ADVANCED METHODS IN DERIVATIVES PRICING with application to volatility modelling M5MF6

ADVANCED METHODS IN DERIVATIVES PRICING wih applicaion o volailiy modelling M5MF6 Dr Anoine Jacquier wwwf.imperial.ac.uk/~ajacquie/ Deparmen of Mahemaics Imperial College London Spring Term 216-217 MSc in Mahemaics and Finance This version: February 23, 217 1

Conens 1 Opion pricing: from super-replicaion o FTAP 6 1.1 Zoology of sochasic analysis.............................. 6 1.1.1 Sopping imes.................................. 6 1.1.2 Non-negaive local maringales......................... 8 1.1.3 Brackes, uniform inegrabiliy and ime changes............... 1 1.1.4 A brief inroducion o Iô processes and sochasic calculus......... 12 1.2 Fundamenal probabilisic resuls for finance..................... 13 1.2.1 Maringale represenaion heorem and hedging................ 13 1.2.2 Change of measure and Girsanov Theorem.................. 17 1.3 The super-replicaion paradigm and FTAP...................... 2 1.3.1 Overure on opimal ranspor problems and model-free hedging...... 26 1.4 Applicaion o he Black-Scholes model........................ 26 1.5 Beyond Vanilla opions: a probabilisic approach................... 3 1.5.1 American opions: sopping he Brownian moion.............. 3 1.5.2 Barrier opions.................................. 33 1.5.3 Forward-sar opions.............................. 36 1.5.4 Variance and volailiy swaps.......................... 36 2 Maringale heory and implied volailiy 39 2.1 Exisence of implied volailiy.............................. 39 2.1.1 Preliminaries: general properies of Call opion prices............ 39 2.1.2 Characerisaion of European opion price funcions............. 4 2.1.3 Implied volailiy................................. 46 2.1.4 A new look a variance swaps.......................... 49 2.2 No-arbirage properies of he implied volailiy surface............... 5 2.2.1 Slope of he implied volailiy.......................... 51 2.2.2 Time asympoics................................ 52 2.2.3 Wing properies................................. 55 2

2.2.4 The SVI parameerisaion............................ 59 3 From SDEs o PDEs 61 3.1 Sochasic differenial equaions: exisence and uniqueness.............. 61 3.1.1 Properies of soluions of SDEs......................... 66 3.1.2 Momen esimaes................................ 66 3.1.3 Iô diffusions and he Markov propery..................... 67 3.1.4 More resuls in he one-dimensional case.................... 69 3.2 The PDE counerpar.................................. 7 3.2.1 From Kolmogorov o Feynman-Kać; from SDEs o PDEs.......... 7 3.2.2 Parabolic PDEs: exisence, uniqueness and properies............ 72 3.2.3 The PDE approach o pah-dependen opions................ 72 3.3 Overure o non-linear PDEs and conrol heory................... 73 3.3.1 Bellman s principle and dynamic programming................ 74 3.3.2 Hamilon-Jacobi-Bellman equaion....................... 75 3.3.3 Financial examples................................ 77 3.4 A rough inroducion o PIDEs............................. 77 3.4.1 Preamble on semimaringales.......................... 77 3.4.2 Inroducion o Lévy processes......................... 78 3.4.3 PIDES for compound Poisson processes.................... 82 4 Volailiy modelling 83 4.1 Local volailiy...................................... 83 4.1.1 Bruno Dupire s framework............................ 84 4.1.2 Local volailiy via local imes......................... 87 4.1.3 Implied volailiy in local volailiy models................... 9 4.1.4 A special example: he CEV model....................... 92 4.2 Sochasic volailiy models............................... 94 4.2.1 Pricing PDE and marke price of volailiy risk................ 95 4.2.2 Arbirage and Equivalen Local Maringale Measures............. 96 4.2.3 The Heson model................................ 99 4.2.4 Oher popular sochasic volailiy models................... 13 4.2.5 Fracional sochasic volailiy models..................... 15 4.2.6 The Uncerain Volailiy Model......................... 17 4.2.7 The new generaion............................... 18 4.3 Hedging: which volailiy o choose?.......................... 19 4.3.1 Applicaion o volailiy arbirage....................... 11 3

4 4.4 Pu-Call symmery.................................... 11 4.4.1 Black-Scholes................................... 11 4.4.2 Sochasic volailiy models........................... 111 4.5 Link wih he Skorokhod embedding problem..................... 112 A Miscellaneous ools 114 A.1 Essenials of probabiliy heory............................. 114 A.1.1 PDF, CDF and characerisic funcions.................... 114 A.1.2 Gaussian disribuion.............................. 115 A.1.3 Miscellaneous ools................................ 115 A.2 Convergence of random variables............................ 116 A.2.1 Cenral limi heorem and Berry-Esséen inequaliy.............. 118 A.3 Uniformly inegrable random variables......................... 118 A.4 Oher sochasic analysis resuls............................ 119 B Kolmogorov equaions 12 C Spanning European payoffs 121

5 Noaions and sandard definiions The noaions below will be used hroughou he noes. We also wish o emphasize some common noaional misakes. N sricly posiive ineger numbers {1, 2,...} R non-zero real numbers: R \ {} R + R + M n,d R S n R N X = X X non-negaive real numbers: [, + sricly posiive real numbers:, + space of n d marices wih enries valued in R space of symmeric n n marices wih real enries cumulaive disribuion funcion of he sandard Gaussian disribuion a process evolving in ime, as opposed o X, which represens he random value of he process X a ime 1 {x A} indicaor funcion equal o 1 if x A and zero oherwise x y minx, y x y maxx, y a.s. almos surely x y + max, x y δ x Dirac measure a x C c R space of real coninuous funcions wih compac suppor

Chaper 1 Opion pricing: from super-replicaion o FTAP 1.1 Zoology of sochasic analysis In hese noes, we shall follow a uiliarian approach, and only inroduce he ools we need when we need hem. Some of hem, being core o everyhing else, shall be inroduced righ away. The riple Ω, F, P will denoe he ambien given probabiliy space, where Ω is he sample space of possible oucomes, F he se of evens, and P a probabiliy, namely a map from F o [, 1]. In his inroducory par, we shall le T denoe a ime index se, which can be eiher counable T = { 1, 2,...} or uncounable T = R +. We recall ha a σ-field on Ω is a non-empy collecion of subses of Ω, closed under counable unions and inersecions, and closed under complemenaion. A filraion F T is defined as a non-decreasing in he sense of inclusion family of σ-fields in F; we shall say ha a process X = X T is adaped o he filraion if X F for each T. The following definiion is sandard in he sochasic analysis / mahemaical finance lieraure, and will always be aken for graned in hese lecure noes. Definiion 1.1.1. A filered probabiliy space Ω, F, F, P saisfies he usual hypoheses if F conains all he P-null ses of F; he filraion is righ-coninuous: for any T, F = F u. s> 1.1.1 Sopping imes We inroduce here sopping imes, which are measurable random variables, and which will be defining ingrediens of local maringales, as we shall see laer. 6

