Local risk minimizing strategy in a market driven by time-changed Lévy noises. Lotti Meijer Master s Thesis, Autumn 2016

Transcription

1 Local risk minimizing sraegy in a marke driven by ime-changed Lévy noises Loi Meijer Maser s Thesis, Auumn 216

2 Cover design by Marin Helsø The fron page depics a secion of he roo sysem of he excepional Lie group E 8, projeced ino he plane. Lie groups were invened by he Norwegian mahemaician Sophus Lie ( ) o express symmeries in differenial equaions and oday hey play a cenral role in various pars of mahemaics.

3 iii Preface Absrac The purpose of his hesis is o sudy he hedging of financial derivaives, using he so-called local risk-minimizing sraegy, which is a popular quadraic hedging sraegy in incomplee markes. The local risk minimizaion aims a perfecly replicaing he derivaive. However such sraegies canno be selffinancing in general, and herefore allowing for a cos. Then a good sraegy should have minimal cos. The problem of finding he local risk minimizing sraegy is in his hesis ackled by wo mehods, via a change of measure using he minimal maringale measure and via backward sochasic differenial equaions. The financial marke model sudied is driven by a ime-changed Lévy noise, where he ime change is independen of he Lévy process. In his hesis wo differen informaion flows are considered. Boh filraions are naurally linked o he noise. Informaion flow G is he large filraion conaining informaion abou he fuure, which capures all saisical properies of he noise. While he smaller informaion flow F is a more realisic informaion flow from a financial modeling perspecive.

4 iv Acknowledgemen I would like o hanks o my supervisor, Giulia, for he guidance and advice she has provided me hroughou my work on his hesis. Thanks o my family for always believing in me. And hanks o Rober for he love and suppor. Loi Meijer Oslo, November 216

5 Conens 1 Inroducion 1 2 The Marke model and he noise Lévy processes Time changed process Subordinaor Absoluely coninuous ime change Time changed Brownian moion The marke model The Local Risk Minimizing sraegy The Local Risk Minimizing problem The LRM under G, via MMM The LRM under F, via MMM Backward Sochasic Differenial Equaion The LRM under G, via BSDEs The LRM under F, via BSDEs Conclusion and furher research Appendix A 69 Appendix B 78 References 84 v

6 vi CONTENTS

7 Chaper 1 Inroducion In a incomplee marke no all claims are perfecly replicable. In such incomplee markes one can apply quadraic hedging. The sraegy in quadraic hedging is o minimize he hedging error in a mean square sense. One of he wo main approaches in quadraic hedging is local risk minimizing. The main feaure of he local risk minimizing approach is ha one works wih sraegies which are no self-financing. So one is looking for a sraegy ha insiss on perfecly replicaing he coningen claim, however can no be self-financing. Then a good sraegy should have minimal cos. Time-changed Lévy noises are in he financial marke considered as a sandard mehod for consrucing financial models. The ime-changed Lévy process, inroduce a new sochasic process rough a random change of ime. Models driven by hese ime-changed Lévy processes are based on he fac ha he volailiy is sochasic and increasing wih he inensiy of rades. So hese models are a good way of describing he business aciviies. The new sochasic process, also called he ime change, can be considered as he new random ime, he so-called business ime clock, and he original clock is considered as he calendar ime. There are several well-known applicaions o financial models driven by imechanged Lévy noises. Models driven by ime-changed Brownian moions appear in financial price modeling in he class of sochasic volailiy. See he papers [BNS2], [Hes93], [HW87] and [SS91]. This hesis aims a sudy he following quesion: Given a coningen claim F how can we characerize he local risk minimizing sraegy in such imechanged markes? We ackle he problem by considering wo mehods. The firs mehod, is o sudy he relaionship beween he Föllmer-Schweizer decomposiion and he 1

8 2 CHAPTER 1. INTRODUCTION Galchouk-Kunia-Waanabe decomposiion. For his we will need a change of measure ha preserves orhogonaliy of local maringales, wih he socalled minimal maringale measure. The necessary background on local risk minimizing sraegies in he seing of a general semimaringale is found in [Sch99] and [FS91]. We exend his heory o he ime-change seing. The second mehod ha we consider uses backward sochasic differenial equaions driven by he ime-changed Lévy noises, whose erminal condiion is he coningen claim F. Then we will look a he relaion beween he backward sochasic differenial equaions and he Föllmer-Schweizer decomposiion. This mehod is based on he work in [DKV15]. In his hesis wo informaion flows are considered. The firs filraion is generaed by he naural filraion of he ime-changed Brownian moion. The second, is generaed by he ime-changed Brownian moion and in addiion has knowledge of he enire fuure of he ime-change process. These informaion flows are denoed by F and G, respecively. From a financial poin of view i is more realisic o consider he informaion flow F. While he informaion flow G, capurers all he saisical properies of he noise. In he seing of he wo differen filraions, one can expec ha he associaed porfolio value of he local risk-minimizing sraegy, will lead o differen soluions depending on he filraions measurabiliy properies. The srucure for he hesis is as follows. In Chaper 2, we sar by inroducing he preliminary heory for he noise, and presen wo popular ime change models, subordinaor and absoluely coninuous ime change. Then we will narrow he heory down o ime-changed Brownian moions. The marke model is also presened. In Appendix A here are more heoreical resuls and definiions o suppor he heory in Chaper 2. In Secion 3.1, we discuss he local risk minimizing problem, under a measure change, wih respec o he wo filraions G and F. In Secion 3.2, we discuss he local risk minimizing problem wih backward sochasic differenial equaions, wih respec o he wo filraions G and F. For boh mehods, we firs consider he informaion flow G, here we give imporan resuls o show ha he heory on local risk minimizing in he seing of a general semimaringale can be exended o he case of a semimaringale driven by a ime-changed Brownian moion. When considering he informaion flow F, we will need o apply some oher echniques o find he soluion o our problem.

9 My work for his hesis was srucured as follows. I sared by sudy he necessary background on ime change echniques, where he focus was kep on ime-changed Brownian moion. This included background on Lévy processes, subordinaors and absoluely coninuous ime change. Then I defined he marke model driven by a ime-changed Brownian moion, and he necessary properies for ime-changed Brownian moion. Nex, I coninued wih acquiring basic knowledge on he local risk minimizing (LRM) sraegy and he minimal maringale measure (MMM). Then, sared ackling he LRM via MMM in he seing of he ime-changed Brownian moion wih respec o he filraion G. I also acquire he basic knowledge on backward sochasic differenial equaions (BSDEs), boh in he seing of general maringales and BSDEs driven by càdlàg maringales. I coninued by ackling he LRM problem via BSDEs wih respec o filraion G. Then I exended he LRM sraegy via MMM and BSDEs o he case of filraion F, using a parial informaion approach. My main conribuion o his hesis is he work in Secion and Secion 3.2.2, when we consider he LRM problem under he informaion flow F, generaed by he ime-changed Brownian moion. In addiion, I conribued wih he following: Chaper 2: Proposiion (The Lévy characerizaion of a ime-changed Brownian moion) and is proof. Proof (i) of Proposiion (The proof of he ime-changed Brownian moion being a maringale wih respec o filraion G). Chaper 3: Theorem (Girsanov s Theorem in he seing of a ime-changed Brownian moion) and is proof. The Explici soluion of he LRM sraegy under filraion G. Proposiion (The represenaion of he predicable quadraic variaion of B wih respec o F, as a projecion of he ime change process) and is proof. My original conribuions are marked wih R. 3

