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3 The Black-Scholes Model

of default. ( good appro%i ation to such an in!est ent is a go!ern ent bond or a deposit in a sound bank. The greatest risk-free return thatone can ake on a portfolio of assets is the sa e as the return if thee"ui!alent a ount of cash were placed in a bank.

The ke$ words in the definition of arbitrage are 2instantaneous+ and2risk-free+0 b$ in!esting in e"uities, sa$, one can probabl$ beat the bank,

but this cannot be certain. If one wants a greater return then one ustaccept a greater risk. Wh$ should this be so Suppose that an oppor-tunit$ did e%ist to ake a guaranteed return of greater agnitude thanfro a bank deposit. Suppose also that ost in!estors beha!e sensi-

bl$. Would an$ sensible in!estor put one$ in the bank when putting itin the alternati!e in!est ent $ields a greater return )b!iousl$ not.

Moreo!er, if he could also borrow one$ at less than the return on thealternati!e in!est ent then he should borrow as uch as possible fro the

bank to in!est in the higher-$ielding opportunit$. In response to the pressure of suppl$ and de and we would e%pect the bank to raise itsinterest rates to attract one$ and4or the $ield fro the other in-!est ent to drop. There is so e elasticit$ in this argu ent because ofthe presence of 2friction+ factors such as transaction costs, differences in

borrowing and lending rates, probles with li"uidit$, ta% laws, etc., but onthe whole the principle is sound since the arket place is inhabited b$arbitragers whose highl$ paid &ob it is to seek out and e%ploitirregularities or ispricings such as the one we ha!e &ust illustrated.

Technical 5oint6 7isk.7isk is co onl$ described as being of two t$pes6 specific and non-

specific. The latter is also called arket or s$ste atic risk. Specific risk isthe co ponent of risk associated with a single asset or a sector of thearket, for e%a ple che icals , whereas non-specific risk is associated

with factors affecting the whole arket. (n unstable anage ent wouldaffect an indi!idual co pan$ but not the arket0 this co pan$ would

show signs of specific risk, a highl$ !olatile share price perhaps. )n

the other hand the possibilit$ of a change in interest rates would be anon-specific risk, as such a change would affect the arket as a whole.

It is often i portant to distinguish between these two t$pes of risk because of their beha!iour within a large portfolio a portfolio is a ter

for a collection of in!est ents . 5ro!ided one has a sensible definition ofrisk, it is possible to di!ersif$ awa$ specific risk b$ ha!ing a portfolio with alarge nu ber of assets fro different sectors of the arket0 howe!er,it is not possible to di!ersif$ awa$ non-specific risk. Market risk can

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3.3 )ption 8alues, 5a$offs and Strategies 39

be eli inated fro a portfolio b$ taking opposite positions in two assets

which are highl$ negati!el$ correlated - as one increases in !alue theother decreases. This is not di!ersification but hedging, which is of theut ost i portance in the anal$sis of deri!ati!es. It is co onl$ saidthat specific risk is not rewarded, and that onl$ the taking of greaternon-specific risk should be rewarded b$ a greater return.

( popular definition of the risk of a portfolio is the !ariance of thereturn. ( bank account which has a guaranteed return, at least in theshort ter , has no !ariance and is thus ter ed riskless or risk-free. )n theother hand, a highl$ !olatile stock with a !er$ uncertain return and thusa large !ariance is a risk$ asset. This is the si plest and co onestdefinition of risk, but it does not take into account the distribution ofthe return, but rather onl$ one of its properties, the !ariance. Thus as

uch weight is attached to the possibilit$ of a greater than e%pectedreturn as to the possibilit$ of a less than e%pected return. )ther, oresophisticated, definitions of risk a!oid this propert$ and attach differentweights to different returns.

3.3 )ption 8alues, 5a$offs and Strategies

:ow we turn to option pricing. ;et us introduce so e si ple notation,which we use consistentl$ throughout the book.

< We denote b$ 8 the !alue of an option0 when the distinction is i por-tant we use S, t and 5 S, t to denote a call and a put respecti!el$.This !alue is a function of the current !alue of the underl$ing asset,S, and ti e, t6 8 = 8 S,t . The !alue of the option also depends onthe following para eters6

< a, the !olatilit$ of the underl$ing asset0< #, the-e%ercise price0< T, the e%pir$0< r, the interest rate.

>irst, consider what happens &ust at the o ent of e%pir$ of a calloption, that is, at ti e t = T. ( si ple arbitrage argu ent tells us its!alue at this special ti e.

If S ? # at e%pir$, it akes financial sense to e%ercise the call option,

handing o!er an a ount #, to obtain an asset worth S. The profit frosuch a transaction is then S - #. )n the other hand, if S @ # at e%pir$,we should not e%ercise the option because we would ake a loss of #-S.

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3A The Black-Scholes Model

)ption price

3

'/9 'C9 'D9

#%ercise price

>igure 3.1 The !alue of a call option at and before e%pir$ against e%ercise price0 option !alues fro >T-S# inde% option data.

In this case, the option e%pires worthless. Thus, the !alue of the call

option at e%pir$ can be written asS,T = a% S - #, . 3.1

(s we get nearer to the e%pir$ date we can e%pect the !alue of ourcall option to approach 3.1 . To confir this we reproduce in >igure3.1 the figure fro hapter 1 which co pares real >T-S# inde% calloption data with the !alue of the option at e%pir$ for fi%ed S. In this figure we show a% S - #, as a function of # for fi%ed S = 'C/'and superi pose the real data for 8 taken fro the >ebruar$ call optionseries. )bser!e that the real data is alwa$s &ust abo!e the predicted line.This reflects the fact that there is still so e ti e re aining before theoption e%pires - there is potential left for the asset price to rise further,

gi!ing the option e!en greater !alue. This difference between the option

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3.3 )ption 8alues, 5a$offs and Strategies 3/

>igure 3.' The pa$off diagra for a call, S,T , and the option !alue,S, t , prior to e%pir$, as functions of S.

!alue before and at e%pir$ is called the ti e !alue and the !alue ate%pir$ the intrinsic !alue.'

If one owns an option with a gi!en e%ercise price, then one is lessinterested in how the option !alue !aries with e%ercise price than withhow it !aries with asset price, S. In >igure 3.' we plot

a% S - #,

as a function of S the bold line and also the !alue of an option at so eti e before e%pir$. The latter cur!e is &ust a sketch of a plausible forfor the option !alue. >or the o ent the reader ust trust that the!alue of the option before e%pir$ is of this for . ;ater in this chapter

we see how to deri!e e"uations and so eti es for ulae for such cur!es.The bold line, being the pa$off for the option at e%pir$, is called a

pa$off diagra . The reader should be aware that so e authors use theter 2pa$off diagra + or 2profit diagra + to ean the difference

between the ter inal !alue of the contract our pa$off and the original pre iu . We choose not to use this definition for two reasons. >irstl$,

' )ther i portant &argon is at-the- one$, which refers to that op tion whose e%-ercise price is closest to the current !alue of the underl$ing asset, in-the-one$,which is a call put whose e%ercise price is less greater than the current asset

price - so that the option !alue has a significant intrinsic co ponent - and out-of-the- one$, which is a call or put with no intrinsic !alue.

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3C The Black-Scholes Model

S#

>igure 3.3 The pa$off diagra for a put, 5 S, T , and the option !alue,5 S, t , prior to e%pir$, as functions of S.

the pre iu is paid at the start of the option contract and the return, ifan$, onl$ co es at e%pir$. Secondl$, the pa$off diagra has a naturalinterpretation, as we see, as the final condition for a diffusion e"uation.

It should now be clear that each option and portfolio of options has itsown pa$off at e%pir$. (n argu ent si ilar to that gi!en abo!e for the!alue of a call at e%pir$ leads to the pa$off for a put option. (t e%pir$it is worthless if S ? # but has the !alue # - S for S @ #. Thus the

pa$off at e%pir$ for a put option is

a% # - S, .

The pa$off diagra for a #uropean put is shown in >igure 3.3, wherethe bold line shows the pa$off function a% # - S, . The other cur!eis again a sketch of the option !alue prior to e%pir$. (lthough the ti e!alue of the call option of >igure 3.' is e!er$where positi!e, for the putthe ti e !alue is negati!e for sufficientl$ s all S, where the option !aluefalls below the pa$off. We return to this point later.

(lthough the two ost basic structures for the pa$off are the call and the put, in principle there is no reason wh$ an option contract cannot bewritten with a ore general pa$off. (n e%a ple of another pa$off isshown in >igure 3. . This pa$off can be written as

B -1 S - # ,

where -l . ,is the Eea!iside function, which has !alue when itsargu ent is negati!e but is 1 otherwise. This option a$ be interpreted

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3.3 )ption 8alues, 5a$offs and Strategies 3D

B -1

S#

>igure 3. The pa$off diagra for a cash-or-nothing call, e"ui!alent to a bet

on the asset price.

as a straight bet on the asset price0 it is called a cash-or-nothing call.)ptions with general pa$offs are usuall$ called binaries or digitals.

B$ co bining calls and puts with !arious e%ercise prices one can con-struct portfolios with a great !ariet$ of pa$offs. >or e%a ple, we show in>igure 3.9 the pa$off for a 2bullish !ertical spread+, which iscon structed b$ bu$ing one call option and writing one call option withthe sa e e%pir$ date but a larger e%ercise price. This portfolio is called2bullish+ because the in!estor profits fro a rise in the asset price, +!erti cal+

because there are two different e%ercise prices in!ol!ed, and 2spread+ because it is ade up of the sa e t$pe of option, here calls. The pa$off

function for this portfolio can be written as

a% S - #l, - a% S - #',

with #' ? #l.Man$ other portfolios can be constructed. So e e%a ples are

2co binations+, containing both calls and puts, and 2horiFontal+ or2calendar+ spreads, containing options with different e%pir$ dates. )thersare gi!en in the e%ercises at the end of this chapter.

The appeal of such strategies is in their abilit$ to redirect risk. Ine%change for the pre iu - which is the a%i u possible loss andknown fro the start - one can construct portfolios to benefit fro!irtuall$ an$ o!e in the underl$ing asset. If one has a !iew on thearket and this turns out to be correct then, as we ha!e seen, one can

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8

# -#--' 1

BB #'

The Black-Scholes Mode

S

>igure 3.9. The pa$off diagra for a bullish !ertical spread.

ake large profits fro relati!el$ s all o!e ents in the underl$ingasset.

