading :StrategiesandRiskvolumeIII:OptionMarketMaking:PartI:AnIntroductionvolumeIV:Trading Implied Volatility :

AnIntroductionvolumeV:Option Greeks for Traders :Part I : Delta, Vega andTheta

1. Introduction tooptionsOptions contracts traded inthe financial markets areultimately just options!A lotof people think of optionsfirstlyascomplexderivatives,but the easiest way is torealise that a financialoption

isjustachoiceaboutwhethertodo something,nodifferenttoanyotheroptionwemighthave.Solet’sthinkaboutanoptionwe might encounter ineveryday life. Suppose yourfriendoffersyoutheoptiontojoin him at a music concert.Let’s consider what thisoptionmightmeantoyouandwhat might affect its value.Firstly, this option may be

intrinsically valuable to you.Maybe the concert involvesyour favourite musicianswhom you have waited yourwhole life to see play. Youwould certainly think anoptiontogotothisconcertisgoing to be valuable to you.Butforhowlongdoyouhavethe option? If the option isonly good for this evening,then that is pretty shortnotice. Maybe you haveanother engagement and

cannot attend. Clearly, itwouldbeabetteroption if itlasted longer; if your friendcan let you join him at theconcertanytimethisweekormonth, it is a lot morevaluabletoyou.Whatifyourfriend cannot be absolutelycertain who is playing at theconcert? This changes thingsagain.Ifyoucannotbesureitis your favourite musiciansplaying, going to the concertmaynotbeasgreatforyouas

you originally thought. Sothat uncertainty about theoutcome of exercising theoption, reduces its value inthiscase.Letustidythisupabit.Whatwe are saying is that if youhave the option to dosomething, the value of thatoption to you is going todepend on a number offactors. Firstly, whether theoutcome from exercising the

optionisintrinsicallygoodorbad. If I give you the optionaboutwhetherornot I stampon your toe, then this optionhas no obvious intrinsicallygood value for you whereasthe option, say, to receive afree lottery ticketprobably isintrinsically interesting toyou. Secondly, if there isuncertainty about theoutcome, that can affect thevalueoftheoption,positivelyand negatively. Suppose I

giveyoutheoptionthatIwilleither stamp on your toe orgive you $100, with a 50:50chance of either outcome.Thisoptionmightbevaluableto you (depending on howhighly you value your toes);sointhiscase,increasingtheuncertainty of the outcomeincreases the value of theoption. Thirdly, having anoption to do something forlonger, is generally morevaluable than having it for

only a short time. Forexample,ifyouhavethe“getstamped on/get $100″ optionforaweek,maybeyoucangoanbuysomesteel-toecappedboots and then exercise theoption? But if the option is‘now or never’, its value isprobablylowertoyou.Financialoptionsarenot justsimilar to options like these;theyareoptionsexactly suchas these. Financial options

give you the option, ratherthan an obligation, to tradesomething. Ifyouareoffereda call option, this just givesyou the option to buy aproductforacertainpricebya certain date. The value ofthisoption toyou isgoing todepend on whether the tradeis intrinsically valuable (i.e.whether trading the productvia the option is better thanbuying the product in themarket), how long you have

to exercise the option andalso how much uncertaintythere is in the price of theproduct. If the price of theproduct never changes andthe option has no intrinsicvaluerightnow,thenitneverwill have intrinsic value(because the price of theproduct will never move tomake the option valuable).But a bit of uncertainty overthefuturepriceoftheproductcould make the option

valuable now even though itis not intrinsically valuable;because the price of theproductmayriseinthefutureabove thepriceatwhichyouhavetheoptiontobuy.So,whatareoptions?Optionsin the financial markets aresimply choices, no differentin essence to the options wehave in everyday life. Theirterms and conditions arespelled out carefully in the

form of a contract and theytypically involve actuallybuying or selling somethingelse. But a lot of the ideasaroundthepricingandriskofoptions can easily beunderstood by simplyrecognising that financialoptions are ultimately justoptions!

Basicdefinitions

Let’s get some basicdefinitionsunderourbelt.Def #1 : An option is theright,butnottheobligationtotrade a certain amount of acertain product at a certainprice on or before a certaindate.Let’sbreakdownDef#1.a certain product....This isthe underlying product or

(spot product) to which theoptionrelates.to trade..... A call optiongives its owner the option tobuy. A put option gives itsownertheoptiontosell.a certain amount.... Thecontract multiplier. Eachoptionwillallowitsownertotradeapreciseamountof theunderlying product; it couldbe that eachoption relates to

100 shares or 1 one futurescontractforexample.a certain price.... This iscalled thestrikeprice of theoption.a certain date.... This is theexpirationdateoftheoption.A European option can onlybeexercisedatexpiration.AnAmerican option can beexercised at anydateor timebeforeoruptotheexpiration.

Note that this has nothing todo with geographywhatsoever; ‘American’ and‘European’ options trade allaroundtheworld!Examples:Example 1.1 The 115 putson company ZXY with amultiplierof50,expiringon16th February (Europeanoptions).

Theowneroftheseputshas the rightbutnot theobligation to sell 50 lots ofZXYsharesatapriceof115,on (and not before) the 16thFebruary.Example1.2The$110 callson crude oil futures,expiring in August(Americanoptions).Theownerofthesecallscanbuycrudeoilfutures

for $110. They can beexercisedanytimebeforetheexpiration date. The exactdate and contract multiplierarenotgiven.

CallandputpayoffsIfyoutradeanoptionnowata certain price, what will bethe profit or loss from thisoption when it expires? This

is the most fundamentalquestion about anoption andit is known as the option'spayoff profile. When theoptionexpires,theunderlyingproduct will be trading at acertain price and we mustunderstandwhat an option isworthatthatprice.Fortunately, this is easy tolearn. An option payoffprofilecanbecalculatedwithbasic arithmetic. Let's start

withacalloptionandlookata simple version of the basicdefinition.Acalloptiongivesitsowner

theoptiontobuytheunderlyingproductata

certainprice.Attheoption'sexpiration,theowner of the call can decidewhetherornottoexercisetheoption.Ifheexercises,hewillpurchase the underlying

product at the strike price ofthe call. Is this a good idea?Of course it depends on theprice of the underlyingproduct compared to thestrikepriceof thecall. If theunderlying product is worthmore than the strike price ofthe call, then exercising thecall optionmakes sense. Theownerofthecallwillthenbebuying the underlyingproduct for a price below itscurrent trading price. But, if

the underlying product iscurrently trading below thestrikepriceofthecall,itdoesnot make sense to exercisethe call.Whywouldhewanttopaythestrikepricefor theunderlying product if he cansimply pay the market pricewhichislower?Example 1.3 : A call optionhasastrikepriceof90.Attheexpiration of the call, theunderlying product is trading

at 100. Should the owner ofthe call option exercise thecall?Yes. By exercising the calloption, the owner of the callpays 90 for the underlyingproduct which is currentlytradingat100,givingaprofitof10.Thevalueof call optionsatexpiration

The call option in example1.3 is in-the-money. Itsstrikeis below the currentunderlying product price.What is thevalue this call atexpiration? The call must beworth 10. If the call wastrading at a price of 9, thenanyonecouldbuy thecall (ata cost of 9) and exercise thecall(atacostof90;thestrikeprice).The result is that theywould own the underlyingproduct of a total cost of 99.

Since the underlying productistradingat100,thismakesarisk-free profit of 1. So thecalls at expiration must beworth 10. If anyone waspreparedtopaymorethan10,they would be foolish. Bypaying say 11 for the callsandthenexercising(atacostof90),theypayatotalof101for the underlying productwhichisonlyworth100.So the value of an in-the-

money call option atexpiration is simply thedifference between the strikeprice and the price of theunderlying product. If a calloption is not in-the-money(i.e.itsstrikeisnotbelowthecurrent underlying productprice),thenitisworthless.AdjustingthepayofffortheoptiontradepriceOf course, no option is

available for free! Anybodywho is prepared to sell anoption(orasitisalsoreferredto,writeanoption),willwantsome compensation for therisk they are taking. Ifsomeonebuysanoption, thatamount must be deductedfrom the payoff, irrespectiveof the underlying productprice.Example 1.4 : John pays$1.50the$120strikecalls.At

expiration, the underlyingproduct is trading at $125.What is John's net profit orloss?First calculate the value ofthe option. Then deduct thepricepaidfortheoption.Thecall option is in-the-money(strike < spot price), so itsvalue is $125 - $120 = $5.John paid $1.50 for the call,sohisnetprofit is$5-$1.50=$3.50.

Profit and loss payoffs fortheoptionsellerThesellerofthecalloptioninExample 1.4 has a precisemirror image payoff profileto John. He collects $1.50from John when he sells thecalls, but at expiration thecalls are worth $5, so theseller has made a loss of$3.50.Anoteon limited/unlimited

payoffprofilesSomething important to notethat is demonstrated byExample 1.4 is that John'sdownside (the most he canlose)islimitedtotheamounthepaysforthecall.Johncannever lose more than the$1.50 he paid because theleast the call can ever beworth is zero. However, forthe seller of the option, hisloss is potentially unlimited.

The most he can make fromsellingthecallisthe$1.50hereceivesfromJohn.Butifthecall expires in-the-money,Johnwillexercisethecallandthe seller’s loss is thedifference between the spotprice and the strike (minusthe $1.50). In other words,the higher the spot price, themore the call seller can lose.This leads to some generalrulesaboutcalloptions.

Thelossfromowningacalloptionislimitedtothe

amountpaidfortheoption.Thegainispotentially

unlimited.Thelossfromsellingacall

optionispotentiallyunlimited.Thegainislimited

tothesaleprice.Putoptionpayoffprofiles

Puts and calls of the samestrike have lots in commonwithoneanother(asweshallseeinChapter2).Intermsofpayoff profiles, the ideas arevery similar. Just rememberthataputoption is related toselling theunderlying insteadof buying, so the interestingpart of the payoff profile iswhen the underlying pricefalls below the put optionstrike.