10 Chaper 2 The Marke model and he noise 2.1 Lévy processes In his secion we formally inroduce Lévy processes. The heory is from he book of Applebaum (29) [App9]. Le (Ω, F, P ) be a complee probabiliy space. Definiion Le X = {X, } be a sochasic process on (Ω, F, P ). Then X is a Lévy process if (i) X = a.s. (ii) X has independen and saionary incremens (iii) X is sochasically coninuous, i.e. for all k > and for all s lim P ( X X s > k) = s Le X = {X, } be a Lévy process on R d. Then he random variable X is characerized by he riple (b, A, ν), for all, called Lévy riple and given by he Lévy-Iô decomposiion 1. Moreover he whole process X is described by he Lévy riple of X 1, ha is (β, A, ν). Denoe he Borel σ-algebra by B. The measure ν on R d is defined by ν(b) = E[# [, 1] : X, X B], B B(R d ) called he Lévy measure. The characerisic funcion of he random variable X, is defined by Ψ X (u) := E[exp(iu X )] = exp(ψ X (u)), u R d for all, where ψ X (u) = ψ X (u) and ψ X : R d R is a coninuous funcion called he characerisic exponen of X. 1 See Appendix A.2 4

11 2.2. TIME CHANGED PROCESS 5 The Lévy-Khinchine represenaion for a d-dimensional Lévy process X = {X, } gives an expression of he characerisic funcion of X in erms of is Lévy riple (β, A, ν). Tha is, he characerisic exponen is given by ψ X (u) = iβ u 1 ( ) 2 u Au + e iu y 1 iu y1 y 1 ν(dy) (2.1) R d (where is he ransposed). Example The (sandard) Brownian moion W = {W, } on R d is a Lévy process, wih Lévy riple (, I d, ). 2.2 Time changed process Time changing is a ransformaion of a sochasic process ino a new sochasic process hrough random change of ime. In finance, applicaions of price dynamics driven by ime-changed noises are based on he fac ha volailiy is increasing wih inensiy of rades. So he ime change process represens he ransiion beween he real-ime clock o he rading clock. Le X = {X, } be a sochasic process on R d and Λ = {Λ, } be a nonnegaive, non-decreasing sochasic process on R, no necessarily independen of X. Then he ime-changed process define by Y = {Y, } wih respec o he filraion G is defined as Y = X Λ Where G = σ{y s, Λ s, s } N and N is he collecion of all P-measure zero ses in F. The process Λ is called he ime change process. Laer on we will only work wih ime-changed Brownian moions. Firs we will inroduce some useful resuls (see e.g. [VW1]). Theorem (Dubins-Schwarz). Every coninuous local maringale M can be wrien as a ime-changed Brownian moion B M. Here M = { M, } is he predicable quadraic variaion of M (see Appendix A.3). Example Le M = σ sdw s, where W is a Brownian moion wih respec o i naural filraion F W, for all and σ a independen nonnegaive càdlàg sochasic process, such ha M is a local maringale. In fac for a sequence of sopping imes T n, n N, wih lim T n =, we have n

12 6 CHAPTER 2. THE MARKET MODEL AND THE NOISE ha M Tn 1 T n> is a square inegrable maringale. Le F W Opional Sampling, = G T. Then by E[M G Ts ] = E[ lim M Tn G Ts ] n = lim E[M Tn G Ts ] = lim M s Tn = M s n n So we have M = σ2 sds =: Λ, and by he Dubins-Schwarz Theorem, B Λ is a ime-changed Brownian moion. Theorem (Monroe). Every càdlàg F-semimaringale S = {S, } can be wrien as a ime-changed Brownian moion, if here is a filered probabiliy space (Ω, F, {F }, P ), an F - Brownian moion W = {W, } and a càdlàg family of sopping imes Λ = {Λ, }, such ha S d = W Λ. For a general ime-changed process Y = {Y, } and W = {W, } a Brownian moion on R, we have ha Y = W Λ is a condiionally Gaussian process. Tha is, P (Y y Λ ) = Φ( y ) Λ Where Φ is he cumulaive probabiliy disribuion funcion of a sandard normal variable. So Y is Gaussian, condiional on he ime change process. There are differen ypes of ime change processes, we will look a subordinaors and absoluely coninuous processes Subordinaor A subordinaor is a Lévy process on R ha is non-decreasing and nonnegaive. So for a subordinaor Λ = {Λ, } we have Λ a.s. for each and Λ 1 Λ 2 a.s whenever 1 2. Le he Lévy riple of Λ be (a,, ρ), hen, for all u, E[exp(uΛ )] = exp(l(u)) where l(u) = au+ (, ) (euy 1)ρ(dy), a and he Lévy measure ρ saisfies ρ(, ) = and (1 y)ρ(dy) < The funcion l( ) is called he Laplace exponen of Λ.

13 2.2. TIME CHANGED PROCESS 7 Example Le N = {N, } be a Possion process of inensiy λ. This can be represened as follows. For a sequence of independen exponenial random variables τ i, i = 1, 2,... wih parameer λ and T n = n i=1 τ i, we have N = n 1 1 Tn The Poisson process is a Lévy process. Also i is nonnegaive, since N is defined on N {} and non-decreasing, since N is a consan on each inerval [T n, T n+1 ) and increases by one a each T n. Hence he Poisson process is a subordinaor. Example Le Λ be a subordinaor, wih Lévy riple (a,, ρ). Then Λ is also a absoluely coninuous ime change if we can wrie i as, Λ = a where a >. In his case he riple is given by (a,, ). In general, we can wrie a subordinaor Λ as a non-decreasing semimaringale (see Definiion A.3.7). Tha is, le Λ be a subordinaor wih Lévy riple (a,, ρ), hen by he definiion of a subordinaor, condiion (i)-(ii) in Definiion A.3.7 are saisfied, also we have B, for all, hence condiion (iii) is saisfied. Finally, le α = a xρ(dx)ds, for all. When α <x 1 is non-decreasing, we have Λ = α + xj(, dx) (2.2) (, ) where J is he jump measure of he possible posiive jumps of Λ. We will use subordinaors o ime-change. The following resul is from Theorem 4.2 in [CT4]. Theorem Le X = {X, } be a Lévy process on R d wih Lévy riple (β, A, ν) and characerisic exponen ψ X ( ) given by (2.1) and le Λ = {Λ, } be a subordinaor, wih Lévy riple (a,, ρ) and Laplace exponen l( ). Le Λ be independen of X. Define a new sochasic process Y = {Y, } by Y := X Λ for each. Then Y is a Lévy process wih riple (β Y, A Y, ν Y ), given by β Y = aβ + ρ(ds) xp X (dx) (, ) x 1

