# T ϕ t ds t + ψ t db t,

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1 16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in Lecure I. Recall from definiion 1.4 ha a general European pay-off wih exercise dae T is an FT W -measurable random variable X. The way o hink abou X is as a fuure cash-flow,o be delivered a ime T, whose fuure value of X depends only on wha will have happened wih he sock price S in beween now =and = T. Since here is a one-o-one relaionship beween S and W namely W =logs /σ µ, his is equivalen o saying ha X only depends on he Brownian W rajecory beween and T, which is he same as saying ha X is FT W -measurable. Sep 1: consrucing a self-financing replicaing porfolio-sraegy. Our firs aim is o find a self-financing porfolio sraegy ϕ,ψ for S,B which replicaes he claim X, in he sense ha he porfolio s value a T coincides wih X: X = V T ϕ, ψ =ϕ T S T + ψ T B T. Since he sraegy is o be self-financing, his is equivalen o asking for φ, ψ such ha 3 X = V ϕ, ψ+ ϕ ds + ψ db, or, afer discouning remembering ha B =1, 31 X = V ϕ, ψ+ ϕ d S ; cf. corollary 1.3 and formula 9. Such a sraegy can be obained from he maringale represenaion heorem, as follows. Firs, using proposiion 1.2, discoun everyhing o presen value by dividing by B,. This doesn affec he self-financing propery, and i herefore suffices o find a self-financing sraegy ϕ,ψ for S, 1, S =:= S /B, which replicaes he discouned claim, X := X/B T. We now firs change Brownian moion and probabiliy measure, using Girsanov s heorem. If we choose γ =µ r/σ in heorem 1.9, hen we find an equivalen measure Q such ha Ŵ = γ+ W is a Q-Brownian moion, and S evolves according o he SDE 32 d S = σ S dŵ, see secion 1.7. Now by he maringale represenaion heorem, heorem 1.5, here exis: aconsanx,and

2 an adaped process h T, such ha: 33 X = X + PRICING II: MARTINGALE PRICING 17 We can rewrie his as follows, using 33: 34 where we have pu X = X + = X + = X + h dŵ. h dŵ h σ S σ S dŵ ϕ d S, ϕ := h 35 σ S. The financial inerpreaion of 34 is ha ϕ is exacly he number of discouned sock which you mus own a ime, o obain he discouned claim a ime T, given an iniial invesmen of X. Comparing wih 31, we see ha 35 is he ϕ-componen of he self-financing sraegy we are looking for. The corresponding ψ-componen is found by solving ϕ S + ψ = Ṽϕ, ψ = X + ϕ u d S u, where we used he self-financing propery again, or 36 ψ = X + ϕ u d S u ϕ S. In paricular, ψ = X ϕ S. Sep 2: Compuing he price as an expecaion. By he law of one price, X should hen be he fair value of he claim a ime. How do we compue X? The essenial poin is ha he Io-inegral in 34 is a Q-maringale: 37 E Q h u dŵu F = h u dŵu. See Mah Mehods I. In paricular, if =, we ge. Hence, aking expecaions in 33, we find ha X = E Q X.

3 18 PRICING II: MARTINGALE PRICING Le us wrie π X for he price a of he European claim X a T. Then, remembering ha X = X/B T = e rt X,wehaveshownha 38 π X =e rt E Q X. This can be generalized o any beween and T : if ϕ, ψ isour replicaing sraegy, hen π X should be equal o he porfolio s value V ϕ, ψa, by absence of arbirage. Hence, afer discouning, π X = B Ṽϕ, ψ. Now, since he porfolio is self-financing, we find ha X = Ṽϕ, ψ+ = Ṽϕ, ψ+ = Ṽϕ, ψ+ dṽuϕ, ψ ϕ u d S u h u dŵu. Taking condiional expecaions, and using he maringale propery of Io inegrals 3, we find ha E Q X F W =Ṽϕ, ψ = π X. B Hence, remembering ha B = e r, π X =B E Q B 1 T X=e rt E Q X. We summarize he discussion up ill now in he following heorem: Theorem 2.1. European Opion Pricing Formula: The value a ime of a European claim X a ime T is, in he Black and Scholes model, given by: 39 π X =E Q X B T = e rt E Q X T. More generally, is price a, <<T, will be given by: 4 B E Q X B T F = e rt E Q X T F W, where F W is he Brownian filraion. 3 explicily, T EQ h udŵu F = E T Q hudŵu F E Q hudŵu F = hudŵu hudŵu =.