Let’sstartabovethestrike.If,when the put expires, theunderlying is trading at apriceabovethestrikepriceofthe put, the put is worthless.It isout-of-the-money.Why?Because exercising the putmeans selling the underlyingat the strike price and this isnot agood idea if the spot istradingabovethestrikeprice.The put owner would bebetter advised to just sell theunderlying product at the

higher price, rather thansellingviaexercisingtheput.The put is therefore onlyvaluable if the underlyingproduct price is below thestrike price at expiration. Insuch circumstance, the put isworth the difference betweenthe strike price and the spotprice. This is exactlyanalogous to the situationwith a call option. Butwhereas a call option is in-

the-moneyifthespotisabovethe strike price, a put is in-the-moneyifthespotpriceisbelowthestrike.AdjustingtheputprofitandlossforthetradedpriceJust as for call options (orindeed any trade), any putoption profit and loss profileneeds to be adjusted for theoriginaltradeprice.Whentheputhasbeenbought,theprice

paid must be deducted fromthe intrinsic value of the putatexpiration.Example 1.5 : John pays $1for the 99 strike puts. Atexpiration, the underlyingproductistradingat95.WhatisJohn’snetprofit-and-loss?The99putsatexpirationare$4 in-the-money. They areintrinsically worth $4. Johnpaid $1 for the puts, so his

netprofitis$3.

Assignment 1.1 :Watch VolcubeQuick Start video #1inLearningFollow the instructions inyour Volcube Confirmationemail to login to Volcube.Click on the Learning tab,then choose

Videos&Audiocasts from theleft hand menu. Choose theQuick Start videos and thenStarter Edition. In here youwillfindQuickStartvideo#1which gives you a 5 minuteintroduction to using theVolcubeapplication.

Assignment 1.2 :Call/PutPayoffMini-game

Click on the Game tab inVolcube. Choose OptionMinigames from the menu.Select the second game,CallandPutPayoffs.Thisstartsanew game which tests yourunderstandingofcallandputpayoffs. Answer at least 25questions and you will thenregisterapercentagescoreforthis minigame. Ideally youshouldpractisethisminigame(and indeed every minigamethatyouwillplayinVolcube)

until you are scoring almost100%.

Assignment 1.3 :Start a Level 1options tradingsimulationFinally, we are just going tostartaLevel1optionstradingsimulation to whet yourappetite for some simulated

options trading! Click on theGame tab, then chooseLevels.Thiswill showyouaLevel 1 simulation profile inVolcube. Don’t worry toomuch about all the data youcan see. All will becomeclearer as you progressthroughthiscourse.Fornow,justpressSinglePlayerontheright hand side and a newLevel 1 trading game shouldstartup.

Exercise11.1 Does the buyer of a putoptionhavetherighttosellorbuytheunderlyingproduct?1.2CanEuropeanoptionsbeexercisedbeforeexpiration?1.3Johnsellsanoption.DoesJohn have the right but nottheobligationtodoanything?1.4 John buys a call option.

Doeshewantthepriceoftheunderlying product to rise orfall?1.5Acalloptionhasastrikeof110.Thespot is tradingat85. Is the call in-the-moneyorout-of-the-money?1.6Canaputandacallofthesame strike both be in-the-moneyatthesametime?1.7 The spot product is

trading at $50 when theoptionsexpire.Whichismorevaluable; the $60 strike callsorthe$60strikeputs?1.8Johnsellsthe$110putsat$3. The spot product istradingat $107when theputoptions expire. What isJohn’sprofit-and-loss?1.9Johnsellsacalloption.Ishis profit-and-loss profilelimitedorunlimited?

1.10 John pays $0.50 for the90putsand$0.50forthe110calls. If the spot is tradingat$112 at expiration, what isJohn’sprofit-and-loss?

2.Optionvaluation

Optionality prior toexpirationIn part 1, we looked at thepayoff profiles of options atthemomentofexpiration.Wefoundthatcallsorputshadavalue of zero if they expiredout-of-the-money (or at-the-money). If they expired in-

the-money we found thattheir value was simply thedifference between theirstrikepriceandthespotprice.Butwhatisanoption’svaluebefore expiration? This infact is the more interestingsidetooptions.Atexpiration,an option is either worth anobvious,intrinsicamountoritis worthless. But whilst itlives and breathes, differentfactorsaffectitsvaluation.

It can be helpful to think ofanotherkindofoption to seewhat factors influence thevalue of options generallyprior to expiration. Supposeyouhavetheoptiontousemyumbrellaandyoumustdecideimmediately whether or notto exercise this option.Well,theoptiontousemyumbrellareally only has any value toyou if you are going outsideat thisverymoment and it israining.Thisoption,whichis

expiring imminently, is easyenoughtovalue.ButwhatifIinsteadofferyoutheoptiontouse my umbrella at any onetimeoverthenext3months?Whatwouldyoupay for thisoption (i.e. what do youconsider its value to be)?Suddenlylotsofextrafactorscome into play. Is it rainingrightnow? Is itgoing to rainevery day over the next 3months? Are you goingoutsideinthenext3months?

The point is that an optionthat expires almostimmediately is easy enoughtovalue;either ithasavalueor it does not. This value isknown as the intrinsic valueof an option. For options(whether regarding umbrellausage or trading financialinstruments) that have alongerlifespan,thevaluecanalso be made up of othercomponents. And it ispossible for anoption to still

have value even when it isnot currently intrinsicallyvaluable. Even if it is notraining right now, the optionto use my umbrella in thenext three months may stillhave a value to you. So it iswith financial options; theymaynotbeintrinsicallyvalueright now, but that couldchangebeforetheyexpire.

Intrinsic & extrinsicvalueThe value of an option ismadeupof two components.An option can have intrinsicvalue and it can haveextrinsicvalue.The intrinsic value just tellsustheamountanoptionmustbe worth by virtue of itsstrike price in relation to thespotprice(andwhetheritisa

calloraput).Forexample,ifthespotistradingat$100,the$90 call gives its owner theright (but not the obligation)tobuythespotat$90.Thisis$10 below the current spotprice, so thecalloptionmusthave an intrinsic value of atleast $10. Again, this is sobecauseifthecallwastradingat less than $10, it would bepossible to buy the call and(in the case of Americanoptions)immediatelyexercise

the call in order to buy thespot for a net price belowwhere it is currently trading.This is a risk-free arbitrageprofitthatsimplycannotexistinthemarket(atleastnotforvery long!). So, if the callwas being offered for sale at$5, we could buy the call(paying $5), exercise it (i.e.pay $90 for the spot) for atotaloutlayof$95.Wecouldthen sell the spotback to themarket at $100, making $5

risk-free profit. (Because inthecaseofEuropeanoptions,wecould simplybuy thecalland sell the spot productshort. When the optionexpires, we exercise the calloption in order to buy backthe spot at the cheaper price.In other words, the principleis the same, we just have towait until expiration for thepositiontobeclosedout).Sothecallmustbeworthatleast$10 otherwise an arbitrage

opportunityexists.If an option has not yetexpired, it can also haveextrinsic value. Extrinsicvalueisthatcomponentofanoption’s value that reflectsthe fact that the option stillhasoptionality.Theoptiontouse my umbrella may nothaveintrinsicvalueifitisnotcurrently raining. But if rainis expected in the next fewmonths, theoption touse the

umbrella may still bevaluable. So it is withfinancial options. If the spotproductistradingat$100,the$90 puts have no intrinsicvalue. (Who wants to sellsomething at $90 when it istrading at $100?). But if theoptions have enough timeuntilexpiration,theymaystillhave value. Why is this?Well, as per the rain thatcomes and goes, the price ofthe spot product may vary.

Perhaps in a few monthstime, the spotwill be tradingat$80, inwhichcase theputoptions are in-the-money andhave $10 of intrinsic value.So the extrinsic valuecaptures the optionalitycomponent of an option’svalue.Itreflectsthevaluethatis owing to the fact that theoption could become moreintrinsically valuable in thefuture, because it still hastimeleftuntilitexpires.

Put-callparityPut-call parity describes animportant relationshipbetween puts and calls.Essentially, it suggests putsand call are fundamentallythe same thing!Butwe needtobemorespecifichereaboutwhich puts and which callsare so very alike. Put-callparity applies to puts andcalls of the same strike andexpiration date, sharing the

same underlying product. (Inits strictest form, it appliesonly to options wheredividends and carry cost arenot a factor; but let’s notworry about this for now).Whatwearesayingisthatthe$100 puts and $100 calls onthe same spot with the sameexpiration share everythingthat is important aboutoptions.To understand why put-call

parity holds, it is easiest torecall the payoff profiles ofcalls and puts. By workingthrough an example of the$100callsand$100putsandconsidering the payoff atexpiration, you should noticea symmetry. If the spot istradingat$110,thecallshave$10 of intrinsic value, theputs have $0 of intrinsicvalue. With the spot tradingat$90,itistheputsthathave$10ofintrinsicvalueandthe

callsthatareworthless.Now,it is possible to syntheticallyre-create the exact payoffprofile of the $100 calls,using the $100 puts and theunderlying. By simplyhedging the $100 puts withthe underlying (which weassumeis tradingat$100fornow), we create a synthetic$100 call. Suppose we ownthe $100 puts and wish tosynthetically transform theseputs into the $100 calls.We

cando thisbysimplybuyingthe spot (assume it is trading$100), one-for-one with theput. Consider what happensto the payoff profile of thehedged put. Versus $90, theputsstillhave$10ofintrinsicvalue, but the spot trade haslost $10 (because we paid$100and it isnow tradingat$90). So overall, the profitandloss iszerowiththespotat$90.Withthespotat$110,the put is worthless but the

spot trade makes a profit of$10;soournetprofitandlossis $10. This is exactly thesame payoff profile as the$100callsalone.Put-call parity is a veryimportant result for optiontraders. It means that insimple cases the onlydifferenceinvaluebetweenacall and a put of the samestrike and expiration is theintrinsicvalue.Intrinsicvalue

is simply the value that thein-the-money option musthave by virtue of being in-the-money(i.e.thedifferencebetweenitsstrikeandthespotprice). Extrinsic valuecontains the remainder of anoption’s value and, due toput-call parity, it will be thesameforputsandcallsharingthe same strike andexpiration.

The three mostimportant factorsthat affect optionvalueMany factors affect anoption’s value before itexpires. However, three ofthese factors usuallydominatethevaluationandsothese are the most crucial tounderstand.

Important factor #1 : Thespot price relative to thestrikeprice.Herewe are justtalking about the intrinsicvalue of an option. If a callhas a strike of $100 and thespot is trading at $110, thecall must be worth at least$10.Butwiththespotat$90,the call would have zerointrinsicvalue.Soclearly thespot price relative to thestrikepriceisimportant.

Important factor #2 : Thelength of time until expiry.An option that expires in 5minutesandisfarout-of-the-money,isprobablyworthless.If an option on the sameunderlying with the samestrike has 12 months until itexpires, it could still bevaluable. So, other thingsbeing equal, the longer anoptionhasuntilitexpires,themoreitisworth.

