Change of measure and Girsanov theorem

Transcription

1 and Girsanov heorem Sochasic calculus I Geneviève Gauhier HEC Monréal

2 Example 1 An example I Le (Ω, F, ff : 0 T g, P) be a lered probabiliy space on which a sandard Brownian moion W P = W P : 0 T is consruced. The sochasic process S = fs : 0 T g represens he evoluion of a risky securiy price and sais es he sochasic di erenial equaion ds = µs d + σs dw P. Le s also assume ha he ineres rae r is consan. discoun facor is herefore The β () = exp ( r) which implies ha d β () = r exp ( r) d.

3 Example 1 Le s se, for all 0 T, Y = β S An example II i.e. Y represens he presen value a ime of he risky securiy. Using Iô s lemma (more precisely he muliplicaion rule), we obain dy = (µ r) Y d + σy dw P. Indeed, dy = d β S = β ds + S d β + d hβ, Si = β µs d + σs dw P + S ( r β d) = (µ r) β S d + σβ S dw P.

4 An example III Example 1 In is inegral form, such a sochasic di erenial equaion becomes Z Z Y = Y 0 + (µ r) Y s ds + σ Y s dws P. 0 0

5 Example 1 Refresher Iô process Le W P be a (ff g, P) Brownian moion. An Iô process is a process X = fx : 0 T g aking is values in R such ha: Z Z X X 0 + K s ds + H s dws P 0 0 wih K = fk : 0 T g and H = fh : 0 T g, processes h adaped o he lraion ff g, R i T P h 0 jk s j ds < = 1 R i T P 0 (H s ) 2 ds < = 1 Damien Lamberon and Bernard Lapeyre, Inroducion au calcul sochasique appliqué à la nance, Ellipses, page 53.

6 Example 1 Recall ha W P is a (ff g, P) Example (suie) I Brownian moion. In a risk-neural world (Ω, F, ff : 0g, Q), he sochasic process Y = fy : 0 T g should be a (ff g, Q) maringale. Thus, under he risk-neural, he rend of Y should be nil, i.e. we wan he drif coe cien o be 0.

7 Example 1 Le s se and noe ha W Q Example (suie) II Z = W P + γ s ds 0 1 W Q is no a P maringale (is expecaion varies in ime) and 2 dw Q = dw P + γ d. As a consequence Z Z Y = Y 0 + (µ r) s ds + σ s dws P 0 0 Z Z Y = Y 0 + (µ 0 r σγ s ) Y s ds + σ Y s dws Q. 0 In order o ge rid of he drif erm, i is su cien o se µ r σγ s = 0, γ s = µ r σ.

8 Example 1 Recall ha Example (suie) III Z Y = Y 0 + σ Y s dws Q 0 Noe ha, under he P, he process W Q is no a sandard Brownian moion since he law of W Q under he µ P is N,. r σ The process Y will no be a (ff g, P) maringale since he sochasic inegral is consruced wih respec o W Q which is no a (ff g, P) maringale. Indeed, varies in ime. h i E P W Q = µ r σ

9 Example 1 Recall ha W P is a (ff g, P) Example (suie) IV Z Y = Y 0 + σ Y s dws Q 0 Brownian moion, where W Q () = W P () + µ r σ. So we wan o nd he probabiliy Q o be placed on he space (Ω, F, ff g) such ha W Q is a Q sandard Brownian moion. By changing he probabiliy on he se Ω, we ransform he drif coe cien so ha he rend becomes zero and we inegrae wih respec o a (ff g, Q) maringale. As a resul, he process Y will be (ff g, Q) maringale.

10 heorem I A way o consruc new probabiliy s on he measurable space (Ω, F) when we already have a probabiliy P exising on ha space is as follows: Le Y be a random variable consruced on he probabiliy space (Ω, F, P) such ha 8ω 2 Ω, Y (ω) 0 and E P [Y ] = 1. For all even A 2 F, δ A denoes he indicaor funcion of ha even: 1 if ω 2 A δ A (ω) = 0 oherwise. For all even A 2 F, le s se Q (A) = E P [Y δ A ]. Then Q is a probabiliy on (Ω, F).