3. 5ut-call 5arit$

(lthough call and put options are superficiall$ different, in fact the$ can be co bined in such a wa$ that the$ are perfectl$ correlated. This isde onstrated b$ the following argu ent.

Suppose that we are long one asset, long one put and short one call.The call and the put both ha!e the sa e e%pir$ date, T, and the sa ee%ercise price, #. Genote b$ E the !alue of this portfolio. We thus ha!e

rI=SH5- ,

where 5 and are the !alues of the put and the call respecti!el$. The pa$off for this portfolio at e%pir$ is

S H a% # - S, - a% S - #, .

This can be rewritten as

SH #-S -B=# if S@#,

or

SH - S-# =# if S?#.Whether S is greater or less than # at e%pir$ the pa$off is alwa$s thesa e, na el$ #.

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3.9 The Black-Scholes (nal$sis 1

:ow ask the "uestion

< Eow uch would I pa$ for a portfolio that gi!es a guaranteed # att = T

This is, of course, the sa e "uestion that we asked in hapter 1, andthe answer is arri!ed at b$ discounting the final !alue of the portfolio.:ote that here we do ha!e to assu e the e%istence of a known risk-freeinterest rate o!er the lifeti e of the option. Thus this portfolio is nowworth #e-++ T-t This e"uates the return fro the portfolio with thereturn fro a bank deposit. If this were not the case then arbitragerscould and would ake an instantaneous riskless profit6 b$ bu$ing andselling options and shares and at the sa e ti e borrowing or lendingone$ in the correct proportions, the$ could lock in a profit toda$ withFero pa$off in the future. Thus we conclude that

S H 5 - = #e- r T -t 3.'

This relationship between the underl$ing asset and its options is called put-call parit$. It is an e%a ple of risk eli ination, achie!ed b$ car-r$ing out one transaction in the asset and each of the options. In thene%t section, we see that a ore sophisticated !ersion of this idea, in-!ol!ing a continuous rebalancing, rather than the one-off transactionsabo!e, allows us to !alue #uropean call and put options independentl$.

3.9 The Black-Scholes (nal$sis

Before describing the Black-Scholes anal$sis which leads to the !alue of anoption we list the assu ptions that we ake for ost of the book.

< The asset price follows the lognor al rando walk '.1 .)ther odels do e%ist, and in an$ cases it is possible to perforthe Black-Scholes anal$sis to deri!e a differential e"uation for the!alue of an option. #%plicit for ulae rarel$ e%ist for such odels.Eowe!er, this should not discourage their use, since an accuratenu erical solution is usuall$ "uite straightforward.

< The risk-free interest rate r and the asset !olatilit$ a are known func-tions of ti e o!er the life of the option.

)nl$ in hapters 1/ and 1C do we drop the assuption ofdeter in istic beha!iour of r0 there we odel interest rates b$ astochastic differential e"uation.

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3.9 The Black-Scholes (nal$sis 3

that II follows the rando walk

dII = aSas - o d% H + Sa8 as H 1,'S' asa'8 ' H at - , oS dt.

3.9(s we de onstrated in Section '. , we can eli inate the rando co -

ponent in this rando walk b$ choosing

as3.A

:ote that ( is the !alue of a84aS at the start of the ti e-step dt.This results in a portfolio whose incre ent is wholl$ deter inistic6

'

dII =

atH ''9' aS dt.3./

We now appeal to the concepts of arbitrage and suppl$ and de and,with the assu ption of no transaction costs. The return on an a ount I Iin!ested in riskless assets would see a growth of rII dt in a ti e dt. If theright-hand side of 3./ were greater than this a ount, an arbitrager couldake a guaranteed riskless profit b$ borrowing an a ount II to in!est inthe portfolio. The return for this risk-free strateg$ would be greater thanthe cost of borrowing. on!ersel$, if the right-hand side of 3./ were lessthan rII dt then the arbitrager would short the portfolio and in!est n inthe bank. #ither wa$ the arbitrager would ake a riskless, no cost,instantaneous profit. The e%istence of such arbitragers with the abilit$ totrade at low cost ensures that the return on the portfolio and on theriskless account are ore or less e"ual. Thus, weha!e

'

rIldt = Hat La'S' a dt.

3.C

Substituting 3. and 3.A into 3.C and di!iding throughout b$ dt wearri!e at

a8at

1 a'8'S'

H a as+' HrS

a8- r8 = . 3.D

as

This is the Black-Scholes partial differential e"uation. With itse%tensions and !ariants, it pla$s the a&or role in the rest of the book.

It is hard to o!ere phasise the fact that, under the assu ptions statedearlier, an$ deri!ati!e securit$ whose price depends onl$ on the current!alue of S and on t, and which is paid for up-front, ust satisf$ theBlack-Scholes e"uation or a !ariant incorporating di!idends or ti e-dependent para eters . Man$ see ingl$ co plicated option !aluation

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The Black-Scholes Model

proble s, such as e%otic options, beco e si ple when looked at in thiswa$. It is also i portant to note, though, that an$ options, for e%a ple

( erican options, ha!e !alues that depend on the histor$ of the asset price as well as its present !alue. We see later how the$ fit into theBlack-Scholes fra ework.

Before o!ing on, we ake three re arks about the deri!ation weha!e &ust seen. >irstl$, the delta, gi!en b$

a! - as,

is the rate of change of the !alue of our option or portfolio of optionswith respect to S. It is of funda ental i portance in both theor$ and

practice, and we return to it repeatedl$. It is a easure of the correlation

between the o!e ents of the option or other deri!ati!e products andthose of the underl$ing asset.

Secondl$, the linear differential operator BS gi!en b$

FBs = at H LaFSF9S,F HrS1DS Nr

has a financial interpretation as a easure of the difference between thereturn on a hedged option portfolio the first two ter s and the return ona bank deposit the last two ter s . (lthough this difference ust beidenticall$ Fero for a #uropean option, in order to a!oid arbitrage, wesee later that this need not be so for an ( erican option.

Thirdl$, we note that the Black-Scholes e"uation 3.D does not con-

tain the growth para eter M. In other words, the !alue of an option isindependent of how rapidl$ or slowl$ an asset grows. The onl$ para e-ter fro the stochastic differential e"uation '.1 for the asset price thataffects the option price is the !olatilit$, a. ( conse"uence of this is thattwo people a$ differ in their esti ates for $ $et still agree on the !alueof an option.

3.A The Black-Scholes #"uation

#"uation 3.D is the first partial differential e"uation that we ha!ede ri!ed in this book. The theor$ and solution ethods for partialdifferen tial e"uations are discussed in depth in hapters and 90

ne!ertheless, we now introduce a few basic points in the theor$ so thatthe reader is aware of what we are tr$ing to achie!e.

B$ deri!ing the partial differential e"uation for a "uantit$, such as anoption price, we ha!e ade an enor ous step towards finding its !alue.

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A The Black-Scholes Model

co on to transfor backward e"uations into forward e"uations before

an$ anal$sis. It is i portant to re e ber, howe!er, that the parabolice"uation cannot be sol!ed in the wrong direction0 that is, we should noti pose initial conditions on a backward e"uation.

3./ Boundar$ and >inal onditions for #uropean)ptions

Ea!ing deri!ed the Black-Scholes e"uation for the !alue of an option, weust ne%t consider final and boundar$ conditions, for otherwise the partialdifferential e"uation does not ha!e a uni"ue solution. >or the o entwe restrict our attention to a #uropean call, with !alue now denoted b$

S, t , with e%ercise price # and e%pir$ date T.The final condition, to be applied at t = T, co es fro the arbitrage

argu ent described in Section 3.3. (t t = T, the !alue of a call is knownwith certaint$ to be the pa$off6

S, T = a% S - #, . 3.1

This is the final condition for our partial differential e"uation.)ur 2spatial+ or asset-price boundar$ conditions are applied at Fero

asset price, S = , and as S -H oo. We can see fro '.1 that if S ise!er Fero then dS is also Fero and therefore S can ne!er change. This isthe onl$ deter inistic case of the stochastic differential e"uation '.1 . IfS = at e%pir$ the pa$off is Fero. Thus the call option is worthless on S

= e!en if there is a long ti e to e%pir$. Eence on S = we ha!e

, t = . 3.11

(s the asset price increases without bound it beco es e!er ore likel$that the option will be e%ercised and the agnitude of the e%ercise price

beco es less and less i portant. Thus as S -H oo the !alue of the option beco es that of the asset and we write

S, t + S as S -O oo. 3.1'

>or a #uropean call option, without the possibilit$ of earl$ e%ercise,3.D - 3.1' can be sol!ed e%actl$ to gi!e the Black-Scholes !alue of acall option. We show how to do this in hapter 9, and at the end of this

section we "uote the results for a #uropean call and put.>or a put option, with !alue 5 S, t , the final condition is the pa$off

5 S,T = a% # - S, . 3.13

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3./ Boundar$ and >inal onditions /

We ha!e alread$ entioned that if S is e!er Fero then it ust re ain

Fero. In this case the final pa$off for a put is known with certaint$ to be#. To deter ine 5 , t we si pl$ ha!e to calculate the present !alue ofan a ount # recei!ed at ti e T. (ssu ing that interest rates areconstant we find the boundar$ condition at S = to be

5 , t = #e-r T-t . 3.1

More generall$, for a ti e-dependent interest rate we ha!e

5 B, t = #e- f,- r r

dr

(s S - oo the option is unlikel$ to be e%ercised and so

5 S, t -H as S - oo. 3.19

Technical 5oint6 Boundar$ onditions at Infinit$.We see later that we can transfor 3.D into an e"uation with constantcoefficients b$ the change of !ariable S = #eF. The point S = beco es %= -oo and S = oo beco es % = oo. (s we also see, a ph$sical analog$ tothe financial proble is the flow of heat in an infinite bar. learl$,

pre scribing the te perature of the bar at % = Poo has no effect whatsoe!erat finite !alues of % unless the te perature is highl$ singular there. Ifthe te perature at infinit$ is well-beha!ed then the te perature in an$ finite region of the bar is go!erned wholl$ b$ the initial data6 it cannot beinfluenced b$ the ends at infinit$. Since ost option proble s can betransfor ed into the diffusion e"uation it is also not strictl$ necessar$ to

prescribe the boundar$ conditions at S = and S = oo. We onl$ need toinsist that the !alue of the option is not too singular.