Important factor #3 : Theexpectedlevelofvolatilityinthe underlying product. Themorevolatilethespotproductis expected to be during thelife of the option, the morevaluable that option, ingeneral, will be.Why is thisthe case? Well, consider anoption that is out-of-the-money and suppose that theunderlying product priceneveraltered.Thisoptionhasnochanceofeverexpiringin-

the-money and so it isessentially worthless.Whereas if the underlyingproduct is very volatile, thechancesarefarhigherthattheoption may finish its life in-the-money. This illustratesprettysimplyhowtheamountof volatility that is expectedduring an options life in thespot product price positivelyaffects option value. Theexpected levelofvolatility ismore commonly known as

theimpliedvolatilitylevel.

Assignment 2.1 :Intrinsic/extrinsicmini-gameLoad the Option Mini-gameIntrinsic/Extrinsic. You willbepresentedwithfourpiecesof information and need tocalculate the missing fielddenoted by ???. Remember

that the total value of anoption is the sum of itsintrinsic andextrinsicvalues.The intrinsic value is eitherzero for out- (or at-)-the-money options or it thedifferencebetweentheoptionstrike and the spot price forin-the-money options.Attemptat least25questionsinordertoregisteryourscoreforthisminigame.

Assignment 2.2 :Study the PricingSheetStart a Volcube simulation(either via Levels or CustomPlay). Consider the PricingSheetinthetopmiddleofthepane. In the centre is acolumnlabelled‘Strike’.Thisshowsthestrikepricesof thedifferentoptionsinthisgame.To the left of this column,

you will see a columnlabelled CALL THEO. Thiscolumnshows the theoreticalvalues of call options in thisgame,giventheirstrikeprice.To the right of the strikecolumn, you will see theequivalent theoretical valuecolumn for puts options.These values have beengenerated by an optionspricing model; amathematical formula thattakes into account the

different factors that affectoptionvaluesandgeneratesavalue.Spend some time studyingthis pricing sheet. Notice inparticular the difference inprice between calls and putsofthesamestrike.Rememberthat when put-call parityholds (as it does inVolcube)theonlydifferencebetweenaput and call’s value is theintrinsic value of one of the

options. The default value ofthe spot in Custom Play andinLevel1gamesis$100.The$90 calls have a value of$10.338. Notice that the $90puts are worth $0.338. Thedifference between the twooptions’values is$10,whichis the intrinsic value of the$90 calls (i.e. the amount bywhich they are in-the-money).By considering the Pricing

Sheet, a great deal can belearntaboutoptionvaluesandthe relationships betweencalls and puts of differing,andthesame,strike.

Exercise22.1 If the spot is trading at$100, what is the intrinsicvalue of the $85 puts? a)$15b)0c)No-onecansay

2.2 If the spot is trading at$50, what is the extrinsicvalue of the $45 calls? a)Zero b) $5 c) No-one cansay2.3 The $75 calls are $5 in-the-money and have $1 ofextrinsic value. What is thevalueofthe$75puts(assumesame expiration and sameunderlyingasthecalls)ifput-callparityholds?

2.4IfIbuythe$100callsandsellthe$100putsatzerocost(i.e.thepricesofthecallsandthe puts are equal), what ismy position equivalent to intermsofthespotproduct?2.5At expiration, do optionshaveextrinsicvalue, intrinsicvalueorboth?2.6 What are the three mostimportant factors that affectan option’s value prior to

expiration?2.7Istheintrinsicvalueofanoptiondependenton the timeremaininguntilexpiry?2.8 Are options with longeruntil expiry more or lessvaluable than options withshorter lifespans but in allotherrespectsidentical?2.9 A long put option pluslong spot position is

equivalent to what callposition, assuming put-callparityholds?2.10 Does higher implied(expected) volatility ingeneral increase or decreasethevalueof anoption?Doesthechangeinvalueaffecttheintrinsic or extrinsiccomponent of the option’svalue?

3. Ways to tradeoptionsOptions are amongst the most flexiblefinancial instruments in the market.This isbecauseoptionscanbeusedbyinvestors and traders to accuratelyreflect theirpreferenceswithrespect toa number of different factors. Forexample,optionscanbeusedbytraderswhoarebullishontheunderlyingassetprice or bearish or even who aredirectionallyneutralonthepriceoftheunderlying. Investors who expect alargemovementintheunderlyingpricebutarenotcertainaboutwhetheritwillbe up or down. Investors who already

own the underlying and are looking toprotect their investment. Traders whowanttouseoptionstocreateanincomestream in addition to owning theunderlying. All of these situations andmany more can be catered for withrelatively simple option strategies andportfolios. And this goes someway toexplain thepopularityofoptions.Hereare some of the most commonmotivationsfortradingoptions.

Using options totradedirectionallyOptionscanbeusedtogainexposureto

the price of the underlying product.This is known as trading optionsdirectionally. For example, if aninvestor is bullish with respect to theprice of the underlying product, hemight decide to buy some call optionstoprofitfromanyrallyinthespotprice.He could, with the same end goal inmind,sellsomeputoptionsshort.Bothoftheseareexamplesoftradingoptionstogainexposure to thedirectionof theprice of the underlying. Combinationsofoptions (knownasoptionstrategies)can be traded to reflect more precisedirectionalbiases.For instancea traderwho is bullish on the underlying, butonlyexpectsarallyuptoacertainpriceand no higher, can trade call spreads(alsoknownasbullspreads)ratherthan

asinglecalloption.Thereasontotradea strategy rather thana simpleoutrightis that themore precisely a trader canspecifyhisexpectations,themorecost-effectively the option portfolio can beconstructed.Inthecaseofabullspread,the trader partially finances thepurchaseofonecallwiththeshortsaleofacallofahigherstrike.Helimitshisupside potential from a rally in theunderlying; but if the upper strike ischosentobebeyondthelevelwhichthetraderreasonablyexpectstosee,thenhecannot be sorry. Option portfolios canbebuilttoreflectandprofitfromalmostanypermutationofspotpriceoutcomesthatthetraderexpects.

Using options tohedge an existingpositionAn extension to using options to tradethe direction of the price of theunderlying, is to protect an existingportfolio against price changes. Forexample,supposetheinvestorownstheunderlyingproductandheisconcernedthat the price will fall. If he sells theunderlying product to reduce hisportfolio,heofcoursewillnotbenefitifthe underlying product pricesubsequentlyrises.An alternative is touse options as a form of priceinsurance.Forexample,bybuyingaput

option on the underlying, the investormay protect his asset in part or in fullagainst the drop in price. If theunderlying product price rises, theinvestorwill still profit by owning theassetbuthewilllosethepremiumpaidfor the put (insurance). So options canbe used to alter the risk profile of aportfolio to suit the preference of theinvestor.

Using options togenerateincomeOptions can be used to generate anincomeor toenhance the returnsofan

existing portfolio. The basic ideabehindthisstrategyisthatoptionshaveextrinsic valuewhichwill all be ‘lost’by the time the options expire. So, thethinkingisthatbysellingsomeoptions,this fall in value can be captured. Ofcourse, there are risks associated withselling options; whilst the extrinsicvalueowingtotimewillindeedfall,theintrinsic value of the option atexpiration may be zero, but it may bevery different from zero. The classicexampleof thiswayof tradingoptionsistheso-called‘coveredcall’.Theideais that the owner of some asset (say astock) can enhance his return on thisasset by selling out-of-the-money calloptionsonthestock.Ifthestockfallsinprice,helosesmoneysinceheownsthe

stock,butthecallswillexpireworthlessand hewill recover some or all of hislosses on the stock via this source ofincome. On the flipside, if the stockrallies,theinvestorprofitsfromtherisein the stock’svalue.Ofcoursehemaythen lose money on the short callposition.As with every trading strategy, this isnotawin-winscenariobutariskversusreward trade-off. Nevertheless, optionscan be used in this way either toenhance a portfolio’s return (withskillful management) or at least withtheintentionofsmoothingthereturns.

Using options totradevolatilityOne of the fundamental reasons fortradingoptionsistogainexposuretoorprotectionagainstvolatility.Recallthatoneofthethreeessentialcomponentsofoption value is the expected level ofvolatility in theunderlyingproduct.Byhedging away the exposure to the spotprice movements (i.e. by delta-hedging),volatilityisleftastheprimarydriveroftheoption’svalue.Whywouldtraders or investors want to ‘trade’volatility? Increasingly, volatility isviewedasanassetclassinitsownright.As volatility impacts on so many

trading and investment strategies, it isnowimperativefortraderstobeabletomanagethisriskor tocapitaliseon theopportunitiesitaffords.Whilst options can be used simply totrade the direction of the spot product,this is rather like only using a car infirstgear.Totradevolatilityviaoptionsrequires traders to have a fullerunderstanding of options’ capabilities,potentialandresponsetochangingriskfactors.Mastering any style of optionstrading requires an essentialunderstanding of options as volatilitytradinginstruments.

Market makingoptionsMarket making is an activity wheretradersarepreparedtoshowabidpriceand an offer price in an instrument.Insteadofonlyshowingapricetheyarepreparedtopayoronlyshowingapriceatwhich theywill sell,marketmakersalways show two-way prices wherethey are prepared to buy or sell. Thebuying price they arewilling to pay isalways lower than the price at whichtheyarewillingtosellandtheyhopetomakeaprofitbyconsistentlybuyingatthelowerpriceandsellingatthehigherprice. This represents their reward for

marketmaking.Theirriskfrommarketmaking is related to the inventory thattheymightacquirewhilsttheyarebusybuying and selling. For example, iftherearea successionof sellingordersfrom the rest of the market to themarketmakers, then themarketmakerwillstarttoaccumulatealongposition.The risk is that the value of theseinstruments will fall; and if marketparticipants are consistently selling,thenafallingprice,economicstellsus,willbetheresult.Sothemarketmakerof any instrument has to manage therisk associated with the inventory heacquiresuntilheisabletoliquidatetheposition.Market making options is a core

business function of many banks andproprietary derivatives trading firms.They will show bid and ask prices inmany different option contracts andmanage the risk of any position thataccrues. The risks relating to optionsaremanyinnumber,buttherearemanyopportunities and techniques formanagingtheseriskseffectively.Successful option market makingrequiresavery thoroughunderstandingofoptions;ofhowtheyarepriced,howthey are affected by changingcircumstances and how they relate tooneanother.Weshall refer to someofthese techniques throughout theremainderof thisbook. If thesecanbefully understood, a very large step

towards mastering options will havebeentaken.

