The Implied Volatility of Forward Starting Options: ATM Short-Time Level, Skew and Curvature

Transcription

1 The Implied Volailiy of Forward Saring Opion: ATM Shor-Time Level, Skew and Crvare Elia Alò Anoine Jacqier Jorge A León May 017 Barcelona GSE Working Paper Serie Working Paper nº 988

2 The implied volailiy of forward aring opion: ATM hor-ime level, kew and crvare Elia Alò Dp d Economia i Emprea Univeria Pompe Fabra and Barcelona GSE c/ramon Tria Farga, Barcelona, Spain Jorge A León Conrol Aomáico CINVESTAV-IPN Aparado Poal México, DF, Mexico Anoine Jacqier Deparmen of Mahemaic Imperial College London London SW7 AZ, UK Abrac For ochaic volailiy model, we dy he hor-ime behavior of he a-he-money implied volailiy level, kew and crvare for forwardaring opion Or analyi i baed on Malliavin Calcl echniqe Keyword: Forward aring opion, implied volailiy, Malliavin calcl, ochaic volailiy model JEL code: C0 1 Inrodcion Conider wo momen ime > A forward-ar call opion wih mariy T > allow he holder o receive, a ime and wih no addiional co, a call opion expirying a T, wih rike e eqal o KS, for ome K > 0 So, he opion life ar a, b he holder pay a ime he price of he opion Some claical applicaion of forward aring opion inclde employee ock opion and cliqe opion, among oher ee for example Rbinein 1991 Under he Black-Schole formla, a condiional expecaion argmen lead o how ha he price of a forward aring opion i he price of a plain vanilla Sppored by gran ECO P and MTM P MINECO/FEDER, UE Parially ppored by he CONACyT gran

3 opion wih ime o mariy T In he ochaic volailiy cae, a change-ofmeare link he price of he forward opion wih he price of a claical vanilla ee, for example, Rbinein 1991, Miela and Rkowki 1997, Wilmo 1998, or Zhang 1998 For ochaic volailiy model, change of nmeraire echniqe can be applied o obain a cloed-form pricing formla in he conex of he Heon model ee Kre and Nögel 005 The implied volailiy rface for forward aring opion exhibi banial difference o he claical vanilla cae ee for example Jacqier and Roome 015 Thi paper i devoed o he dy of he a-he-money ATM hor ime limi of he implied volailiy for forward aring opion More preciely, we will e Malliavin Calcl echniqe o compe he ATM hor-ime limi of he implied volailiy level, kew and crvare In pariclar, we will ee ha -conrary o he claical vanilla cae- he ATM hor-ime level depend on he correlaion parameer ee Lemma 6 and Theorem 7 We will alo prove ha he kew depend of he Malliavin derivaive of he volailiy proce in a imilar way a for vanilla opion, while he crvare ee Theorem 14 and 15 i of order OT The paper i organized a follow Secion i devoed o inrodce forward aring opion and he main noaion ed rogho he paper In Secion 3 we obain a decompoiion of he opion price ha will allow o compe, in Secion 4, 5 and 6, he limi for he ATM implied volailiy level, kew and crvare Forward ar opion We will conider he Heon model for ock price on a ime inerval 0, T nder a rik neral probabiliy P : ds = ˆrS d σ S ρdw 1 ρ B, 0, T, 1 where ˆr i he inananeo inere rae ppoed o be conan, W and B are independen andard Brownian moion defined on a probabiliy pace Ω, F, P and σ i a poiive and qare-inegrable proce adaped o he filraion generaed by W In he following we will denoe by F W and F B he filraion generaed by W and B, repecively Moreover we define F := F W F B I will be convenien in he following ecion o make he change of variable X = log S, 0, T Now, we conider a poin 0, T We wan o evalae he following opion price: V = e ˆrT E e X T e α e X, where E denoe he condiional expecaion given F and α i a real conan Noice ha if hi i he payoff of a call opion, while in he cae < hi define a forward ar opion We will make e of he following noaion