14 8 CHAPTER 2. THE MARKET MODEL AND THE NOISE ν Y (B) = aν(b) + A Y = aa (, ) P X (B)ρ(ds), B B(R d ) where P X is he probabiliy disribuion of X. Moreover, he Lévy characerisic funcion is given by Ψ Y := E[exp(iuY )] = exp(l(ψ X (u))), u R d. (2.3) Proof. (i) Y has independen incremens. Denoe F Λ = FT Λ o be he filraion generaed by Λ a he finie ime horizon T and le < 1 < < n be a pariion of [, T ]. Then by he independen incremens of X and Λ, E[ n n exp(iu i (X Λi X Λi 1 ))] = E[E[ exp(iu i (X Λi X Λi 1 )) F Λ ]] i=1 i=1 i=1 n = E[ E[exp(iu i (X Λi X Λi 1 )) F Λ ]] ( ) = E[ = n E[exp((Λ i Λ i 1 )ψ X (u i ))] = i=1 n exp((λ i Λ i 1 )ψ X (u i ))] i=1 n E[exp(iu i (X Λi X Λi 1 ))] Where in (*) we applied he independence beween X and Λ. Tha is, for a bounded funcion f, we have E[E[f(X Λ ) F Λ ]] = E[g(Λ )], where g() = E[f(X )]. i=1 (ii) Saionariy of incremens. Similar o (i). (iii) For every ɛ >, δ >, we have P ( X Λ X Λs > ɛ) P ( X Λ X Λs > ɛ Λ Λ s < δ) + P ( Λ Λ s δ) Since X is a Lévy process, i is uniformly coninuous in probabiliy, so he firs erm can be made arbirarily small. The second erm ends o zero when s, because Λ is coninuous in probabiliy. Hence P ( Y Y s > ɛ) as s Hence by Definiion Y is a Lévy process. Moreover, he funcion (2.3) is obained by condiioning on F Λ, E[exp(iuY )] = E[E[exp(iuX Λ ) F Λ ]] = E[exp(Λ (ψ X (u)))] = exp(l(ψ X (u))) The proof of finding he Lévy riple (β Y, A Y, µ Y ), is aken from he proof of Theorem 3.1 in Sao (1999). Define he measure γ by γ({}) = and γ(b) = P X (B)ρ(ds), B B(R d ) (, )

15 2.2. TIME CHANGED PROCESS 9 We need o show ha x 2 γ(dx) < x 1 and x 1 x 2 γ(dx) < By Lemma 3.3 in [Sa99], here is a C 1 = C 1 (ɛ) such ha for any, and here is a C 2 such ha for any, P ( X > ɛ) C 1 (2.4) E[ X 2 X 1] C 2 (2.5) Le D = {x : x 1}. Then (2.4) and (2.5) yields, respecively γ(dx) = P ( X s > 1)ρ(ds) < D x >1 x 2 γ(dx) = (, ) (, ) Furhermore, we have l(ψ X (u)) = aψ X (u) + ρ(ds) (, ) D x 2 P X (dx) < (e ψ X(u)s 1)ρ(ds) Le g(u, x) = e iux 1 iux1 D (x), hen (e ψx(u)s 1)ρ(ds) = ρ(ds) (e iux 1)P X (dx) (, ) (, ) R d = ρ(ds) g(u, x)p X (dx) + i ρ(ds) ux1 D (x)p X (dx) (, ) R d (, ) R d = g(u, x)γ(dx) + iu ρ(ds) xp X (dx) R d (, ) D Hence aψ X (u) + (, ) (e ψx(u)s 1)ρ(ds) = iβau 1 2 u2 aa + a + g(u, x)γ(dx) + iu ρ(ds) xp X (dx) R d (, ) D = iu(βa + ρ(ds) xp X (dx)) (, ) D 1 2 u2 aa + g(u, x)(aν(dx) + γ(dx)) R d R d g(u, x)ν(dx)

16 1 CHAPTER 2. THE MARKET MODEL AND THE NOISE Example Le X in Theorem be a d-dimensional Brownian moion (w.r. is naural filraion) wih Lévy riple (,A,), such ha ψ X (u) = 1 2 u Au, u R d Le Λ be a subordinaor, independen of X wih Laplace exponen l( ) and riple (a,, ρ). Then Y = X Λ has characerisic funcion Ψ Y (u) = exp(l(ψ X (u))) = exp(l( 1 2 u Au)), u R d and he Lévy riple (β Y, A Y, ν Y ) given by β Y = ρ(ds) ν Y (B) = (, ) (, ) x 1 xp X (dx) A Y = aa (2.6) P X (B)ρ(ds), B B(R d ) where P X is he probabiliy disribuion of he Brownian moion. Example Le Λ be a subordinaor, Λ = and le W be a sandard Brownian moion wih respec o is naural filraion, independen of he subordinaor. Le G = σ{λ s, W Λs, s }. We wan o show ha he Lévy process X = µλ + σw Λ (µ, σ R) is a ime-changed Brownian moion wih respec o G, for all. We find he characerisic funcion of X, E[e iux ] = E[E[e iux F Λ T ]] = E[E[e iu(µλ+σw Λ ) F Λ T ]] = E[e iuµλ 1 2 u2 σ 2 Λ ] = e (l(iuµ 1 2 u2 σ 2 )) (2.7) where F Λ is he σ-algebra generaed by Λ and T he finie ime horizon. The Lévy riple (β X, A X, ν X ) of X, is given by β X = aµ + ρ(ds) xp W (dx) ν X (B) = (, ) (, ) x 1 A X = aσ 2 (2.8) P W (B)ρ(ds), B B(R d ) where P W is he probabiliy disribuion of he Brownian moion, wih drif µ and volailiy σ 2. Hence X is a ime-changed Brownian moion wih drif µ and volailiy σ 2 wih respec o G, for all.

17 2.2. TIME CHANGED PROCESS Absoluely coninuous ime change The second ype of ime change we wan o look a is he absoluely coninuous ime change. Le Λ = λ sds be a sochasic process on R, where λ is a posiive and inegrable process. Noe ha λ may exhibi jumps. The sochasic process Y = X Λ, where X is a Lévy process on R d wih characerisic funcion Ψ X and Λ absoluely coninuous ime change, has for all, u R d he characerisic funcion Ψ Y (u) = E[exp(iu X Λ )] = E[E[exp(iu X s ) Λ = s]] If we assume independence of X and Λ, we obain he characerisic funcion Ψ Y (u) = E[exp( Λ ψ X (u))] Then we can wrie i in erms of he Laplace ransform of Λ where L Λ (u) := E[exp( u λ sds)]. Ψ Y (u) = L Λ (ψ X (u)) Remark. The wo classes of ime change, subordinaors and absoluely coninuous ime-change, are wo differen classes, bu hey inersec. Le Λ be he subordinaor from Example 2.2.5, i.e. Λ = a. Then he subordinaor is also a absoluely coninuous process, as we can wrie Λ = a = λ s ds On he oher hand, le Λ be a absoluely coninuous ime-change, and suppose λ is consan, hen we can wrie Λ as in Example The advanage of ime change processes ha are absoluely coninuous is ha we can deermine he characerisic funcion and he Laplace funcion. In view of he paper by Carr and Wu (24), we know ha absoluely coninuous ime-change lead o Affine models, and hese models are racable. There are several ways o specify he affine funcion of Λ and he corresponding Laplace funcion, see Table 2 in [CW4]. The Affine model is a class of processes X where, X is affine wih respec o a Markovian process Y, in he sense ha we can wrie X = A B Y

