Approximation for Option Prices under Uncertain Volatility

Transcription

1 Approximaion for Opion Price under Uncerain Volailiy Jean-Pierre Fouque Bin Ren February, 3 Abrac In hi paper, we udy he aympoic behavior of he wor cae cenario opion price a he volailiy inerval in an uncerain volailiy model (UVM) degenerae o a ingle poin, and hen provide an approximaion procedure for he wor cae cenario price in a UVM wih mall volailiy inerval. Numerical experimen how ha hi approximaion procedure perform well even a he ize of he volailiy band i no o mall. Inroducion o UVM Since heir inroducion in [3 and [3, uncerain volailiy model have received inenive aenion in mahemaical finance. In a imple UVM, i i aumed ha he marke ha wo ae: one rikle ae and one riky ae. Their price procee are denoed a (B ) and (X ). I i alo aumed ha he price proce of he rikle ae (B ) ha dynamic db = rb d, where r i a conan. The price proce of he riky ae (X ) olve he following ochaic differenial equaion (SDE) dx = rx d + α X dw, where (W ) i a Brownian moion on he probabiliy pace (Ω, F, F, Q) and he volailiy proce (α ) A which i a family of progreively meaurable and [σ, σ-valued procee. For each ochaic volailiy proce α A, one ha a general ochaic volailiy model for (X ). In a UVM, we only know ha he rue model lie in he above family of general ochaic volailiy model. Noe ha we do no have a prior belief (a probabiliy diribuion) over he family of general ochaic volailiy model. Therefore, we ue ambiguiy o diinguih hi ype of uncerainy. Inuiively, we can conider he ize of he volailiy inerval a he degree of model ambiguiy. Due o he preence of model ambiguiy (or abence of a prior diribuion), he wor cae cenario analyi i applied in derivaive pricing under a UVM. Suppoe ha χ i a European derivaive wrien on he riky ae wih mauriy T and payoff ϕ(x T ). I i known ha i wor cae cenario price a ime < T i given by V (, X ) := up E [ϕ(x T ), α A Deparmen of Saiic & Applied Probabiliy, Univeriy of California, Sana Barbara, CA 936-3, fouque@pa.ucb.edu. Work uppored by NSF gran DMS Deparmen of Saiic & Applied Probabiliy, Univeriy of California, Sana Barbara, CA 936-3, ren@pa.ucb.edu.