4 BS, x, K, σ will repreen he price of a Eropean call opion nder he claical Black-Schole model wih conan volailiy σ, crren log ock price x, ime o mariy T, rike price K and inere rae ˆr Remember ha in hi cae BS, x, K, σ = e x Nd e ˆrT KNd, where N i he cmlaive probabiliy fncion of he andard normal law and x ln K ˆrT d ± := σ ± σ T T L BS σ will denoe he Black-Schole differenial operaor in he log variable wih volailiy σ : L BS σ = 1 σ ˆr 1 σ ˆr I i well known ha L BS σ BS, ; K, σ = 0 G, x, K, σ := xx x BS, x, K, σ H, x, K, σ := x xx x BS, x, K, σ BS 1 a := BS 1, x, K, a denoe he invere of BS a a fncion of he volailiy parameer α := ˆrT We recall he following rel, ha can be proved following he ame argmen a in ha of Lemma 41 in Alò, León and Vive 007 Lemma 1 Le 0, < T and G := F FT W here exi C = Cn, ρ ch ha Then for every n 0, a If E n G, X, M, v G CE e X G σd b If E n G, X, M, v G CE e X G σd 1 n1 1 n1 3

5 3 A decompoiion formla for forward opion price We will and for M := E e α e X, 0, T We oberve ha M = e α eˆr e X0 v := 0 0 σ 1 0, e α eˆr e X ρdw 1 ρ db = M 0 e α σ eˆr e ρdw X 1 ρ db Y T 1, wih Y := σ 1,T d = Noice ha, if <, v T = v T σ d We will need he following hypohee: H1 There exi wo conan 0 < c < C ch ha c < σ < C, for all 0, T, wih probabiliy one H σ, σe X L, L 1,4 Now we are in a poiion o prove he following heorem, ha allow o idenify he impac of correlaion in he forward opion price Theorem Conider he model 1 and ame ha hypohee H1 and H hold Then, for all 0 T, V = E expx BS, 0, e α, v where Λ W := ρ DW e ˆr H, X, M, v σ Λ W ρ G, 0, eα, v σ θ dθ d e ˆr e X σ Λ W Proof Remember ha M := E e α e X and noice ha V = e ˆrT E e X T e α e X = e ˆrT E e X T M T = e ˆrT BST, X T, M T, ν T 4

6 Therefore, he anicipaing Iô formla for he Skorohod inegral ee for example Nalar 006 allow o wrie e ˆrT BST, X T, M T, v T E = E e ˆr BS, X, M, v ˆr 1 ˆr e ˆr e, X, M, v d, X, M, v e ˆr BS, X, M, v d ˆr σ d e ˆr BS, X, M, v σd e ˆr BS K, X, M, v d X, M e ˆr BS K, X, M, v d M, M e ˆr e ˆr K e ˆr BS, X, M, v v σ 1,T d BS, X, M, v σ ρλ W BS, X, M, v σ e α eˆr e X ρλ W 1 0, d 5

7 Tha i, ing, V = E BS, X, M, v e ˆr L BS v BS, X, M, v d e ˆr BS, X, M, v σ v d e ˆr BS K, X, M, v d X, M e ˆr BS K, X, M, v d M, M e ˆr e ˆr e ˆr K Th, we ge, for, V = E BS, X, M, v e ˆr L BS v BS, X, M, v d BS, X, M, v v σ 1,T d BS, X, M, v σ ρλ W BS, X, M, v σ e α eˆr e X ρλ W e ˆr BS, X, M, v σ σ 1,T d e ˆr BS K, X, M, v d X, M e ˆr BS K, X, M, v d M, M e ˆr e ˆr K Now, aking ino accon he fac ha L BS v BS, X, M, v = 0, BS, X, M, v σ ρλ W d M, X = σ e α eˆr e X 1 0, d, BS, X, M, v σ e α eˆr e X ρλ W 6