11 heorem II Proof. We mus verify ha (P1) Q (Ω) = 1, (P2) 8A 2 F, 0 Q (A) 1, (P3) 8A 1, A 2,...2 F such ha A i \ A j = si i 6= j, Q S i1 A i = i1 Q (A i ).

12 heorem III Veri caion of (P1). Bu, since for all ω, δ Ω (ω) = 1 and because we have assumed ha E P [Y ] = 1, Q (Ω) = E P [Y δ Ω ] = E P [Y ] = 1, which esablishes condiion (P1).

13 heorem IV Veri caion of (P2). The second condiion is jus as easy o prove: since Y is a posiive random variable, Y δ A is a posiive random variable oo, and Q (A) = E P [Y δ A ] 0. Moreover, Q (A) = E P [Y δ A ] E P [Y δ Ω ] = E P [Y ] = 1.

14 heorem V Veri caion of (P3). As we have esablished in an exercise in he rs chaper, 8A 1, A 2,...2 F such ha A i \ A j = if i 6= j, As a consequence, δ S i1 A i! Q [ A i i1 = δ Ai. i1 = E P h Y δ S i1 A i i # = E "Y P δ Ai i1 = E P [Y δ Ai ] i1 = Q (A i ). i1

15 heorem VI De niion Two probabiliy s P and Q consruced on he same measurable space (Ω, F) are said o be equivalen if hey have he same se of impossible evens, i.e. P (A) = 0, Q (A) = 0, A 2 F.

16 heorem VII Quesion. Given wo equivalen probabiliy s P and Q, does here exis a non-negaive valued random variable Y such ha Q (A) = E P [Y δ A ]? Noe he di erence beween such a problem and he resul we have jus proven. In he laer, Y and P were given o us and we have consruced Q. In his case, P and Q are given o us and we need o nd Y, which is less easy. The exisence of such a variable is esablished in he nex heorem which is a version of he famous heorem.

17 heorem VIII Theorem heorem. Given wo equivalen probabiliy s P and Q consruced on he measurable space (Ω, F), here exiss a posiive-valued random variable Y such ha Q (A) = E P [Y δ A ]. Such a random variable Y is ofen denoed by d Q d P. Such a heorem sill does no ell us how o nd our risk-neural. Acually, i is he nex resul ha will provide us wih he recipe o consruc our and i involves he derivaive.

18 heorem IX A few houghs abou he discree case Assume ha Ω only conains a nie number of elemens. Le Y = β T X be he presen value of he aainable coningen claim X. Si F 0 = fω, g, hen is price a ime = 0 is E Q [Y ] = Y (ω) Q (ω) ω2ω = Y (ω) Q (ω) P ω2ω (ω) P (ω) = E P Y Q P

19 heorem X Consider he binomial marke model: S (1) represens he evoluion of he riskless asse and S (2) models a risky asse. The unique risk-neural is denoed by Q, P being he real. ω S (1) 0 (ω) S (2) 0 (ω)! S (1) 1 (ω) S (2) 1 (ω)! S (1) 2 (ω) S (2) 2 (ω)! P Q d Q d P ω 1 (1; 2) 0 (1, 1; 2) 0 (1, 21; 1) , 360 1, 44 ω 2 (1; 2) 0 (1, 1; 2) 0 (1, 21; 3) , ω 3 (1; 2) 0 (1, 1; 4) 0 (1, 21; 1) , 015 0, 06 ω 4 (1; 2) 0 (1, 1; 4) 0 (1, 21; 5) , 34 The Nykodym derivaive is somewha he memory of he change of. For each pah, i remembers how we have changed weighs.

20 Girsanov heorem I Example 1 Le s focus on a bounded ime inerval: 2 [0, T ]. Le W = fw : 2 [0, T ]g represen a Brownian moion consruced on a lered probabiliy space (Ω, F, ff g, P) such ha he lraion ff g is he one generaed by he Brownian moion, plus i includes all zero-probabiliy evens, i.e. for all 0, F = σ (N and W s : 0 s ). The nex heorem will enable o consruc our risk- neural s.