18 12 CHAPTER 2. THE MARKET MODEL AND THE NOISE where A, B are deerminisic funcions. The Markovian process Y is said o be of affine erm srucure. Recall ha a Markov process Y = {Y, } on R d, is a sochasic process wih respec o F he σ-algebra generaed by Y, such ha, for s, E[f(Y ) F s ] = E[f(Y ) Y s ] where f is a bounded Borel funcion from R d o R. In finance affine models are commonly used o relae zero-coupon bond prices o a spo rae model. Tha is, he zero-coupon bond prices process P is affine wih respec o a spo rae model. For example he Vasicek model dr = (b ar)d + σdw is a spo rae model wih affine erm srucure. We will give a summary of he Affine aciviy rae model. Le Λ = λ sds and le λ be a funcion of a Markov process Z = {Z, } on R d. The dynamics of Z is given by dz = µ(z )d + σ(z )dw, Z = z where W is a d-dimensional Brownian moion and µ he drif vecor and σ is he diffusion marix, boh d-dimensional. All echnical condiions are assumed o hold. The Laplace funcion of Λ is an exponenial-affine funcion of Z if L Λ (u) := E[exp( uλ )] = exp(b z + c ), b R d, c R (2.9) Moreover if λ, µ(z) and σ(z)σ(z) are all affine wih respec o Z, hen we obain ha L Λ is exponenial-affine in z. Tha is, if we can wrie he processes in he following form, λ = b λz + c λ µ(z ) = a κz { [σ(z )σ(z ) α i + β i Z, if i = j ] ij =, if i j (2.1) where c λ, α R, b λ, a, β R d and κ R d R d, hen (2.9) is saisfied. Moreover, b and c from (2.9) is given by he following ordinary differenial equaions b = ub λ κ b βb2 2, b = c = uc λ b a b αb 2, c =

19 2.2. TIME CHANGED PROCESS 13 The affine process is racable, in ha he coefficiens b, c are known explicily by he ordinary differenial equaion. Hence, he Laplace funcion and hus he characerisic funcion of he ime-changed Lévy process remains compleely deermine.

20 14 CHAPTER 2. THE MARKET MODEL AND THE NOISE 2.3 Time changed Brownian moion Now we narrow he resuls o he ime changed Brownian moions. Tha is, we consider he Lévy process X of he previous secions o be a Brownian moion. Hence X Λ = W Λ =: B for all. In his hesis we will consider he case where d = 1. We also assume for convenience ha Λ =. Le T be he finie ime horizon. We need o inroduce he filraion we will work under. Le F Λ = σ{λ, [, T ]} (2.11) be he σ-algebra generaed by Λ, [, T ] and F = σ{b s, s } N he filraion generaed by he ime-changed Brownian moion B a ime. Le F = {F, [, T ]}. Then we define H = {H, [, T ]} by and G = {G, [, T ]} by H := F F Λ N G := F F Λ T N Here H is he filraion generaed by he noise wih iniial knowledge of he ime-change process and G is generaed by he noise and he whole imechange process. Moreover, we see ha G T = H T = F T, G = F Λ T while F and H are rivial. By Theorem in [App9] we conclude he following resul Lemma The filraion G is righ-coninuous. So we can say ha he filraion G saisfies he usual condiions, i is complee (i.e. conains all he P-zero ses of F) and righ-coninuous. We can in a way regard he informaion flow F as parial wih respec o he informaion flow G, even hough he filraion is of perfec informaion of he ime-changed Brownian moion. We have ha he filraion G includes informaion abou he fuure since we have knowledge of he whole ime change process, so we will see ha he marke under G is in some sense complee in Chaper 3. From a financial poin of view, filraion F will be more ineresing for he our sudy. However i can be difficul o find he local risk minimizing sraegy under F as we will see in Chaper 3, herefore we will also consider he sraegy under filraion H.

21 2.3. TIME CHANGED BROWNIAN MOTION 15 Remark. Noe ha if a process is F-adaped i is also H- and G-adaped. Moreover, if a process is H-adaped i is G-adaped. For simpliciy we will denoe K o be he filraion for resuls where we can apply boh F, H and G. Assume now ha Λ = {Λ, [, T ]} is eiher a subordinaor or an absoluely coninuous ime-change. Denoe Λ Θ, Θ a Borel se on [,T], o be a posiive random measure generaed by Λ (s,] (ω) := Λ (ω) Λ s (ω), ω Ω (see e.g. [KF61] form more informaion on measures). Moreover, we assume E[Λ T ] <. Now we inroduce he noise driving he sochasic dynamics in he marke model we shall consider. Definiion Le B be a signed random measure on he Borel ses of [, T ], such ha (i) P (B(Θ) x FT Λ) = P (B(Θ) x Λ Θ) = Φ( x ΛΘ ), x R, Θ [, T ] (ii) B(Θ 1 ) and B(Θ 2 ) are condiionally independen given FT Λ for any disjoin ses Θ 1, Θ 2 [, T ]. Here Φ sands for he cumulaive probabiliy disribuion funcion of a sandard normal variable and "condiional independen given FT Λ " means ha we have a.s. P (B(Θ 1 ) Γ 1, B(Θ 2 ) Γ 2 F Λ T ) = P (B(Θ 1 ) Γ 1 F Λ T )P (B(Θ 2 ) Γ 2 F Λ T ) for Γ 1, Γ 2, Borel ses on [,T] and Θ 1, Θ 2 [, T ] such ha Θ 1 Θ 2 =. We have ha he random measure B is a Gaussian random measure condiional on Λ. Moreover, B is relaed o he ime-change Brownian moion. Tha is, given Λ, he characerisic funcion of B(Θ) for any Θ [, T ] is given by E[exp(iuB(Θ)) F Λ T ] = exp( 1 2 u2 Λ Θ ), u R (2.12) The following resul is from Theorem 3.1 [Ser72], Theorem Le W = {W, [, T ]} be a Brownian moion on (Ω, F, P ) independen of Λ. Then B saisfies (i)-(ii) in Definiion and (2.12) if and only if, for any [, T ], B := B((, ]), is such ha B d = W Λ (2.13)

22 16 CHAPTER 2. THE MARKET MODEL AND THE NOISE Proposiion B is a maringale wih respec o (i) G, (ii) F, and (iii) H Proof. (i) R For all s, E[B B s G s ] = E[B((s, ]) G s ] = E[B((s, ]) F s F Λ T ] ( ) = E[B((s, ]) F Λ T ] ( ) = Where we used: (**): from condiion (i) in Definiion 2.3.2, we have E[B((s, ]) F Λ T ] = for all s (*): from condiional independen (ii) in Definiion we have, for F F s, G F Λ T, E[1 F 1 G E[B((s, ]) F s F Λ T ]] = E[1 F 1 G B((s, ])] = E[1 G E[1 F B((s, ]) F Λ T ]] (ii) = E[1 G E[1 F F Λ T ]E[B((s, ]) F Λ T ]] = E[E[1 G 1 F E[B((s, ]) F Λ T ]] = E[1 G 1 F E[B((s, ]) F Λ T ]] By a monoone class argumen, his exends o E[1 A E[B((s, ]) F Λ T F s ]] = E[1 A E[B((s, ]) F Λ T ]] for A G s = F s F Λ T. (ii) By he ower propery and he fac ha B s is F s -measurable, we have for all s, E[B F s ] = E[E[B G s ] F s ] = E[B s F s ] = B s (iii) The proof is similarly o (ii). B is H-adaped, so by he ower propery and (i) he resul follows. Proposiion R Le X be a coninuous sochasic process defined on he probabiliy space (Ω, F, P ), wih X =. Suppose ha (i) X is a (G, P )-maringale, (ii) X = Λ, for all [, T ]. Then here is a Brownian moion W wih respec o F W, independen of Λ, such ha for each [, T ], X = W Λ. Hence X is a ime-changed Brownian moion wih respec o G.