8 and d M, M = σe α e ˆr e X 1 0, d, BS K, x, K, σ = 1 BS K, x, K, σ BS 1 BS, x, K, σ = K K, x, K, σ, i follow ha V = E BS, X, M, v 1 e ˆr 1 K and = E e ˆr BS, X, M, v Since we have BS e ˆr e ˆr e ˆr K BS, X, M, v σ ρλ W BS, X, M, v σ e α eˆr e X ρλ W BS, X, M, v σ ρλ W BS, X, M, v σ ρλ W BS, X, M, v σ M ρλ W, x, k, σ = ex N d σ 1 T d σ T BS K, x, k, σ = ex N d Kσ d T σ, T hen i i eay o ee ha V = E BS, X, M, v ρ ρ e ˆr H, X, M, v σ Λ W d e ˆr G, X, M, v σ Λ W 7

9 Hence, de o and, for all <, BS, X, M, v α ˆrT = expx N v T v T α ˆrT e α expx e ˆrT N v T v T = expx BS, 0, e α, v he proof i complee e X N αˆrt v T G, X, M, v = v T = e X G, 0, e α, v, v T Remark 3 We have choen Hypohee H1 and H for he ake of impliciy, which can be bied by adeqae inegrabiliy condiion Remark 4 Noice ha, if = we recover he decompoiion formla for call opion price preened in Alò, León and Vive 007 Remark 5 If he volailiy proce i conan σ = σ, for ome poiive conan σ, hen v = σ and Λ W 0, which implie ha, for, V = expx BS, 0, e α, σ 3 Coneqenly, we recover he well-known opion pricing formla for forward ar opion nder he Black-Schole model ee for example Rbinein 1991, Wilmo 1998, or Zhang The ATM hor-ime limi of he implied volailiy For 0,, we define he implied volailiy I, ; α a he F -adaped proce aifying V = expx BS, 0, e α, I, ; α 4 Noice ha, from 3, in he conan volailiy cae σ = σ, I, ; α = σ For he ake of impliciy, we will conider fir he ncorrelaed cae ρ = 0 In hi cae, we have he following rel 8

10 Lemma 6 Conider he model 1 wih ρ = 0 and ame ha hypohee in Theorem hold Then, for all <, Proof We know ha I, ; α lim T I, ; α = E σ = BS 1 E BS, 0, expα, v = E BS 1 BS, 0, expα, v BS 1 E BS, 0, expα, v BS 1 BS, 0, expα, v = E v BS 1 E BS, 0, expα, v BS 1 BS, 0, expα, v Noice ha a direc applicaion of Clark-Ocone formla give ha wih BS, 0, expα, v = E BS, 0, expα, v U r dw r, D U = E W T, 0, expα, v σ rdr 5 T v Then, applying Iô formla o he proce BS 1 A, where A := E BS, 0, expα, v U r dw r and aking expecaion, we ge BS 1 E BS, 0, expα, v BS 1 BS, 0, expα, v Hence E = 1 E lim I,, T α = E σ 1 lim T E Now, conidering ha BS 1 E r BS, 0, expα, v U r dr BS 1 E r BS, 0, expα, v U r dr BS 1 E r BS, 0, expα, v 1 BS 1 = N d 1 T E r BS, 0, expα, v T, 4 9

11 where d 1 := BS 1 E rbs,0,expα,v T, we ge ha 1 lim T E BS 1 E r BS, 0, expα, v U r dr = 0 Therefore, he proof i complee A a coneqence of he previo rel, we have he following limi in he correlaed cae Theorem 7 Conider he model 1 and ame ha hypohee in Theorem hold Then, for all <, lim I, ; T α = E σ ρ 1 e X E e ˆr e X σ D W σd Proof We know, by Theorem, ha I, ; α = BS 1 E BS, 0, expα, v ρ exp X E ρ exp X E σ e ˆr H, X, M, v σ Λ W G, 0, expα, v e ˆr e X σ Λ W By he mean vale heorem, we can find θ beween E BS, 0, expα, v and BS, 0, expα, v E ch ha ρ exp X ρ exp X G, 0, expα, v e ˆr H, X, M, v σ Λ W d e ˆr e X σ Λ W I, ; α BS 1 E BS, 0, expα, v π exp BS 1 θ 8 T ρ = T exp X T E e ˆr H, X, M, v σ Λ W ρ E G, 0, expα, v e ˆr e X σ Λ W = I 1 I 6 10