4 PRICING II: MARTINGALE PRICING 19 Remark 2.2. Formulas 39, 4 are he simples examples of he Risk-Neural Pricing Principle, which can be saed as: 41 Price of an asse a = E Q discouned fuure cash-flows F, Q being a risk-neural probabiliy measure, ha is, one wih respec o which he discouned asse prices are maringales. Observe ha we have pu he discouning under he expecaion-symbol, since in general he discouning facor may be sochasic: you will be asked o work ou an example of his in exercise 2.11 below. We pause wih he general discussion o consider he classical example. Example 2.3. If he claim X is of he form: X = gs T, for some given funcion g, hen he claims value a is: 42 π X =e rt E Q gs T = E Q gs e /2T +σŵt, since S, wih respec o he Q-measure, follows he SDE ds = rs d + σs dŵ, whose soluion is S = S expr σ 2 /2 + σŵ cf. Mah. Mehods I. Remembering he pdf of ŴT, and he formula for compuing expecaions of funcions of random variables wih known pdf, we find ha 42 equals: 43 π X = More generally, since we find ha π X = E Q = E Q g = R g S e 2 T +σw e w2 /2T S T = S e 2 T +σw T W, g S e 2 T +σw T W S e 2 T +σw T W g S e 2 T +σw e w2 /2T dw 2πT. F W dw, 2πT where we used ha W T W is independen of F W and N,T - disribued. Observe ha in his case of an X of he form gs T, he price is of he form π X =fs,,

5 2 PRICING II: MARTINGALE PRICING wih f given by 44 fs, = R g Se 2 T +σw e w2 /2T dw. 2πT In he special case of a European call wih srike E, gs T =maxs T E,, his inegral can be explicily evaluaed, which gives he famous Black and Scholes formula. Wriing he call s price as CS,, one compues ha 45 CS,=SΦd + e rt Φd, where 4 1 d± = logs σ /E+r ± T. T 2 Here is a simple way o memorize d ± : since σ T is he oal volailiy of he sock-reurn over he remaining life-ime [, T ] of he opion, we have: sock price a vol over remaining life-ime2 log ± discouned exercise price 2 d ± =. vol over remaining life-ime Sep 3: Replicaion and hedging. For financial pracice, we no only need o know he price, bu also how o se up he replicaing porfolio for his is wha he wrier of he opion should immediaely do, afer having sold he opion, in order o be able o mee his obligaion of providing he pay-off a T o he buyer of he opion. We will discuss his for pay-offs of he form X = gs T. In his case we have seen ha he opion s price can be wrien as π X =fs,, for some suiable funcion fs,. From wha you already know, from Pricing I, abou opion pricing using he PDE-mehod, you may guess ha he amoun of sock we should hold is ϕ = f S S,, he opions, bu how can we see his using he presen maringale formalism? The rick is o use self-financing propery of our replicaing porfolio ϕ, ψ which we know exiss on absrac grounds, bu which we do no know how o consruc ye. In fac, fs,=π X =V ϕ, ψ. We also know, since ϕ, ψ is self-financing, ha dv = ϕ ds + ψ db = ϕ ds + rψ B d. 4 Here, and elsewhere, log sands for he naural logarihm wih base e: log=ln.

6 PRICING II: MARTINGALE PRICING 21 Therefore we should have ha 46 df S,=ϕ ds + rψ B d. Now he lef hand side can be evaluaed using Io s lemma 5. This gives 47 df S, = f f d + S ds f 2 S ds 2 2 f S 2 2 f S 2 = f S ds + d, all derivaives evaluaed in S,. Comparing coefficiens of ds wih 46, we see ha 48 ϕ = f S S,, as expeced. The number of savings bonds ψ o hold hen simply follows from ϕ S + ψ B = V ϕ, ψ =fs, ψ = fs, f/ SS, B ; which simply amouns o puing he balance fs, ϕ S inoabank accoun. We herefore know exacly how many sock and how much money in savings o hold, a any fuure ime, as a funcion of he price S of he underlying. Wha abou comparing he coefficiens of d in 46, 47? I urns ou ha his does no really give any new informaion, bu allows us o re-encouner an old acquainance: in exercise 2.12 you will be asked o show ha his leads o he Black and Scholes PDE for fs,, hus making he connecion wih he PDE-pricing mehod from Pricing I. However, see also exercise Pricing exoics. One of he big advanages of he presen pricing mehodology over he one using PDE s is, ha i is raher sraighforward o price exoic 6 claims, like asian opions, look-back opions, barrier opions like knock-in s and knock-ous s, ec. One jus has o wrie he pay-off as a funcion of he S s, T and plug i in in he risk-neural pricing formula 39, while recalling ha we have an explici formula for S in erms of he risk-neural Q-Brownian moion Ŵ : S = S expr σ 2 /2 + σŵ. This will give us he price as an expecaion, which in general canno be evaluaed in closed form, bu which can be reaed numerically by Mone Carlo and simulaion of he rajecories of Ŵ which is simply a 5 The funcion fs, given by 44 can be shown o be arbirarily many imes differeniable wih bounded derivaives, for nice funcions g = gx, for example hose no growing faser han apowerofx as x. 6 basically, everyhing which is neiher a call or a pu; he laer are ofen called vanillas