1.1. Zoology of sochasic analysis 7 Definiion 1.1.2. A random variable τ aking values in T {sup T } is called a sopping ime resp. opional ime if he even {τ } resp. {τ < } belongs o F for all T. Consider for insance T = [,, a coninuous, adaped, real-valued process X a sandard Brownian moion for example and fix a poin x R. Then he firs hiing ime of he level x, τ := inf{ : X = x}, is a sopping ime. However, he las hiing ime of he level x, τ := sup{ : X = x}, is no a sopping ime. Inuiively, he informaion available a ime is no sufficien o deermine wheher he process will reach he level x a some poin in he fuure. The following properies of opional and sopping imes are lef as an exercise: Proposiion 1.1.3. The following asserions hold: i Le τ be a random variable in T {sup T }; i is a sopping ime if and only if {τ > } F for all T ; ii every sopping ime is opional; iii an opional ime is a sopping ime if he filraion is righ-coninuous; iv if τ is an opional ime and a sricly posiive consan, hen τ + is a sopping ime; v if τ 1 and τ 2 are sopping imes, hen so are τ 1 τ 2, τ 1 τ 2 and τ 1 + τ 2 ; Proof. Saemen i follows direcly since {τ > } = {τ } c. Regarding ii, le τ be a sopping ime, and assume ha T = N. Then he even {τ < } is equal o {τ 1} F 1 F. Consider now he general case T = [, and fix some T. Le n n N be a sricly increasing sequence in T converging o as n increases, so ha {τ < } = n 1{τ n } F. Indeed, for each n N, he even {τ n } belongs o F n, which is iself a subse of F. Consider now iii: assume ha he filraion is righ-coninuous and τ a random ime. For any T, consider a sricly decreasing sequence n n N converging o ; for any m >, we can wrie {τ } = {τ < n }; n m now, for any n m, {τ < n } F n F m. Therefore {τ n } F + = F. We leave iv and v as exercises. For every F T -sopping ime τ, we can associae he σ-field F τ := {B F : B {τ } F, T }.

1.1. Zoology of sochasic analysis 8 1.1.2 Non-negaive local maringales We now consider he filered probabiliy space Ω, F, F T, P, for some index se T. All he processes in his secion shall be consider as aking values in R d, for some d N. Definiion 1.1.4. An R d -valued, adaped process S on Ω, F, F T, P is called a maringale respecively supermaringale, submaringale if, for all T, E S is finie or S L 1 P and ES F u = S u resp. ES F u S u, ES F u S u for all u [, ] T. Example 1.1.5. Le W = W be a sandard Brownian moion, F W i.e. F := {σw s : s }, and T = [,. is naural filraion, W is an F -maringale; he processes W 3 3W and W 4 6W 2 + 3 2 are rue F -maringales. he soluion o he sochasic differenial equaion ds = σs dw, saring a S >, corresponds o he Black-Scholes model see Secion 2.1.1, which is he canonical model in mahemaical finance. Then, for any u, S = S u exp 1 2 σ2 u + σw W u, and he process S is clearly a rue maringale. Unless oherwise specified, all processes here will be adaped o he filraion F T. The following lemma shows how sub/super maringale properies are preserved under ransformaions: Lemma 1.1.6. Le S be an F-maringale and f : R d R a convex funcion such ha fs is inegrable i.e. fs PS ds <, hen fs is a submaringale. R d Useful examples in mahemaical finance of such convex funcions are fx x 2, fx x +. Proof. The proof follows direcly from Jensen s inequaliy: for any T and u, fs u = fes F u EfS F u, almos surely. Remark 1.1.7. One could naurally wonder wheher he converse holds, namely wheher any submaringale can be generaed from a rue maringale via a convex funcion. This is no rue in general, bu some resuls hold in paricular cases: evey non-negaive submaringale is he absolue value of some maringale [72], see also [8] and [124]. Definiion 1.1.8. A process S on Ω, F, F T, P is a local maringale if here exiss a sequence of F T -sopping imes τ n n saisfying lim n τ n = sup T almos surely, and such ha he sopped process S τ n T := S τn T is a rue maringale wih respec o he filraion F T, for any n. Such a sequence of sopping imes is called a localising sequence for he process S.

1.1. Zoology of sochasic analysis 9 The following space of processes is of primary imporance in sochasic analysis, and we will make use of i repeaedly. Of paricular ineres will be he cases n = 1 and n = 2. Definiion 1.1.9. For any n N, he space L n loc Ω, F, P denoes he space of progressively measurable wih respec o F T processes φ such ha P φ u n du < = 1, for all T, while L n Ω, F, P denoes he space of progressively measurable processes φ such ha E φ u n du < <. The following proposiion allows us o consruc large classes of local maringales: Proposiion 1.1.1. For a given Brownian moion W, define pahwise he process X by X := u s dw s, for all. If u L 2 Ω, F, P, hen X is a coninuous square inegrable maringale; if u L 2 loc Ω, F, P, hen X is a coninuous local maringale. The heory of local maringales is very profound, and we refer he avid reader o he excellen monograph [95]. True maringales are of course local maringales ake τ n = sup T for all n N, bu he converse is no rue in general. For insance as firs indicaed in [93], if W is a hreedimensional Brownian moion no saring a he origin, hen he inverse Bessel process defined by W 1 is a sric local maringale e.g. a local maringale, bu no a rue maringale. However, he following always holds: Lemma 1.1.11. Every bounded local maringale from below is a rue maringale. Proof. Le X be a bounded local maringale, and τ n n N a localising sequence of sopping imes for X. Poinwise in ω Ω, i is clear ha he sequence X τn ω n N converges o X ω for any T. By dominaed convergence, we obain EX F s = E lim n Xτ n F u = lim E X τ n F u = lim X τ n u = X u, n n for any u, and he lemma follows. We shall see some implicaions on opion prices of he difference beween sric local maringales and rue maringales in Chaper 2. Proposiion 1.1.12. A non-negaive local maringale S is a super-maringale and ES F u is finie for all u [, ] T.