12 From Lemma 1, we have lim I exp X T 1 C lim T T T Finally, 6 and Lemma 6 imply lim T I, ; α = E σ ρ expx = E σ ρ expx E E e X G T 1 Λ W d C exp X lim T T 1/ E e X G 1/ = 0 lim T E 1 Therefore, he proof i complee σ v exp T 8 v T e ˆr e X σ D W σd e ˆr e X σ Λ W Remark 8 Noice ha, in he cae =, we recover he rel by Drrleman 008 for claical vanilla opion Alo, conrary o he claical vanilla cae, he ATM hor-ime limi depend on he correlaion parameer We will need he following lemma laer on Lemma 9 Conider he model 1 and ame ha E exp a T I, ; α T 0 0 σ < Then Proof We know ha, in hi cae, V = e ˆrT E e X T eˆrt e X, which converge o zero a T Indeed, we have e X T eˆrt e X 0, a T wih probabiliy 1 Moreover, by Novikov heorem ee for example Karaza and Shreve 1991, we have e X T eˆrt e X e X T eˆrt e X and E e X T eˆrt e X = eˆrt eˆr 11

13 a T Th, he dominaed convergence heorem implie ha V 0 a T, wih probabiliy one On he oher hand, V = expx N I, ; α T = expx 1 N I, ; α T which allow o complee he proof N I, ; α T, 5 An expreion for he derivaive of he implied volailiy Eqaion 4 lead o ge where k V α = expx k, 0, eα, I, ; α expx := ln K Then, from Theorem we are able o wrie = = E, 0, eα, I, ; α I, ; α, α I, ; α α V α expx k, 0, eα, I, ; α expx, 0, eα, I, ; α k E, 0, eα, v, 0, eα, I, ; α H e ˆr k, 0, eα, I, ; α k, X, e α e X, v σ Λ W ρ expx, 0, eα, I, ; α ρ E G k, 0, eα, v ex e ˆr σ Λ W expx, 0, eα, I, ; α d 7 Remark 10 Noe ha in he ncorrelaed cae ie, ρ = 0, eqaliy 7 implie I α, ; α = E k, 0, eα, v k, 0, eα, I, ; α, 0, eα, I, ; α 1

14 In hi cae, for α = α, we ge, by Theorem, E k, 0, expα, v = E N v T = E N v T N BS, 0, expα = E, v 1 and = V exp X 1 So, we conclde ha E k, 0, expα, v and, coneqenly, I α, ; α = 0 v T k, 0, expα, I, ; α = N I, ; α T = V exp X 1 1 k, 0, expα, I, ; α = 0 We will need he following hypohei H3 There exi wo conan C > 0 and δ 0 ch ha, for all θ < < r E D W σ r Cr δ 8 and E D W θ D W σ r C r δ r θ δ 9 Theorem 11 Conider he model 1, and ame ha hypohee H1, H and H3 hold and ha here exi a F -mearable random variable D σ ch ha E E D W σθ D σ 4 0 a T 10 Then, for all <, p θ T I lim T α, ; α = ρ exp ˆr E e X D σ 4 expx σ 13

15 Proof From 7 we obain I α, ; α = E k, 0, expα, v k, 0, expα, I, ; α, 0, expα, I, ; α ρ E ρ E = T 1 T T 3 H e ˆr k, X, M, v σ Λ W d expx, 0, expα, I, ; α G k, 0, expα, v ex e ˆr σ Λ W expx, 0, expα, I, ; α Proceeding a in Remark 10 and ing Theorem,we ge E k, 0, expα, v k, 0, expα, I, ; α BS, 0, expα = E, v 1 V exp X 1 { = exp X E ρ T e ˆr H, X, M, v σ Λ W α=α Then, where and exp X ρ G, 0, expα, v 4 exp X ρe I 1 = lim T 1 = lim I 1 lim I, T T T } e ˆr σ e X Λ W e ˆr H, X, M, v σ Λ W α=α 4, 0, expα, I, ; α I = exp X ρe G, 0, expα, v e ˆr σ e X Λ W 4, 0, expα, I, ; α I i eay o ee ha, nder H3, lim T I 1 = 0 de o Lemma 1 On he oher hand, G, 0, expα, v = G k, 0, expα, v, which yield I = ρ exp X E G k, 0, expα, v e ˆr e X σ Λ W, 0, expα, I, ; α = T 3 14