23 2.3. TIME CHANGED BROWNIAN MOTION 17 Proof. By he maringale represenaion heorem (see Theorem 3.5 in [DS14]) here exiss a φ such ha X = φ s db s where B is a ime-changed (G, P )-Brownian moion. By assumpion X = Λ Apply Iô s formula for coninuous maringales o f(x) = e iux, u R, d(e iux ) = iue iux φ db 1 2 u2 e iux dλ Fix θ, e iux e iux θ = e iu(x Xθ) 1 = θ θ iue iuxs φ db s 1 2 iue iu(xs X θ) φ db s 1 2 θ u 2 e iuxs dλ s Then aking condiional expecaion given FT Λ, we obain E[e iu(x X θ) F Λ T ] 1 = E[ By he ower propery, E[ θ iue iu(xs X θ) φ db s F Λ T ] 1 2 u2 E[ iue iu(xs X θ) φ db s F Λ T ] = E[E[ θ θ Clearly Λ is FT Λ -measurable, so θ θ u 2 e iu(xs X θ) dλ s e iu(xs X θ) dλ s F Λ T ] iue iu(xs X θ) φ db s G θ ] F Λ T ] = Le so we have E[e iu(x Xθ) FT Λ ] = u2 E[e iu(xs Xθ) FT Λ ]dλ s Z = E[e iu(x Xθ) FT Λ ], [, T ] dz = 1 2 u2 Z dλ, Z θ = 1 θ Hence by Iô s formula for coninuous processes, we have, for all θ, Z = exp( 1 2 u2 (Λ Λ θ ))

24 18 CHAPTER 2. THE MARKET MODEL AND THE NOISE This is he characerisic funcion of he condiional normal disribuion given F Λ T, hus X is normal disribued wih mean and variance Λ given F Λ T. This means ha here exiss a sandard Brownian moion W, s.. W Λ = X. Furhermore, le Q be he law of X and le B wih law P be given by Definiion Since X and B has he same characerisic funcion, we ge ha P = Q (see e.g. Theorem in [Dud2]). Hence X is condiional independen given F Λ T. So X saisfies boh condiions (i)-(ii) in Definiion and equaion (2.12), so by Theorem we ge he resul.

25 2.4. THE MARKET MODEL The marke model Le (Ω, F, P ) be a complee probabiliy space as before. The marke model is given by he riskless asse ds () = r S () d, S () = 1 and one risky asses, wih dynamics given by ds (1) = α S (1) d + σ S (1) db, S (1) = x > 1 (2.14) where B is a ime-changed Brownian moion from (2.13), i.e. B = W Λ. Marke models driven by ime-changed Brownian moions are considered a useful ool o describe he sochasic volailiy in he marke. A class of popular sochasic volailiy models driven by ime-changed Brownian moions can be found in [BNS2], [Hes93], [HW87] and [SS91]. In common hese volailiy models have he following dynamics of he price process ds = µ S d + σ λ S dw (1) dλ = α λ d + β dw (2) where W (1), W (2) are Brownian moions. In he case where W (1) and W (2) are independen, he process B = λ sdw s (1) is a condiional Brownian moion as in Definiion 2.3.2, and our framework can be applied. To ensure he exisence of a srong soluion and o allow furher analysis, we assume for he marke model ha he processes α, σ and r are càglàd F-adaped and sochasic, σ P d a.e. and E [ ( α r d + σ 2 d B K ) ] < Here K represens any filraion F, H, G. From (2.14) we see ha S (1) is a K-semimaringale. Noe ha he processes α, σ and r are also H- and G- adaped. Here we se B K o be he predicable quadraic variaion of he K-maringale B wih respec o he filraion K (see Appendix A.3). Noe ha B G = B H = Λ Here we have used Doob-Meyer Theorem, ha is B 2 B K is a unique K-maringale. While B F is more difficul o calculae, since Λ is no a F- adaped, and hence B 2 Λ is in general no a F-maringale. We will however see in Secion 3.1.2, ha we can wrie B F as a projecion of Λ.

26 2 CHAPTER 2. THE MARKET MODEL AND THE NOISE From now on we will use discouned prices. Tha is, we use he riskless asse as numéraire, so he discouned riskless asse has price 1 a all imes and he discouned risky asse is given by S = S1. So he dynamics of he S K-semimaringale S is given by ds = (α r )S d + σ S db, S = x (2.15) Proposiion The soluion o (2.15) wih respec o K, is given by (i) S = xexp ( 1 ) (α s r s )ds 2 σ2 sd B K s + σ s db s when Λ is a absoluely coninuous ime change. (ii) S = xe Y where dy = { (α r ) + σ βy 1 } 2 σ2 A 2 Y d + σ A Y dw + log[1 + σ x]n(d, dx), Y = 1 R\{} when Λ is a subordinaor. Proof. (i) Le Λ be a absoluely coninuous ime change, hen S is a coninuous K-semimaringale. So we can apply Iô s formula 1 for coninuous semimaringales, and we ge he resul. (ii) Le Λ be a subordinaor. From Example we know ha B has he Lévy riple (β Y, A Y, µ Y ) given by (2.6) and by he Lévy-Iô decomposiion we have db = β Y d + A Y dw + xn(d, dx) + xñ(d, dx) x 1 x <1 where W = {W, [, T ]} is a sandard Brownian moion. The dynamics of B can be rewrien in a simpler form, in he case when B is square inegrable (see p.133 Appelbaum (24)). We have E[B 2 ] = E[E[B 2 F Λ T ]] = E[Λ ] < So db = β Y d + A Y dw + xn(d, dx) R\{} 1 See Appendix A.5

27 2.4. THE MARKET MODEL 21 where Le where β Y = β Y xµ Y (dx) x <1 ds = (α r )S d + σ S db = S dx dx = (α r )d + σ db Assume inf{ X, [, T ]} > 1 a.s.. Then by Proposiion A.5.4, where G = (α r ) + β Y F = σ A Y H(, x) = σ x we ge ha he soluion of he K-semimaringale S is he Doléans-Dade exponenial of X, E X (see Appendix A.5.3).

28 Chaper 3 The Local Risk Minimizing sraegy In an incomplee marke perfec hedging is in general no possible. Local risk-minimizing (LRM) is a way o rea he problem of hedging in such markes. The main feaures of he LRM sraegy is finding a porfolio ϕ ha is no self-financing such ha he discouned value process a ime of mauriy T is equal o he discouned coningen claim 1, i.e. V T (ϕ) = F. This we can do by allowing a small cos process. Then he local risk minimizaion aims o minimize such cos. This measure of riskiness by a quadraic crierion of a sraegy was firs described by Föllmer and Sonderman (1986) in he case where he discouned price process is a maringale, and laer exended by Schweizer (1988) o he case of general semimaringales. Denoe ϕ = (η, ξ) o be he porfolio, where η is he number of unis in he riskless asse and ξ is he number of unis in he risky asse. Define he sochasic process V o be he discouned value of he porfolio a ime, given by V (ϕ) = η + ξ S, [, T ] (3.1) Risk Minimizing If he discouned asse S is a local K-maringale, we wan o find an admissible hedging sraegy for he coningen claim F. 1 In his marke we consider a coningen claim whose payoff is given by an F T - measurable random variable F such ha E[ F 2 ] < (or equivalen F L 2 (Ω, F T, P )). 22