21 Example 1 Girsanov heorem II Theorem Cameron-Marin-Girsanov heorem. Le γ = fγ : 2 [0, T ]g be a ff g predicable process such ha Z 1 T E exp P γ 2 2 d <. 0 There exiss a Q on (Ω, F) such ha (CMG1) Q is equivalen o P h (CMG2) d Q d P = exp R T 0 γ dw R i 1 T 2 0 γ2 d n o (CMG3) The process fw = W f : 2 [0, T ] de ned as fw = W + R 0 γ s ds is a (ff g, Q) Brownian moion. (ref. Baxer and Rennie, page 74; Lamberon and Lapeyre, page 84)

22 Girsanov heorem III Example 1 The condiion E hexp P 1 R i T 2 0 γ2 d < is a su cien bu non-necessary condiion. I is know as he Novikov condiion.

23 Girsanov heorem IV Example 1 Consider he sochasic di erenial equaion dx = b (X, ) d + a (X, ) dw where W represens a Brownian moion on he lered probabiliy space (Ω, F, ff g, P). We assume ha he drif and di usion coe ciens are such ha here exiss a unique soluion o he equaion, which we denoe X. We wan o nd a probabiliy Q, such ha, on he space (Ω, F, ff g, Q), he drif of X is eb (X, ) insead of b (X, ).

24 Example 1 Le s go! dx = b (X, ) d + a (X, ) dw = eb (X, ) d + a (X, ) Girsanov heorem V b (X, ) eb (X, ) a (X, ) +a (X, ) dw provided ha a (X, ) is di eren from 0.! Z b (X s, s) eb (X s, s) = eb (X, ) d + a (X, ) d W + ds 0 a (X s, s)! d = eb (X, ) d + a (X, ) d fw where Z fw = W + 0 γ s ds and γ = b (X, ) eb (X, ). a (X, )

25 Example 1 Girsanov heorem VI If E hexp P 1 R i T 2 0 γ2 d < hen by he and Cameron-Marin-Girsanov heorems, Z T Z Q (A) = E exp P 1 T γ dw γ d δ A, A 2 F 0 n o and fw = W f : 2 [0, T ] is a (F, Q) Brownian moion. In pracice, we don need o deermine he Q. I is su cien for us o know i exiss, and o know he sochasic di erenial equaion of he process of ineres on he space (Ω, F, ff g, Q).

26 Example 1 I Girsanav heorem Example 1 Le s go back o he Black-Scholes marke model. The sochasic process Y = fy : 0 T g consruced on he space (Ω, F, ff g, P) used o consruc he Brownian moion represens he evoluion of he presen value of a risky securiy where dy = (µ r) Y d + σy dw P.

27 Example 1 Example 1 II Girsanav heorem Bu, in a risk-neural world (Ω, F, ff g, Q), he rend of Y should be zero, i.e. we wan he drif coe cien o be zero. Thus dy = (µ r) Y d + σy dw P µ r = σy d + σy dw P = σy d σ W P + µ r σ = σy dw Q where W Q W P + µ r σ Z = W P + 0 µ r σ ds.

28 Example 1 III Girsanav heorem Example 1 In he presen case, 8s, γ s = µ r σ.

29 Example 1 IV Girsanav heorem Example 1 Recall he Cameron-Marin-Girsanov heorem. Le γ = fγ : 2 [0, T ]g be a ff g predicable process such ha Z 1 T E exp P γ 2 2 d <. 0 There exiss a Q on (Ω, F) such ha (CMG1) Q is equivalen o P h (CMG2) d Q d P = exp R T 0 γ dw R i 1 T 2 0 γ2 d n o (CMG3) The process fw = W f : 2 [0, T ] de ned as fw = W + R 0 γ s ds is a (ff g, Q) Brownian moion.

30 Example 1 Example 1 V Girsanav heorem Le s verify ha he condiion on he process γ is indeed sais ed: Z 1 T E exp P γ 2 2 d 0 Z = E "exp P 1 T µ r 2 d!# 2 0 σ! 1 µ r 2 = exp T <. 2 σ

31 Example 1 Le s apply Girsanov heorem : Z dq T dp = exp γ dw P 0 " Z T µ r = exp σ = exp " 0 µ r σ W T P dw P Example 1 VI Girsanav heorem Z 1 T γ 2 2 d 0 Z T µ 1 µ 2 1 r 2 d# 2 0 σ # r 2 T. σ This implies ha Q [A] = E P "exp µ r σ W T P 1 µ 2 σ! # r 2 T δ A, A 2 F.