We can distinguish between

< prescribing a boundar$ condition in order to ake the solution uni"ue,and

< deter ining the solution in the neighbourhood of the boundar$, per-haps to assist or check a nu erical solution.

The boundar$ conditions 3.11 and 3.1' contain ore infor ation than isstrictl$ athe aticall$ necessar$ see Section .3.' . :e!ertheless, the$are financiall$ useful6 the$ tell us ore infor ation about the be-ha!iour of the option at certain special parts of the S-a%is and can beused to i pro!e the accurac$ of an$ nu erical ethod. It can be shownthat an e!en ore accurate e%pression for the beha!iour of as S oois

S, t - S - #e- r T -t . 3.1A

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C The Black-Scholes Model

This is a si ple correction to 3.1' which accounts for the discounted

e%ercise price.

Throughout the book we gi!e boundar$ conditions to show the local beha!iour of the option price.

3.C The Black-Scholes >or ulae for #uropean )ptions

Eere we "uote the e%act solution of the #uropean call option proble3.D - 3.1' when the interest rate and !olatilit$ are constant0 in hap-ter 9 we show how to deri!e it s$ste aticall$. In hapter A we drop theconstraint that r and a are constant and find ore general for ulae.

When r and a are constant the e%act, e%plicit solution for the #uro peancall is

S,t = S: dl - #e- r T- t : d' , 3.1/

where : . is the cu ulati!e distribution function for a standardisednor al rando !ariable, gi!en b$

1 $'

: % ' r Q .B eN'

Eere

d$.

dl

and

d'

log S4# H r H La' T - t

a T-t

- log S4# H r - 1 ,' T - t

a T-t>or a put, i.e. 3.D , 3.13 , 3.1 and 3.19 , the solution is

5 S, t = #e- r T- t : -d' - S: -dl . 3.1C

It is eas$ to show that these satisf$ put-call parit$ 3.' .The delta for a #uropean call is

-9 -S : di , 3.1D

and for a put it is

aS : di -1.

The latter follows fro the for er b$ put-call parit$.

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3.C The Black-Scholes >or ulce D

S#

>igure 3.A The #uropean call !alue S, t as a function of S for se!eral !alues ofti e to e%pir$0 r = .1, or = .', # = 1 and T - t = , .9, 1. and 1.9.

5

#

>igure 3./ The #uropean put !alue 5 S, t as a function of S for se!eral !alues ofti e to e%pir$0 r = .1, or = .', # = 1 and T - t = , .9, 1. and 1.9.

)ther deri!ati!es of the option !alue with respect to S, t, r and acan pla$ i portant roles in hedging and are discussed briefl$ at the end ofthis chapter.

In >igures 3.A and 3./ we show plots of the #uropean call and put!alues for se!eral ti es up to e%pir$. :ote how the cur!es approach

the pa$off functions as t -H T. In >igure 3.C we show the #uropeancall delta as a function of S, again for se!eral ti es up to e%pir$. Thedelta is alwa$s between Fero and one, and approaches a step function

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9 The Black-Scholes Model

(

S

#>igure 3.C The #uropean call delta as a function of S for se!eral !alues ofti e to e%pir$0 r = .1, or = .', # = 1 and T - t = , .9, 1. and 1.9.

as t -? T. 7ecall that the writer of a call option will be re"uired todeli!er the asset if S ? # at e%pir$, and not otherwise. If he follows thedelta-hedging strateg$, with a portfolio - (S, he will auto aticall$hold the correct a ount one or Fero of the asset at e%pir$. This is to

be e%pected, since delta-hedging is a risk-free strateg$ right up toe%pir$. If the option e%pires in-the- one$, the re"uired asset will ha!e

been bought o!er the lifeti e of the option, firstl$ in setting up theinitial hedge, and secondl$ in a series of transactions as S changes. The

cost of these purchases and4or sales, less the e%ercise price #, is e%actl$ balanced b$ the initial pre iu and bank interest. on!ersel$, if theoption e%pires out-of-the- one$, the initial hedge is graduall$ sold. Itshould also be noted that if the !alues of the asset &ust before e%pir$ areclose to #, the hedge a$ change fro nearl$ Fero to nearl$ one an$ti es. This is awkward to handle in practice, since each transactionincurs costs. We discuss transaction costs further in hapter 1A.

#"uations 3.1/ and 3.1C for the !alues of #uropean call and putoptions are interesting in that the$ contain the function for the cu ulati!enor al distribution : % . Thus the !alue of an option is related to the

probabilit$ densit$ function for the rando !ariable log S. It can beshown, and we discuss this in hapter 9, that the !alue of an option has a

natural interpretation as a certain discounted e%pected !alue of the pa$off at e%pir$. This leads to the sub&ect of the 2risk-neutral !aluation+ ofcontingent clai s, a phrase which is e%plained there.

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3.D Eedging in 5ractice 91

3.D Eedging in 5ractice

Eedging is the reduction of the sensiti!it$ of a portfolio to the o!e-

ent of an underl$ing asset b$ taking opposite positions in differentf inancial instru ents. Two e%tre e cases ha!e been introduced abo!e0in both cases the sensiti!it$ of the portfolio was reduced to Fero. Thef irst e%a ple was in the de onstration of put-call parit$ for #uropeanoptions and the second was in the Black-Scholes anal$sis with delta-hedging. These are, howe!er, funda entall$ different hedging strategies.The for er in!ol!es a one-off transaction in three products a call, a putand the underl$ing 0 the resulting portfolio can then be left unattendedwith the riskless return locked in. The latter is a d$na ic strateg$0 thedelta hedge is onl$ instantaneousl$ risk-free, and it re"uires a continu-

ous rebalancing of the portfolio and the ratio of the holdings in the assetand the deri!ati!e product. The delta-hedge position ust be onitoredcontinuall$, and in practice it can suffer fro losses due to the costs oftransacting in the underl$ing.

)ne use for delta-hedging is for the writer of an option who also wishesto co!er his position. If the writer can get a pre iu slightl$ abo!ethe fair !alue for the option then he can trade in the underl$ing or afutures contract on the underl$ing, since this is usuall$ cheaper to tradein because the transaction costs are lower to aintain a delta-neutral

position until e%pir$. Since he charges ore for the option than it wastheoreticall$ worth he akes a net profit without an$ risk - in theor$.This is onl$ a practical polic$ for those with access to the arkets at

low dealing costs, such as arket akers. If the transaction costs aresignificant then the fre"uent rehedging necessar$ to aintain a delta-neutral position renders the polic$ i practical. We discuss this pointfurther in hapter 1A.

The delta for a whole portfolio is the rate of change of the !alue ofthe portfolio with respect to changes in the underl$ing asset.3 Writing IIfor the !alue of the portfolio,

arlas O

Thus, when delta hedging between an option and an asset, the positiontaken is called 2delta-neutral+, since the sensiti!it$ of the hedged portfolio toasset price changes is instantaneousl$ Fero. >or a general portfolio

3 This definition, which is standard, is not "uite consistent with our pre!ious use of and fl which is also standard . There, was the sensiti!it$ of a single optionto asset price changes and fI was the hedged portfolio. There should be little riskof confusion.

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9' The Black-Scholes Model

the aintenance of a delta-neutral position a$ re"uire a short position in

the underl$ing asset. This entails the selling of assets which are notowned - so-called short selling. ( broker a$ re"uire a argin to co!eran$ o!e ents against the short seller but this argin usuall$ recei!esinterest at the bank rate.

There are ore sophisticated trading strategies than si pledelta hedging, and here we ention onl$ the basics. In delta-hedgingthe largest rando co ponent of the portfolio is eli inated. )ne can beore subtle and hedge awa$ s aller order effects due, for instance, tothe cur!ature the second deri!ati!e of the portfolio !alue with respect tothe underl$ing asset. This entails knowledge of the ga a of a

portfolio, defined b$a

r asThe deca$ of ti e !alue in a portfolio is represented b$ the theta, gi!en

b$ail

>inall$, sensiti!it$ to !olatilit$ is usuall$ called the !ega and is gi!en b$

anao, ,

and sensiti!it$ to interest rate is called rho, where

an5

or Eedging against an$ of these dependencies re"uires the use of anotheroption as well as the asset itself. With a suitable balance of theunderl$ing asset and other deri!ati!es, hedgers can eli inate the short-ter dependence of the portfolio on o!e ents in ti e, asset price,!olatilit$ or interest rate.

3.1 I plied 8olatilit$

We ha!e suggested in the abo!e odelling and anal$sis that the wa$ touse the Black-Scholes and other odels is to take para eter !alues esti-ated fro historical data, substitute the into a for ula or perhapssol!e an e"uation nu ericall$ , and so deri!e the !alue for a deri!ati!e

product. This is no longer the co onest use of option odels, atleast not for the si plest options. This is partl$ because of difficult$ ineasuring the !alue of the !olatilit$ of the underl$ing asset. Gespite

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3.1 I plied 8olatilit$ 93

our assu ption to the contrar$, it does not appear to be the case that!olatilit$ is constant for long periods of ti e. >urther ore, it is notob !ious that the historic !olatilit$ is independent of the ti e series frowhich it is calculated, nor that it accuratel$ predicts the future !olatilit$that we re"uire, o!er the lifeti e of an option.

( direct easure ent of !olatilit$ is therefore difficult in practice.Eowe!er, despite these difficulties it is plainl$ true that option pricesare "uoted in the arket. This suggests that, e!en if we do not know the!olatilit$, the arket 2knows+ it. Take the Black-Scholes for ula for acall, for e%a ple, and substitute in the interest rate, the price of theunderl$ing, the e%ercise price and the ti e to e%pir$. (ll of these are!er$ si ple to easure and are either "uoted constantl$ or are de-

f ined as part of the option contract. (ll that re ains is to specif$ the!olatilit$ and the option price follows. Since a call option price increasesonotonicall$ with !olatilit$ this is eas$ to show fro the e%plicit for-ula and, as we ha!e alread$ entioned, is clear financiall$ there is aone-to-one correspondence between the !olatilit$ and the option price.Thus we could take the option price "uoted in the arket and, working

backwards, deduce the arket+s opinion of the !alue for the !olatilit$o!er the re aining life of the option. This !olatilit$, deri!ed fro the"uoted price for a single option, is called the i plied !olatilit$.