7 22 PRICING II: MARTINGALE PRICING Brownian moion. We ll ake a brief look a some of he mos common exoics. Example 2.4. Asian opions The pay-off of for example an Asian call depends on he mean of he underlying s price over he enire life-ime of he opion, insead of jus is value a ime T. For example, he pay-off of a fixed-srike Asian call is 1 49 A T =max T S d E,, while ha of a floaing srike Asian call is 1 T 5 A T =max S d S T,, T allowing o buy a he mean-price over [,T], insead of he spo-price a T. Asian pus are defined similarly. In pracice one of course replaces he coninuous mean by a discree one, by discreizing he ime-inerval: 1 N max S jt/n E,. N j=1 For example, one can ake he mean over daily closing prices; N hen would be he number of days ill expiry T. The price of for example he laer opion is given by: 1 N E Q S expr σ 2 /2T + σŵjt/n E,. N j=1 There is lile hope of being able o evaluae his explicily, bu i is prey sraighforward o evaluae using Mone-Carlo. However, for he opion wih he inegral pay-off 49, mahemaicians have made subsanial effors o obain a heoreical undersanding of he price, and here are semi-explici pricing formulas formula available, he mos famous being he one due o Geman and Yor, which gives an explici expression for wha is basically he Laplace ransform wih respec o ime of he Asian opion s price a π A T : see [BK] or [MR] for furher precisions and references. To use his semi-explici formula for pricing and hedging purposes one has o numerically inver he Laplace ransform, which can be delicae. Example 2.5. Barrier opions Single barrier opions become worhless whenever he sock-price crosses a cerain level H, where i also maers wheher H is crossed from above or from below. For example, a down-and-ou call wih srike E and barrier H will have he same pay-off as an ordinary call, provided S says above H for he enire life-ime [,T] of he opion. If i falls below H for even one momen, he opion becomes worhless. I s pay-off can be wrien as: maxs T E, I {min T S H },

8 PRICING II: MARTINGALE PRICING 23 where I A is he indicaor-funcion of he even A. Similarly, he pay-off for adown and in call, which only becomes alife once he underlying s price dips below H, is: maxs T E, I {min T S H }. Noe, ha he sum of a down and ou and a down and in has he pay-off of a ordinary vanilla call, maxs T E,O. Up and ou and an up and in calls have pay-offs: maxs T E, I max T S H, maxs T E, I max T S H, respecively. Explici prices for hese producs can be derived by using he explicily known join probabiliy disribuion funcions of W T, min T W and W T, max T W : see for example [BK] or [MR]. Example 2.6. Look-back opions Look-back opions allow you o, a poseriori, buy an asse S a is low, and sell i a is high. For example, a look-back call has pay-off: S T min S, T while a look-back pu pays max S S T, T a mauriy. Their price a = is, according o he risk-neural pricing formula 39: e rt E Q S T min S, T respecively e rt E Q max S S T. T Plugging in S =expr σ 2 /2 + σw hese can be evaluaed using Mone-Carlo, bu one can also give explici formulas: cf. [BK] and [MR] Exercises. Exercise 2.7. Digial opions A European binary call wih srike E has a pay-off of 1 if S T E, and oherwise. I s pay-off funcion can be wrien as: I {ST E}. Similarly, a binary pu has pay-off I {ST E}. Find an explici expression for he price of hese insrumens, in erms of he cumulaive normal disribuion funcion Φ.