Assignment 3.1 Put-callparityMini-gameLoad the put-call parity Mini-game.This will test your understanding ofput-call parity in practice. There areinstructions on how to play the gamealongside each question as well asdetailed examples atwww.volcube.com/support/volcube-tutorials/. The basic idea is that whenput-call parity holds, the difference

between the value of a call and put ofthe same strike, is simply the intrinsicvalue. The intrinsic value is also thedifference between the spot price andthe strike price. So these two factsallow us to always calculate either thecallvalue,ortheputvalue,orthestrikeprice or the spot pricewhenwe knowtheotherthreevalues.

Assignment 3.2 Playa Level 1 game andentersomequotesLoad a Level 1 trading simulation in

Volcube.YouwillbetakentotheMainGamescreenfor thissimulation.InthetoprightistheMessengerwindow.Thebroker should be asking your for aquote in some calls or some puts;something like “Jan 90 puts?” or “Sep109calls?”.InthemiddlesectionoftheMessenger window you will see theoptions being quoted and the word“Theo:” with a value alongside. Thisshows you a theoretical value for theoptions (more on this in the nextchapter). In the lower half of theMessenger pane is a padwhich allowsyou to adjust your bid and ask pricesandquantities.Thisallowsyoutomakeaquoteresponsetothebroker,showinghimapriceyouarepreparedtopayfortheoptionsandapriceatwhichyouare

prepared tosell.Fornow,youcan justmakethebidpriceacoupleofpenniesbelowthetheoreticalvalueandtheaskpriceacoupleabove. Ifyou then tradeon these prices you will make atheoretical profit because you arebuying below theoretical value andselling above theoretical value. Entersomequotes,perhapsmakesometradesand get used to the whole process.Welcometooptionmarketmaking!

Assignment 3.3WatchBlitzVideo#2:Makingamarketin

optionsThis video covers some of these basicideasaboutbidsandoffersandmarketmaking in general.Afterwatching, tryplayingmoreLevel1games.

Exercise33.1 A hedge fund is short crude oilfutures. If they want to hedge theposition using crude oil futures calloptions,wouldtheybuyorsellcalls?3.2 Are these directional option playsbullishorbearish?a) longputb) shortcallc)shortputandlongcall

3.3 In general, are non-delta hedgedoptionpositions(e.g.asimple,outrightput position) exposed to changes inimpliedvolatility?3.4 What are the primaryrisks/exposurestoadelta-hedgedoptionposition?3.5 How can options be used togenerateincome?3.6Atraderbuysacallandaputofthesame strike, equal to the current spotprice.What is his immediate exposuretoa)thepriceofthespotproductb)theexpectedvolatilityofthespotproduct?

3.7Which statement best characterisestheroleofamarketmaker?a)Takessignificantdirectionalpositionsb)Tradesonlyseldomlyandonlywhentheywanttoc)Tradesfrequentlyandhopestominimisetheiroverallposition3.8Willanoptionmarketmaker’soffer(ask) price ever be lower than his bidprice?

4.PricingoptionsPrior to the 1960s and 1970s, optionswere valued largely by intuition aloneandtheywerepricedsimplybymarketforces. The price of individual optionswas determined by the supply anddemand in the particular contracts andthe risks associatedwith those optionswere not fully understood. Fordirectional traders of options (such asagricultural firms looking to hedgepricemovementsingrains)thiswasnottooproblematic;becausethepayoffsatexpirationoftheoptionswereknownaswell as the current price, the optionswerestillperfectlyusableasahedgingorspeculativetool.

This changed with the widespreadintroduction of mathematical models,most notably theBlack Scholesmodelof 1973 which remains an industrystandard. Although it is not vital tounderstand the inner workings of suchmodels in order to successfully tradeoptions,itisimportanttounderstand,atleastinpart,howtheyareused,someoftheassumptionsonwhichtheyarebuilt,aswellassomeoftheirlimitations.

Valuing options withmodels

Anoption pricingmodelwill typicallytake a number of factors that mayinfluence the option’s value andcombine this mathematically togenerate a single theoretical value forthe option. As discussed in Chapter 2,the most important factors tend to bethe price of the spot product, the timeremaining until the option’s expirationand the volatility in the spot price thatwill occur over the option’s life. Nowthis latter factor (unlike the others) isunknown, so traders use the level ofvolatility that they expect to occur.Other factors such as the cost of carryor expected dividend yields may beincluded if appropriate (for examplewith stock options).By plugging thesenumbers into the model and by

assuming that the price changes in thespot product will, in the long run,resemble certain probabilitydistributions,avalue foranoptioncanbe computed. Options in the Volcubesimulation games are all valued usingtheBlackScholespricingmodelinthisveryway.Various option pricing toolsare available online and it is worthplaying around with one to see howoptionvaluesvaryastheinputschange.It is important to understand the keydistinctionbetween thedifferent inputsto the model. Some are generallyknownwithcertainty,suchasthepriceof the underlying or the date ofexpiration. Others are not known for

certainandthemostimportantfactorinthis category tends to be the level ofexpected volatility. Traders maydisagreeoverthe‘correct’valuetouseand this can be a reason to trade.Traderscanlookatthepriceofoptionsandusingthemodeltheycanimplytheexpected level of volatility that themarketispricingintotheoptions.Fromthis, they may value options as toocheap or too rich and hence decide totrade.

Things to considerwhen pricing options

totradePricing options means different thingstodifferentpeople.Marketmakerswillhave a different approach to saydirectional traders. Nevertheless,manyoftheconsiderationsarecommontoall.Soherearesomethingstoconsider:The theoretical value of the option.Thiswillbetheresultofamathematicalmodel and, remember this, it issubjective and personal to whoeverbuilds/uses the model. Some optiontraderswillmakecomparisonsbetweentheir theoretical value and the currentmarket value and arrive at tradingdecisions. Such as, “Given my

theoretical valueof this option, I thinkthe market price is too cheap andtherefore I will buy”. For marketmakers, the theoreticalvalue isusuallytaken as a guide and a reference pricefor previous trades. When marketmakersmaketwo-wayprices(bidsandoffers) in an option, they usuallyconsider the theoreticalvalue that theirmodel suggests and then considerwhether this should be adjusted in anywaybecauseofwhattheyhavealreadyseentradinginthemarket.Therewardoftradingtheoptionatacertain price. When consideringtrading an option at a certain price(eitherbyinitiatingatradeorbyacting

asamarketmakerandshowingtradablebidandaskprices),thepotentialrewardfrom the trade is a key factor. Formarket makers, this is typicallyencapsulated by the amount of ‘edge’theyarereceivingforthetrade,i.e. thedifferencebetweenthetheoreticalvalueand thepriceatwhich theyarehopingtotrade.Forexample,ifamarketmakershows a 6 bid in an option that hetheoretically values at 7, then hispotentialrewardfromtradingonhisbidprice is 1 tick. For other traders, theymay look at the potential reward interms of expiration payoff profilesversusthetradedprice.Suchas,“Icansellthiscallat6ticksandifthespotistrading below the strike price atexpiration,Iwillwinthewhole6ticks.

Thatismypotentialreward.”The risk associated with trading theoption.The flip side to reward is risk.Howmuchcouldwelosefromatrade?Ifthemarketmakerpays6whenhehasa theoreticalvalueof7,his risk is thatthevaluefallsbelow6.Ifthishappens,hisprofitevaporates.Howfarmightthevaluefallfrom7beforethetraderhasachance to liquidate the position? Andwhatmightcause thevalue tochange?For options, these are reasonablycomplexissuesthatonlyhardworkandpractice can really teach.The essentialpointaboutriskthoughisalwaysworthbearing in mind when pricing options.What should concern us and guide an

assessmentofriskarethefactorswhichcan influence an option’s future value.MoreonthisinChapter5.The option in the context of anyexisting position. When a trader isconsidering a trade, or amarketmakershows prices on which he mayreasonably be expected to trade, hisexisting inventory has to be aconsideration.Does the tradeadd toorreduce the position? Will the overallriskof theportfoliobehigheror lowerafterthetrade?Formarketmakers,thisis an essential consideration whenpricing options. It also reflects widerissuesaroundmarketorder flow. If theflowofordershasgenerallybeentosell

options (tomarketmakers), it is likelythat options have become cheaper andthatmarketmakersaresittingonoptionportfolios that are broadly long innature. Traders will therefore have toweightheriskofaddingtothispositionagainsttheextrareward(intheformofcheaper options). Adjusting pricing tosuitatrader’scurrentbookandinawayappropriate for the current flow ofordersisacrucialskillthattradersmustlearntobesuccessful.Notice that these ideas,ataconceptuallevel, really apply to almost any assetthat one might consider trading. Anotion of ‘fair’ or theoretical value,some ideaof therisk/rewardandone’s

currentexposuretotheassetormarket,are all sensible things to assess beforepricing or deciding to trade. Optionshavetheirownidiosyncrasiesofcourse,butthisbasicchecklistisworthkeepinginmindasonebegins.

Assignment 4.1Buy/SellMini-gameLoad up the Buy/Sell Mini-game inVolcube via the Game tab. This is asimplebuteffectivegametogiveyouagood grasp of market terminology.People new to trading often strugglewiththedifferencebetweenbids,offers,

asking prices and people shouting‘Sold!’, ‘Take ‘em!’ or ‘Yours!’. Thisgame will make sure you becomefamiliar with these essential basics.After20or so tries,yourcurrent scoreshould appear. Play until you arescoring100%!

Assignment 4.2 StartasimpleCustomPlaygame. Use VolcubeMarket Mentor togive you advice on

pricingStart a simulation in Volcube viaCustom Play in the Game tab. In theMessengerpaneofyourgameyouwillsee a green dialogue bubble labelled‘MM’. This is the Volcube MarketMentor which gives you intelligentadvice during your game that reflectsthe current state of play, your existingposition and the current market orderflow. Use the Market Mentor to gainhelponpricingoptions.Youwillseeaselection of help ideas in the MarketMentor menu, covering what to thinkaboutwhen pricing or even suggestingasensiblepriceforyouinthegameyouareplaying.

Assignment 4.3WatchBlitzVideo#3: The differencebetween theoreticalandmarketvalueThis video explains the differencebetween theoretical and market value.Understanding this difference inpractice is particular important foroption market makers or other optionvolatility traders. It also applies moregenerallytoanytraderswhousemodelsto generate theoretical values forfinancialinstruments.