There are ore ad!anced wa$s of calculating the arket !iew of!olatilit$ using ore than one option price. In particular, using op tion

prices for a !ariet$ of e%pir$ dates one can, in principle, deduce the

arket+s opinion of the future !alues for the !olatilit$ of the underl$ingthe ter structure of !olatilit$ .

)ne unusual feature of i plied !olatilit$ is that the i plied !olatil-it$ does not appear to be constant across e%ercise prices. That is, if the!alue of the underl$ing, the interest rate and the ti e to e%pir$ are f i%ed,the prices of options across e%ercise prices should reflect a unifor !aluefor the !olatilit$. In practice this is not the case and this high-lights a flaw in so e part of the odel. (lso, puts and calls tend togi!e slightl$ different i plied !olatilities. Which part of the odel isinaccurate is the sub&ect of a great deal of acade ic research. We illus-trate this effect in >igure 3.D, which shows the i plied !olatilities as afunction of e%ercise price using the >T-S# inde% option data in >igure

1.1. )bser!e how the !olatilit$ of the options deepl$ in-the- one$ isgreater than for those at-the- one$. This cur!e is traditionall$ calledthe 2s ile+, although depending on arket conditions it a$ be lopsided asin >igure 3.D, or e!en a 2frown+.

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9 The Black-Scholes Model

I plied!olatilit$.3 -

.' -

R R R

B.1 - urrentasset price

.'A '/ 'C 'D 3

#%ercise price

>igure 3.D I plied !olatilities as a function of e%ercise price. Gata is takenfro >T-S# inde% option prices.

Technical 5oint6 Trading 8olatilit$.In practice !olatilit$ is not constant, nor is it predictable for ti escales ofore than a few onths. This, of course, li its the !alidit$ of an$ odel

that assu es the contrar$. This proble a$ be reduced b$ pricingoptions using i plied !olatilit$ as described abo!e. Thus one tradingstrateg$ is to calculate i plied !olatilities fro prices of all options onthe sa e underl$ing and the sa e e%pir$ date and then to bu$ the onewith the lowest !olatilit$ and sell the one with the highest. The hope isthen that the prices o!e so that i plied !olatilities beco e ore or lessco parable and the portfolio akes a profit.

More sophisticated odelling in!ol!es describing !olatilit$ itself as arando !ariable satisf$ing so e stochastic differential e"uation. Thisresults in a two-factor odel. If the !olatilit$ is rando then it is nolonger possible to construct the perfect hedge, in which a portfolio grows

b$ a deter inistic a ount, using the asset alone. Eowe!er, it is in principle possible to use other options, but the details are too cople%to go into here.

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#%ercises 99

>urther 7eading

< arefull$ read the original papers of Black Scholes 1D/3 and Mer-

ton 1D/3 .< o pare the bino ial ethod for !aluing options with the differen-

tial e"uation approach. The bino ial ethod can be found in, fore%a ple, o% 7ubinstein 1DC9 . We discuss it in hapter 1 .

< Qarrow 7udd 1DC3 and o% 7ubinstein 1DC9 describe 2&u p-diffusion+ odels and 2constant elasticit$ of !ariance+ odels. In thefor er the asset price rando walk need not be continuous but canha!e rando discontinuous &u ps0 in the latter the !olatilit$ can bea function of S.

< Eull 1DD3 considers the esti ation of !olatilit$ using the i plied

!olatilities of se!eral options. Eull White 1DC/ discuss the !aria-tion of !olatilit$ with ti e.

< There has been a great deal of work done on testing the !alidit$ ofthe Black-Scholes for ulae in practice0 see Eull 1DD3 . >or detailsof how the call option for ula stands up in practice see MacBeth Mer!ille 1D/D and for a test of put-call parit$ see le kosk$ 7esnick 1D/D .

< Ue ill 1DD' gi!es a practical e%a ple illustrating the practicalshortco ings of the purel$ theoretical approach to hedging.

< More sophisticated hedging strategies are described in o% 7ubin-stein 1DC9 .

#%ercises

1. Toda$+s date is D Qanuar$ ' and KVL+s share price stands at 1 . )nC :o!e ber ' there is to be a 5residential election and $ou belie!ethat, depending on who is elected, KVL+s share price will either riseor fall b$ appro%i atel$ 1 X. onstruct a portfolio of options which

will do well if $ou are correct. alls and puts are a!ailable with e%pir$dates in March, Qune, Septe ber, Gece ber and with strike prices of1 plus or inus 9 Y. Graw the pa$off diagra and describe the pa$offathe aticall$.

'. Graw the e%pir$ pa$off diagra s for each of the following portfolios6

a Short one share, long two calls with e%ercise price # this co - bination is called a straddle0

b ;ong one call and one put, both with e%ercise price # this isalso a straddle6 wh$ 0

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9A The Black-Scholes Model

c ;ong one call and two puts, all with e%ercise price # a strip 0 d;ong one put and two calls, all with e%ercise price # a strap 0 e;ong one call with e%ercise price #l and one put with e%ercise

price #'. o pare the three cases #, ? #' known as astran gle , #l = #' and #l @ #'.

f (s e but also short one call and one put with e%ercise price #

when #l @ # @ #', this is called a butterfl$ spread .

se the arket data of >igure 1.1 to calculate the cost of an e%a ple ofeach portfolio. What !iew about the arket does each strateg$e%press

3. Show b$ substitution that two e%act solutions of the Black-Scholese"uation 3.D are

a 8 S, t = (S,b 8 S,t = (ert,

where ( is an arbitrar$ constant. What do these solutions representand what is the ( in each case

. Show that the for ulae 3.1/ for a call and 3.1C for a put also satisf$3.D with the rele!ant boundar$ conditions one at each of S = andS = oo and final conditions at t = T. Show also that the$ satisf$

put-call p arit$.

9. Sketch the graphs of the ( for the #uropean call and put. Suppose thatthe asset price now is S = # each of these options is at-the- one$ .on!ince $ourself that it is plausible that the delta-hedging strateg$ isself-financing for each option, in the two c as es that the option e%piresin-the- one$ and out-of-the- one$0 look at the contract fro the pointof !iew of the writer.

A. >ind the ost general solution of the Black-Scholes e"uation that hasthe special for

a 8 = 8 S 0b 8 =( t B S .

These are e%a ples of 2si ilarit$ solutions+, which are discussed furtherin hapter 9. Ti e-independent options as in a are called perpetualoptions.

/. se arbitrage argu ents to pro!e the following si ple bounds on #u-ropean call options on an asset that pa$s no di!idends6

a @ S0b ? S - #e- r T-t 0

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#%ercises 9/

c If two otherwise identical calls ha!e e%ercise prices #l and #'

with #l @ #', then

@ S,t0#1 - S,t0#' @#'-#,0

d If two otherwise identical call options ha!e e%pir$ ti es Tl andT' with TI @ T'i then

S,t0T& @ S,t0T' .

Geri!e si ilar restrictions for put options.

C. Geri!e e"uation 3.1A .

D. Suppose that a share price S is currentl$ 1 , and that to orrow itwill be either 1 1, with probabilit$ p, or DD, with probabilit$ 1 - p.

( call option, with !alue , has e%ercise price 1 . Set up a Black-Scholes hedged portfolio and hence find the !alue of . Ignore interestrates.

:ow repeat the calculation for a cash-or-nothing call option with pa$off 1 if the final asset price is abo!e 1 , Fero otherwise. Whatdifference do $ou notice

This !er$ si ple discrete odel is the basis of the bino ial ethod,described in hapter 1 .

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5artial Gifferential #"uations

.1 IntroductionThe odelling of hapter 3 cul inates in the for ulation of the pricing

proble for a deri!ati!e product as a partial differential e"uation. Wenow take a break fro the financial odelling to discuss, in this and thene%t chapter, so e of the theor$ behind such differential e"uations. Inthis chapter we describe the ele entar$ theor$ and the nature of

boundar$ and initial conditions. In hapter 9 we deri!e so e e%plicitsolutions, including the original Black-Scholes for ulae. ;ater, in hap ter/, we describe in detail the special proble s arising when there are free

boundaries. This chapter is of particular iportance when consid ering the!aluation of ( erican options.

The stud$ of partial differential e"uations in co plete generalit$ is a!ast undertaking. >ortunatel$, howe!er, al ost all the partial differ entiale"uations encountered in financial applications belong to a uch oreanageable subset of the whole6 second order linear parabolic

e"uations. These technical ter s are discussed below0 ore detailedtreat ents of the areas be$ond the scope of this te%t are gi!en in so e ofthe references at the end of the chapter.

We begin this chapter with a re!iew of second order linear parabolice"uations6 their ph$sical interpretation, athe atical properties of theirsolutions, and techni"ues for obtaining e%plicit solutions to specific

prob le s. Then, we e%ploit this knowledge in the conte%t of financialodels to deri!e e%plicit solutions to so e option !aluation proble s,and we set the scene for the nu erical ethods of hapters C and D.

Before doing this, though, it is helpful to step back and consider ingeneral ter s the "uestions we should ask when considering a partial

9C

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A 5artial Gifferential #"uations

and widel$ used odels of applied athe atics, and a considerable bod$ of

theor$ on its properties and solution is a!ailable. It is often helpful as aguide to intuition to bear in ind the ph$sical situations that leadto the heat e"uation, and we ention the where!er it is appropriate.Thus, we recall that e"uation .1 odels the diffusion of heat in onespace di ension, where u %, T represents the te perature in a long,thin, unifor bar of aterial whose sides are perfectl$ insulated so thatits te perature !aries onl$ with distance % along the bar and, of course,with ti e T.

We begin with a list of so e of the ele entar$ properties of thedif fusion e"uation.

< It is a linear e"uation. That is, if ul and u' are solutions, then so is

1 1 Hc'u' for an$ constants cl and c'.< It is a second order e"uation, since the highest order deri!ati!e

occurring is the second, in the ter a'u4a%e.< It is a parabolic e"uation. Its characteristics are gi!en b$

T = constant. The ter s 2parabolic+ and 2characteristic+ are discussedfurther in Technical 5oint 1 at the end of this section. Thus, infor-ation propagates along these lines in %, T space, and if a change isade to u at a particular point, for e%a ple on the boundar$ of the

solution region, its effect is felt instantaneousl$ e!er$where else.< Uenerall$ speaking, its solutions are anal$tic functions of %. This

eans that for each !alue of T greater than the initial ti e, u %, Tregarded as a function of % has a con!ergent power series in ter s of

% - %o for each %o awa$ fro spatial boundaries. >or practical pur- poses, for T ? we can think of a solution of the diffusion e"uationas being as s ooth a function of % as we could e!er need, but discon-

tinuities in ti e a$ be induced b$ the boundar$ conditions. Thisis again a conse"uence of the fact that infor ation propagates with

infinite speed along the characteristics T = constant.