9 24 PRICING II: MARTINGALE PRICING Exercise 2.8. From he risk-neural opion pricing formula 39 derive he Pu-Call Pariy relaion: CS,E,T P S,S,E=S Ee rt. Exercise 2.9. a Consider a European derivaive wih pay-off gs T. Is price a ime is a funcion of fs,ofs and cf. example 2.3. Show ha = fs S = e rt g S e /2T +σw e /2T +σw e w2 /2T = e rt E Q g S T S T /S F dw 2πT b is he number of sock we need o hold a ime in our replicaing porfolio see nex lecure, and is herefore an imporan quaniy o evaluae. Par a suggess how o evaluae his quaniy using Mone- Carlo. Anoher way of evaluaing would be o approximae i by a finie difference: fs fs + h, fs, h, S h for some small h. Which of he wo mehods is more aracive from a numerical poin of view i.e. is likely o lead o he smalles approximaion and round-off errors? Exercise 2.1. Show ha he Dela of a European call a ime is given by: =Φd +. Hin: do no sar differeniaing he Black and Scholes formula unless you like manipulaing complicaed formulas; raher, use par a of he previous exercise. Exercise We now exend he Black and Scholes world by making he bond B sochasic also: db = rb d + ρb dw. ha is, we suppose ha he ineres rae is no longer fixed rd, bu sochasic also rd + ρdw. a Show ha d S =µ r + ρ 2 ρσ S d +σ ρ S dw. b Show ha here exiss an equivalen probabiliy-measure, Q, and a new Brownian moion, Ŵ wih respec o Q, suchha: d S =σ ρ S dŵ.

10 PRICING II: MARTINGALE PRICING 25 c Le X T be an arbirary European claim a T. Prove ha is value a ime will be given by: XT E Q. B T Generalize o arbirary. d Find he value of a digial call in his model. e Find he value of an ordinary call. Exercise Show ha by equaing he coefficiens of d of 46 and 47, we obain he Black and Scholes PDE for fs,: f S 2 2 f S + r f 2 S = rf. Hin: You will need o use ha f ψ B = fs, S S S,. Exercise In he PDE-approach o derivaive pricing, one ses up a risk-free hedging porfolio V consising of 1 opion long, and f/ SS, underlying shor, where fs, is he opion price a when he price of he underlying is S. The ime- value of his porfolio is clearly f V = fs, S S S,, and one derives he Black and Scholes PDE saring off from he relaion 51 dv = df S, f S S,dS, and using Io s lemma and absence of arbirage: cf. Pricing I. Now 51 is equivalen o saying ha he edging sraegy is self-financing. When one checks his a poseriori, his urns ou o be false, as his exercise will show! So here seems o be a fundamenal problem wih he Risk-Free Porfolio Mehod, which here isn wih he Replicaing Porfolio Mehod. See however he discussion a he end of his exercise. To simplify he formulas, we will denoe parial differeniaion by subscrips: f S = f S, f = f, 2 f S 2 = f SS, ec. a Use Io s lemma o show ha if ds = µs d + σs dw,hen df S =f S + µs f SS + 2 S2 f SSS d + σs f SS dw, all parial derivaives evaluaed in S,, as usual.

11 26 PRICING II: MARTINGALE PRICING b Show, ha if V = fs, f S S,S, hen he hedging sraegy is self-financing iff S df S S,+dSdf S S,=, ha is, iff 52 S f S + µs f SS + 2 S2 f SSS + σ 2 S f SS d + σs 2 f SS dw =. Hin: dv = df f S ds S df S ds df S. c Use he fac ha fs, saisfies he Black and Scholes equaion o show ha he lef and side of 52 is equal o µ r σs 2 f SS σ d + dw, which, using he definiion of our risk-neural Brownian moion, can also be wrien as: σs 2 f SS dŵ. Conclude ha he hedging porfolio is no self-financing, unless f SS is idenically Wha would his mean for he opion?. Discussion: On firs sigh, his is a nasy surprise, which sheds doub on he Risk-Free Porfolio Mehod. However, noe ha he oal addiional capial which we would have o injec beween and T and which we normally sould ave added o he price, σs 2 uf SS S u,u dŵu, has risk-neural expecaion, condiional o F W, by he maringale propery of Io inegrals once more. This explains wy he Risk-Free Mehod sill leads o he correc answer.

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