1.1. Zoology of sochasic analysis 1 Proof. For any localising sequence τ n n Appendix A.1.3: for S, he proposiion follows from Faou s lemma E u S = E u lim inf n Sτ n lim inf n E us τ n = lim inf S u = S u. n In paricular, he proposiion implies ha any non-negaive local maringale is a supermaringale. This will be he exac framework of Chaper 2. A blind applicaion of he reverse Faou lemma would yield ES F u = E lim sup n S τn F u lim sup n E S τn F u = lim inf S u = S u, n which, in combinaion wih Proposiion 1.1.12, would imply ha a local maringale is always a maringale. There is obviously a conradicion here, which comes from he fac he he reverse Faou lemma does no apply, since he sequence of funcions f n s s is no bounded above by an inegrable funcion. Remark 1.1.13. This propery in paricular implies see [86, Chaper III, Lemma 3.6] ha, for any coninuous non-negaive maringale S, if here exiss > such ha PS S = >, if =, hen S = almos surely for all. This seemingly rivial propery is fundamenal when considering discreisaion schemes for sochasic differenial equaions. As a moivaing example, consider he CEV Consan Elasiciy of Volailiy process, defined as he unique srong soluion, saring a S = 1, o ds = σs 1+β dw, where W is a sandard Brownian moion, σ > and β R. We refer he reader o [3, Chaper 6.4] for full deails. The process S is a local maringale, and is a rue maringale if and only if β. Noe ha when β =, his is nohing else han he Black-Scholes sochasic differenial equaion, and S is sricly posiive almos surely for all. Compuaions involving Bessel processes show ha, for any, PS = > if and only if β [ 1/2,. Consider now a simple Euler scheme for he CEV dynamics, along a given pariion = < 1 < < n = T, as S = 1 and, for any i =,..., n 1, S i+1 = S i + σs 1+β i ñ i+1 i, where ñ is a Gaussian random variable wih mean zero and uni variance. Suppose ha along some simulaed pah, here exiss i =,..., n 1 such ha S i > and S i+1 <. Then, in order for he simulaed pah o be a rue approximaion of he original one he soluion of he SDE, he only possibiliy is o se S i+1 o zero and leave i here unil ime T. Noe ha, economically speaking, his absorpion propery also makes sense, as required by no-arbirage argumens. 1.1.3 Brackes, uniform inegrabiliy and ime changes Uniform inegrabiliy is a key propery in probabiliy heory, and conrols how much he ail of he disribuion of random variables accouns for he expecaion.

1.1. Zoology of sochasic analysis 11 Definiion 1.1.14. A family X of random variables is said o be uniformly inegrable if lim sup E X 1 X K =. K X X The following resul, he proof of which is omied, provides an easy-o-check characerisaion: Proposiion 1.1.15. The family X is uniformly inegrable if and only if here exiss a Borel funcion ϕ : R + R + saisfying lim x 1 ϕx = for which sup E ϕx is finie. x X X Theorem 1.1.16. Le M be coninuous local maringale saring from zero. Then here exiss a unique coninuous increasing process M, called he quadraic variaion process, null a zero such ha M 2 M is a coninuous local maringale. If M and N are wo coninuous local maringales saring from he origin, hen here exiss a unique unique coninuous process M, N, null a zero, called he bracke process, such ha MN M, N is a coninuous local maringale. Remark 1.1.17. The bracke process saisfies he polsarisaion ideniy M, N = 1 2 M + N M N. Example 1.1.18. For a sandard Brownian moion W, W = for all ; φsdw s = φs2 ds for φ L 2 loc Ω, F, P. Remark 1.1.19. Oher versions exis, wih sronger assumpions, and hence more precise resuls. In paricular, if M is a coninuous square-inegrable maringale saring from zero, hen here exiss a unique coninuous increasing process M null a zero such ha M 2 M is a uniformly inegrable maringale. See for example [134, Theorem 3.1, Chaper IV, Secion 5]. Remark 1.1.2. A relaed process, he square bracke process [M], exiss in he lieraure, and is defined as [M] := X 2 X 2 2 X s dx s; is consrucion follows from he Doob-Meyer decomposiion see Theorem 1.5.6 below. In he case where he process M is coninuous, he wo brackes and [ ] however coincide. The las echnical ool we shall need in order o move on is he echnique of ime change. We sae he following fundamenal resul wihou proof: Theorem 1.1.21 Dubins-Schwarz. Every coninuous local maringale M can be wrien as a ime-changed Brownian moion W M. In paricular, for any Brownian moion W and any independen, non-negaive, càdlàg process σ, he coninuous local maringale M := σ sdw s can be wrien as M = W σ2 s dw, for all. s

1.1. Zoology of sochasic analysis 12 This heorem, and he refined version by Monroe [118], have been used exensively in mahemaical finance. One moivaion, as deailed in [5], is ha rading ime and real ime are no synchronised, namely ha more rading aciviies ake place during he day, and none during he nigh. 1.1.4 A brief inroducion o Iô processes and sochasic calculus In Chaper 3.1 below, we shall discuss in deail exisence, uniqueness and properies of sochasic differenial equaions. The objecive of his secion here is o quickly go hrough he main ingrediens of Iô s heory, as we shall need i in he nex pages. Again, Ω, F, F, P is a given filered probabiliy space, supporing an R d -valued Brownian moion W. Definiion 1.1.22. An R n -valued Iô process X is a sochasic process of he form X = X + µ s ds + σ s dw s, [, T ], 1.1.1 where X is an F -measurable random variable, µ L 1 loc Rn R n and σ L 2 loc Rn M n,d R. The differenial noaion dx = µ d + σ dw, wih X R n is a useful shorcu, even hough he noise erm dw should be handled wih care. Corollary 1.1.23. The quadraic variaion of an Iô process of he form 1.1.1 is given by X i, X j = σ s σ s ij ds =: C ij s ds. For any [, T ], he marix C M n R represens he covariance marix of he n-dimensional random variable X. Theorem 1.1.24 Iô s formula. Le X be an R n -valued Iô process of he form 1.1.1 and le f C 1,2 R + R n R. Then he following formula holds almos surely for every : f, X = f + s f s ds + dx s, f s + 1 dx s f s dx s 2 { = fx + s f s + µ s, f s + 1 2 Tr σs } f s σ s ds + where we used he shor-hand noaion f s fs, X s. σ s dw s, f s, Here, he gradien operaor only acs on he space variable, and is a vecor in R n : f, x = xi f, x i=1,...,n, and he angle bracke, is simply he Euclidean produc in R n. Remark 1.1.25. In coordinaes, we can re-wrie Iô s formula as n f, X = f + s f s ds + µ i s xi f s + 1 n n Cs ij xi x 2 j f s ds + i=1 i,j=1 d i=1 j=1 σ ij s xi f s dw j s.