>ro the ph$sical point of !iew, diffusion is a s oothing out process6heat flows fro hot to cold and so e!ens out te perature differences.The properties abo!e go so e wa$ towards showing that solutions ofthe diffusion e"uation, which is a athe atical odel of the ph$sical

process, ha!e the sa e tendenc$. (nticipating so e results fro Sec-

tion .3, it can be shown further that e!en though the initial !alues of ua$ be rather irregular or &agged, for an$ T ? the solution of the

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.' The Giffusion #"uation

initial !alue problehu a' u

aT - C%'

with initial data

A1

- @%@oo,

u %, = u %

and

u --? as % - Poo

is anal$tic for all T ? . This s oothness, which is characteristic of allforward linear parabolic e"uations, is !er$ helpful when it co es tonu erical solution.

(n illustration of all these points is the following special solution,which is deri!ed in Section 9.'6

uA %+T

' 84/ -%'

4 r - W @ % @ , T ? . .'

>or T ? this is a s ooth Uaussian cur!e, but at T = it is 2e"ual+ tothe delta function hence our notation 6

uA %, = A % .

(t r = , uA %, !anishes for % 9 0 at % = it is 2infinite+, but itsintegral is still 1. This is to be interpreted as follows6 since for allT ? , f ua %, T d% = 1, the li it as T tends to Fero fro abo!e ofthe integral is still 1. >urther infor ation on delta functions is gi!enin Technical 5oint ' below. We show uA %, T in >igure .1 for se!eral!alues of T0 note how the cur!e beco es taller and narrower as T getss aller.

The delta function initial !alue for uA %, T sa$s that all the heat isinitiall$ concentrated at % = . This function odels the e!olution of an

idealised 2hotspot+, a unit a ount of heat initiall$ concentrated into asingle point, and it is called the funda ental solution of the diffusione"uation. It also illustrates the infinite propagation speed entionedabo!e. (t T = , the solution .' is Fero for all % , but for an$ T ?, howe!er s all, and an$ %, howe!er large, uA %, T ? 6 the heat

initiall$ concentrated at % = i ediatel$ diffuses out to all !alues of%. :ote, though, that uA falls off !er$ rapidl$ as I%Q -? oo.

>inall$, note that the right-hand side of e"uation .' is &ust thenor al distribution of probabilit$ theor$, with ean Fero and !ariance'T. This solution of the diffusion e"uation can be interpreted as the

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A' 5artial Gifferential #"uations

ub

>igure .1. The funda ental solution of the diffusion e"uation.

probabilit$ densit$ function of the future position of a particle thatfollows a constant coefficient rando walk along the %-a%is. The deltafunction initial condition si pl $ sa$s that the particle is initiall$ known to

be at the origin.

Technical 5oint 16 haracteristics of Second )rder ;inear 5ar-tial Gifferential #"uations.

We can think of the characteristics of a second order linear e"uation ascur!es along which infor ation can propagate, or as cur!es across whichdiscontinuities in the second deri!ati!es of u can occur. Suppose thatu %, r satisfies the general second order linear e"uation

a ' F FHb %,T

a %+T &/K ' a T H %+T f/T'

Cu a CuHd %,T a% He %+T CT Hf %,T uHg %,T = .

The idea is to see whether the deri!ati!e ter s can be written in ter sof directional deri!ati!es, so that the e"uation is partl$ like an ordinar$

differential e"uation along cur!es with these !ectors as tangents. Thesecur!es are the characteristics. If we write the as % = % e , T = T e ,

where is a para eter along the cur!es, then % e and T 10 satisf$

a %, T'

d

'

- b %, T @ H c %, T = .@ /B

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.' The Giffusion #"uation A3

There now arises the "uestion whether this e"uation, regarded as a "uad-ratic in d%4d. 4 dT4@ , has two distinct real roots, two e"ual real roots, orno real roots at all. These cases correspond to the discri inant b' - ac beinggreater than Fero, Fero, or less than Fero. The first case, two real fa iliesof characteristics, is called h$perbolic, and is t$pical of wa!e-

propagation proble s. These do not often occur in finance. The secondcase, an e%act s"uare, is called parabolic0 the diffusion e"uation, whichhas b = c = , is the si plest e%a ple. (ll the second order e"uations inthis book are parabolic. The final case, with no real characteristics, iscalled elliptic, and is t$pical of stead$-state proble s such as perpetual

options in ulti-factor odels which are be$ond the scope of this book.

:ote that the definitions gi!en here are pointwise6 the h$perbolic4 parabolic4elliptic distinction is specified at each point. It is possible foran e"uation to change t$pe as a %, T , b %, T and c %, T !ar$, if thediscri inant changes sign. In particular, the Black-Scholes e"uation in Sand t rather than % and T ,

a! Hat 1 c$'s''

a'8as' as

is parabolic for S ? it is in fact h$perbolic at S = , where it reduces toan ordinar$ differential e"uation with characteristic S = . This fact hasi portant financial i plications6 the line S = is a barrier across whichinfor ation cannot cross.

Technical 5oint '6 The Gelta >unction and the Eea!iside>unction.The Girac delta function, written C % , is not in fact a function in thenor al sense of the word, but is rather a 2generalised function+. >or tech-nical reasons, its definition is as a linear ap, but it is reall$ oti!ated b$the need for a athe atical description of the li it of a function whose

effect is confined to a s aller and s aller inter!al, but $et re ains finite.Suppose, for e%a ple, that I recei!e one$ at the rate f t dt in a ti e

dt where f is e"ual to the following function6

f t 14'#, Itl @ #, Itl ?#.

This function is drawn in >igure .' for se!eral !alues of #. (s c getss aller the graph beco es taller and narrower. It is clear that the total

pa$ ent is

Q MM f t dt

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AA 5artial Gifferential #"uations

and so M t satisfies the differential e"uation

dMdt =GA9 t-to .

The discontinuit$ in M t gi!es a delta function in dM4dt at t = to.on!ersel$, when we see a differential e"uation with a delta function onthe right-hand side, there ust be a corresponding delta function in thehighest order deri!ati!e on the left-hand side in order to aintain a

balance. This in turn eans that the ne%t highest order deri!ati!e has a &u p discontinuit$ of agnitude e"ual to the coefficient of the deltafunction. These &u p conditions can be used to &oin together s oothseg ents of the solution across discontinuities. The delta function ine%a ples like this can be ultiplied b$ a s ooth function of % or t, butcare ust be taken to a!oid products like 9 % D-l % or A % ', for whichno sensible definition can easil$ be gi!en.

.3 Initial and Boundar$ onditions

We now consider what initial and boundar$ conditions are appropriatefor solutions of the diffusion e"uation, first in a finite region, then in aninfinite one.

.3.1 The Initial 8alue 5roble on a >inite Inter!al

Suppose we wish to sol!e au 4 T = a'u4a%e in the finite inter!al-; @ % @ ; and for T ? , representing heat flow in a bar of finitelength ';.

)b!iousl$ we should specif$ the initial te perature u %, = uo % for-; @ % @ ;. With the heat flow analog$ in ind, it see s reasonable on

ph$sical grounds that we ha!e enough infor ation to deter ine u %, Tuni"uel$ if we specif$ either the te peratures at the ends of the bar orthe heat flu%es at both ends, but not both. This turns out to be thecase0 in fact both the following state ents of the proble can be shownto be well-posed6

au C'ui aT a%e -; @ % @ ;, with u %, = uo % ,

u -;, -r = gN T , u ;,T = gH T 0

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.3 Initial and Boundar$ onditions A/

au ',ii-aT

' , -; @%@ ;, with u %, = uo % ,a%

-+au -;, T = h- T ,a% ;,T = hH T .

In the first case it is the te perature and in the second case the heatf lu%es that are specified at % = -; and % = ;.

.3.' The Initial 8alue 5roble on an Infinite Inter!al

Suppose now that we consider heat flow in a !er$ long bar, b$ taking theli it ; ---H oo in the e%a ple abo!e. When the bar is infinitel$ long, itis still i portant to sa$ how u beha!es at large distances, but we do notha!e to be as precise in our specification of u at the 2boundaries+ % = Pas we were in the finite case. There are so e technical difficulties here,associated with the notion of infinit$, but roughl$ speaking as long as u isnot allowed to grow too fast, the solution e%ists, is uni"ue, and dependscontinuousl$ on the initial data uo % . To be specific, the solution tothe initial !alue proble

au a'u= - @K@ , r? , .3

aT 1D%'

with

u %, = uo % , .

where

i uo % is sufficientl$ well-beha!ed, .9

ii li uo % e -1D%' = for an$ a ? , .AI%I-.

and lastl$ where

li u %,T e -1D%' = for an$ a ? , T ? , ./I%I-.

is well-posed. The precise definition of the phrase2sufficientl$ well- beha!ed+ here is be$ond the scope of this book, but certainl$ an$ func-

tion that has no worse than a finite nu ber of &u p discontinuities isacceptable. We also note that although it is necessar$ to prescribe the

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AC 5artial Gifferential #"uations

beha!iour at infinit$, in practice the li itations abo!e are not too se!ere.(ll the initial !alue proble s in this book satisf$ the growth conditions"uite co fortabl$.

We so eti es need to consider initial !alue proble s defined on ase i-infinite inter!al, for e%a ple in the anal$sis of barrier options. Inthis case we re"uire a co bination of the two sets of conditions abo!e. If, fore%a ple, we need to sol!e .3 for @ % @ oo, T ? , then gi!en suf ficientl$ s ooth initial data uo % for @ % @ oo, a sufficientl$ s ooth

boundar$ !alue at % = , u , T = go / , and the growth conditions.A , ./ as % -O oo, the proble is well-posed.