32 Example 1 Example 1 VII Girsanav heorem Moreover, under he Q, he evoluion of he presen value of he risky securiy sais es he equaion where W Q is a Q dy = σy dw Q Brownian moion. We can also deduce he sochasic di erenial equaion sais ed by he evoluion of he risky securiy price S: ds = µs d + σs dw P = µs d + σs d W Q µ r σ puisque W Q W P + µ r σ = µs d + σs dw Q µ r σs σ = rs d + σs dw Q. d

33 Example 1 VIII Girsanav heorem Example 1

34 Example 1 Example 1 IX Girsanav heorem Noe ha we don really need o calculae Q, we simply need o know ha i exiss hen oesablish wha is he equaion sais ed by he processus of ineres, i.e. he evoluion of he risky securiy price. Indeed, on (Ω, F, ff g, Q), where fw is a Q ds = rs d + σs dw Q Brownian moion. Bu he unique soluion o ha equaion is S = S 0 exp r σ 2 + σw Q. 2

35 Example 1 Example 1 X Girsanav heorem Since he price of a call opion, he srike price of which is K and he mauriy of which is T, is given by E Q [exp ( rt ) max (S T K ; 0)] = E exp Q ( rt ) max S 0 exp r = E max Q σ 2 S 0 exp = Z max S 0 exp 2 T + σw Q T σ 2 2 T + σz σ 2 2 T + σw Q T K exp ( rt ) ; 0 Ke rt ; 0 f Z (z) dz. K ; 0 where f Z () represens he probabiliy densiy funcion of a normal random variable wih zero expecaion and variance T. The res of he calculaion is a pure applicaion of he properies of he normal law.

36 Example 1 Since σ 2 S 0 exp 2 T + σz > Ke rt Example 1 XI Girsanav heorem, σ2 Ke rt T + σz > ln = ln K 2 S 0 rt ln S 0, z > ln S σ 0 ln K + r 2 2 T σ p d 2 T

37 Example 1 = = Z Z Z = S 0 Z d 2 p T 1 p 2π 1 p T exp Example 1 XII Girsanav heorem σ 2 max S 0 exp 2 T + σz Ke rt ; 0 f Z (z) dz σ 2 p S 0 exp d 2 T 2 T + σz Ke rt f Z (z) dz σ 2 Z p S 0 exp d 2 T 2 T + σz f Z (z) dz p Ke rt f Z ( d 2 T z 2 2T σz + σ 2 T 2 dz Ke rt Z d 2 p T 1 p 2π 1 p T exp 2T z 2 2T dz

38 Example 1 = S 0 Z d 2 p T 1 p 2π 1 p T exp Example 1 XIII Girsanav heorem (z σt ) 2! dz 2T dz Z Ke rt 1 1 z 2 p p p exp d 2 T 2π T 2T Le s se u = z p σt and v = p z T T Z 1 u 2 = S 0 d 2 σ p p exp du T 2π 2 Z Ke rt 1 v 2 p exp dv d 2 2π 2 = S 0 1 N d 2 σ p T Ke rt (1 N ( d 2 ))

39 Example 1 XIV Girsanav heorem Example 1 where N () is he cumulaive disribuion funcion of a sandard normal random variable.

40 Example 1 XV Girsanav heorem Example 1 Bu he symmery of N implies ha 1 N (x) = N ( x). Then S 0 1 N d 2 σ p T Ke rt (1 N ( d 2 )) = S 0 N d 2 + σ p T Ke rt (N (d 2 ))

41 Girsanov (coninued) Example 3 I Le s assume ha W P and fw P represen wo sandard Brownian moions consruced on he lered probabiliy space (Ω, F, ff g, P). n Noe ha B e P : 0o where q eb ρw P + 1 ρ 2f W P is a sandard Brownian moion such ha Corr P W P, eb P = ρ. Exercise: prove i.