Exercise44.1Anoptionexchangeisshowingthelivemarket in an option as 6 bid, at 7offered.Are these theoretical prices ormarketprices?4.2Will all participants in the markethave the same theoretical value for anoption?4.3 Which matters more whenconsideringanopportunitytotrade:theriskofanoptionorthereward?4.4Underwhat circumstanceswill thetheoretical value of an option prove tobe the true, accurate and fair value of

theoption?4.5 For market makers, how is thetheoretical profit and loss from a tradeinanoptioncalculated?4.6Whatistheactual(ormarket)profitfromthesametrade?4.7Amarketmakeriscurrentlyshortalot of options (i.e. he has sold optionshe does not already own). Will hegenerallyshowa)higherbidsb)higheroffers c) higher bids and higher offersd)lowerbidsandloweroffers?4.8 What is the reward that a marketmakerhopestocapture?

5. Managing optionrisk:theGreeksThe risk to a trader with aportfolio of assets is that thevaluesofthoseassetschange.This is justas trueforoptiontraders as it is for stocktraders, investment fundmanagers or indeed asset-

owningbusinesses.Whatcancause the value of options tochange? The value of assetschange when there is achange in the factors thataffect their value. The valueof a house changes when itslocation becomes more orless desirable or when itsowners renovate or let it falltoruin.Foroptions,thereareanalogouschangesthatmatterand these are changes to thefactors that affect option

values (see Chapter 2). Forexample, when the price ofthe underlying productchanges,thiscanaffectaffectoption values (positively ornegatively). If the spot pricefalls,generallyspeakingcallsbecome less valuable andputsmorevaluable.Option traders are able toquantify this risk. What thismeans is that they candetermine how a change in

onefactorwill,indollarsandcents, affect the value of anoption. This is essential forriskmanagementpurposes. Itcanguidethemontheoverallmagnitude of their risk andalso on how to mitigate thisriskbyhedgingappropriately.

What are theGreeks?

Option risk measures areknown collectively as theGreeks. They are derivedfrom the same mathematicalmodels that generatetheoreticaloptionvalues.ThebasicGreeks typically tell ushow much an option’s valuewill change when a certainother factor changes. Forexample, option delta tellshow much an option’s valuechanges when the spot pricechanges.Itisoftengivenasa

percentage number. So wemighthaveacallwitha50%(or0.5)delta.Thismeansthatforeverypennythespotrisesin price, the call option’svalue will increase by half apenny. There are Greeks forall of the factors that affectoptionvaluessuchasthespotprice, the implied volatility,thetimetoexpiryetc.With options, things arerarely constant, simple or

linear! So an addedcomplication is that theGreeks themselves are notusuallyconstant.Forexamplethe option delta is not likelyto stay at 50% if somethingelse changes; the ‘somethingelse’ could be any of theusual important factors, suchas the spot price, the time toexpiry, the implied volatilityetc. There areGreeks knownas ‘higher order’Greeks thattell us by how much lower

order Greeks will changewhen something elsechanges.To summarise, we measureoption risk using theGreeks.Thesegiveusactualnumberstelling how sensitive ouroption’s values are to certainchangeable factors. TheseGreeks are themselves alsoliable to change and we canknow about the change in aGreek by using another,

higherorderGreek.Thismaysound convoluted when firstencountered, but it is thefundamental basis of goodoption risk management.Now let’s take a closer lookat some important Greeks inparticular.

Delta,vega,thetaandgammaDelta tells us the change in

option value for a change inthe spot price. Calls havepositive delta and puts havenegative delta. If the spotprice falls, generally calloptionsareworthlessandputoptions worth more, becauseof the sign of call and putdeltas.Delta is an invaluableGreek which tells optiontraders about their greatestexposure and also informsthem on how to hedge thisrisk away if they so wish.

Optionmarketmakersusetheoption delta to delta-hedge,whichmeanstoneutralisetherisk associated with the spotmarketmoves.Forexampleifatraderbuys100calloptionswith a 50% delta, then todelta hedge hemust sellhalfas many lots of theunderlying (50 lots in thiscase, assumingaone foronecontract multiplier i.e. thatone option contract pertainstoone lot of theunderlying).

If the spot market now fallsby say 10 cents, the optionvaluewill decline by 5 cents(because it has a 50%delta).But the hedged trader iscoveredforthislossbybeingshorthalfasmanylotsofthespot product which havefallen 10 cents. Thus, he hasneutralised the effect of thespot price movement bydelta-hedging.Vega tells us the change in

option value for a change inthe expected level ofvolatility.Vegaispositiveforboth calls and puts. This isbecauseif thespotproduct isexpected to bemorevolatile,all options have a betterchance of finishing in-the-money. (Tosee this, imaginean out-of-the-money optionon a product whose priceneverreallyalterednorwasitever expected to; the optionwouldbeworthless.Butifthe

expected level of volatilityincreased, the option couldnow have a chance ofexpiringin-the-money).Vegais an important risk measurefor traders, particularly forvolatility traders or optionmarket makers. Spot pricemovements can be relativelyeasilyhedgedaway(bydelta-hedging),whereaschangesinimpliedvolatilitycanonlybehedged using other options.Vega is additive for options

struckonthesameunderlyingwiththesameexpirationdate.This means the vega of twosuch options that a traderownscansimplybeadded togive the total,portfoliovega.The vega from any shortoption position would besubtracted. Vega is oftenshownasanormalized,dollaramount. For example aportfolio might have a totalvega of $1500. This meansthat if the implied volatility

moves by 1% (and tradersusually mean 1% as in from25% implied volatility to26%) the value of theportfolio will increase by$1500. A fall in impliedvolatility by 0.5% wouldresultinlossesof$750.Theta tells us the change inoption value for a change inthe time remaining to anoption’s expiration. Noticethat delta, vega and theta

cover the three mostimportant factors that affectoptionvalue.Thetaisusuallynormalized to show us thechangeinoptionvalueforthepassing of a single day. Forexample, an option worth$1.55 might have 30 daysuntil it expires and a thetavalue of 5 cents. Thismeansthat other things being equalthe option tomorrow will beworth $1.50. This fall invalue is a result of the

option’s time value (or itsextrinsicvalue)declining.Astime passes, an option’soptionality reduces. Thetatells us the dollar-amount ofthis fall. Theta is additiveacross options (and notsimply options struck on thesame product with the sameexpiration).Gamma is perhaps the mostimportant higher orderGreek. Gamma tells us how

much an option’s deltachanges,withachangeinthespot price. All options havepositive gamma. This meansthatthedeltaofanoptionwillvary positively with changesin the spot price. Forexample,supposeacallhasa50% delta, given the currentspot price. If the spot pricerallies, will the call deltaincrease or decrease? Thedelta will increase becausethe call has positive gamma,

meaning an increase in spotprice leads to an increase indelta.For put options, we have tobe a little careful becausetheir delta’s are negative tobegin with. So, as with thecalloption,anincreasingspotprice leads to an increasingput delta, but remember thismeans a less negative putdelta.Forexampleaputmayhave a delta of (-)10% at acertain spot price. Suppose

thespotpricerallies.Theputdelta will now be smaller inabsolute terms (for example(-)5%) although strictlyspeakingthedeltaislarge(innominalterms).Gammaisimportanttooptiontraders, especially those whoare generally aiming to bedelta-neutral. An optionportfolio canbedelta-hedgedto become delta-neutral. Butif the portfolio has gamma,

this delta-hedgemay only betemporarilyeffective.Gammais therefore the mostimportant higher-orderGreekfor option traders tounderstand.

Assignment 5.1 PlayaLevel1game.Enteraquoteresponseandmake a trade.

Analyseyourrisk.Start a Level 1 Volcube simulation.Respond to the quote request from thebrokerbyshowingabidandanask intheoptionstrategyheisquoting.Trytotrade whatever order he shows you. Ifthereisnoorderinthisiteration,goonto the next iteration and try to tradethere instead. Now try to analyse theRiskDetail pane. TheRiskDetail is amatrix which would be familiar to alladvanced option traders. It shows thevalueofvariousoptionGreeksforyourcurrentportfolio.Thehorizontalrowofnumbers across the top of the riskmatrix shows various different spotprices. The central column therefore

shows the value of the variousGreeksversus thecurrent spot price.Columnsto the left of this show the value ofGreeks for lower spot prices andcolumnstotherightshowGreekvaluesforhigherspotprices.Rednumbersarenegativeandgreenarepositive.Traders will use a risk matrix such asthistogivethemadetailedinsightintotheirportfolio’sriskandcharacter.Startto familiarise yourself with the limit,askyourselfwhat thenumbersactuallymean in practise and notice how theychangewhen other different trades areaddedtotheportfolio.

Assignment 5.2 UseMarket Mentor togiveyouadviceaboutoptionrisk.The Volcube Market Mentor is an in-game intelligent help system that canprovide you with real-time adviceconcerning your current game. Asmentioned in part 4, Market Mentorprovides advice regarding pricing, riskmanagement, trade selection andunderstanding market flows. Makesome trades in the current game andthen study the Risk Detail alongsidestudying the advice of the Market

Mentor. This should increase yourunderstanding of the interpretation andmeaningoftheoptionGreeks.

Exercise55.1A call option has a deltaof +40%. If the spot productfalls by 30 cents, by howmuch does the option rise orfallinvalue?5.2 An option has atheoretical value of $1.55.The option has a theta of 5

cents. Other things beingequal , what will be theapproximate option value 24hoursfromnow?5.3 Implied volatility is 25%and an option has a value of$1.55and10vega.Ifimpliedvolatility increases to 26%,what is the approximatechangeintheoption'svalue?5.4Anoptionhas a valueof$1.55 and 40% delta. It has10 gamma, defined as thechangeindeltafora$1move

in the underlying. If theunderlying rallies 20 cents,whatisthenewoptiondelta?5.5Adeep in-the-moneycallhas a 100% delta and eachcallrefersto100contractsofthe underlying. If a traderbuys 200 lots of the calls,howwouldshedelta-hedge?5.6Atraderowns100lotsofsomeputswitha-50%delta.Thesearedelta-hedgedwithalongpositionof50lotsofthespot product. The puts have

10gamma.Whatistheirdeltaif the underlying rallies by$1?5.7Followingtherallyin5.6,what is the net delta of theportfolio? How would thetraderrestoredelta-neutrality?5.8Followingthere-hedgein5.7, theunderlyingfallsback$1. Does the trader need tore-hedge?Whatistheoverallresultofhishedgingactivity?5.9Istheowneroftheputsin5.6 – 5.8 long or short theta

(i.e. is he paying theta orcollecting theta on hisposition)?

6. UnderstandingImpliedVolatilityTomovebeyond thesimplereadingofoption values in relation to the spotpriceandstrikeprice,itisnecessarytounderstand one of the fundamentaldrivers of option value. Impliedvolatilityisakeyfactorthatliesbehindmuch of the trading that occurs inoptionmarkets.Indeedinsomemarket,impliedvolatilityinpercentagepointsisspokenofinpreferencetooptionprices

in dollars and cents, such is itsimportancetooptionsprofessionals.