. >orward !ersus Backward

In all the abo!e we ha!e discussed the forward e"uation

)n a ' u)T - a%'+

with conditions gi!en at T = . The reader a$ ask, what is wrong withthe e"uation

)n a'u

TT %' .C

with the sa e initial and boundar$ conditions This e"uation ight, fore%a ple, arise if in a forward proble we had replaced -r b$ To - T forso e constant To, whereupon a u4aT becoes -au4aT. It turns out thatthis backward proble is ill-posed6 for ost initial and boundar$ datathe solution does not e%ist at all, and e!en if it does e%ist, it is likel$to blow up for e%a ple, u a$ tend to oo within a finite ti e. ( goode%a ple is the funda ental solution of the diffusion e"uation .' . (tti e To this solution is e"ual to

1 e-%'4 To

' 1r To

which is as s ooth and well-beha!ed as we could wish. If we use thisfunction as our initial data uo % for e"uation .C , then the solution is

u %,T = 1 e-%'4 T B-T' ir T) - T

and this beco es singular blows up at T = To, when it is e"ual to thedelta function A % . Moreo!er, it cannot be continued be$ond this ti e atleast, not as a 2nor al+ function .

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#%ercises AD

5h$sicall$ this distinction akes good sense. If the forward diffusion

e"uation odels the e!olution of the te perature fro its initial !alues,the backward e"uation poses the "uestion of deter ining the te peraturefro which the initial distribution could ha!e e!ol!ed0 this is clear frothe ti e-re!ersal argu ent abo!e. Since forward diffusion s ooths out

&agged te perature distributions, backward diffusion akes s ooth initialdata beco e ore &agged. (nother wa$ of seeing this is to note thatunder forward diffusion heat flows fro hot to cold, whereas under

backward diffusion it flows fro cold to hot, and so the hot places becoe e!er hotter, leading to blow-up.

There are, howe!er, so e well-posed proble s for e"uation .C 0 in particular the final !alue proble for the backward diffusion e"uation iswell-posed. Thus, we can sol!e .C for @ T @ To with u %, To

gi!en. This is easil$ shown b$ con!erting .C to a forward proble b$replacing T b$ To - T.

>urther 7eading

< >or further infor ation about first order partial differential e"uationsand their solution see Willia s 1DC , Strang 1DCA , eener 1DCCand e!orkian 1DD .

< Three books de!oted wholl$ to the diffusion e"uation are those b$rank 1DCD , Eill Gew$nne 1DD and arslaw Qaeger 1DCD .

< More details about delta functions, and about other generalised func-

tions or 2distributions+ are gi!en b$ 7ichards Voun 1DD .

#%ercises

1. Show that the solution to the initial !alue proble is uni"ue pro!ided

that it is sufficientl$ s ooth and deca$s sufficientl$ fast at infinit$, as

follows6Suppose that u1 %, T and u' %, T are both solutions to the initial

!alue proble .3 - ./ . Show that ! %,T = u1-u' is also a solution of.3 with ! %, = .

Show that if

# T =

Q !' d%, BB# T ? , # = ,

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/

and, b$ integrating b$ parts, that

d#dr@ 0

thus # r = , hence ! %, r = .

5artial Gifferential #"uations

:ote, though, that as $et we ha!e no guarantee that u %, r e%ists,nor that the abo!e anipulations can be &ustified.

'. Show that sin n% a-,''T is a solution of the forward diffusion e"uation,and that sin n% e/''T is a solution of the backward diffusion e"uation.

:ow tr$ to sol!e the initial !alue proble for the forward and backward

e"uations in the inter!al -/r @ % @ Fr , with u = on the boundariesand u %, gi!en, b$ e%panding the solution in a >ourier series in % withcoefficients depending on r. What difference do $ou see between the

two proble s Which is well-posed The for er is useful for doubleknockout options0 see #%ercise 3 in hapter 1'.

3. 8erif$ that uA %, r does satisf$ the diffusion e"uation for r ? .

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9 The Black-Scholes >or ulae

9.1Introduction

In this chapter we describe so e techni"ues for obtaining anal$tical so-lutions to diffusion e"uations in fi%ed do ains, where the spatial bound-aries are known in ad!ance. >ree boundar$ proble s, in which the spa-tial boundaries !ar$ with ti e in an unknown anner, are discussed inhapter /. We highlight in particular one ethod6 we discuss si ilarit$solutions in so e detail. This ethod can $ield i portant infor ationabout particular proble s with special initial and boundar$ !alues, and itis especiall$ useful for deter ining local beha!iour in space or in ti e. It isalso useful in the conte%t of free boundar$ proble s, and in hap-ter / we see an application to the local beha!iour of the free boundar$for an ( erican call option near e%pir$. Be$ond this, though, we can

also use si ilarit$ techni"ues to deri!e the funda ental solution of thediffusion e"uation, and fro this we can deduce the general solution forthe initial-!alue proble on an infinite inter!al. This in turn leads i -ediatel$ to the Black-Scholes for ulae for the !alues of #uropean calland put options. >inall$, we e%tend the ethod to so e options withore general pa$offs, and we discuss the risk-neutral !aluation ethod.

9.' Si ilarit$ Solutions

It a$ so eti es happen that the solution u %,+r of a partial differentiale"uation, together with its initial and boundar$ conditions, depends onl$on one special co bination of the two independent !ariables. In suchcases, the proble can be reduced to an ordinar$ differential e"uation inwhich this co bination is the independent !ariable. The solution to thisordinar$ differential e"uation is called a si ilarit$ solution to the

/1

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/' The Black-Scholes >or ula

original partial differential e"uation. The athe atical reasons for thee%istence of this reduction are subtle and be$ond the scope of this book,

although the Technical 5oint at the end of this section, which dealswith the echanics of finding si ilarit$ solutions, does hint at the .We si pl$ gi!e two e%a ples here.

#%a ple 1. Suppose that u %, T satisfies the following proble on these i-infinite inter!al % ? 6

au a'uK , 9 .1

aTa% -8

with the initial condition

u %, = , 9.'

and a boundar$ condition at % = ,

u , T = 10 9.3

we also re"uire that

u-Oas%-?oo. 9.

These e"uations odel the e!olution of te perature in a long bar,initiall$ at Fero te perature, after the te perature at one end is suddenl$

raised to 1 and held there.>ollowing the argu ents suggested in the Technical 5oint below, we

look for a solution in which u %, T depends onl$ on % and r through theco bination l0 = %4 f, so that u %, r = t0 . Gifferentiation showsthat

NNNN t

aT 'T S and

a'uN 1a%' * B1

where d4dl0. Substitution into e"uation 9.1 shows that all theter s in!ol!ing r on its own a$ be cancelled, and satisfies thesecond order ordinar$ differential e"uation

* HI , = . 9.9

>ro the initial and boundar$ conditions 9.' - 9. ,

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9.' Si ilarit$ Solutions /3

) = 1, oo = . 9.A

The second of these incorporates both 9.' and 9. , since as T -H fro abo!e, L -? oo.

Separating the !ariables, we find that

+ = e-e'4

for so e constant . Integrating,

f L0 = Q e-D'4 ds H G

where G is a further constant. (ppl$ing the boundar$ conditions 9.A ,

writing fo = f / - f , and using the standard result'

e-D 4 ds ,1BBB

we find that

) =

that is,

u %,T N

e- 9'4 ds81N/Nrf

1 f

Q e-D'4 ds.It is eas$ to !erif$ that this function does satisf$ the proble state ent9.1 - 9. , so that the solution does indeed depend onl$ on %4.,fT-.

#%a ple '. >or our second e%a ple we deri!e the funda ental solutionuA %, i- , which we introduced in hapter . We again look for a solution ofthe diffusion e"uation that depends on % onl$ through the co bination

= %4 f, but now we tr$ the for

us %,T = T-14' A .

The r-14' ter ultipl$ing A is there to ensure that f u %, rr d% isconstant for all r, which can be shown b$ direct calculation. (co putation si ilar to the e%a ple abo!e shows that A L0 satisfiesthe ordinar$ differential e"uation

A H L A + = .

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/ The Black-Scholes >orrnul .#

The general solution of this, obtained b$ integrating twice, the secondti e with the help of the integrating factor e '4 , is

A = e- '4 H G

for constant and G. hoosing G = and nor alising the solution b$ setting = 14 ' f , so that f * ud% = 1, $ields the funda entalsolution

uA % T =

as re"uired.

1

' /rT4 -r

e2'

The si ilarit$ solution techni"ue is rarel$ successful in sol!ing aco plete boundar$ !alue proble , because it re"uires such special

s$ e tries in the e"uation and the initial and boundar$ conditions. )nthe other hand, it co es into its own in local anal$ses in space or inti e, for e%a ple the initial otion of a free boundar$ in an ( ericanoption proble and the !alue of an at-the- one$ option shortl$ beforee%ercise, which are hard to resol!e nu ericall$.

Technical 5oint6 Uroup In!ariances and Si ilarit$ Solutions. Theke$ to the si ilarit$ solutions abo!e is that both the e"uations and theinitial and boundar$ conditions are in!ariant under the scal-ings % -O (%, T (+/- for an$ real nu ber (. Such a scaling is called aone-para eter group of transfor ations. This in!ariance is readil$!erified using the new !ariables K = (%, T = (+/-, whereupon u is eas-

il$ seen to satisf$ au4aT = a'u4aK'. >urther ore, in #%a ple 1, theinitial and boundar$ conditions beco e u K, = , u , T = 1 for an$(. :ow %4,4 = K4! is the onl$ co bination of K and T which isindependent of (, and so the solution ust be a function of %44 onl$.It is essential that the e"uation, the boundar$ conditions and the initialconditions should all be in!ariant under the scaling transfor ation for

the ethod to work. In #%a ple ', the function of r, in this case T -14',ultipl$ing A t0 is present because the diffusion e"uation, being linear,

is also in!ariant under the one-para eter group u +--f \u. ( good practi-

cal test for si ilarit$ solutions is to tr$ u = ra f %4/- in the hope that% and T will re ain in the e"uations onl$ in the co bination A = % 4Tp.

In #%a ple 1 abo!e, the result of doing this is a = fro the bound-

ar$ condition at % = and , = F fro the diffusion e"uation, while in#%a ple ', a = - a because we want the integral of u %, r o!er % to beindependent of T, and again J = 1 '

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9.3 (n Initial 8alue 5roble /9

9.3 (n initial !alue proble for the diffusion e"uation

The funda ental solution of the diffusion e"uation can be used to deri!e

an e%plicit solution to the initial !alue proble .3 - ./ , in which weha!e to sol!e the diffusion e"uation for -oo @ % @ oo and T ? , witharbitrar$ initial data u %, = uo % and suitable growth conditions at% = Poo. The ke$ to the solution is the fact that we can write the initialdata as

uo % = f uo b - % dl0 BBBB

where A . is the Girac delta function. We recall that the funda entalsolution of the diffusion e"uation,

sub S,T

' !4/ r/t1

has initial !alue

uA s, = A s .