2 where E [ i he condiional expecaion given F wih repec o he meaure Q, ee [3 [3. I i proved in [8 ha he eller of he derivaive χ can uper-replicae χ wih iniial wealh up α A E [ϕ(x T ) whaever he rue volailiy proce i. The imporance of he wor cae cenario price i no only becaue of i uper-replicaion propery, bu alo due o i relaionhip wih coheren rik meaure [ [5. Following he argumen in ochaic conrol heory, V (, x) aifie he following Hamilon-Jacobi-Bellman (HJB) equaion (in mah-finance i i called Black-Schole-Barenbla (BSB) equaion) [ V + r(x x V V ) + up x α xv =, (.) α [σ,σ V (T ) = ϕ. If ϕ i convex (like he payoff of a European call), i i known ha he wor cae cenario price of χ i equal o i Black-Schole price wih conan volailiy σ. For concave ϕ, we have a imilar reul, ee [7 for deail. However, he fully nonlinear PDE (.) doe no have a cloed form oluion, like Black-Schole formula, for a general erminal payoff funcion ϕ. In order o evaluae he wor cae cenario price, we have o reor o numerical mehod [3 and [5. Similarly, he be cae cenario price of χ can be defined a inf α A E [χ. Moreover, i i hown in [7 ha given any price beween inf α A E [χ and up α A E [χ, he marke i arbirage free. I i clear ha he wor cae cenario price i larger han any Black-Schole price wih a conan volailiy σ [σ, σ. In hi paper, we hall conider how he wor cae cenario price behave a he volailiy inerval [σ, σ degenerae o a ingle poin σ [σ, σ. Inuiively, if he model ambiguiy i reduced, hen he exra price (which i included in he wor cae cenario price) paid for ha hould be le. A i i hown in he equel, he wor cae cenario price of χ will converge o i Black-Schole price wih conan volailiy σ. Indeed, Con uggeed in [5 a meaure of impac of model uncerainy on he wor cae cenario price for any derivaive χ: µ(χ) = up E [χ inf E [χ, α A α A which vanihe a he volailiy inerval hrink o a ingle poin. In addiion, our udy parially anwer he eniiviy problem of he wor cae cenario o he degree of model ambiguiy which i propoed in [5. In fac, we obain he rae of convergence of he wor cae cenario price a volailiy inerval hrink o a ingle poin. Therefore, hi reul give u an approximaion of he wor cae cenario price when he inerval i ufficienly mall. Along he paper we denoe he Black-Schole price a V and he rae of convergence a V, which are he oluion o linear parial differenial equaion. Conequenly, he fir order approximaion V + (σ σ)v of he wor cae cenario price i achieved. Of coure, he approximaed price V + (σ σ)v doe no have he propery of uper-replicaion. Wha did we gain in he approximaion procedure? Fir, he problem of olving a fully nonlinear BSB equaion i reduced o olving wo Black-Schole like PDE. The numerical example alo how ha he approximaion procedure i able even wih reaonably large volailiy inerval. Second, we are able o ee how a linear expecaion urn ino a ublinear expecaion. In order o udy he aympoic behavior of wor cae cenario price, we re-parameerize our uncerain volailiy model and aume ha he riky ae price proce (X α,ε ) ha a dynamic dx α,ε = rx α,ε d + α X α,ε dw, (.) where α := (α ) A ε, he family of progreively meaurable, [σ, σ + ε-valued procee and (W ) i a Brownian moion on he probabiliy pace (Ω, F, F, Q). If ε = and no danger of confuion, we hall ue (X ) o denoed (X α, ) which i indeed a geomeric Brownian moion wih conan volailiy σ, dx = rx d + σx dw.

3 We define he wor cae cenario price a a value funcion of a ochaic conrol problem J ε (, x, α) := E x [ϕ(x α,ε T ), V ε (, x) = up α A ε [J ε (, x, α), where he condiional expecaion E x [ i aken wih repec o he law of X α,ε T given X α,ε = x. The wor cae cenario opion price when ε = i a Black-Schole price. We alo repreen i a a value funcion of a rivial ochaic conrol problem V (, x) = J(, x, σ) := E x [ϕ(x T ), where he ubcrip in E x [ alo mean ha X = x. Thi paper i rucured a follow. In Secion, we briefly recall ome well-known reul from ochaic conrol heory. The coninuiy of he wor cae cenario price wih repec o he parameer ε i dicued in Secion 3. In Secion 4, we heuriically derive he equaion for he convergence rae of V ε a ε vanihe. Secion 5 i devoed o udying he wor cae cenario ae price proce for he clam χ. In Secion 6, he analyi of error erm i developed. In Secion 7, 8 we perform a numerical experimen and conclude he paper. Preliminary reul from ochaic conrol Becaue we are only inereed in he cae where ε i cloe o, i i legiimae o aume ha all ε. In paricular, α σ +. In order o inroduce he reul ha are needed in hi paper, we borrow he noaion from [. Given a SDE wih random coefficien: if here exi a conan K > uch ha dx = b (X )d + σ (X )dw, (.) b (x) b (y) K x y, σ (x) σ (y) K x y for all [, T, ω Ω, and x, y R, hen we ay ha (L ) condiion i aified. If for all [, T, ω Ω, and x R here exi ome K > uch ha b (x) K x + h, σ (x) K x + r for ome ochaic procee (h ) and (r ), we ay ha (R) condiion i aified. I can be een ha he condiion (L ) implie he condiion (R), by noicing ha Noe ha b (x) b (x) b () + b () σ (x) σ (x) σ () + σ (). rx ry r x y, α x α y α x y (σ + ) x y. Therefore, i i clear ha he SDE of (X α,ε ) aifie he condiion (L ). According o Corollary in he ecion.5 [, we have he following univeral eimae of he momen of (X α,ε ) E [ up X α,ε q Ne N ( + x ) q, (.) [, for all α A ε, [, T and q >, where N = N(q, σ, r) (we aumed ha ε < ) and X α,ε = x. 3