What is impliedvolatility?We have seen how one of the mostimportant factors determining optionvalue is the expected volatility in theprice of the spot product. In thesimplest cases, this is a particularlyimportant factorbecause it isgenerallyunknown with any certainty. Thecurrent price of the spot product isusually public knowledge. Likewise,theexpirationdateofoptionsispartof

the option contract and there is noconfusion or uncertainty. The level ofvolatility in the spot price that willoccur in thefuture ishoweveramatterof opinion and a reason for traders todisagree about the value of an option.When two traders disagree about thevalue of an instrument, they have areason to trade; one thinks theinstrumenttoocheap,theothertoorich.Expected volatility, and difference ofopinion thereon, is thus a primaryreasontotradeoptions.The expected volatility is knownmorecommonly as the implied volatility.This is because the value of an option(givenbya theoreticalmodel)mustbe

associated with a single level ofexpected volatility. So it should bepossible to take the price of an option(indollarsandcents)andusethemodeltoimplytheexpectedlevelofvolatilitythat was plugged into the model toarrive at the valuation. Consider thisanalogy.Supposeacar isknown tobeabletodrive100milesgivenafulltankoffuel.Ifwesetoffandthecardrivesdrives50milesbeforerunningdry,thisobviouslyimplies therewashalfa tankoffuelwhenwedeparted.Implied volatility is useful in at leasttworespects.Firstly,itgivesagaugeofwhattheoptionmarketisimplyingwithrespecttoexpectedvolatilityinthespot

product. If options on a stock aretrading at 25% implied volatility (alsoknownas“impliedvol” forshort) thenthis impliestradersexpect thatstocktomove around with an annualizedvolatilityofabout25%over the lifeofthe options. Traders may have a viewonwhetherthisseemstoohigh,toolowor about right and they can trade theoptions to create an exposure thatmatchestheirview.Secondly,itactsasauniquereferencepointforthepriceofoptions,asweshallnowsee.

Impliedvolatilityandthepriceofoptions

How does implied volatility affectoption valuations and option prices? Itis always worth considering extremecaseswhentryingtoassesstheeffectonoptions of a change in a key variable.Suppose the implied volatility is zero.Inotherwords, the spot product is notexpectedtochangepriceatalloverthelife of the option. Here, the optionswould have no extrinsic valuewhatsoever. Out-of-the-money optionswill have no value because they areexpected to always remain out-the-moneysincetheunderlyingpriceisnotexpected tomove in order for them tobecome in-the-money.Options that arecurrentlyin-the-moneywouldbeworththeirintrinsicvaluebutnomore.

Let’s consider the other extreme.Suppose impliedvolatility isextremelyhigh(saymanyhundredsof%).Inthiscaseoptionsoveraverywiderangeofstrikes will be valuable because theirchance of expiring in-the-money isexpected tobehigh.Theunderlying isexpected to be highly volatile andtherefore strikes currently out-of-the-moneystillhavea strong likelihoodofexpiringvaluably.The relationship between changes inimplied volatility and option prices isencaptured by an option’s vega. Vegatellsusbyhowmuchanoption’svalue(in dollars and cents) changes for a

change in implied volatility inpercentageterms.Forexample,supposean option has a value of $1.50 givenimplied volatility of 25%.Suppose theoptionhas10vega.Whatistheoption’svalue if implied volatility increases to25.5%? The vega of an option istypically normalized so that it showsthechangeinoptionvalue(inticks)fora1%changeinimpliedvolatility.Sointhis instance, the option will be worth$1.55 after the increase in impliedvolatility;a fivecents increasegive10vegaanda½%change in impliedvol(10 * ½ = 5 cents). As we wouldexpect, higher implied volatility leadsto a higher option price (as discussedabove). It should be mentioned thatoption vega is not a constant (few

things are in the options world!) andcan change due to several factors.However, for small moves in theunderlyingorovershorttimeperiods,itisoftenfine toassumevegais roughlyconstant when computing changes tooptionvalues.

Assignment 6.1 Playthe vega/vol/valuemini-gameThis mini-game ensures you arecompletely comfortable with therelationshipbetweenanoption’svalue,

its implied volatility and its vega, byasking you to calculate either thestarting or the finishing level of theoption’s theoretical value or impliedvolatility. The critical formula torememberanduseis:

Optionvega=Changeinoptionvalue/Changeinimplied

volatilityThiscanofcourseberearrangedintwootherusefulways.Changeinimpliedvolatility=Change

inoptionvalue/Optionvegaand…

Changeinoptionvalue=Optionvega*Changeinimpliedvolatility

Adetailedexampleofthisoptionmini-game is available atwww.volcube.com/support/volcube-tutorials/.

Assignment 6.2 PlayaLevel 1 game.Aimfor good pricingaccuracy.

In this assignment you will playthrough a simulation aiming for goodpricingaccuracy.Whatdoesthismean?Your Pricing Sheet in theMainGamescreen displays the values of calls andputs calculated using your model andyour implied volatilities (displayed onthePricingSheetinthefarleftcolumn).At the start of thegame, these impliedvolatilities accurately reflect themarket’s perception of impliedvolatilities in your game. However, asthegameproceeds,themarketmaystartto value implied volatility differently.This will be noticeably if for examplethe broker tries to continually buyoptions from you (implied volatility is‘going bid’ ie increasing). You shouldaim to adjust your quote responses to

the broker by an amount that reflectshowfaryouthinkimpliedvolatilityhasmoved. This will make your pricingmore accurate. For example, supposeyou have the $100 strike callstheoreticallyworth3.15with12.6vegaand a theoretical implied volatility of25%.Ifthebrokersellsthesetoyouat3.09,youhavepaidroughly½avolie24.5% for the calls (from the aboveformulae,thechange(i.e.thedifferencebetween your value and the pricetraded) inoptionvalue isabout6 ticksand the vega is about 12, so thedifference in implied vol is roughly½%).StartaLevel1Volcubesimulation.Try

to play the game for at least 25 quoterequestsandnoteyourVolcubePricingAccuracyMetricintheAnalysisscreen.In thePerformance tabof theAnalysisscreen you will see your accuracy forevery quote entered. Play some moregamesand see ifyoucan improve thisscore.

Exercise66.1Other thingsbeingequal,if two spot products areidenticalexceptthatspotAisexpected to be less volatile

thanspotB,wouldweexpectoptions on A to be more orless valuable than optionsstruckonB?6.2 If a product’s optionvaluesareverylow,allthingsconsidered, is the spotproduct expected to move agreat deal? Is impliedvolatility in the optionstherefore likely tobehighorlow?

6.3 Far out-of-the-moneyoptions have low levels ofvega. Intuitively, why is thisthecase?6.4 An option has a vega of20andistheoreticallyvaluedat $4.50 with an impliedvolatility of 23%. If impliedvolatility increases to 24%,whatisthenewoptionvalue?6.5 An option was valued at$4.50 and had a vega of 20

and an implied volatility of23%. However, it is nowvalued at $4.40.Has impliedvolatility risen or fallen andtowhatlevel?6.6 An option was worth $2with implied volatility at10%.Impliedvolhasrisento10.25%andtheoptionisnowworth $2.30. What is theoption’svega?

7. Trading optionsandvolatility

Trading impliedvolatilityviaoptionsRecall the threemain driversofanoption’svalue:

Now, unless an option isgetting close to expiration,thentheimpactofthepassingof time happens relativelyslowly. An option that hasplentyoftime(weeksorevenmonths) until it expires, willnot see its value changedramatically for smallchanges in time. This iscertainly not true for very

short-dated options (such asthosewithonlydaysorhoursuntil expiration), but for anylongertermoptionthanthese,time’spassing isnot amajordriverofchangesinvalue.Thisleavesthespotpriceandthe expected volatility as thekeyfactors.Buthereagain,itis possible to, at leasttemporarily, minimise theimpact of one of thesefactors.Itispossibletohedge

the option such that changesinthespotpricedonotcausethe portfolio as a whole tomakeorlosemoney.Inotherwords,toallbuteliminatetherisk associated with changesin the spot price. This isknown as delta-hedging.Delta-hedging a portfolioeliminates the short-term riskto an option (or options)owing to the spot pricechanging.Thenetresult is toconcentrate most of the

exposure towards thevolatilitycomponent.This ismosteasilyshownbyan example. Consider a calloption with around 1 monthuntil it expires. Suppose thespotistradingat$100.00andthe $100 strike call (i.e. the50%delta,at-the-moneycall)is theoretically worth $3.15,with an implied volatility of25%.Let’ssupposeanoptiontraderpays$3.15forthecall;

but really it is the 25%implied volatility level he isinterestedin.Maybehethinksthis level is cheap comparedto the actual volatility heexpectsinthespotpriceoverthe coming weeks. Orperhapshe thinkshe can sellthecallslaterwithanimpliedvolatility greater than 25%.Whatever the reason, hewillneedtoeliminatesomeoftheother risks to focus hisexposure on the implied

volatility level. The mostpressing of these risks is thedelta risk. It is the mostpressing because the spotprice is generally the mostvolatile of the all the riskfactors. A one-month optionwill typically not lose valueminute-by-minutesimplydueto time passing (theta risk;indeed these optionsdescribed only lose about 4cents per night due to timeerosion). But in one minute

the spot can moveconsiderably. So this isusually the most urgent riskfactor to address. To delta-hedge and ‘lock-in’ the 25%vol level, the trader needs tosell the spot in a certainquantity at $100.00. Thisraises three questions; whydoeshesell the spot,what isthe‘certainquantity’andwhyatpriceof$100.00?

Whydoeshesellthespot?The value of the call is apositive function of the spotprice. If the spot rallies, thecall’s value will increase. Itsvalue falls, if the spot falls.Owning the call optiontherefore brings a positivedelta exposure to the spotproduct price. To eliminatethis risk, the tradermust sellthe spot product. If he hadsold the call short, he would

need to buy the spot. If hadbought a put, hewould needto buy the spot to be delta-hedged. The direction of thehedge trade (i.e. whether tobuyor sell)must simply aimto offset the exposure of theoption.