4 T

:ow note that because uA s - %, T = uA % - s, T ,

A S - K, /- = 1 e- s-% '4 T' /rT

is a solution of the diffusion e"uation using either s or % as the spatialindependent !ariable, and its initial !alue is

uA s - %, = b s - % .

Thus, for each s, the function

uo s ub S - K, T ,

regarded as a function of % and T with s held fi%ed, satisfies the diffusione"uation au 4aT = a'u 4a%e, and has initial data uo s b s - % . Because thediffusion e"uation is linear, we can superpose solutions of this for .Going so for all s b$ integrating fro s = -oo to s = oo, we obtain afurther solution of the diffusion e"uation,

u %,T

which has initial data

' T f uo s e- %-s '4 Tds9./

u %, = uo s A s - % ds = uo % .BBf

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/A The Black-Scholes >or ul$

This, therefore, is the e%plicit solution of the initial !alue proble .3./ . It can be shown #%ercise 1 of hapter that this solution isuni"ue. The deri!ation abo!e is not the onl$ wa$ of finding it6 the>ourier transfor is an alternati!e, but we do not describe it here seean$ of the books referred to in hapter for treat ents .

The solution 9./ can be interpreted ph$sicall$ as follows. 7ecallthat the funda ental solution of the diffusion e"uation describes thespreading out of a unit 2packet+ of heat which, at T = , is all concen-trated at the origin. Mathe aticall$, this 2packet+ is represented b$ adelta function. :ow i agine the initial te perature distribution uo %as being ade up of an$ s all packets, the packet at % = s ha!ingagnitude uo s ds. #ach of these e!ol!es to gi!e a te perature distri-

bution e"ual to the funda ental solution, ultiplied b$ uo s and with% replaced b$ % - s. Because the diffusion e"uation is linear, we obtainthe whole te perature distribution b$ superposing adding the e!olu-tions of these indi!idual packets0 in the li it, this su is replaced b$the integral 9./ .

9. The Black-Scholes >or ulae Geri!ed

The Black-Scholes e"uation and boundar$ conditions for a #uropeancall with !alue S, t are, as described in Sections 3.9 and 3.A,

'

at H Za+LS 9S H rS ac

- r = ,

with

, t = , S, t - S asS -H oo,

and

S,T = a% S - #, .

9.C

#"uation 9.C looks a little like the diffusion e"uation, but it has oreter s, and each ti e is differentiated with respect to S it is ultiplied b$S, gi!ing nonconstant coefficients. (lso the e"uation is clearl$ in

backward for , with final data gi!en at t = T.The first thing to do is to get rid of the awkward S and S' ter s

ul tipl$ing a 4aS and C' 4aS'. (t the sa e ti e we take theopportu nit$ of aking the e"uation di ensionless, as defined in theTechnical 5oint below, and we turn it into a forward e"uation. We set

S = #e%, t = T - T4J',

' = #! %, T . 9.D

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9. The >or ulr Geri!ed

This results in the e"uation

a ! a '! a!aT a%' H k - 1 a% - k!where k = r4 a'. The initial condition beco es

'! %, = a% e% - 1, .

//

9.1

:otice in particular that this e"uation contains onl$ one di ensionless para eter, k = r4 a', although there are four di ensional para eters, #,T, a' and r, in the original state ent of the proble . There is in fact '

another, 'a'T, the di ensionless ti e to e%pir$, and these two are theonl$ genuinel$ independent para eters in the proble 0 the effect of allother factors is si pl$ brought in b$ in!erting the abo!e transfor ations,

i.e. b$ a straightforward arith etical calculation.#"uation 9.1 now looks uch ore like a diffusion e"uation, and

we can turn it into one b$ a si ple change of !ariable. If we tr$ putting

! = ea%H)ru %,T ,

for so e constants a and 43 to be found, then differentiation gi!es

au a '4&u H aT = a ' u H 'aa% H

a%'H k - 1 au H

TK I - ku.

We can obtain an e"uation with no u ter b$ choosing 4

43=a'H k-1 a-k,

while the choice

= 'a H k - 1s

eli inates the au4a% ter as well. These e"uations for aand 43 gi!e

a=- k-1 , I3=- kH1 '.

! = e-' k-1 %-a kH1 'Tu % T

au = a'ufor -oo@%@oo, r ?B,

aT a%'

u %, = uo % = a%e'k H1 %

- e' k-1 % 9.11

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/C The Black-Scholes >or ulae

This a$ see like a long wa$ to tra!el fro the original for ulation,

but we ha!e reached the pa$off. The solution to the diffusion e"uation proble is &ust that gi!en in e"uation 9./ 6

u %,T ' /rT BBuo s e- %- e ' 4 rds 9.1'

where uo % is gi!en b$ 9.11 .It re ains to e!aluate the integral in 9.1' . It is con!enient to ake

the change of !ariable %+ = s - % 4 'T, so that

u %, r 1 l o %+ 'T H % e-'%+'d%+'/r N

BB1 e'// N%4 'r 'ck

H1? %H%,

'T e-'T

1

'//

Il - 1',

sa$.

e' k-1 %H 'r e-I %4 'r $,'d%,'

We e!aluate I1 b$ co pleting the s"uare in the e%ponent to get astandard integral6

I1 = e 1 -,kH1 ,%H%+ 'r '%i'd%+'// #%

e'4 'r

kH1 %e

'// N%4 'r

FkH 1 're- 'r

']%-' kH1 d%+

e ' k H1 % H k H1 Fr fo 1e '5dp '

'// %4 'r- kH1 'r '

e 1A kH 1 %H kH1 'r: d1 ,'

where N %

d1 '/

and

: d1 =

H' kH1 '/-,

rd l1' //Q e-'S'ds BB

is the cu ulati!e distribution function for the nor al distribution.

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9. The >or ulee Geri!ed /D

The calculation of I' is identical to that of Il, e%cept that k H 1 is

replaced b$ k - 1 throughout.;astl$, we retrace our steps, writing

! % T = e-' k- i %- 9.13a kHi LTu %,T

and then putting % = log S4# , r = J' T - t and = #! %,T , to

reco!er '

S, t = S: dl - #e-++ T-t : d' , 9.1

where

dl

d'

log S4# H r H !' T - t '

a T - t

- log S4# H r - J' T - t '

J T - t

The corresponding calculation for a #uropean put option follows si -ilar lines. Its transfor ed pa$off is

u %, = a% e

' k- 1 a N e' kHi %, , 9.19

and we can proceed as abo!e. Eowe!er, ha!ing e!aluated the call, asi pler wa$ is to use the put-call parit$ for ula

- 5 = S - #e- + T -t

for the !alue 5 of a put gi!en the !alue of the call. This $ields

5 S,t = #e-r T-t : -d' - S: -di ,

where we ha!e used the identit$ : d H : -d = 1.The deltas of call and put options are calculated b$ differentiation6

for the call,

acas

: dl H S a : dl - #e-+

TS as : dl H S:+ dl

as - #e-++ T-t :I d' a ad' s

: di H S:+ dl - #e-T T-t :+ d' 4 So T- --t

: di ,

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C The Black-Scholes >or ulae

since a rather painful calculation shows that S:+ dl = #e- r T- t :+ d'

di!ide both sides b$ : d' = 14 '/r e -,d' first . Then, the delta forthe put is

- C5

as : di - 1,

again using put-call parit$. These "uantities are !ital if an option posi-tion is to be hedged correctl$.

So e co puter algebra packages offer a li ited range of financialroutines. Maple, for e%a ple, has a Black-Scholes call co and. It has to

be loaded b$ t$ping

?readlib finance 0

and then the co and

?blackscholes #,T-t,S,r,sig a 0

returns the Black-Scholes !alue of the call with e%ercise price #, ti e toe%pir$ T-t, current asset price S, interest rate r and !olatilit$ sig a. Inthis e%a ple the s$ bols ha!e to be replaced b$ their nu erical !alues,

but the routine can also be used as a function. >or e%a ple,

?plot blackscholes 1 , .9,S, .1, .' , S= ..' 0

generates a plot of the call !alues for @ S @ ' , with the other

para eters held fi%ed at the !alues indicated. )ther Maple features canalso be used0 for e%a ple

?plot diff blackscholes 1 , .9,S, .1, .' ,S ,S= ..' 0?plot diff blackscholes 1 , .9,S, .1, .' ,S ' ,S= ..' 0

plot the delta and ga a respecti!el$. (lthough there is no separate put routine, it is eas$ to write one using the call routine and put-call parit$.

Technical 5oint6 Gi ensionless 8ariables.The differential e"uations used to odel ph$sical and financial processesoften contain an$ para eters0 these ight be aterial properties of the

substances in!ol!ed, for e%a ple, ther al conducti!it$, or constantsof the underl$ing stochastic processes, such as their rate of return or

!olatilit$. (n earl$ step in ost solutions is to scale the dependent and

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9.9 Binar$ )ptions C1

independent !ariables with 2t$pical !alues+ in order to collect these pa-

ra eters together as far as possible. Thus abo!e we scaled S and 8 with #,the onl$ a priori t$pical !alue a!ailable. (lthough S ight be ea-sured in # or , or GM, or an$ other units , % has no units, and nor does

!. This is i portant, since an e%pansion of the for es = 1H9H S'Hurther ore, the pa$off need not be a finite co bination of calls and puts6 we can consider an$ function of Sthat we wish. )ptions with pa$offs ore general than !anilla calls and

puts are known as binar$ options or digital options.Suppose that the pa$off at ti e T is ( S , and that the !alue of the

option is 8 S,t , so 8 S,T = ( S . We first work out the functionuo % corresponding to ( S after the transfor ations that we used

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9.A 7isk :eutralit$ C3

function is Fero, and therefore close to e%pir$ we e%pect that we shouldnot ha!e to hedge the option. Eowe!er, if S is close to # near e%pir$there is a high p robabilit$ that the asset price will cross the !alue #,

perhaps an$ ti es, before e%pir$. #ach ti e this !alue is crossed thedelta goes fro nearl$ Fero to !er$ large and back to nearl$ Fero. TheBlack-Scholes odel assu es that the option is continuousl$ hedgedwith a nu ber of assets e"ual to the delta0 this is clearl$ i practical if,at one o ent, the portfolio contains no assets, then is rehedged tocontain a large nu ber of the assets onl$ for that position to be li"ui-dated shortl$ afterwards. Vet, if this rehedging is not done, the pa$off ate%pir$ is either Fero or B, and cannot be known for certain.+ It istherefore open to "uestion whether options with discontinuous pa$offs

can be !alued according to the si ple Black-Scholes for ula 9.1A .