42 II Girsanov (coninued) Example 3 The insananeous exchange rae dc = µ C C d + σ C C dw P enables us o model he number of Canadian dollars per uni of foreign currency a any ime. Suppose also ha he sochasic di erenial equaion ds = µ S S d + σ S S d eb P = µ S S d + σ S ρs dw P + σ S q 1 ρ 2 S d fw P models he evoluion of a foreign risky asse price.

43 Girsanov (coninued) Example 3 III Lasly, he Canadian insananeous ineres rae r and he foreign insananeous ineres rae v are assumed o be consan. As a consequence, he discoun facor is β = exp ( r). and he value in foreign currency of an iniial invesmen equal o one foreign currency uni is B = exp (v).

44 IV Girsanov (coninued) Example 3 Le s pu ourselves in he shoes of a Canadian invesor. C S gives us he Canadian dollar value of a risky asse a ime, C B gives us he Canadian dollar value, a ime, of one foreign currency uni invesed in a foreign bank accoun U = β C S gives us he Canadian dollar presen value of he risky asse a ime. V = β C B gives us he Canadian dollar presen value, a ime, of one foreign currency uni invesed in a foreign bank accoun. We wish o nd a Q, such ha he sochasic processes U and V are (ff g, Q) maringales.

45 Girsanov (coninued) Example 3 Recall: V dc = µ C C d + σ C C dw P, db = vb d d β = r β d Firs, le s deermine he sochasic di erenial equaion sais ed by he Canadian dollar presen value of one foreign currency uni invesed in a foreign bank accoun V = βcb under he P. Iô s lemma allows us o wrie dc B = C db + B dc + d hb, C i = C (vb d) + B µ C C d + σ C C dw P = (v + µ C ) C B d + σ C C B dw P.

46 VI Girsanov (coninued) Example 3 Thus, dv = d β C B = β dc B + C B d β + d hβ, CBi = β (v + µ C ) C B d + σ C C B dw P +C B ( r β d) = (µ C + v r) β C B d + σ C β C B dw P = (µ C + v r) V d + σ C V dw P.

47 VII Girsanov (coninued) Example 3 Recall: ds = µ S S d + σ S ρs dw P dc = µ C C d + σ C C dw P, db = vb d, d β = r β d. + σ S q 1 ρ 2 S d fw P,

48 Girsanov (coninued) Example 3 VIII Second, le s deermine he sochasic di erenial equaion sais ed by U = βcs under he P. Iô s lemma allows us o wrie dc S = C ds + S dc + d hs, C i q = C µ S S d + σ S ρs dw P + σ S 1 ρ 2 S d fw P +S µ C C d + σ C C dw P + σ S ρs σ C C d = (µ S + µ C + σ S σ C ρ) C S d q + (σ S ρ + σ C ) C S dw P + σ 1 ρ 2 C S d fw P.

49 Girsanov (coninued) Example 3 Applying Iô s lemma again, we obain du = d β C S IX = β dc S + C S d β + d hβ, CSi (µ = β S + µ C + σ S σ C ρ) C S d + (σ S ρ + σ C ) C S dw P p + σ S 1 ρ 2 C S d fw P +C S ( r β d) = (µ S + µ C + σ S σ C ρ r) β C S d q + (σ S ρ + σ C ) β C S dw P + σ S 1 ρ 2 β C S d fw P = (µ S + µ C + σ S σ C ρ r) U d q + (σ S ρ + σ C ) U dw P + σ S 1 ρ 2 U d fw P.

50 Girsanov (coninued) Example 3 We have X dv = (µ C + v r) V d + σ C V dw P, du = (µ S + µ C + σ S σ C ρ r) U d + (σ S ρ + σ C ) U dw P which allows us o wrie + σ S q 1 ρ 2 U d fw P dv = (µ C + v r σ C γ ) V d Z +σ C V d W P + γ s ds 0 µ du = S + µ C + σ S σ C p ρ r (σ S ρ + σ C ) γ σ S 1 ρ 2 U eγ d Z + (σ S ρ + σ C ) U d W P + γ s ds 0 Z +σ S q1 ρ 2 U d fw P + eγ s ds. 0