29 The porfolio ϕ is a K-rading sraegy if ξ is a K-predicable processes such ha ξ L 2 K (S)2 and η is a K-adaped process such ha he value process V (ϕ) is righ-coninuous and E[V (ϕ) 2 ] <. Definiion Define he cos process by C (ϕ) := V (ϕ) such ha i is righ-coninuous and square-inegrable. Remark. The erm ξ uds u is called gain process. 23 ξ u ds u (3.2) Thus he cos process represens he difference beween he value process of an K-rading sraegy and he self-financing evaluaion of he gain process. A K-rading sraegy ϕ = (η, ξ) is called K-self-financing if he cos process is consan over ime. I is called K-mean self-financing if he cos process is a K-maringale under P. Since we wan o minimize he cos process, we define he K-risk process by R (ϕ) := E[(C T (ϕ) C (ϕ)) 2 K ] The idea behind risk minimizing is o look among all rading sraegies wih V T (ϕ) = F for he one which minimizes he risk process. Definiion An K-rading sraegy ϕ is called K-risk-minimizing if for any K-rading sraegy ϕ such ha V T ( ϕ) = V T (ϕ) P-as, we have R (ϕ) R ( ϕ) P-as for every [, T ] 2 The space L 2 K (S) denoes all K-predicable processes ξ such ha (E[ ξ2 d S K ]) 1 2 <.

30 24 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY 3.1 The Local Risk Minimizing problem In he case where he discouned price process S is no a local K-maringale, bu a K-semimaringale, Schweizer (1988) has proved ha he coningen claim F does no admis a risk-minimizing hedging sraegy. Therefore Schweizer exended he heory o LRM sraegies. Before inroducing he formal definiion of local risk minimizing under boh he filraion G and F, we inroduce some useful conceps where we can apply all hree filraions G, H and F. Assumpion. We will now assume ha he ime-change process Λ is absoluely coninuous, i.e. Λ = λ sds. Hence he discouned price process S is coninuous (recall Proposiion (i)). From he semimaringale decomposiion of Definiion A.3.6, we consider he wo processes A K = {A, [, T ]} and M K = {M, [, T ]}, where A K is K-predicable of finie variaion wih A K =, and M K is a locally square inegrable local K-maringale, wih M K =, such ha S can be wrien as S = S + M K + A K (3.3) We say ha he semimaringale S saisfies he srucure condiions under he filraion K (for shor (SC) K ) if here exiss a K-predicable process θ = {θ, [, T ]} such ha da K = θ d M K and We define he mean-variance radeoff process by K = θ 2 d M K < P-a.s. θ 2 sd M K s We need o define he space of K-predicable processes φ such ha Θ(K) = {φ K-predicable : E[ φ 2 d M K + ( φ u da ) 2 ] < } where he processes M and A are he decomposiion of he K-semimaringale S. When ξ Θ(K), he sochasic inegrals in (3.2) is well defined.

31 3.1. THE LOCAL RISK MINIMIZING PROBLEM The LRM under G, via MMM Consider he filraion G, i.e. full informaion of he ime change process. In his case he discouned price process S is a coninuous G-semimaringale. The main maerial and he general definiions in his secion are aken from [FS91] and [Sch99] if no oherwise specified. Recall he discouned price process, given by (2.15), ha is S = S + S s (α s r s )ds + S s σ s db s Moreover, S admis he G-semimaringale decomposiion (3.3), i.e., S = S + M G,P + A G,P By comparing he wo equaions above, we obain Then A G,P := M G,P := S s (α s r s )ds S s σ s db s (3.4) d M G,P G = (σ S ) 2 d B G = σ 2 S 2 dλ For simpliciy we denoe quadraic variaion by = G and also A G,P = A P, M G,P = M P for he res of his secion. If S saisfies he (SC) G, here exiss a G-predicable process θ such ha (α r )S d = σ 2 S 2 θ dλ Recall he previous assumpion on he marke model, σ P d a.e., he soluion of S given by Proposiion (i), is nonnegaive, so S > for all [, T ] and Λ = λ sds > for (, T ], Λ =. Then we obain, for [, T ] θ = α r σ 2 S λ (3.5) wih θ = and (θ σ S ) 2 dλ = (α r ) 2 d < P-a.s. σ 2 λ Noaion. Le X be a K-semimaringale and H a bounded K-predicable process, hen we denoe he Iô-ype he sochasic process H X by H X = ( H s dx s ) [,T ]

32 26 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY Definiion The porfolio ϕ = (η, ξ) is a G-rading sraegy if ξ is a G-predicable process, ξ Θ(G) and η is G-adaped such ha V = ξs + η has righ-coninuous pahs and E[V (ϕ) 2 ] < for every [, T ]. A G-rading sraegy ϕ is called G-local risk-minimizing if he remaining risk R (ϕ) is minimal under all infiniesimal perurbaions of he sraegy a ime. Definiion A rading sraegy = (δ, ɛ) is called a small perurbaion if i saisfies he following condiions (i) δ is bounded (ii) δ sda P s is bounded (iii) δ T = ɛ T = and for any (s, ] [, T ], (s,] := (δ1 (s,], ɛ1 [s,) ) Definiion Le ϕ be a rading sraegy, a small perurbaion and τ a pariion of [,T], and define r τ (ϕ, ) := i, i+1 τ R i (ϕ + (i, i+1 ]) R i (ϕ) E[ M P i+1 M P i G ] 1 ( i, i+1 ]() Then ϕ is called locally risk-minimizing if lim inf n rτn (ϕ, ) P-a.e. on Ω [, T ], for every as defined above and lim n τ n = [, T ]. The above definiion can be shown o be equivalen o he following properies of he associaed cos process (see [FS91]). Recall, if he cos process is a (G, P )-maringale, hen he G-rading sraegy ϕ is mean-self-financing. Theorem Assume S saisfies he (SC) G and E[K T ] <. Le F be a coningen claim in L 2 (Ω, F T, P ) and le ϕ be a G-rading sraegy. Then ϕ is a G-locally risk-minimizing sraegy if and only if ϕ is mean-selffinancing and he maringale C(ϕ) is orhogonal o M P. Definiion The G-rading sraegy ϕ is called a G-opimal sraegy if ϕ is mean-self-financing and he maringale C(ϕ) is orhogonal o M P.