4 For anoher ε (,, i i aumed ha ε < ε wihou loing generaliy. We conider he proce (X α (σ+ε ),ε ) which aifie he SDE (.) wih volailiy proce (α (σ + ε )) for ome (α ) A ε. I i alo aumed ha X α,ε = x. By Theorem 9 in he ecion.9 [ and he eimae of he momen, we can conclude ha E [ up X α,ε [, [ N q e N E X α (σ+ε ),ε q X α,ε q α α (σ + ε ) q d N q e N N e N ( + x q )(ε ε ) = N q e N N e N ( + x q )(ε ε ) = N q e N ( + x q )(ε ε ), (.3) for any q, where N = N (q, σ, r), N = N (q, σ, r), and N = max N N, N + N }. 3 Coninuiy of V ε in ε In hi ecion, we mainly analyze he coninuiy of V ε wih repec o ε and he main reul i: Theorem 3.. Given ϕ which i Lipchiz coninuou wih Lipchiz conan K and for any ε [, ) lim ε ε V ε (, x) = V ε (, x) for all (, x) [, T R. Proof. Fir, recall ha A ε i he family of progreively meaurable and [σ, σ + ε-valued procee. If ε < ε <, we have ha [ V ε (, x) = E x [ϕ(x α,ε T ) = up E x ϕ(x α (σ+ε),ε T ). α A ε up α A ε Therefore, by he eimae (.3) V ε (, x) V ε (, x) Ex up [ϕ(x α,ε T ) E x[ϕ(x α (σ+ε),ε T ) α A ε ( X α,ε K up E x T X α (σ+ε),ε T ) / α A ε K (N(T )e N(T ) ( + x ) ε ε ) /. I can be een ha for any fixed poin (, x) [, T R, V ε (, x) V ε (, x) a ε approache ε from above. I can be proved imilarly ha lim ε ε V ε (, x) V ε (, x) = for ε >. In paricular, when ε = we have one-ided convergence of V ε (, x)} ε> which i aed in he following corollary. Corollary 3.. When he condiion in Theorem 3. are aified, for all (, x) [, T R. lim V ε (, x) = V (, x), ε 4

5 A he volailiy inerval [σ, σ + ε become maller, he above corollary ell u ha he Black-Schole price V of χ wih conan volailiy σ i he main conribuor o i wor cae cenario price relaive o he exra price paid for he model ambiguiy, i.e. V would be he leading erm in he approximaion of V ε. We now inveigae he fir order correcion erm in hi approximaion, which will help u underand he eniiviy of he wor cae cenario price o he degree of model ambiguiy. 4 Heuriic derivaion of he fir order correcion erm To implify he noaion, we aume r = in he equel, bu all he reul ill hold when r. Recall ha V ε olve he following BSB equaion V ε + up α [σ,σ+ε α x xv ε =, V ε (T ) = ϕ. To udy he aympoic behavior of V ε, we alo re-parameerize he BSB equaion a follow V ε + up g [, (σ + εg) x xv ε =, (4.) V ε (T ) = ϕ. Noe ha V i he oluion of he following Black-Schole equaion V + σ x xv =, (4.) V (T ) = ϕ. In hi heuriic derivaion, we aume he differeniabiliy of V ε wih repec o ε and he inerchangeabiliy of parial differenial operaor ε, and x. We differeniae he equaion (4.) wih repec o ε } ε V ε + σ x xv ε + ε up g [, g x xv ε + ε up gσx xv ε =. g [, Le V = ε V ε ε=, and according o Corollary 3. we alo noe ha V ε ε= = V. Therefore, V i he unique oluion of he following linear PDE wih ource and zero erminal condiion V + σ x xv + up gσx xv =, (4.3) g [, V (T ) =. Noe ha he ource erm in he above equaion i known from V and i in fac equal o σx xv I x V >}. I i nonlinear in V and can be een a he fir manifeaion of he nonlineariy of he full problem (4.). In he above heuriic argumen, we obained he equaion characerizing V. I will be verified ha V which olve (4.3) i he fir order derivaive of V ε wih repec o ε a ε =. Tha i, we hall prove ha he error erm V ε (V + εv ) i of order o(ε). More preciely, he following heorem i achieved wih more echnical condiion impoed on ϕ. Theorem 4.. Aume ha he payoff funcion ϕ C 4 p(r + ) (p for polynomial growh), ϕ i Lipchiz and i derivaive up o order 4 have polynomial growh. Moreover, we alo aume ha he econd derivaive of ϕ ha a finie number of zero poin. Then, poinwie V ε (V + εv ) lim =. ε ε 5