9.A 7isk :eutralit$

( rather different !iew of option !aluation fro that presented abo!e isthe risk-neutral approach. This ste s fro the obser!ation that thegrowth rate p does not appear in the Black-Scholes e"uation 3.D .Therefore, although the !alue of an option depends on the standardde!iation of the asset price, it does not depend on its rate of growth.

Indeed, different. in!estors a$ ha!e widel$ !ar$ing esti ates of thegrowth rate of a share $et still agree on the !alue of an option. Moreo!er,the risk preferences of in!estors are irrele!ant6 because the risk inherentin an option can all be hedged awa$, there is no return to be ade o!erand abo!e the risk-free return. Whether for !anilla options or other

products, it is generall$ the case that if a portfolio can be constructedwith a deri!ati!e product and the underl$ing asset in such a wa$ that therando co ponent can be eli inated - as was the case in our deri!ationof the Black-Scholes e"uation in hapter 3 - then the deri!ati!e producta$ be !alued as if all the rando walks in!ol!ed are risk-neutral.This eans that the drift ter in the stochastic differential e"uation forthe asset return for our e"uit$ odel, u is replaced b$ r where!er it

appears. The option is then !alued b$ calculating the present !alue ofits e%pected return at e%pir$ with this odification to the rando walk.The process works as follows.

We begin b$ recalling that the present !alue of an$ a ount at ti e T is that a ount discounted b$ ultipl$ing b$e- r T -t Then, we set up a risk-neutral world6 we pretend that the rando walk for the return on S has drift r of t. >ro this, we can calculate the probabilit$ densit$ function of future !alues of S6 we use e"uation '.1with \ replaced b$ r. It is ost i portant to realise that the new probabilit$ densit$ function is not that of S.

:e%t, we calculate the e%pected !alue of the pa$off ( S using this probabilit$ densit$ function. That is, weultipl$ ( S b$ the risk-neutral probabilit$ densit$ function and inte- grate o!er all possible future !alues of theasset, fro Fero to infinit$. >inall$, we discount to get the present !alue of the option. The resulting for ula is, a

before,

8 S,t =e -r T-t

%r c 'ir T-t

i 9.1 bg SIS - r- ' T-t '4'o' T-t ( Si

S+

I

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This e%pression can be shown b$ direct differentiation to satisf$ e"uation

3.D . When the pa$off is si ple, it can be integrated e%plicitl$ to gi!ethe Black-Scholes for ula for for e%a ple a #uropean call option.

The idea of replacing \ b$ r is !er$ elegant. It does, howe!er, ha!eso e a&or drawbacks. >irst, it re"uires us to know the probabilit$ den-sit$ function of the future asset !alues under the risk-neutral assu p-tion . This is eas$ enough for our constant-coefficient rando walks, but ifwe want to use an$ ore co plicated odel, we ust first find thedistribution before integrating to calculate the e%pected return. )ften,the calculation of the probabilit$ densit$ function in!ol!es sol!ing a par-

tial differential e"uation e"ui!alent to that satisfied b$ the option, and

the subse"uent integration ust in general be carried out nu ericall$ aswell. It is usuall$ "uicker to sol!e the option pricing e"uation directl$.

Moreo!er, when we co e to e%otic options or ( erican options, it isuch ore difficult to see how to i ple ent the risk-neutral approach,while as we show the direct approach !ia the partial differentiale"uation for the option can be e%tended in a clear-cut wa$.

( further drawback is that risk neutralit$ can lead to confusion. >ore%a ple, it is so eti es said that

< *It can be shown that t = r,*

or that

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#%ercises C9

< *The delta of an option is the probabilit$ that it will e%pire in the

one$.*

Both of these state ents are wrong. If the first state ent were correctthen all assets would ha!e the sa e e%pected return as a bank depositand no one would in!est in e"uities see the Technical 5oint on riskin hapter ' . If p were e"ual to r then the second state ent would

be correct. The probabilit$ that S ? # at t = T can be found b$calculating the e%pected !alue of f S - # . This necessaril$ in!ol!esthe para eter p.

>inall$, risk-neutralit$ is far fro eas$ to grasp intuiti!el$, whichis perhaps the source of the confusion abo!e. The ke$ steps in thederi!ation of the Black-Scholes e"uation, na el$ no arbitrage and thatrisk-free portfolios earn the risk-free rate, are intuiti!el$ clear.

>urther 7eading

< >or a discussion of si ilarit$ solutions of the diffusion e"uation see

rank 1DCD and Eill Gew$nne 1DD .< The risk-neutral ethod is set out in the papers b$ o% 7oss 1D/A

and Earrison 5liska 1DC1 . >or so e details of option !aluationunder risk neutralit$, see Earrison reps 1D/D and Eull 1DD3 .

#%ercises

1. >ind a si ilarit$ solution to the proble

au a'u -aT - a%e+

with

@%@ , T? ,

u %, = Ti % .

Show that au4a% is the funda ental solution ub %, -r , either b$ di-rect differentiation or b$ constructing the initial !alue proble that itsatisfies. ,

'. Suppose that u %, T satisfies the following initial !alue proble on ase i-infinite inter!al6

au a'uaT =a%e %? , T? ,

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CA The Black-Scholes >or ulc

with

u %, = uo % , % ? , u ),T = , T ? .

Gefine a new function ! %, T b$ reflection in the line % = , so that

! %, T = u %,T if % ? ,

! %,T = - u -%,T if % @ B.

Show that ! ), r = , and use 9./ to show that

u % T = 1L //

!o s QThe function ultipl$ing uo s here is called the Ureen+s function forthis initial-boundar$ !alue proble . This solution is applicable to

barr ier options.

3. >ind si ilarit$ solutions to

au a'u

or = a%e H > % , % ? , r ? ,with

u %, = , % ? , u , t = , r ? .

in the two cases a > % = %0 b > % = 1.

#%tend case b b$ letting u , r = T. ( related si ilarit$ solu-tion pla$s an i portant role in the free boundar$ proble s studied inhapter /.

. Suppose that a and b are constants. Show that the parabolic e"uation

a,u au

aT a %' H a H bucan alwa$s be reduced to the diffusion e"uation. se a change of ti e!ariable to show that the sa e is true for the e"uation

)n a'uT

Tr = a%'

where c r ? . Suppose that J' and r in the Black-Scholes e"uation are both functions of t, but that r4 , ' is constant. Geri!e the Black Scholesfor ulae in this case.

9. Suppose that in the Black-Scholes e"uation, r t and ,' t are both non-constant but known functions of t. Show that the following procedurereduces the Black-Scholes e"uation to the diffusion e"uation.

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CC The Black-Scholes >or ula

See Earper 1DD3 for further e%a ples of this ingenious procedure ap- plied to other e"uations.

A. Show that e"uation 9.1 can also be reduced to the diffusion e"uation b$ writing

! %, T = e- k/ 8 S, /- ,

where

10=%H k-1 T.

What disad!antages ight there be to this change of !ariables

/. If S, t and 5 S, t are the !alues of a #uropean call and put withthe sa e e%ercise and e%pir$, show that - 5 also satisfies the Black-

Scholes e"uation 9.C , with the particularl$ si ple final data - 5 =S - # at t = T. Geduce fro the put-call parit$ theore that S -#e-++ T-t is also a solution0 interpret these results financiall$.

C. se the e%plicit solution of the diffusion e"uation to deri!e the Black-

Scholes !alue for a #uropean put option without using put-call parit$.

D. alculate the ga a, theta, !ega and rho for #uropean call and putoptions.

1B. se Maple or an$ other co puter algebra package to plot out thefunctions of #%ercise D. se the plot3d co and to generate three-di ensional plots of call and put options as functions of two !ariables,for e%a ple, S and t or S and a.

11. What is the rando walk followed b$ a #uropean call option1'. If u %, in the initial !alue proble for the heat e"uation on an infinite

inter!al e"uations .3 - ./ is positi!e, then so is u %, r for r ? .Show this, and deduce that an$ option whose pa$off is positi!e alwa$shas a positi!e !alue.

13. onsider the following initial !alue proble on an infinite inter!al6

a,uaT = %' Hf %, T

with

u %, = and u --? as % -Poo.

It can be shown for e%a ple b$ the Ureen+s function representation of thesolution that if f %, T ? then u %, T ? . Wh$ is this ph$sicall$reasonable se this result to show that if , S, t and ' S, t are the

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#%ercises CD

!alues of two otherwise identical calls with different !olatilities -1 and

a' @ al, then ' @ 1. Is the sa e result true for puts

1 . In di ensional !ariables, heat conduction in a bar of length ; is od-elled b$

) C' pc C/T k)KF

for @ K @ ;, where K, T is the di ensional te perature, p isthe densit$, c is the specific heat, and k is the ther al conducti!it$.Suppose also that ) is a t$pical !alue for te perature !ariations,ei ther of the initial te perature K , or of the boundar$ !alues atK = , ;0 ake the e"uation di ensionless.

19. What is the !alue of an option with pa$off /-1 #-S What is the !alueof a supershare

1A. The #uropean asset-or-nothing call pa$s S if S ? # at e%pir$, andnothing if S @ #. What is its !alue

1/. What is the probabilit$ that a #uropean call will e%pire in-the- one$

1C. (n option has a general pa$off ( S at ti e T, and its !alue is 8 S, t .Show how to s$nthesise it fro !anilla call options with !ar$ing e%ercise

prices0 that is, how to find the 2densit$+ f # of calls, with the saee%pir$ T, e%ercise price # and price S, t0 # , such that

48 S, t = Q f # S, t0 # d#.

8erif$ that $our answer is correct

a when ( S = a% S - #, a !anilla call 0b when ( S = S. What is the s$nthesiFing portfolio here

7epeat the e%ercise using cash-or-nothing calls as the basis.

1D. Suppose that #uropean calls of all e%ercise prices are a!ailable. 7ega