33 3.1. THE LOCAL RISK MINIMIZING PROBLEM 27 To find a soluion o he LRM problem, Föllmer and Schweizer inroduced in [FS91] he Föllmer-Schweizer (FS) decomposiion. Proposiion Le F L 2 (Ω, F T, P ), hen F admis a G-opimal sraegy ϕ, wih V T (ϕ) = F if and only if F admis he following decomposiion, F = F + ξ F S ds + L F T S, P-a.s. (3.6) where F L 2 (Ω, F Λ T, P ), ξf S Θ(G) and L F S is a G-maringale orhogonal o M P, wih L =. The sraegy ϕ = (η, ξ) is given by ξ = ξ F S, C (ϕ) = F + L F S wih V (ϕ) = F + ξ F S ds + L F S The decomposiion (3.6) is called he FS decomposiion of F Since he G-semimaringale S is coninuous, he Föllmer-Schweizer decomposiion of F under P can be obained as he Galchouk-Kunia-Waanabe (GKW) decomposiion (see Appendix A.6) under he minimal maringale measure (MMM). This comes from he fac ha he minimal maringale measure "preserves orhogonaliy" when S is coninuous. By preserving orhogonaliy we mean ha given any square inegrable maringale under P orhogonal o he maringale par of S, M P, i is sill a maringale under MMM, orhogonal o S. Remark. If he semimaringale S is disconinuous he only way o obain he LRM sraegy is by calculaing direcly he FS decomposiion as in Theorem in [Van1]. Minimal Maringale Measure We exend he heory on MMM from Chaper 3 in [FS91], o our framework. Tha is, o consruc a MMM Q, for a given coninuous marke model wih a discouned price, we inroduce he Radon-Nikodym densiy process Z ha describes a change of measure from P o an equivalen maringale measure Q. Such change of measure applied o Lévy noises is srucure preserving (e.g. see Di Nunno and Karlsen [DK16]). Then we will show ha for he

34 28 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY equivalen maringale measure Q wih Radon-Nikodym densiy process Z, here exiss a G-predicable process θ such ha da P = θd M P Finally we find a explici soluion o he densiy process Z, such ha he associaed equivalen maringale measure Q becomes minimal. Recall ha for an equivalen maringale measure Q wih respec o P, we can define he process Z = {Z, [, T ]} by and, for all [, T ], Z T = dq dp Z = E[ dq dp G ] which is posiive (G, P )-maringale and i is he so-called Randon-Nikodym densiy process. Theorem (Girsanov Theorem). R Le S be a G-semimaringale under P wih decomposiion S = S + M P + A P, and le B be he imechanged (G,P)-Brownian moion. Le Z = {Z, [, T ]} be a posiive (G,P)-maringale wih E[ZT 2 ] <, and define he probabiliy measure Q by Define he processes dq dp = Z T M Q := M P B Q := B 1 Z s d Z, M P s 1 Z s d Z, B s Then M Q is a coninuous (G, Q)-maringale, and B Q is a coninuous (G,Q)- maringale and a ime-changed (G,Q)-Brownian moion. Proof. We know ha M P and Z are (G,P)-maringales. By inegraion by par we have, for all [, T ], d(m Q Z ) = M Q dz + Z dm Q + dm Q dz = M Q dz + Z dm P

35 3.1. THE LOCAL RISK MINIMIZING PROBLEM 29 So M Q Z is a (G,P)-maringale, hence M Q is a (G,Q)-maringale. In he same way we can show ha B Q is a (G,Q)-maringale. We have, for all [, T ], d(z B Q ) = B Q dz + Z db Hence B Q is a coninuous (G,Q)-maringale. Also B Q = B = Λ, for all [, T ] Hence since B Q is a coninuous (G,Q)-maringale and since B Q = Λ, by Proposiion 2.3.5, B Q is a ime-changed (G,Q)-Brownian moion. In wha follows we wan o deermined he Radon-Nikodym densiy process Z such ha Q is a minimal maringale measure. By Girsanov s heorem, we have, for all [, T ], S = S + M P + A P = S + M Q + 1 Z s d Z, M P s + A P Since Q ia an equivalen maringale measure, S is a (G, Q)-maringale, hence A P = 1 Z s d Z, M P s From Kunia-Waanabe inequaliy (see Proposiion [Pha9]) we obain ha Z, M P M P. Tha is, for a bounded process α = {α, [, T ]}, α s d Z, M P s Z α s d M P s so if α sd M P s =, hen α sd Z, M P s =, for all [, T ]. Hence by Radon-Nikodym heorem, here exiss a G-predicable process β = {β, [, T ]} such ha d Z, M P = βd M P Moreover, le he G-predicable process θ = {θ, [, T ]}, be given by θ = β Z such ha da P = θd M P (3.7)

36 3 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY Remark. Le he G-predicable process φ Θ(G), hen noe ha E[ φ 2 d M P ] E[ φ 2 d M P + ( φ da ) 2 ] < Moreover, E[ φ2 d M P ] < implies φ 2 d M P <, P-a.s. Now suppose he predicable process in (3.7) is in he space Θ(G), which implies Then S saisfies he (SC) G. θ 2 sd M P s < P-a.s. Definiion Suppose S saisfies (SC) G, and le Q be he equivalen maringale measure, ha is dq dp L2 (Ω, F T, P ). Then Q is called minimal if (i) Q = P on G, where G = FT Λ (ii) for any square inegrable (G, P )-maringale L, wih L =, orhogonal o M P, L is sill a maringale under Q. Moreover also orhogonaliy is preserved in his case, namely any square inegrable (G,P)-maringale, orhogonal o M P under P is also orhogonal o S under he minimal maringale measure (see e.g. p.22 [Van1]). Theorem Suppose Q is a minimal maringale measure. Then Q exiss if and only if Z = exp( θ dm P θ 2 d M P ) (3.8) is a square-inegrable maringale under P (in ha case, Q is given by dq dp = Z T ), where he G-predicable process θ is given by (3.5). Moreover, Q preserves orhogonaliy, i.e. any maringale under P orhogonal o M P is also a maringale under Q orhogonal o S. Proof. Suppose Q exiss, and le S = S + M P + A P be he Doob Meyer decomposiion under (G, P ). Le Z be a square-inegrable (G, P )-maringale associaed o he equivalen maringale measure, hen by he GKW decomposiion under P, Z = Z + β s dm P s + L

37 3.1. THE LOCAL RISK MINIMIZING PROBLEM 31 where L is a square-inegrable maringale under P orhogonal o M P, L = and β Θ(G). This gives 1 d Z Z, M P s = s By Girsanov s Theorem, he process 1 Z s β s d M P s M P 1 Z s d Z, M P s = M P 1 Z s β s d M P s is a (G, Q)-maringale. On he oher hand, we know ha S is a maringale under Q. Define Since Z > and β Θ(G), we have which implies E[ θ = 1 Z β ( 1 Z β ) 2 d M P ] E[ β 2 d M P ] < θ 2 sd M P s <, P a.s. From he definiion of he MMM, condiion (i) gives ha Z = 1 and condiion (ii) says ha L is a maringale under Q, so L, Z = and hence L. Thus Hence Z = Z. Z = 1 + Z s θ s dm P s Now assume Z is given by (3.8) and Z T = dq. We wan o show ha Q is dp minimal. Le L be a square-inegrable (G, P )-maringale wih L, M P = under P. We hen ge L, Z =, since for all [, T ], L, Z = Z s θ s d L, M P Thus L is a local maringale under Q. Moreover, since sup L L 2 (Ω, F, P ) T

38 32 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY and Z L 2 (Ω, F, P ), we have sup L L 1 (Ω, F, Q) T So L is a local Q-maringale and E Q [sup T L ] < for every [, T ], hence L is a maringale under Q. Now we wan o show ha L also saisfies L, S = under Q. Since S and A P are coninuous, we have L, S = [L, S] = [L, M P ] + [L, A] = [L, M P ] under Q. Bu M P is coninuous, so we have [L, M P ] = L, M P = under P. This implies ha [L, M P ] = also under Q (see he proof of Theorem 1 in [FS91]). Le us reurn o he problem of compuing he G-opimal sraegy (see Definiion 3.1.5). Define he process ˆV = { ˆV, [, T ]} by ˆV := E Q [F G ], [, T ] (3.9) where he noaion E Q [ G ] denoes he condiional expecaion wih respec o G under Q. Since F L 2 (Ω, F T, P ) and dq dp L2 (Ω, F T, P ), we obain ha F L 1 (Ω, F T, Q), hus he process V Q is well-defined. The process ˆV admis he following GKW decomposiion under S and Q, ˆV = ˆV + ˆξ u ds u + ˆL, Q-a.s. (3.1) where ˆV L 2 (Ω, F Λ T, Q), ˆξ L 2 G (S) and ˆL is a (G,Q)-maringale orhogonal o S under Q wih ˆL =. Theorem Suppose S saisfies (SC) G. Le ˆV be given by (3.9) and le Q be he MMM. If (i) F admis he FS decomposiion, given by (3.6) or