6 5 Ae price proce in he wor cae cenario and eimae of i momen I i known from [8 and [6 ha if ϕ i locally Lipchiz coninuou, ϕ and ϕ have polynomial growh, hen he vicoiy oluion V ε of (4.) belong o Cp, ([, T ) R) and here exi κ (, uch ha xv ε i Hölder-κ coninuou. 5. Exience and uniquene of (X,ε ) The equaion (4.) would produce he wor cae cenario volailiy proce α,ε = σ + εg,ε for he claim χ, where g,ε (, x) = xv ε (, x), xv ε (, x) <. From he equaion (4.3) of V, we would have anoher choice of he volailiy proce for he claim χ: ᾱ = σ + εḡ, where ḡ(, x) = xv (, x), xv (, x) <. Therefore, he ae price proce in he wor cae cenario for he claim χ i a ochaic proce which aifie he SDE (.) wih α = α,ε and r =, i.e. Define a ranformaion dx,ε Y,ε which i well-defined for any < τ δ where τ δ = inf > X,ε = inf > Y,ε = α,ε X,ε dw. (5.) := log X,ε, = δ or X,ε = /δ } = log δ or Y,ε = log δ }, for any δ >. By Iô formula, he proce (Y,ε ) aifie he following SDE dy,ε = (α,ε ) d + α,ε dw. (5.) I i noed ha he coefficien in (5.) are bounded and progreively meaurable. Moreover, he diffuion coefficien i bounded away from zero: α,ε σ >. Therefore, hank o Theorem in ecion.6 in [ or he reul in [6, he SDE (5.) ha a unique weak oluion. Tha i, we have a unique oluion o he SDE (5.) unil τ δ for any δ >. In order o prove ha he SDE (5.) ha a unique oluion for all (, ), i uffice o how ha for any T > lim Q ( τ δ < T ) =. (5.3) δ Indeed, by Chebyhev inequaliy, i hold ha ( which implie (5.3). lim Q ( τ δ < T ) lim Q δ δ E lim δ up [,T ) Y,ε > log δ [ up [,T Y,ε log δ = 6

7 5. Eimae of momen and exi probabiliy of (X,ε ) A a pecial cae of (.), we have ha for all q >, N = N(q, σ) and X,ε = x. Given ρ >, define a opping ime E x [ up X,ε q Ne N(T ) ( + x q ), (5.4) [,T τ ρ := inf [, T, uch ha X,ε ρ}. Convenionally, inf =. By uing he eimae of momen and Chebyhev inequaliy, Q x (τ ρ < T ) Q x ( up [,T X,ε ρ) NeN(T ) ( + x ), (5.5) ρ where N = N(σ). Thi conrol on he exi probabiliy enable u o ue localizaion argumen in he equel. Remark 5.. I i imporan o noice ha he eimae in hi ecion are independen of ε, due o our aumpion ε. 6 Analyi of he error erm Define he error erm of he uggeed approximaion Z ε = V ε (V + εv ). Le L(σ) := + σ x x. Nex, we apply he operaor L(α,ε ) o he error erm: L(α,ε )Z ε = L(α,ε )(V + εv ) = (L(α,ε ) L(σ)) V ε (L(α,ε ) L(σ)) V εl(σ)v = ( σ (σ + εg,ε ) ) x xv + ε ( σ (σ + εg,ε ) ) x xv +εḡσx xv ( ) = ε (g,ε ḡ) σx xv ε (g,ε ) x xv + g,ε σx xv ε 3 ( (g,ε ) x xv ). (6.) Noe ha he erminal condiion of Z ε i Z ε (T ) = V ε (T ) V (T ) εv (T ) =. From now on, we impoe more regulariy condiion on he erminal daa ϕ, i.e. polynomial growh condiion on he fir four derivaive of ϕ(x): ϕ (x) K, ϕ (x) K ( + x m ), ϕ (x) K 3 ( + x n (6.) ), ϕ (4) (x) K 4 ( + x l ), where K i for i,, 3, 4}, and m, n, and l are poiive conan. 7