1.2. Fundamenal probabilisic resuls for finance 13 Remark 1.1.26. In he case X = W and f C 2 R d, Iô s formula reads, in differenial form: dfw = dw, fw + 1 2 fw d. The following corollary is lef as a sraighforward exercise. Corollary 1.1.27. For wo R n -valued Iô processes X and Y, he following produc rule holds: X Y = X Y + X s dy s + Noe ha rearranging he erms in he corollary yields X s dy s = X Y X Y Y s dx s + Y s dx s which can be seen as a sochasic inegraion by pars formula. Example 1.1.28. Prove he following represenaions: i Show ha he process Y defined by Y := 2 W 3 saisfies 2Ys Y = s + 34/3 Ys 1/3 ds + 3 dx s dy s. dx s dy s, sy s 2/3 dw s. ii For any α, σ R, show ha X := e α X + σ eαs dw s saisfies X = X α X s ds + σw. iii Consider he couple X, Y := cosw, sinw. Show ha i saisfies X = 1 1 2 Y = 1 2 X s ds Y s ds + Y s dw s, X s dw s. 1.2 Fundamenal probabilisic resuls for finance We inroduce in his secion wo key resuls from sochasic analysis ha are used exensively in mahemaical finance, and in he res of hese noes: he maringale represenaion heorem and Girsanov s heorem. 1.2.1 Maringale represenaion heorem and hedging Le Ω, F, F, P be a probabiliy space where he filraion F is he sandard filraion generaed by a Brownian moion W, i.e. for any, F = σ W s, s. Unless oherwise saed, T > will be a fixed finie ime horizon. We shall need he following space of admissible inegrands in order for he Iô inegral o make sense:

1.2. Fundamenal probabilisic resuls for finance 14 Definiion 1.2.1. We le V denoe he space of funcions f from [, Ω R ha saisfy he following condiions: he map, ω f, ω is B F-measurable, wih B being he Borel sigma-algebra on [, ; for any, he funcion f, is F -adaped; T E f, ω2 d is finie. Theorem 1.2.2 Maringale Represenaion Theorem. Le M be a F -adaped squared inegrable maringale wih respec o P. Then here exiss a unique process φ V such ha Remark 1.2.3. M = EM + φ s dw s almos surely for any. 1.2.1 We have saed here he heorem in dimension one. A similar resul holds in any finie dimension, and we leave his exension o he keen reader. The heorem has a converse resul, namely ha he Iô inegral φ sdw s is a F - maringale whenever φ is adaped and square inegrable. The financial consequence of his resul is ha he only source of uncerainy comes from he Brownian moion, which can be removed by hedging. The heorem only assers exisence of a process φ. Explici represenaions hereof can be deermined using Malliavin calculus, bu his falls ouside he scope of hese lecures. The proof follows from he following lemmas. Recall ha a subse D of F is dense if for every X in F, hen D B for every neighbourhood B of X. Lemma 1.2.4. Theorem 1.2.2 holds if he represenaion 1.2.1 holds for any random variable in some dense subse of L 2 Ω, F T, P. For some ineger n, and some sequence 1 n T, we consider he subse D T L 2 Ω, F T, P consising of all he random variables of he form hw 1,..., W n, where h is some bounded coninuous funcion from R n o R. Lemma 1.2.5. The space D T is dense in L 2 Ω, F T, P. Lemma 1.2.6. Le X and Y be wo random variables on Ω, F, F, P aking values in R n and R m. Le G be a sub-σ-field of F such ha Y is G-measurable and X independen of G. Then, for any measurable funcion f : R n+m R such ha E fx, Y is finie, he equaliy holds, where by := R n fx, ypx dx. E [fx, Y G] = by, of

1.2. Fundamenal probabilisic resuls for finance 15 Proof of Lemma 1.2.4. Le X be a random variable in L 2 Ω, F T, P, and consider a sequence X n n 1 in a dense subse D L 2 Ω, F T, P, converging o X in L 2. By assumpion, here exiss a sequence of adaped and square inegrable funcions g n such ha, T X n = EX n + g n sdw s, for any n 1. Iô s isomery applied o he cenered random variables X n := X n EX n yields E Xn X 2 T m = E [g n s g m s] 2 ds. Being convergen, X n n 1 is a Cauchy sequence, and hence he sequence g n n 1 is convergen in L 2 ω [, T ], dp d, which is a complee space by he Riesz-Fischer heorem. Therefore, here exiss a limiing funcion g such ha he expecaion E T g ns gs 2 ends o zero as n becomes large, and he represenaion 1.2.1 holds wih his very funcion g. Proof of Lemma 1.2.5. Consider a sequence i i 1, which forms a dense subse of he closed inerval [, T ] and define he increasing sequence F i i 1 as he sigma-algebras generaed by {W j } 1 j i, for each i 1. The maringale convergence heorem Theorem A.4.2 implies ha, for any φ L 2 F T, P, he poinwise limi φ = Eφ F T = lim i Eφ F i holds P- almos everywhere and in L 2 F T, P. Doob-Dynkyn Lemma Lemma A.4.1 hen yields Eφ F i = φ i W 1,..., W i for some Borel measurable funcion φ i : R i R, which can be approximaed in L 2 F T, P by smooh and bounded funcions h i W 1,..., W i, and he lemma follows. Proof of Lemma 1.2.6. The saemen of he lemma is clearly equivalen o showing ha he equaliy E [fx, Y Z] = E[ZbY ] holds for any G-measurable random variable Z. Le hen µ XY Z denoe he law of he riple X, Y, Z aking values in R n+m+1. By independence, we clearly have µ XY Z dx, dy, dz = µ X dxµ Y Z dy, dz, so ha E [fx, Y Z] = zfx, yµ XY Z dx, dy, dz = R n+m+1 = zbyµ Y Z dy, dz = E[ZbY ]. R n+1 R m+1 z fx, yµ X dx µ Y Z dy, dz R n We now move on o he proof of he Maringale Represenaion heorem for Brownian moion: Proof of Theorem 1.2.2. I is clear from he lemmas above ha i is sufficien o show ha he random variable hw 1,..., W n has he represenaion propery 1.2.1 for any n 1 and h : R n R coninuous and bounded. We prove he case n = 2, bu he proof exends easily o he general case. Le n ; m, σ 2 denoe he densiy of a Gaussian random variable wih mean m and variance σ 2 : nz; m, σ 2 := 1 } { σ 2π exp z m2 2σ 2, for all z R,

1.2. Fundamenal probabilisic resuls for finance 16 and inroduce he funcion v : R + R n R m as v, x, y = hx, znz; y, 2 dz, for all [, 2. R One can immediaely check ha i saisfies he parial differenial equaion v, x, y + 1 2 v, x, y =, 2 y2 for all [, 2, x, y R, wih boundary condiion v 2, x, y = hx, y. Iô s formula hen yields hw 1, W 2 = v 2, W 1, W 2 = v 1, W 1, W 1 + 2 1 = E [hw 1, W 2 F 1 ] + by Lemma 1.2.6, since E [hw 1, W 2 F 1 ] = R hw 1, zpw 2 v y s, W 1, W s dw s dz. Consider now he funcion v : [, 1 ] R R defined by v 1, x, x, if = 1, v, x = v 1, z, znz; x, 1 dz, if, 1, R 2 1 v y s, W 1, W s dw s, Noe in paricular ha he equaliy v, x + 1 2 xxv, x = holds for all [, 1 and x R. Iô s formula hen yields and hence v 1, W 1 = v 1, W 1, W 1 = v, + 2 hw 1, W 2 = v 1, W 1, W 1 + 2 = v 1, W 1 + 1 1 1 v y s, W sdw s, v y s, W 1, W s dw s v y s, W 1, W s dw s 1 v = v, + y s, W sdw s + = v, + 2 ψsdw s, 2 wih he obvious definiion y vs, W s, if s < 1, ψs := y vs, W 1, W s, if 1 s < 2. The heorem hen follows since v, = E [hw 1, W 2 ]. 1 v y s, W 1, W s dw s Example 1.2.7. Fix some ime horizon T > and consider he F -maringale M defined pahwise by M := E e W T F. Now, M = E e W T W e W F = e W E e W T W = exp W + 1 2 T =: f, W.