16 Thi give ha I T 3 = 0 On he oher hand, ing Lemma 1 and 9, and he anicipaing Iô formla again, i follow ha ρ T E H e ˆr k, X, M, v σ Λ W lim T = lim T T = ρ E lim T = ρ lim T E = expx, 0, expα, I, ; α H k, X, M, v e ˆr T ρ expx lim T E Now, 10 allow o wrie lim T = T and now he proof i complee σ Λ W expx, 0, expα, I, ; α exp X exp ˆr T expx vt 3 σ Λ W exp X exp ˆr T σt 3 ρ exp ˆr E 4 expx D σ σ, σ Λ W Remark 1 If =, hi formla agree wih he a-he-money hor-ime limi kew proved in Alò, Leòn and Vive The econd derivaive of he implied volailiy In hi ecion we figre o I α, ; α Toward hi end, noe ha implie V α = expx, 0, expα, I, ; α k expx I, 0, expα, I, ; α, ; α α V α = expx BS k, 0, expα, I, ; α expx BS I, 0, expα, I, ; α, ; α k α expx BS I, 0, expα, I, ; α, ; α α expx, 0, expα, I, ; α I, ; α 11 α 15

17 61 The ncorrelaed cae In hi becion we ame ha ρ = 0 We will need he following rel, imilar o Theorem 5 in Alò and León 015 Lemma 13 Conider he model 1 wih ρ = 0 and ame ha hypohee in Theorem 11 hold Then where, 0, expα, I, ; α I α, ; α = 1 T Ψ E a E BS, 0, expα, v Ud, Ψa := BS k, 0, expα, BS 1 a and U i given in 5 Proof Thi proof i imilar o ha in Alò and León 015 Theorem 5, o we only kech i Noice ha 11 and Remark 10 give V α α=α = expx BS k, 0, expα, I, ; α expx, 0, expα, I, ; α I α, ; α Then, aking ino accon Theorem and he fac ha I, ; α = BS 1 exp X V, we are able o wrie expx, 0, expα, I, ; α I α, ; α = V α α=α expx BS k, 0, expα, I, ; α = expx E BS k, 0, expα, v BS k, 0, expα, I, ; α = expx E BS, 0, expα k, BS 1 BS, 0, expα, v BS, 0, expα k, BS 1 E BS, 0, expα, v Now, ing 4, applying Iô formla o he proce A := BS k, 0, expα, BS 1 E BS, 0, expα, v and aking expecaion, he rel follow 16

18 Theorem 14 Conider he model 1 wih ρ = 0 and ame ha hypohee in Theorem are aified Then lim T I T α, ; α = 1 D 4 E E W σ E σ 3 d Proof Lemma 13 yield lim T I T α, ; α = lim T T lim T T E = lim T T 1 T E σ Ψ a E BS, 0, expα, v Ud, 0, expα, I, ; α Ψ a E BS, 0, α, v Ud, 0, expα, I, ; α Remember ha we are aming ha here exi wo poiive conan c, C > 0 ch ha c σ C Th, he fac ha BS, 0, eˆrt, i an increaing fncion, ogeher wih H3 and Ψ a E BS, 0, α, v BS π exp 1 E BS,0,eˆrT,v T 8 = BS 1 E BS, 0, eˆrt, v 3, T 3/ allow o dedce ha, conidering ha C i a conan ha may change from line o line, T CT 0 < T CE exp 8 c 3 U C T T d E U T CT 1/ which implie ha lim T T = 0 Finally, he dominaed convergence heorem and Lemma 6 give ha lim T 1 = π lim T T = 1 4 lim T E = 1 4 E exp I,;α T 8 I, ; α 3 T E 1 I, ; α 3 T D W σr E σ U d E E σ 3 d DW σrdr d v T 17