8 6. Feynman-Kac repreenaion of he error erm Z ε Given he equaion (6.) for Z ε ogeher wih he exience and uniquene of (X,ε ), we have he following probabiliic repreenaion of Z ε : Z ε = εe x where (g,ε ε E x ε 3 E x (g,ε (g,ε } ḡ ) σ (X,ε ) xv (, X,ε ) d ) (X,ε ) xv (, X,ε ) + g,ε σ (X,ε ) xv (, X,ε ) (X,ε ) xv (, X,ε } ) d } ) = εi ε I ε 3 I 3, (6.3) I = E x (g,ε I = E x I 3 = E x (g,ε (g,ε } ḡ ) σ (X,ε ) xv (, X,ε ) d, ) (X,ε ) xv (, X,ε ) + g,ε σ (X,ε ) xv (, X,ε ) (X,ε ) xv (, X,ε } ) d. } ) We hall derive he bound of I and I 3 in he nex ecion. The erm I will be deal wih in Secion 6.3. d d, 6. Conrol of he erm I and I 3 A a pecial cae of (.), he proce (X ) which olve dx = σx dw (6.4) ha he following eimae of momen [ E x up X q Ne N(T ) ( + x q ), (6.5) [,T for all q >, where N = N(q, σ). Becaue V i he oluion of he Black-Schole equaion(4.), he following lemma hold. Lemma 6.. Given ϕ(x) which aifie he condiion (6.), here exi conan M, M, and M 3 which only depend on σ, T, m, n, and l, uch ha x V M ( + x m ), 3 x V M ( + x n ), and xv M3 ( + x p ), where p = max m, n +, l + }. Proof. We hall ue he probabiliic repreenaion of V V (, x) = E x [ϕ(x T ), (6.6) where he proce (X ) i he geomeric Brownian moion (6.4). Then, aring a x a ime, we have X T = xe σ (T ) +σ(w T W ). Therefore, uing he noaion τ = T, we obain x V = e σ τ+σ τy ϕ (xe σ τ +σ τy ) e y dy R π K e σ τ+σ ( τy + x m e mσ τ +mσ ) τy e y dy π R M ( + x m ), 8

9 wih M = M (K, m, T, σ). Similarly, we obain conrol of xv 3 and xv 4 : 3 x V = e 3 σ τ+3σ τy ϕ (xe σ τ +σ τy ) e y dy R π K 3 e 3 σ τ+3σ ( τy + x n e nσ τ +nσ ) τy e y dy π 4 x V = R M ( + x n ), e σ τ+4σ τy ϕ (4) (xe σ τ +σ τy ) e y dy R π K 4 e σ τ+4σ ( ) τy + x l e lσ τ +lσ τy e y dy π R M 4 ( + x l ), wih M = M (K 3, n, T, σ) and M 4 = M 4 (K 4, l, T, σ). ( From he PDE of V, i can be een ha xv = x ) σ x xv ( ) = σ xv + 4x xv 3 + x xv 4. Therefore, he conrol of he mixed hird parial derivaive xv i obained hrough he conrol on he parial derivaive only wih repec o x. Conequenly, here exi wo conan M 3 = M 3 (M, M, M 4, m, n, l) and p = maxm, n +, l + } uch ha xv σ M ( + x m ) + 4 x M ( + x n ) + x ( M 4 + x l )} M 3 ( + x p ). Given V which olve (4.), we define The fir order parial derivaive of f are f(, x) := σx xv (, x). f (, x) = σx xv (, x), f x (, x) = σ [ x xv (, x) + x 3 xv (, x). Due o he lemma 6., i can be oberved ha for a conan K 5 = K 5 (M, σ) and ha f(, x) σx M ( + x m ) K 5 ( + x m+ ) (6.7) f (, x) + f x (, x) σ ( x M 4 ( + x p ) + x M ( + x m ) +x M ( + x n ) ) K 6 ( + x p ), (6.8) where K 6 = K 6 (M, M, M 4, p, m, n, σ) and p = maxp +, m +, n + }. In order o analyze he parial derivaive of V, we ue (X,x ) o denoe he oluion of he SDE (6.4) wih he iniial condiion X,x = x. Similarly, (X +h,x ) denoe he oluion of he ame SDE wih he iniial condiion X +h,x = x. We alo define X = X,x+h 9 h X,x. (6.9)