1.2. Fundamenal probabilisic resuls for finance 17 On he oher hand, Iô s lemma yields f, W = f, W + = f, W + w fu, W u dw u + M u dw u 1 2 M u du + 1 2 u fu, W u du + 1 2 M u du = M + ww fu, W u d W, W u M u dw u, which corresponds o he represenaion 1.2.1. Noe ha he filraion generaed by M is he same as ha from he Brownian moion W. 1.2.2 Change of measure and Girsanov Theorem The second fundamenal heorem in mahemaical finance is he Girsanov 1 Theorem, which allows o change probabiliy measures, and ofen allows for nea simplificaions. Change of measure We firs inroduce he concep of change of measure, which shall be used laer in a dynamic version o sae Girsanov Theorem. Theorem 1.2.8. Le Ω, F, P be a probabiliy space and Z a non-negaive P-almos surely random variable saisfying EZ = 1. Define he probabiliy measure P by PA := ZωdPω, for any A F. A Then he following hold: P is a probabiliy measure; if X is a non-negaive random variable hen ẼX = EXZ; if Z > P-almos surely, hen EY = ẼY/Z for any non-negaive random variable Y. Proof. We only prove he firs saemen in he heorem, he oher wo being sraighforward. In order o prove ha P is a probabiliy measure, we need o show ha PΩ = 1 and ha i is counably addiive. By definiion, PΩ = Ω ZωdPω = EZ = 1. Now, le A k k 1 be a sequence of disjoin ses, and se, for any n, n B n := A k. k=1 1 Igor Vladimirovich Girsanov 1934-1967 was a Russian mahemaician, who firs worked in he group led by Eugene B. Dynkyn a Moscow Sae Universiy, and laer became an advocae of quaniaive mehods in mahemaical economics, and of applicaions of opimal conrol o chemisry. He died of a hiking acciden a he age of 33.

1.2. Fundamenal probabilisic resuls for finance 18 Since B n is an increasing sequence, hen 1 B1 1 B2..., B = k 1 A k, and he monoone convergence heorem implies PB = 1 B ωzωdpω = lim 1 Bn ωzωdpω Ω n Ω n n = lim 1 Ak ωzωdpω = lim 1 Ak ωzωdpω n n = lim n k=1 which proves he saemen. Ω k=1 n PA k = PA k, k=1 Exercise 1.2.9. Le X be a cenred Gaussian random variable wih uni variance under he probabiliy P, and, for θ >, define Y := X + θ. Using he variable } Z := exp { θx θ2, 2 and inroducing he probabiliy Q defined via is Radon-Nikodym derivaive dq dp is a cenred Gaussian random variable wih uni variance under Q. k=1 Ω := Z, show ha Y Soluion. Define he random variable Z := exp { θx 1 2 θ2}, and a new probabiliy measure P as in Theorem 1.2.8, which is well defined since Z is sricly posiive almos surely wih uni expecaion. For any y R, we can now wrie considering he se A = {ω Ω : Y ω y} F PY y = ZωdPω = 1 {Y ω y} ωzωdpω = = = {ω Ω:Y ω y} Ω 1 {Xω y θ} ω exp { θxω θ2 2 } 1 1 {x y θ} exp { θx θ2 exp 2π R y θ Girsanov Theorem 1 2π exp Ω 2 } { θx θ2 exp 2 } dpω x2 2 x2 dx = 2 dx y θ 1 exp x2 dx. 2π 2 We presen he general muli-dimensional version of Girsanov heorem below, bu will resric he proof mainly for noaional convenience o he one-dimensional case. Theorem 1.2.1 Girsanov Theorem. Consider an n-dimensional Brownian moion W defined on some filered probabiliy space Ω, F, F, P. Fix a ime horizon T >, inroduce he n-dimensional adaped process Θ := Θ 1,..., Θ n, and define { Z := exp Θ u dw u 1 } Θ u 2 du and W := W + Θ u du. 2 T If E Θ u 2 Zudu 2 is finie, hen EZ T = Z = 1 and W is a P-Brownian moion, where PA := Z T ωdpω, for any A F. A

1.2. Fundamenal probabilisic resuls for finance 19 The oher way of wriing he new measure, which we shall always do here, is d P := Z dp T. FT Remark 1.2.11. Novikov s condiion [121] ensures ha a given process is a maringale: le X is a real-valued adaped process on some filered probabiliy space supporing a Brownian moion W ; 1 if E exp 2 X s ds 2 is finie, hen he Doléans-Dade exponenial E X s dw s := exp X s dw s 1 Xs 2 ds 2 is a rue maringale. Oher condiions exis in he lieraure, in paricular Kazamaki s condiion, bu Novikov s crierion is he mos widely used. Recen echnical developmens can be found in he works of Mijaović and Urusov [117] and of Ruf [138]. Before proving he heorem, we need a few ools. Theorem 1.2.12 Lévy s characerisaion of Brownian moion. Le M be a coninuous F -maringale such ha M = and M = for all, hen M is a Brownian moion. Now, on he probabiliy space Ω, F, P, if Z is a non-negaive random variable wih expecaion equal o one, hen one can define a new probabiliy measure P via PA := ZωdPω, for all A F. 1.2.2 A Adding a filraion F on [, T ], we can define a process Z as Z := EZ F for all [, T ]. This process is called he Radon-Nikodým derivaives process and i clearly a P- maringale. The following lemma hen holds see for insance [144, Lemma 5.2.2]: Lemma 1.2.13. For any u T, and any F -measurable random variable Y, Z u Ẽ Y F u = E Y Z F u. Proof of Theorem 1.2.1. We only prove Girsanov heorem in dimension one, he general case being analogous, albei wih more involved noaions. From Lévy characerisaion heorem Theorem 1.2.12, since he process W sars a zero and has quadraic variaion equal o, all is lef o prove is ha i is a maringale under P. I is easy o see, using Iô s lemma, ha dz = Θ Z dw, so ha Z = Z Θ u Z u dw u, and Z is a maringale wih EZ T = EZ = 1. Now, he Iô produc rule Corollary 1.1.27 and simple manipulaions yield d W Z = 1 Θ W Z dw, so ha W Z is also a maringale under P, which implies ha Ẽ W Fu = Zu 1 E W Z F u = Zu 1 W u Z u = W u, by Lemma 1.2.13, and he heorem follows.