51 Girsanov (coninued) Example 3 So, we wan o solve he linear sysem XI µ C + v r σ C γ = 0 µ S + µ C + σ S σ C ρ r (σ S ρ + σ C ) γ σ S q1 ρ 2 eγ = 0 he unknowns of which are γ and eγ. In marix form, we wrie σ C p 0 γ µ σ S ρ + σ C σ S 1 ρ 2 = C + v r. eγ µ S + µ C + σ S σ C ρ r The soluion is γ = µ C + v r σ C eγ = µ S v + σ S σ C ρ σ S p 1 ρ 2 ρ (µ C + v r) σ C p 1 ρ 2

52 Girsanov (coninued) Example 3 So, le s se W Q fw Q = fw P + R 0 eγ s We can hen wrie = W P + R 0 γ s ds where γ s = µ C + v r σ C eγ s = µ S v + σ S σ C ρ σ S p 1 ρ 2 XII ds and ρ (µ C + v r) p. σ C 1 ρ 2 dv = σ C V dw Q du = (σ S ρ + σ C ) U dw Q + σ S q 1 ρ 2 U d fw Q. Is i possible o nd a Q such ha W Q and fw Q are (ff g, Q) Brownian moions simulaneously?

53 Girsanov (coninued) Example 3 Girsanav heorem I Le W = W (1),..., W (n) be a Brownian moion wih dimension n, i.e. is componens are independen sandard Brownian moions on he lered probabiliy space (Ω, F, ff g, P) Theorem Cameron-Marin-Girsanov heorem. For all i 2 f1,..., ng, γ (i) = γ (i) : 0 T is a ff g predicable process such ha Z 1 T E exp P γ (i) 2 d <. 2 0 There exiss a Q on (Ω, F) such ha (CMG1) Q is equivalen o P

54 Girsanav heorem II Girsanov (coninued) Example 3 (CMG2) dq exp dp = n i=1 R T 0 γ(i) dw (i) R 1 T 2 0 n i=1 (CMG3) For all i2 f1,..., ng, he process fw (i) = W f (i) : 0 T de ned as fw (i) = W (i) moion. γ (i) 2 d + R 0 γ(i) s ds is a (ff g, Q) Brownian (ref. Baxer and Rennie, page 186)

55 (coninued) I Girsanov (coninued) Example 3 Since he funcions γ and eγ are consan, he Novikov condiion is sais ed and Girsanov heorem (mulidimensional version) allows o conclude here exiss a maringale Q such ha W Q and fw Q are (ff g, Q) Brownian moions.

56 Girsanov (coninued) Example 3 (coninued) II An ineresing fac, on he space (Ω, F, ff g, Q), we have he sochasic di erenial equaion sais ed by he insananeous exchange rae dc = µ C C d + σ C C dw P = µ C C d + σ C C d W Q = (r v) C d + σ C C dw Q. µ C + v r σ C The di erence beween he domesic and foreign insananeous ineres raes can be recognized in he drif coe cien.

57 Girsanov (coninued) Example 3 (coninued) III Again on he space (Ω, F, ff g, Q), he sochasic di erenial equaion for he evoluion of he risky asse Canadian dollar price is dc S = (µ S + µ C + σ S σ C ρ) C S d q + (σ S ρ + σ C ) C S dw P + σ S 1 ρ 2 C S d fw P = (µ S + µ C + σ S σ C ρ) C S d + (σ S ρ + σ C ) C S d W Q +σ S q 1 ρ 2 C S d fw Q µ C + v r σ C µ S v + σ S σ C ρ σ S p 1 ρ 2 = rc S d + (σ S ρ + σ C ) C S dw Q + σ S q 1 ρ 2 C S d fw Q!! ρ (µ C + v r ) p σ C 1 ρ 2

58 Girsanov (coninued) Example 3 (coninued) IV where he las equaliy is obained by simplifying he drif coe cien (µ S + µ C + σ S σ C ρ) (σ S ρ + σ C ) µ C + v r σ C! q σ S 1 ρ 2 µ S v + σ S σ C ρ ρ (µ p C + v r) p. σ S 1 ρ 2 σ C 1 ρ 2

59 Girsanov (coninued) Example 3 (coninued) V Again under he risk-neural Q, he Canadian dollar value of one foreign currency invesed in a foreign bank accoun sais es dc B = (v + µ C ) C B d + σ C C B dw P = (v + µ C ) C B d + σ C C B d = W Q µ C + v r σ C µ v + µ C σ C + v r C C B d + σ σ C C B dw Q C = rc B d + σ C C B dw Q.