What is the size of the deltahedge?Thesizeofthehedgedependson a few factors. The option

tobehedgedhasadelta.Thisreflects its sensitivity tochanges in the spot price. Acall option with a delta of50% (½) needs to be hedgedwith 1 lot of the spot forevery2options.Butthismustalso be modified for theoption contract multiplier. Ifeachoptioncontractgivestheright to trade say 100 lots ofthe underlying (contractmultiplier = 100), then thedelta hedge must be

multiplied by 100. Sosupposea traderbuys50 lotsof the -25%deltaputswhichhave a contract multiplier of10. The delta hedge involvesbuying 50*0.25*10=125lotsoftheunderlying.

Whyapriceof$100.00?Remember that an option’simplied volatility mapsuniquely to its theoreticalvalue. In other words, an

option with a theoreticalvalue of x, must have asingle, associated impliedvolatilityof sayy.And thesetwo also map uniquely to asingle spot price. So we cansay that the calls aretheoretically worth $3.15with the spot trading at$100.00 and that this meansthe calls have an impliedvolatility of 25%. With thespot trading at $100.01, thecalls will be worth $3.155

(becausethecallshavea50%deltaandthespotistrading1cent higher, so the calls areworthanextra0.5ofacent).The implied volatility willstill be 25%; nothing hashappened to alter it. So, inimplied volatility terms, wecan say that trading the callsat$3.15withthespotat$100or trading the calls at $3.155with the spot at $100.01,amounts to the same thing.However, trading the calls at

$3.15 with the spot at$100.01 does not equate tothe same level of impliedvolatility. In fact trading thecallsat$3.15withthespotat$100.01 means trading aslightly lower impliedvolatility level (<25%). Thecalls are cheaper, in impliedvolatility terms. This can beseen using an option pricingcalculator (or in a Volcubesimulation by deselecting theLive Price checkbox in the

Pricing Sheet and entering adifferent spot price in thewhiteboxinthefarright).Tocalculatethedifferenceinimpliedvolatilitybetweenthetwo different levels of thespot, we can use arearrangement of the vegaformula. The difference inimplied vol is the differenceintheoptionvaluedividedbytheoptionvega.Ifthesecallshaveavegaofsay12.6,then

trading them at prices 0.5cents apart is (0.5/12.6 = )0.04% vols different. Sobuying the $100 strike callsfor $3.15 with the spot at$100.01, is equivalent tobuying 24.96% implied vol,rather than 25%. This maynot be immediately obvious,so be sure to work throughthis example several timesand the Exercise below. Thekey point is that the spotprice,optionpriceandoption

implied volatility matchuniquely. So trading anoption at a price of p andhedging with the spot at aprice of s, ‘locks-in’ theimpliedvolatilityleveltradedatoneuniquelevel,i.Changeanyoneof thesethreeandatleast one of the other twomustalsochange.This can be easier to portraythroughsimpleexamplesthanprose. So the exercise at the

end of this chapter is wellworth working through andhopefully all will becomeclear.

Delta-neutraltradingA delta-neutral tradingstrategy is one that aims toeliminate the (short-term)exposure to changes in thespotprice, thus focussing theexposure on other option-

relatedfactors.Inparticular,adelta-neutral trading strategyusually means in impliedvolatility trading strategy.Examples would be optionmarket making, so-calledvolatility ‘arbitrage’, orrelative-value implied volstrategies or ‘dispersiontrading’ (trading impliedvolatilityofabasketofstocksagainstanindex).Theroleofactual,realizedvolatilitymayor may not feature in these

strategies. It is possible totrade implied volatility as afactor in its own right. Or itmay be traded against actualvolatility. For more on thistopic, see the VolcubeAdvance Options TradingGuide volume IV, TradingImpliedVolatility.We can very briefly outlinethepremisebehindanoptionsstrategy to illustrate impliedvol trading. An options

market maker is prepared tobuy or sell options at anygivenmoment. His bid pricewill be below his selling (oroffer) price.Hehopes tobuylow, sell highandpocket thedifference. His risk is thatbuyers are not equallymatched by sellers and heends up with inventory;inventorywhosevaluemovesagainsthim.For instance, suppose a

market maker shows a bidand an offer in some calls,when his theoretical impliedvol for the calls is 25%.Thebroker hits the marketmaker’s bid and they trade.So the market maker hasbought some options.Suppose he delta-hedgesthese calls. This locks-in acertain level of impliedvolatility, say 24.5%. Nowsuppose a different brokerrequests a price in some

similar options. Perhaps putoptionswith a strike close tothat of the calls and a verysimilar theoretical impliedvolatility; perhaps 25.2%.Themarketmakerwill againshow a bid and an offer, buthe will probably do somindful of the level inimplied volatility he hasalready traded. If he can selltheputsatsay25.5%,hewilloffset some of the optionGreek risks from thecallshe

bought,andhewillbetradingfor a profit, in impliedvolatility terms.Buying0.5%vols under theoretical in thecalls and selling a similarstrike put 0.3% overtheoretical, creates a spreadpositionforatheoreticalgainof0.8%vols.Thepremiseofthestrategyisthatchangesinimplied volatility will,generally, be similar forclosely related options.Ultimately the trader will

look to sell the calls back tosomeone and buy back theputs from someone else. Butinthemeantime,thispositionis nicely spread off and itwould require dramaticmovements in impliedvolatility across the curve ofoptions for this position tolose money. The risk wouldbe that instead of someonekindly buying the offsettingputs after the market makerhas bought the calls, more

sellers of options appear. Inthiscase,thegeneralpriceofoptions is likely to fall.Without the offsetting shortput position, the marketmakeristhereforeexposedtolosses; hewill losemoney ifthevalueofimpliedvolatilityfalls below 24.5%, since thisisthepricehepaid.

Assignment 7.1 Play

a complete Level 1game.InVolcubeLevel1youmusttrade as a market maker,showingbidandofferprices.Asyoutradewiththebroker,try to estimate the impliedvolatility level you havetraded. This is simplified inLevel1becausethespotpriceis fixed at $100.00. Youroption trades areautomatically delta-hedged.

So if you trade the at-the-moneycalls(whichhave25%theoretical implied volatility)for say3.10, this is5.6cents(ticks)belowtheir theoreticalvalue. 5.6 divided by theoption vega of 12.6 (whichyou can see in the PricingSheetontheright)gives0.44.This means that 3.10 isequivalent to trading atroughly 24.56% vol. Byquickly calculating in thisway,youcanprice and trade

optionsaccurately.

Assignment 7.2 Usethe Volcube Analysisscreen to analyseyourperformanceTheVolcubeAnalysis screenoffersamplefeedbackforyouto review and improve. TheTPI (Trader PerformanceIndex) gives your

performanceanoverallscore.Ifyoucanscore100ormoreon a regular basis, that isgood progress. Clicking thecircled ‘R’ will reveal aReport, giving more detailabout your performance.There are plenty of othermetrics available to assessyour pricing accuracy,competitiveness and speed.Use the Risk/Reward tab tocompare your profitabilitywith the degree of risk you

took during the game. ThePerformancetabcangiveyoumore insight into theCompetitiveness andAccuracyofallyourquotes.

Exercise77.1Anoption isworth$1.50with 25% theoretical impliedvol and 6 vega. What is its

new value if the theoreticalimpliedvolatilityisincreasedto26%?7.2Anoption is theoreticallyworth $3.20 with 20%implied volatility and 10vega. If a trader pays $3.10for the option, whattheoretical value for impliedvolatilityhashebought?7.3A tradermakes amarketin some puts worth $2.00,withimpliedvolat10%.Theoptions have 8 vega. The

trader shows $1.98 bid, at$2.02 offered. Whatequivalent implied vol levelsishebiddingandoffering?7.4Atradercanpay$1foranoption theoretically worth$1.05with15vega.Orhecanpay $1.50 for an optiontheoretically worth $1.60with 10 vega. Both optionshave theoretical impliedvolatilities of 20%. Whichtrade is cheaper in impliedvolatilityterms?

7.5 The spot is trading at$100.00whenthetraderpays$3.10 for some at-the-moneycall options he hastheoretically valued at $3.15withimpliedvolof25%.Thecallshave10vega.Thetraderdelta hedges the trade. Laterintheday,thespotistradingat $99.90. Someone is tryingtobuythecalls,paying$3.10.If the trader sells his calls at$3.10,canhemakeaprofit?7.6 What implied volatility

levelswouldthetraderin7.5buy and sell if he executesbothtrades?

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SolutionstoExercisesExercise11.1Sell.1.2No.1.3No.Theowneroftheoptionhastheright but not the obligation to dosomething.Johnmay(ormaynot)havethe obligation to do something (if theowner of the option exercises hisoption).Ifyouareshortanoption,itisoutofyourhands!

1.4Rise.Viathecall,Johnhastherightto buy the spot at a fixed price (thestrike price of the call). So, the higherthe spot price rallies, themoremoneyJohnwillmake.1.5Out-of-the-money.1.6No. If thecall is in-the-money, theputisout-of-the-moneyandviceversa.1.7 The puts are worth $10. The callsareworthless.1.8Zero.Johnbreaksevenbecausetheputsareworth$3,whichisthesameasthepricehesoldthemfor.1.9Intheory,itisunlimited.Thehigherthespotpriceisatexpiration,themorethecallsareworthandthemoremoneyJohncanpotentiallylose.1.10 $1 of profit. Combinations ofoptionsareknownasoptionstrategies.