1.3. The super-replicaion paradigm and FTAP 2 1.3 The super-replicaion paradigm and FTAP The roue we choose o follow in his firs approach o mahemaical finance may sound unorhodox a firs sigh, saring wih a more absrac framework abou sub/super-replicaion, before venuring back ino he realm of more classical mahemaical finance and opion pricing. We believe, however, ha his logic, borrowed from [77], follows he pracical inuiion more closely. Our aim is o answer he following quesion: wha is he correc price he buyer or seller of an opion should pay? We fix a priori a filered probabiliy space Ω, F, F, P, where P is he hisorical probabiliy measure. A marke model is hen defined as a pair process B, S aking values in,, n, for some n N. The firs componen denoes he money-marke accoun and saisfies he sochasic differenial equaion db = r B d, B = 1, 1.3.1 where r denoes he insananeous risk-free rae process. Noe ha his ordinary differenial equaion can be solved in closed form as B = exp r sds. The remaining n-asse vecor S = S 1,..., S n is he unique srong soluion o he sochasic differenial equaion ds = b, S d + σ, S dw, S, n, wih W a d-dimensional sandard Brownian moion, and where he drif b : R +, n, n and he diffusion σ : R +, n M n,d R + are assumed o be bounded and globally Lipschiz coninuous his in paricular guaranees exisence and uniqueness of he soluion, as we shall show in Chaper 3. Consider now a porfolio Π consising of invesed cash and he n asses: Π = π B + π S, where π = π 1,..., π n represens he vecor of quaniies of each asse in he porfolio, which can be readjused as ime passes. Definiion 1.3.1. The porfolio Π is self-financing if, for any, dπ = π db + π ds ; he variaion of he porfolio only comes from he evoluion of B, S, and no from exogenous ransfer in or ou of money. Assuming ha he porfolio Π is self-financing and denoing by D s, := B s B 1 facor beween ime s and ime in paricular, D, = B 1 he discoun saisfies dd, = r D, d, we have d Π := dd, Π = d π D, B + π D, S = π d D, B + d π D, S = π d S, where he noaion denoes discouning Π := D, Π, and hence, for any, Π = Π + π u d S u. 1.3.2

1.3. The super-replicaion paradigm and FTAP 21 Consider an asse S wih no drif b, and volailiy coefficien σ, adaped o he filraion of he Brownian moion W. We can rewrie he previous equaion as Π = Π + D,u π u σ u dw u, which is srongly reminiscen of he Maringale Represenaion Theorem 1.2.2 wih φ = D, π σ. Remark 1.3.2. Le Z denoe he sochasic process defined pahwise as Z := π B, for all. Then, for any, Π = π B + π S = Z + π S = Π + dπ u = Π + = Π + π udb u + r u Z u du + π u ds u π u ds u, so ha dz = ds +r Z d, wih S := π u ds u π S, which we can solve Example 1.1.28ii as D, Z = π + D,udS u, or equivalenly, using inegraion by pars Corollary 1.1.27, D, Z = π = π + = Π + D, S + D,u ds u = π + D, S D, S S u dd,u, r u D,u S u du. 1.3.3 This implies ha he weigh π is an Iô process ha can be chosen such ha Π is self-financing. Definiion 1.3.3. A porfolio π, π 1,..., π n [,T ] is said o be admissible over he horizon [, T ] if, for any [, T ], Π is bounded from below P-almos surely, e.g. here exiss M R such ha P inf Π M = 1. 1.3.4 T We shall denoe by A he space of admissible porfolios. Definiion 1.3.4. A self-financing admissible porfolio is called an arbirage if Π = and Π T, P-almos surely, and PΠ T > >. In plain words, an arbirage occurs if i is possible o obain a sricly posiive gain ou of a sraegy wih zero iniial cos. Inuiively, absence of arbirage in a marke is relaed o he noion of equilibrium, and mahemaically, his is saed in erms of condiions on admissible porfolios. One could wonder abou he necessiy of Condiion 1.3.4 in Definiion 1.3.3. This condiion imposes a limi acceped by crediors. As he following example shows, one canno do wihou i: Example 1.3.5. Wihin a fixed ime horizon [, T ], consider a marke wihou ineres rae B = 1 for all [, T ] and consising of one asse S saisfying ds = dw, wih S =. Define now he process Y as Y := T s 1/2 dw s. By ime-change echniques Theorem 1.1.21, here exiss a Brownian moion Ŵ such ha Y = Ŵ Y, where Y = ds = logt, for all [, T. T s

1.3. The super-replicaion paradigm and FTAP 22 Fix a consan M >, and define he sopping imes τ M := inf{ : Ŵ = M} and τ Y M := inf{ : Y = M}. As τ M is finie almos surely Lemma 1.5.9 below and equal o logt τ Y M, so ha τm Y < T almos surely. We now inroduce he weigh process π 1 T 1/2, if [, τm Y :=,, if [τm Y, T ], and le π be given by 1.3.3 such ha he corresponding porfolio Π has zero value a incepion: π = S = π 1 uds u π 1 S, for [, T ]. Indeed, Π = π + π 1 S = if and only i π = π 1. The value of he porfolio herefore reads Π = π 1 uds u = τ Y M and in paricular, a mauriy T, he porfolio is worh Π T = τ Y M dw u T u, dw u = T u Ŵτ M Y = M almos surely. This implies ha, saring from zero iniial value, he porfolio hus consruced can reach any sricly posiive value almos surely, leading o an arbirage. Noe, however, ha, for any [, T ], Π is no bounded below almos surely, hus violaing Condiion 1.3.4. The following lemma, despie is simpliciy, is a core resul for opion pricing. Lemma 1.3.6. Assume ha here exiss a probabiliy measure Q equivalen o P under which he discouned asse prices S are Q-local maringales. Then he marke does no admi arbirage. For wo probabiliy measures P and Q on Ω, F, P is said o be absoluely coninuous wih respec o Q, which is denoed by P Q, if for any A F such ha QA =, hen PA =. In paricular, if P Q, hen here exiss a random variable Z L 1 dp such ha dq/dp = Z and E P Z = 1. The wo probabiliy measures are said o be equivalen, and we wrie P Q, if P Q and Q P. In ha case, hey have he same null ses and dp/dq = dq/dp 1. This analysis is reminiscen of Girsanov s heorem Theorem 1.2.1. Proof of Lemma 1.3.6. Assume here exiss an arbirageable sraegy π,..., π n. By consrucion, he corresponding discouned porfolio is worh see 1.3.2, a ime, Π = Π + π u d S u, and hence is a Q-local maringale. Being bounded below by zero, i is a Q-supermaringale by Proposiion 1.1.12, so ha E Q Π T Π =. However, since Π T P-almos surely, hen i is also non-negaive Q-almos surely. Since is expecaion is null, hen Π T = Q-almos surely, hence P-almos surely, which yields a conradicion. Definiion 1.3.7. Any measure Q equivalen o P on Ω, F such ha S is a Q-local maringale is called an Equivalen Local Maringale Measure ELMM, or a risk-neural measure.