10 From D. in [9, we have ha E x [ X N, (6.) for any [, T, where N = N(σ). Due o (6.8), we can derive ha f(, X,x+h ) f(, X,x ) h fx (, X λ ) X dλ ( K 6 + X,x p + X,x+h p) X, (6.) where X λ = ( λ)x,x hold ha + λx,x+h f( + h, X +h,x for λ (, ). Moreover, by he mean value heorem, i ) f(, X +h,x ) ( h K 6 + X +h,x p). (6.) If we denoe he ource erm in he equaion of (4.3) a f + (, x), hen i i noed ha } } f + (, x) := up g [, gσx xv = max σx xv, = max f, }. Therefore, i i alo clear ha f + (, x) f(, x), f + (, x) f + (, y) (6.3) f(, x) f(, y), for any (, x), (, y) [, T R. Noe ha he fir order difference quoien of V wih repec o he ime variable can be repreened a follow V ( + h, x) V (, x) h = E x T h T f + ( + h, X +h,x ) f + (, X +h,x h ) T h f + (, X +h,x ) f + (, X,x ) +E x d h h E x T h d f + (, X,x )d. (6.4) Lemma 6.. Given ϕ(x) which aifie he condiion (6.), here exi a conan M 5 which depend on K 5, K 6, T, p, m, and σ uch ha V M 5 ( + x p ). Proof. Due o (6.3) and (6.), he fir erm in (6.4) can be eimaed a follow T h E f + ( + h, X +h,x ) f + (, X +h,x ) x d h T h f( + h, X +h,x ) f(, X +h,x ) E x d h T h ( K 6 E x + X +h,x p) d A ( + x p ), where A = A (K 6, T, p, σ). For he econd erm in (6.4), by noicing ha he hifed proce (X,x ) i idenical o (X +h,x ) up o ime T h, we herefore can conclude ha T h E f + (, X +h,x ) f + (, X,x ) x d h =.

11 Due o (6.3) and (6.7), he hird erm in (6.4) can be conrolled a follow, T h E x f + (, X,x )d T h E x f + (, X,x ) d T h T h h K 5E x T T h A ( + x m+ ), ( + X,x m+) d where A = A (K 5, T, m, σ). Then, afer ummarizing he above hree eimae he lemma follow. Propoiion 6.. Given ϕ(x) which aifie he condiion (6.), here exi wo conan p = maxp, m+} and M 6 which depend on K 5, M 5, T, m, p, and σ, uch ha x xv M 6 ( + x p ). Proof. By noicing ha V + σ x xv = f +, we have x xv ( σ V + f + ). The above fac ogeher wih (6.7), (6.3) and he lemma 6. reul in he propoiion. Theorem 6.. Given ϕ(x) which aifie he condiion (6.), here exi a conan D which depend on M, M 6, T, m, p, and σ, uch ha I and I 3 in (6.3) aify I + I 3 D ( + x p ). Proof. Due o he propoiion 6. and (5.4), i follow ha I 3 = E x (g,ε T ) (X,ε ) xv (, X,ε ) d M ( 6 E x + X,ε p ) d [ M T 6 + N(σ, p )e N(σ,p )(T ) ( + x p )d = M 6 Similarly for I, we have I E x +E x M T E x [ T + N(σ, p )e N(σ,p )(T ) ( + x p )(T ) (X,ε ) x V (, X,ε ) d σ (X,ε ) x V (, X,ε ) d X,ε T ( ( + X,ε m ) d + M 7 σe x M (T )N(σ)eN(σ)(T ) ( + x ) + M (T )N(m +, σ)en(m+,σ)(t ) ( + x m+ ) +M 6 σ(t )N(p, σ)e N(p,σ)(T ) ( + x p ).. + X,ε p ) d