60 Girsanov (coninued) Example 3 To summarize, (coninued) VI dc = µ C C d + σ C C dw P dc = (r v ) C d + σ C C dw Q dc S = (µ S + µ C + σ S σ C ρ) C S d q + (σ S ρ + σ C ) C S dw P + σ S 1 ρ 2 C S d fw P dc S = rc S d q + (σ S ρ + σ C ) C S dw Q + σ S 1 ρ 2 C S d fw Q dc B = (v + µ C ) C B d + σ C C B dw P dc B = rc B d + σ C C B dw Q

61 Girsanov (coninued) Example 3 (coninued) VII For he foreign currency securiies, we have he equaion, under probabiliy Q, which characerizes he evoluion of he risky asse foreign currency price: ds = µ S S d + σ S ρs dw P = ds = µ S S d + σ S ρs d +σ S q 1 ρ 2 S d fw Q q + σ S 1 ρ 2 S d fw P W Q µ C + v r σ C µ S v + σ S σ C ρ! ρ (µ C + v r) p σ S 1 ρ 2 q = (v σ S σ C ρ) S d + σ S ρs dw Q + σ S 1 ρ 2 S d fw Q σ C p 1 ρ 2 and he equaion of he evoluion of a foreign currency bank accoun db = vb d.

62 Example 3 I Non-uniqueness of he maringale Girsanov (coninued) Example 3 Le W P and fw P be wo independen sandard Brownian moions consruced on he lered probabiliy space (Ω, F, ff g, P). Assume he risky asse price evolves according o he SDE ds = µs d + σs dw P + σs d fw P and ha he insananeous ineres rae r is consan.

63 Girsanov (coninued) Example 3 Example 3 II Non-uniqueness of he maringale Le s se, for all 0 T, Y = β S i.e. Y represens he presen value a ime of he risky securiy. Using Iô s lemma (more precisely he muliplicaion rule), we obain Indeed, dy = (µ r) Y d + σy dw P + σy d fw P. dy = d β S = β ds + S d β + d hβ, Si = β µs d + σs dw P + σs d fw P = (µ r) β S d + σβ S dw P + S ( r β d) + σβ S d fw P.

64 Example 3 III Non-uniqueness of he maringale Girsanov (coninued) Example 3 dy = (µ r) Y d + σy dw P + σy d fw P = (µ r σγ σeγ ) Y d +σy d W P + Z 0 γ s ds + σy d Z fw P + eγ s ds 0 Le s force he drif coe cien o cancel ou: µ r σγ σeγ = 0, eγ = µ r σ So here is an in niy of soluions. γ.

65 Girsanov (coninued) Example 3 Recall: Example 3 IV Non-uniqueness of he maringale eγ = µ r σ γ. If we decide ha he process fγ : 0 T g does no depend on ime, hen he same will be rue for eγ. Since γ and eγ are deerminisic and consan, he Novikov condiion is sais ed. As a consequence, for all γ 2 R, here exiss a maringale Q γ such ha W γ = W P + γ and fw γ = fw P + eγ = fw P + µ σ r γ are (ff g, Q γ ) Brownian moions.

66 Example 3 V Non-uniqueness of he maringale Girsanov (coninued) Example 3 For all γ 2 R, he process Y, dy = σy dw γ + σy d fw γ is a (ff g, Q γ ) maringale.

67 Girsanov (coninued) Example 3 Consequences Example 3 VI Non-uniqueness of he maringale The marke is incomplee Some coningen claims canno be replicaed. In such a case, he expecaion, under a risk-neural, of he presen value of he coningen claim will give us A price, bu no THE price. How o deermine wheher a coningen claim is aainable? The answer can be found in he nex series of slides!

68 Marin Baxer and Andrew Rennie (1996). Financial Calculus, an inroducion o derivaive pricing, Cambridge universiy press. Chrisophe Bisière (1997). La srucure par erme des aux d inérê, Presses universiaires de France. Damien Lamberon and Bernard Lapeyre (1991). Inroducion au calcul sochasique appliqué à la nance, Ellipses.