1.3. The super-replicaion paradigm and FTAP 23 Remark 1.3.8. In he classical Black-Scholes framework, see also below in Secion 1.2, here exiss a unique probabiliy measure under which he discouned sock price n = 1 is a rue maringale. However, here may be markes, as will be he case for sochasic volailiy models for insance, where an infiniy number of such probabiliy measures exis. The las resul we wish o sae in his framework is a necessary and sufficien condiion ensuring ha a marke has no arbirage. I also makes absence of arbirage easier o check han he general condiions above. Theorem 1.3.9. i Assume ha here exiss a process u [,T ] V Definiion 1.2.1 such ha σ, S u = b, S r S almos surely for all [, T ], 1.3.5 { 1 T and such ha E exp 2 u2 d} is finie. Then he marke does no admi any arbirage. ii If he marke has no arbirage, here exiss an F -adaped process u such ha 1.3.5 holds. Proof. We only prove he sufficien condiion i and assume for simpliciy ha r = almos surely for all. The probabiliy measure Q on F T, defined via he Radon-Nikodym derivaive { dq T = exp u dw 1 } T u 2 d, dp 2 FT is equivalen o P and Girsanov s heorem Theorem 1.2.1 ensures ha he process W defined as W := u s ds + W, is a Q-Brownian moion, and furhermore ds = σ, S d W. Therefore S is a local Q-maringale and he heorem follows from Lemma 1.3.6. Example 1.3.1. Consider he marke given by B = 1 almos surely for all, and ds = 2 d + 1 dw. 1 1 1 Show ha he marke does no admi arbirage. Consider now he given by B = 1 almos surely for all, and ds = 2 d + 1 1 dw. 1 1 1 Show ha he marke admis an arbirage and explain wha happens if one considers he sraegy π 1, 1.

1.3. The super-replicaion paradigm and FTAP 24 The above framework guaranees ha marke paricipans will be able o buy or sell derivaives a a fair price. Tha said, i is no clear wha his price should be and, inuiively, here is no reason why a buyer should be willing o pay he same price as a seller 2. Assume ha, a ime, he buyer buys a price p, a European opion wih mauriy T and payoff P T which may depend on all he asses, and hedges i unil mauriy. A mauriy, using 1.3.2, he discouned value of is porfolio is worh T T Π T = D,T D 1,T p + π u d S u + D,T P T = D, p + π u d S u + D,T P T. 1.3.6 This yields he following naural definiions of he buyers and he seller s price: Definiion 1.3.11. A ime, he buyer s B and he seller s S prices are defined by { } B P T := sup S P T := inf { p F : π A : Π T = D, p + p F : π A : Π T = D, p + T T π u d S u + D,T P T, P-a.s π u d S u D,T P T, P-a.s namely he buyer s price resp. seller s price is he larges resp. smalles iniial amoun o pay in order o obain a non-negaive value of he porfolio a mauriy. The following heorem is a key resul o deermine bounds for hese prices. Theorem 1.3.12. If here exiss a local maringale measure Q equivalen o P, hen },, B P T E Q D,T P T F S P T. Proof. Using he definiion of he buyer s price Definiion 1.5.1, here exiss an admissible porfolio π such ha T D, p + π u d S u + D,T P T, P-a.s. By consrucion, he inegral is a local maringale, which is bounded below since he porfolio is admissible, and herefore is a supermaringale, saring from zero a ime zero, and hence, aking Q-expecaions condiional on F, we obain D, p E Q D,T P T F, or p E Q D,T P T F. Taking he supremum over all p F yields one inequaliy in he heorem; he proof for he seller s price is analogous. This resul provides arbirage-free bounds for he price of any European opion. However, from a pracical poin of view, i migh seem limied in he sense ha i he equivalen local 2 hese are no mere heoreical consideraions, and can be clearly observed hrough limi order books and bid-ask spreads.

1.3. The super-replicaion paradigm and FTAP 25 maringale measure Q may no be unique, and ii he range of admissible prices migh be very wide. Uniqueness or absence hereof of Q is relaed o he noion of complee marke, in which hese bounds collapse o a single arbirage-free price. Definiion 1.3.13. A payoff P T is said o be aainable if here exiss a self-financing admissible porfolio π A and a real number p such ha a p + T π d S D,T P T =, P-almos surely; b he process π d S is a rue Q-maringale wih Q P. Noe as a comparison o a, ha Equaion 1.3.6 represens he value of he discouned porfolios, from he seller s poin of view. Definiion 1.3.14. A marke is said o be complee if every payoff is aainable. Theorem 1.3.15. In a complee marke, he double equaliy B P T = E Q D,T P T F = S P T holds for any equivalen local maringale measure Q. Proof. Since he marke is complee, here exis p R and an admissible porfolio π such ha T D,T P T = p + π u d S u, P-almos surely, and herefore T D,T P T = D, Z + π u d S u, Q-almos surely, where Z := D, 1 p + π ud S u is an F -measurable random variable for any [, T ]. The inegral is a Q-maringale, so ha, aking Q-expecaions condiional on F on boh sides yields Z = E Q D,T P T F, and, idenifying Z = p, S P T E Q D,T P T F and he heorem follows from he arbirage-free bounds in Theorem 1.3.12. The same argumens apply o he buyer s price, and he heorem follows. The following heorem, he proof of which unforunaely 3 falls ouside he scope of hese noes, provides a new characerisaion of he buyer s and he seller s prices in erms of he equivalen local maringale measures: Theorem 1.3.16. Le M e P denoes he se of P-equivalen local maringale measures. Then B P T = inf E Q D,T P T F and S P T = sup E Q D,T P T F, Q M e P Q M e P The proof of he following corollary, however, is wihin reach and is lef as a edious exercise: 3 The expression unforunaely here refers o he fac ha is proof is a beauiful, ye edious, exercise in convex dualiy. The hard-working ineresed reader can consul [132] for full deails.

1.4. Applicaion o he Black-Scholes model 26 Corollary 1.3.17. A marke is complee if and only if here exiss a unique equivalen local maringale measure. Similarly o Theorem 1.3.9, we provide wihou proof an easy o check crierion for marke compleeness. Theorem 1.3.18. The marke is complee if and only if σ admis a lef inverse almos surely. Noe in paricular ha if σ is inverible almos surely, hen clearly he process u in Theorem 1.3.9 is well defined, and hence he marke is free of arbirage. The converse does no hold in general, hough. Example 1.3.19. 1 1 Show ha he marke characerised by B 1 and ds = 2 d + 1 1 dw, W 2 3 1 1 is complee. The marke given by B 1 and ds = dw 1 + dw 2 is incomplee. In layman erms, he compleeness of he marke is due o he fac ha he number of Brownian moions driving he sysem, quanifying he randomness, is equal o he number of asses. The Black-Scholes model is he obvious example of a complee model, as is Dupire s local volailiy model, which we will sudy in Secion 4.1. Sochasic volailiy models, on he oher hand, are classical examples of incomplee marke models Secion 4.2, since he sysem is driven by wo Brownian moions, bu only one asse he sock price is radable. In a complee marke, Corollary 1.3.17 ensures he exisence of a unique equivalen local maringale measure Q such ha he unique arbirage-free price reads B P T = S P T = E Q D,T P T F see Theorem 1.3.16. As an example, in he Black-Scholes model, here is one source of randomness only one Brownian moion, driving a single asse. The marke is hen complee. We shall see laer some examples of incomplee markes, in paricular sochasic volailiy models, where he number of Brownian moions driving he sysem is larger han he number of raded asses. 1.3.1 Overure on opimal ranspor problems and model-free hedging 1.4 Applicaion o he Black-Scholes model We now assume ha a probabiliy P is given, under which he sock price process is he unique srong soluion o he following sochasic differenial equaion: ds = S µd + σdw, S >, 1.4.1