19 and hi allow o complee he proof 6 General cae Now we are ready o analyze he crvare in he correlaed cae So we ame ρ 0 Theorem 15 Under he ampion of Theorem, we have lim T I T α, ; α = 1 D 4 E E W σ σ r E σ 3 d 1 E σ 1 E σ ρ exp X E 1 σ e ˆr e X σ D W σd ρ 1 exp X E e ˆr e X σ D W d Proof By Theorem and 11, we dedce σ 3 expx, 0, expα, I, ; α I, ; α α = expx E BS k, 0, expα, v BS k σ, 0, expα, I, ; α expx BS k, 0, expα, I, ; α I, ; α α expx BS I, 0, expα, I, ; α α, ; α ρ T E e ˆr H k, X, M, v σ Λ W α=α G k, 0, expα, v e ˆr e X σ Λ W 1 Th we can wrie 18

20 T I α, ; α E = T E T BS k, 0, expα, v BS k, 0, expα, I 0, ; α BS k, 0, expα, I, ; α, 0, expα, I 0, ; α BS k, 0, expα, I, ; α, 0, expα, I, ; α T BS k, 0, expα, I, ; α I α, ; α, 0, expα, I, ; α BS T, 0, expα, I, ; α I α, ; α, 0, expα, I, ; α ρ exp X E e ˆr H k, X T, M, v σ Λ W, 0, expα, I, ; α α=α ρ exp X E G k, 0, expα, v T e ˆr e X σ Λ W, 0, expα, I, ; α = T 1 T T 3 T 4 T 5 T 6 Here I 0, ; α denoe he implied volailiy in he ncorrelaed cae ρ = 0 I i eay o ee e definiion of BS lead o lim T T 3 T 4 T 5 = 0 Moreover, we can eaily ee ha lim T Then, he proof of Lemma 13 yield By comping BS k, 0, expα, I, ; α, 0, expα, I 0, ; α = 1 lim T 1 = lim T I 0,, α T T k i i eay o ee ha lim T = lim T T 1 I 0, ; α 1 I, ; α Therefore, we can e Lemma 6 and Theorem 7 o figre o hi limi On he oher hand lim T T 6 = ρ E G exp X k, 0, expα, v lim T e ˆr e X σ Λ W T, 0, expα, I, ; α = ρ 1 exp X E e ˆr e X σ D W σ d σ 3 19 d

21 Finally, he rel i a coneqence of Lemma 6, and Theorem 7 and 14 Aknowledgemen Elia Alò acknowledge nancial ppor from he Spanih Miniry of Economy and Compeiivene, hrogh he Severo Ochoa Programme for Cenre of Excellence in R&D SEV Jorge A León hank Univeria Barcelona, Univeria Pompe Fabra and he Cenre de Recerca Maemàica of Univeria Aònoma de Barcelona for heir hopialiy and economical ppor Reference 1 Alò, E, León, JA: On he econd derivaive of he a-he-money implied volailiy in ochaic volailiy model Working paper UPF, 015 Alò, E, León, JA and Vive, J: On he hor-ime behavior of he implied volailiy for jmp-diffion model wih ochaic volailiy Finance and Sochaic 11, , Drrleman, V Convergence of a-he-money implied volailiie o he po volailiy Jornal of Applied Probabiliy 45 0: , Hll, J C and Whie, A: The pricing of opion on ae wih ochaic volailiie Jornal of Finance 4, , Jacqier, Aand Roome, P: Aympoic of forward implied volailiy SIAM Jornal on Financial Mahemaic, 61, , Kre, S and Nögel, U: On he pricing of forward aring opion in Heon model on ochaic volailiy Finance and Sochaic 9, 33-50, Miela, M, Rkowki, M: Maringale mehod in financial modeling Berlin, Heidelberg, New York: Springer Nalar, D: The Malliavin Calcl and Relaed Topic Second Ediion Probabiliy and i Applicaion Springer-Verlag, Rbinein, M: Pay now, chooe laer Rik 4,