39 3.1. THE LOCAL RISK MINIMIZING PROBLEM 33 (ii) ˆξ Θ(G) and ˆL is a (G,P)-maringale orhogonal o M P under P, hen (3.1), for =T, gives he FS decomposiion of F and ˆξ deermines he G-opimal sraegy for F. Proof. Assume (i). Since he (G, P )-maringale L F S of he FS decomposiion is orhogonal o M P under P, and Q is he MMM, L F S is a (G, Q)-maringale orhogonal o S under Q. Then by projecion of G under Q Hence E Q [F G ] = F + ˆV = F + ξu F S ds u + L F S, Q-a.s. (3.11) ξu F S ds u + L F S, P-a.s. (3.12) By he uniqueness of he GKW decomposiion (3.12) coincides wih (3.1). Now, assuming (ii). Then clearly (3.1) is given for P-a.s., and for = T, i admis he FS decomposiion of F. Hence, for boh (i) and (ii), by Proposiion 3.1.6, we ge he G-opimal sraegy ϕ = (η, ξ), given by ξ = ξ, η = V ξ S In a summary, we have shown ha finding he LRM sraegy is deduce o finding he GKW decomposiion of he claim F under he MMM Q. This is very useful since he densiy process Z of Q wih respec o P, given by (3.8), can be wrien explicily in erms of he dynamics of he price process S, given ha (SC) G is saisfied. Moreover, by he unique soluion of he porfolio we can obain he cos process (recall Definiion 3..1). Tha is, he associaed cos process, for all [, T ], is given by C = V (ϕ) ξ s ds s = V + L F S where V is a FT Λ -measurable random variable and L is a G-maringale, boh orhogonal o M P.

40 34 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY Now we can see ha he local risk minimizing sraegy wih respec o he filraion G, is "in a sense complee ". To see his, le use firs consider he case when we are in a complee marke and S is a G-maringale, hen he claim F admis an Iô represenaion F = F + ξs F S ds s where F is G -measurable orhogonal o he inegrand and ξ F Then he sraegy is given by L 2 G (S). ξ = ξ F S, η = V ξ S, V = F + ξ F s ds s Moreover, he sraegy is self-financing, i.e. he cos process is known from he sar since i is G -measurable, C = C T = F Thus, his represenaion leads o a sraegy which produces he claim H from he iniial amoun C = F. Now back o he LRM wih respec o G. Recall ha G = FT Λ informaion. We have F = F + ξs F S ds s + L F T S is he iniial where F is a G (= F Λ T )-measurable random variable and LF a G-maringale orhogonal o S. The sraegy is ξ = ξ F S, η = V ξ S wih V = F + ξs F S ds s + L F S On he oher hand, his means ha C = F + L F η = F + L Hence in realiy since η is he asse of he riskless asse, his means ha he maringale L F vanishes. So his gives C = C T = F

41 3.1. THE LOCAL RISK MINIMIZING PROBLEM 35 where F is a G (= FT Λ)-measurable random variable. Bu F is a known process a ime =, since we have knowledge of he whole ime change process. Hence, he sraegy is in a sense self-financing, ha is, he cos processes is a know process, hence he marke is in a sense complee under he filraion G. Explici soluion R In a special case where we can wrie he value process of a porfolio ϕ in he form V (ϕ) = f(, S ) =: V for a funcion f(, x) C 1,2 ([, T ] R), we can find explici soluion for V and he soluion of he associaed porfolio ϕ = (η, ξ). Proposiion Le F be he coningen claim, F L 2 (Ω, F T, P ). Then he local risk minimizing sraegy ϕ = (η, ξ) is given by where ξ = f x (, S ) + ζ η = f(, S ) ξ S L = ζ S Proof. Since f(, x) C 1,2 ([, T ] R) we can use Iô s formula V = V + f (u, S u)du + f x (u, S u)ds u f x 2 (u, S u)s 2 uσ 2 uλ u du Moreover, V is a (G, Q)-maringale, so we can deduce ha f saisfies a parial differenial equaion (PDE) and V = V + f x (u, S u)ds u Hence from he GKW decomposiion of V under Q, we ge ξ = f x (, S ) + ζ L = ζ S Using Iô s formula, we deduce ha he funcion f(, x) is he soluion o he following PDE { f (, S ) f (, S 2 x 2 )S 2 σ 2 λ = (3.13) f(t, S T ) = F

42 36 CHAPTER 3. THE LOCAL RISK MINIMIZING STRATEGY Example If we have a European pu opion, wih sick price K and payoff funcion F = (K S T ) +, hen f(, x) = E Q [(K x S T S ) + ]

43 3.1. THE LOCAL RISK MINIMIZING PROBLEM The LRM under F, via MMM We consider he local risk minimizing sraegy in he case of F, i.e. he filraion generaed by he ime-changed Brownian moion. We will use a parial informaion approach. This approach is discussed by he auhors in [CCR14a] and [CCR14b]. The main idea in he papers by Ceci e al. (214), is ha for he wo filraions H and F, F H, for all [, T ], we can for he H-maringales M and L and for F L 2 (Ω, H T, P ), find a GKW decomposiion of F, i.e. F = F + ξ s dm s + L T, P-a.s. where ξ is a F-predicable process. This is achieved by projecing he resuls obained for he H-predicable case ono he F-predicable one, using he so-called predicable dual projecion. Moreover, we will also see ha he quadraic variaion B F can be obained by projecion Λ. Under parial informaion, given he porfolio ϕ = (η, ξ), we have ha ξ is F-adaped and η is H-adaped. This means ha he invesors can only access he informaion flow F abou rading in he risky asse. From his, he echnical definiions on he porfolio ϕ = (η, ξ) such as rading and opimal sraegies, will depend on boh he wo filraions H and F. For example we can apply parial informaion when working wih incomplee informaion credi risk models where invesors may have a delayed observaion of he process driving he defaul risk. Then F = H ( τ) + where τ (, T ) is a fixed delay. In he papers [CCR14a] and [CCR14b] he auhors aim a providing he exisence and uniqueness of he BSDEs under parial informaion, while in his secion, we will apply his heory o find he LRM sraegy wih a measure change. Tha is, under he MMM wih respec o H, we will find a GKW decomposiion under parial informaion of he claim F. We recall ha he coningen claim F L 2 (Ω, F T, P ). Under he filraion F he claim F only admis he FS decomposiion and he GKW decomposiion under parial informaion, herefore we will see ha he FS decomposiion under P is no equivalen o he GKW decomposiion under a MMM. We will sill provide he GKW decomposiion under parial informaion under he MMM Q, and will herefore need o assume ha he claim F L 2 (Ω, F T, P ) L 2 (Ω, F T, Q). We need o clarify ha, even hough, we apply he parial informaion approach, our model is compleely differen han in he papers [CCR14a]