1.4. Applicaion o he Black-Scholes model 27 where σ is a sricly posiive real number, and W a Brownian moion on Ω, F, F, P. We are ineresed in evaluaing, a incepion, a derivaive on S, wih payoff H T a mauriy T >. The random payoff H T is assumed o be F T -measurable and square inegrable. Le now Υ denoe he following se of sraegies: { } Υ := π [,T ] adaped : T π 2 d < almos surely, 1.4.2 and, for each π Υ, le Π denoe he soluion o he SDE dπ = π ds + D, Π π S db, saring a Π = x. Noe ha Υ is nohing else han he almos sure version of V from Definiion 1.2.1. In he conex of Secion 1.3, he process Π corresponds o he porfolio associaed o he sraegies π and π = D, X π S, which is clearly self-financing. The discouned process Π := D, Π saisfies he SDE d Π = π D, S µ rd + σdw =: σπ S d W, where S is he discouned sock price process and W a P-Brownian moion. The probabiliy P here is defined via he Girsanov ransformaion d P = exp 12 dp γ2 T + γw T, FT and W := W γ for [, T ], wih γ := r µ/σ. Under P, he sock price process S saisfies ds = S rd + σd W, and hence P is a risk-neural probabiliy measure. The Maringale Represenaion Theorem implies ha here exiss an adaped square inegrable process ϕ such ha T D,T H T = E P D,T H T + ϕ d W. Leing π = ϕ /σd, S and π as above, and x = E PD,T H T, he rading sraegy π, π, x generaes a porfolio Π such ha Π T = H T almos surely, i.e. a replicaing porfolio for he coningen claim H T, and herefore he opion value a incepion is equal o x. Theorem 1.4.1. If here exiss a sricly posiive consan A π such ha Π A π almos surely for all [, T ], hen he se Υ does no conain any arbirage opporuniy. Proof. Suppose here exiss π Υ and ha x =. The discouned process Π is a P-local maringale bounded below, and herefore he process Π + A π is a supermaringale, and so is Π; in paricular, for any, E P Π E P Π =. If Π P-almos surely for any and PΠ > > hen E P Π > since P and P are equivalen, which yields a conradicion. We can now derive he celebraed Black-Scholes formula:

1.4. Applicaion o he Black-Scholes model 28 Theorem 1.4.2. Under he risk-neural probabiliy P, he European Call price wih srike K and mauriy T, wih payoff H T := S T K + is worh, a any ime [, T ], C BS S, K,, T, σ := E P [D,T S T K + F ] = S N d + KD,T N d, 1 where d ± := σ T log S ± 1 D,T K 2 σ T. Corollary 1.4.3. For a European Pu price, we have he following: P BS S, K,, T, σ := E P [D,T K S T + F ] = KD,T N d S N d +, Proof. Under P, he sock price saisfies ds = S rd + σdw ; hence Iô s formula yields S T = S exp {r 12 } σ2 T + σw T W, for any T, and he heorem follows by direc inegraion of he Gaussian random variable W T W. The proof for he Pu price is analogous. We now inroduce some noaions o simplify compuaions. Rewrie Theorem 1.4.2 as KD,T C BS S, K,, T, σ = E [D,T S T K + F ] = S BS log, σ 2 T, for any [, T ], where he funcion BS : R R + R is defined as N d + k, v e k N d k, v, if v >, BSk, v := 1.4.3 1 e k, if v =, + wih d + k, v := k/ v ± v/2, where N denoes he cumulaive disribuion funcion of he Gaussian disribuion. The Black-Scholes model has independen and saionary incremens, so ha he price of he Call opion only depends on ime hrough he remaining ime o mauriy T ; noe ha i also only depends on he volailiy hrough he oal variance: v := σ 2 T. Therefore, from now on, we shall prefer he noaion C BS k, T, σ, or even C BS k, v, o he over-parameerised C BS S, K,, T, σ, where k := S /D,T K is he log-forward moneyness. S Robusness of he Black-Scholes sraegy The way we proved he Black-Scholes formula in Theorem 1.4.2, via he maringale represenaion heorem, does no follow he original proof of he auhors [2]. They assumed ha here exiss a smooh funcion C : R 2 + R describing he value of he opion, and such ha C, s converges o he opion payoff h as approaches he mauriy T. This funcion saisfies he Black-Scholes parial differenial equaion + rs S + σ2 2 S2 SS r C, S =, 1.4.4

1.4. Applicaion o he Black-Scholes model 29 for all [, T and S, wih boundary condiion CT, S = hs. In he Black-Scholes framework, assuming ha we believe ha σ is he rue volailiy, hen we hedge ourselves by buying/selling an amoun S C, S of he sock price. This is called dela hedging. Suppose now ha he rue dynamics of he underlying sock price are in fac given by ds = S α d + β dw, where α and β are wo adaped processes. As seen in he previous secion, we can consruc a self-financing replicaing porfolio Π saisfying dπ = S C, S ds + Π S C, S S rd. Now, using Iô s formula, he Call price funcion saisfies dc, S = S C, S ds + + β2 S 2 2 SS C, S d. The hedging error E := Π C hen saisfies de = rπ + β2 S 2 2 SS + rs S C, S d σ 2 β 2 S 2 = rπ SS r C, S d, 2 = re 1 2 σ2 β 2 S 2 SS C, S d, where, in he second line, we used he fac ha he Call price saisfies he Black-Scholes PDE 1.4.4. An applicaion of Iô s formula yields, a mauriy, E T = 1 2 T e rt s S 2 Γ 2 σ 2 β 2 d. This formula is fundamenal for hedging, and indicaes ha he error in he hedging sraegy is due o i he under/overesimaion of he acual volailiy he sign of σ 2 ξ 2, and ii he amoun of posiive convexiy Γ of he opion price wih respec o he underlying. Taking derivaives of he Call opion price wih respec o he parameers yields he so-called Greeks, which, as seen above, are fundamenal ools for hedging purposes: k BSk, v = e k N d k, v, [ kk BSk, v = e k N d k, v nd ] k, v = k BSk, v + ek nd k, v, v v v BSk, v = nd +k, v 2, v vv BSk, v kv BSk, v = nd +k, v 16v 5/2 4k 2 v 2 4v, = nd + k, v v d k, v = nd +k, v 4 2k v v 3/2, 1.4.5 where n N denoes he Gaussian densiy. The proof of he equaliies 1.4.5 is lef as an exercise.