Their payoff profiles can be calculatedbyassessingtheindividual‘legs’ofthestrategyandsumming.Inthiscase, theputs are worthless (out-of-the-money)andJohnpaid$0.50,soheloses$0.50.The calls are worth $2 and John paid$0.50forthese,soonthecallsheshowsanetprofitof$1.50.Overall,hisprofitis a 50 cents loss on the puts and a$1.50 profit on the calls, hence a netp&lof$1.Exercise22.1 Zero. The puts are out-of-the-money and therefore have no intrinsicvalue.2.2 No-one can say. Extrinsic value

dependsonseveralfactors,butthepriceof the spot relative to the strike pricedoesnotrevealit.2.3 $1. The puts must have zerointrinsic value because they must beout-of-the-moneywhenthecallsarein-the-money. The extrinsic value of theputandthecallmustbesamesincetheyhave the same strike and we are toldput-callparityholds.2.4 Equivalent to being long theunderlyingfromapriceof$100.Toseethis,drawapayoffgraph,withthespotprice on the x-axis and the profit andloss on the y-axis, for the $100 strikeput (short position) and call options(long position), assuming they weretraded at a price of zero. Thecombinationofthetwopositionsshould

just resemble an upward sloping line,which is identical to the payoff profileof a long position in the underlying,fromapriceof$100.2.5 At expiration, options have noextrinsic value remaining. They mayhave intrinsic value if they are in-the-money.Otherwise, they have no valuewhatsoever.2.6Thespotprice,thetimeuntilexpiryandtheexpected(implied)volatility.2.7 No. The intrinsic value onlydependsuponthestrikepriceversusthecurrentspotprice.2.8 Longer dated options are morevaluable than shorter dated options,otherthingsbeingequal.2.9 Use payoff graphs to prove this ifrequired. But in simple terms, owning

the put and the underlying issynthetically equivalent to owning thecall(ofthesamestrike).2.10 Other things being equal, higherimplied volatility means higher optionvalue. Changes to implied volatilityalter the extrinsic value of an option.Only a change in the spot price canaffecttheintrinsicvalue.Exercise33.1Buycalls.Shortfuturesisabearishposition.3.2a)bearishb)bearishc)bullish3.3Ingeneral,yes.Impliedvolatilityisone of the factors that determinesoptionvalues.Changes in its levelwill

generallyaffectoptionvalues.3.4 Changes in implied volatility andchangesinthetimeremainingtoexpiryare probably themost significant risksto adelta-hedgedoptionposition.Spotmoves can have an impact via gammaeffects(seechapter5formoreonthis).3.5 Because the time-value (extrinsic)value of options diminishes over time,sellingoptionscanbeawaytogeneratean income-yielding position. As thetime-value erodes gradually, theposition profits from being short anassetwhose value is falling. There areof course significant risks to such aposition which must be fullyunderstood.3.6 a) His immediate exposure tochanges in the spot price is small/very

close to zero. This is because thechangeinspotpriceaffectsthecallandput values in opposite, and almostequal,ways.So,arisingspotcausesthecalltoriskinvalue(makingaprofitforitsowner)buttheputofthesamestrikewill fall in value by a similar amount(making a loss for its owner).Owningboth options therefore results in aposition with minimal exposure to(small)changesinthespotprice.b) Here the exposure is significant.Sinceputsandcallsarebothaffectedbychanges in expected volatilitypositively, any change in expectedvolatilitywillresultinboththeputandthe call changing value in the sameway. A rise in expected volatility willincrease the value of the both the put

andthecall.Thereisnohedgingeffectas in part a). This option strategy isknownasa‘straddle’; thepurchase(orsale)ofaputandcallofthesamestrikeonthesameunderlyinganditisknownmoreasavolatilityplaythanasashort-termdirectionalplay.3.7c.3.8Never.Else thebroker couldmakemoney from the market maker bytrading on both sides. If the marketmaker offers at 5 but bids 7 forexample,thebrokercouldbuyfromthemarketmaker paying 5 and sell to themarket maker at 7. By paying 5 andselling 7 simultaneously, the broker isleft with a risk-free arbitrage. Hence,themarketmaker’s bidwill always belowerthanhisask(offer).

Exercise44.1Thesearemarketprices.4.2 It is very unlikely. Even ifparticipants use the same model, theymay well use different inputs andthereforearriveatadifferentoutput.4.3 The question does notmakemuchsense.Riskandrewardmustalwaysbeconsidered relative to one another;knowledge of either is insufficient byitselftoinformatradingdecision.4.4Whentheassumptionsofthemodelthatgeneratethetheoreticalvalueprovetoholdperfectly.4.5 The theoretical profit is thedifference between the traded price of

theoptionanditstheoreticalvalue.Ifamarketmakerpays6centsforanoptionhe theoretically values at 7 cents, histheoreticalprofitis1cent.4.6 The actual profit is the differencebetweenthemarketvalueoftheoptionand the traded price. If the marketmaker pays 6 cents and the marketvalues the options at 7.5 cents, theactual profit is 1.5 cents. The marketvalueisoftentakentobethemid-priceofthebestbidandofferpricecurrentlyinthemarket.4.7Generally speaking, he is likely toshow c) higher bids and higher offers.Higher bids, because he will probablybebiased towardsbuyingbackoptionswhenheisshort.Higheroffers,becauseheisalreadyshortoptionsandtherefore

willrequiremorerewardthanbeforeinordertosellmore.4.8Thebid-askspread.Exercise55.1Afallof12cents.5.2$1.50.5.3Increaseof10centsto$1.65.5.4 The new delta is 42%. 20 cents isonefifthof$1,sothedeltaincreasesbyone fifth of 10.Gamma is positive forputsandcalls.5.5 By selling 20,000 lots of theunderlying.5.6 -40%. Gamma is positive so thedelta increases (i.e. becomes lessnegative)witharallyintheunderlying.5.7 The trader is equivalently net long

10lotsoftheunderlying.Heislong50lots of the spot and synthetically short40lotsviatheputs(whichheownsandwhich now have a -40% delta). So torestoredelta-neutralityhemust sell the10 lots (which is better than having tobuy10 lots,given that the spotmarketisnow$1higherthanitwas).5.8 If thespot re-tracesbackdown$1,the putswill again have a -50% delta.But thetrader isnowonlylong40lotsof the spot (because he sold 10 of theoriginal 50 that he owned when re-hedging his delta following the rally).Sonowthetraderisnetshort10lotsofthe underlying (short 50 lotssyntheticallyviathe50%deltaputsandlong40 lotsof actual spot).The traderneedstobuybackthe10lotshesoldto

restoredelta-neutrality.Thenet result is that the traderhas thesamepositionhestartedwithbeforetherallyandretracementofthespot.Buthehaslocked-ina$10profit,byselling10lots of the spot $1 higher and thenbuyingthembackversustheunchangedspotprice.This isknownasgammahedging.Thetrader was long gamma via his delta-hedged puts and was able to make aprofitfrommovementsinthespotpriceviare-hedging.5.9Thetraderwillbelongtheta,whichis to say he is paying theta. He putswill,otherthingsbeingequal,beworthlesstomorrowthantheyaretoday.Thisistheflip-sidetohavingtheopportunityto gamma hedge profitably. Long

optionsbringlonggammawhichcanbeprofitablyre-hedged,butthecostofthisgamma is the theta time-decay orerosion in the portfolio's value as timepasses.

Exercise66.1 Less valuable. The higher theimpliedvolatility,themorevaluabletheoption,otherthingsbeingequal.6.2 No. Options are more valuablewhen the underlying is expected to bevolatile as they have more chance ofexpiring in-the-money. Impliedvolatilityislikelytobelow.6.3 Low vega means that the optionvaluesdonotchangemuchforchangesinimpliedvolatility.Thisisbecauseforfar out-of-the-money options, a smallchange in implied (ie expected)volatility isunlikely togreatly increase

theirchancesofexpiringin-the-money.Vega is themeasure of responsivenessof an option’s value to changes inimpliedvolatility,hencefarout-of-the-moneyoptionshavelowvega.6.4 Assuming the vega is constant forsmallchanges in impliedvolatility, thenewvalueis$4.70.6.5 Implied volatility has fallen to22.5%. A 10 cent decline in optionvalue,withavegaof20mustmeana½%pointdropinimpliedvolatility.6.6 The vega is 120. Since the optionhas risen by 30 cents for a¼% pointincrease in implied volatility, it wouldincrease by 120 cents for a 1% pointincrease (which is the definition ofnormalizedoptionvega).

Exercise77.1$1.56.7.2 19%. 10 cents (below value)dividedby10vega=1%difference inimpliedvol.7.3 Themarket is 0.5% of a vol wide(2.02-1.98=4ticks.Dividethisbythe8 vega). The market is symmetricaroundthevalueof$2.00(sincethebidis2ticksbelowthevalueandtheoffer2 ticks above). Hence the market isequivalent to showing 9.75% bid, at10.25% offered, in implied vol terms.Note that in some option markets,quotes are given in implied vol termsratherthanindollarsandcents.

7.4 The second trade is cheaper. Thefirst trademeans buying 5 cents undervaluewith15vega,whichis⅓ofavolpointundervalue;19.66%.Thesecondtrade means buying 10 cents undervalue with 10 vega, which is 1% volpointundervalue;19%.7.5Yes,thetradercanmakeaprofit.Itmay seem counter-intuitive because hepaid$3.10andnowhewillsellat$3.10whichclearlyisp&lneutral.Theprofitis realised via the delta-hedge trades.When the trader bought the calls, hesoldthespotat$100.00.Whenhesellsthe calls, he will take off the delta-hedge; in other words buy back thespot,paying$99.90.Essentiallyhehasbought at one implied volatility level,andsoldatahigherlevel.

7.6Thetraderpaid$3.10,whichwas5cents below the value of $3.15. Thetheoreticalimpliedvolwas25%andthevega 10, so the trader paid ½ a volbelow the theoretical (5 cents/10vega)= 24.5%. When the spot drops to$99.90,wecanassumethecallwillbewortharound$3.10, theoretically.Thisis because the call was at-the-moneyand so had a 50% delta. Hence a 10centdropinthespotwillcausea5centdrop in option value. Remember, theimplied volatility does not changesimply because the spot has movedslightly. So selling $3.10 when theoptionswere theoreticallyworth$3.10,issimplysellingthetheoreticalimpliedvol level;25%. In summary, the traderbought 24.5% vol and sold 25%,

making0.5%ofavolinprofit.0.5%ofa vol is a 5 cent profit (on a 10 vegaoption).Thetradermade10centsprofiton his delta hedge.But remember thathisdelta-hedgewasonlyhalfthesizeofthe option trade, since the delta was50%.

Glossaryask - The selling price.Alsoknown as the offer price. Atrader whose ask is 7, isprepared to sell at a price of7.at-the-money - An optionwhosestrikepriceisequal tothecurrentspotprice.bearish-Toexpectapricetofall.

bid - The buying price. Atrader whose bid is 6, ispreparedtopay6.bullish-Toexpectapricetorise.contract multiplier - Thenumber of lots of theunderlyingtowhichanoptioncontract pertains. Forinstance, a multiplier of 100means that each option

confers the right to trade to100lotsoftheunderlying.long - Owning. To be longsomethingistoownit.lot-Aunitofaproduct.100options might be referred toas100lots.Selling10futuresmightbereferredtoasselling10 lots. The contractmultiplierofanoptionwillbespecifiedinanumberof‘lots’oftheunderlying.

offer - See ‘ask’. Offer andaskaresynonymous.out-of-the-money - Anyoptionthatisnotintrinsicallyvaluablegivenitsstrikepriceandthecurrentspotprice.Forcalls,thisisanyoptionwithastrike above the current spotprice. For puts, it is anyoption whose strike is belowthecurrentspotprice.

short - To own negativeamounts of something. Tosellshort is tosellsomethingwhichonedoesnotcurrentlyown. Short positions areusually reflected withnegative amounts ofinventory.