12 By ummarizing he above conrol on I and I 3, here exi D = D (M, M 6, T, m, p, σ) uch ha I + I 3 < D ( + x p ). 6.3 Convergence of he erm I Given he conrol of I and I 3 in he heorem 6., o prove Z ε (, x) o(ε) for a fixed poin (, x) [, T R, i uffice o prove ha lim I =. ε Le K ρ := [ ρ, ρ and g d,ε := g,ε ḡ which depend on ε hrough g,ε a follow g d,ε (, x) = xv ε (, x) xv (, x) <, xv ε (, x) xv (, x). Inuiively, a ε approache, V ε and i derivaive will ge cloer and cloer o V and i correponding derivaive. In order o lay ou he analyi of hi inuiion, we decompoe he range of X,ε for each [, T ino wo par: a compac e K ρ and i complemen. Therefore, I can be wrien a he expecaion of a um of (i) he compac par (when X,ε lie in he compac e) and (ii) he ail par (when X,ε lie ouide of he compac e): I = E x +E x g d,ε σ (X,ε ) xv (, X,ε ) I Kρ (X,ε ) d g d,ε σ (X,ε ) xv (, X,ε ) I K c ρ (X,ε ) d = E x [i + E x [ii. (6.5) In order o achieve lim ε I =, we hall ue localizaion argumen o deal wih E x [ii and he problem i reduced ono a compac e. On he compac e, i will be proved ha V ε and i parial derivaive converge o V and i correponding derivaive. Then, i i followed by he convergence of E x [i o Conrol of he ail par We apply Hölder inequaliy o he econd erm of (6.5) where E x [ ii σm [ E x T T (X,ε E x T (X,ε ) 4 ( + X,ε m ) d ) 4 ( + X,ε m ) d / [Q x (τ ρ < T ) / N(4, σ)e N(4,σ)(T ) ( + x 4 ) + N(m + 4, σ)e N(m+4,σ)(T ) ( + x m+4 ) +N(m + 4, σ)e N(m+4,σ)(T ) ( + x m+4 )d M 7 ( + x m+4 ), (6.6) for ome conan M 7 = M 7 (T, m, σ).

13 From (6.6) and he exi probabiliy eimae of he proce (X,ε ) in (5.5), i i concluded ha E x [ ii σm M7 ( + x m+4 ) D ( + x m+3/ ) ρ, N(σ)e N(σ)(T ) ( + x )(T ) ρ for ome conan D = D (M, T, m, σ). A ρ increae, i.e. he compac e K ρ become larger and larger, he proce will be le and le likely o deviae ouide of he e K ρ. Then, we would expec he ail par i mall enough for ufficienly large ρ. Thi reul i ummarized in he following propoiion. Propoiion 6.. Given ϕ(x) which aifie he condiion (6.), E x [ ii a ρ Conrol of he compac par In order o prove ha he compac erm i negligible when ε i ufficienly mall, we need he convergence of xv ε o xv which would imply ha g d,ε gradually vanihe a ε end o. Recall he reul regarding he regulariy of he oluion of he BSB equaion in [8 [6. Theorem 6.. If ϕ i locally Lipchiz coninuou and ϕ, and ϕ have polynomial growh, hen he oluion V ε of (4.) belong o Cp, ([, T ) R). Moreover, xv ε (, x) i Hölder coninuou in x wih an exponen κ (, for any [, T ). Remark 6.. All he conan in he polynomial conrol and Hölder coninuiy only depend on he bound of he volailiy inerval [σ, σ + ε. Since we are only inereed in he cae where ε are mall, we can chooe hee conan including Hölder exponen o be univeral, i.e. independen of ε; ee [8 [6 or [. We recapiulae he heorem a follow V ε (, x) B ( + x b ), x V ε (, x) B ( + x b ), x V ε (, x) B ( + x b ), x V ε (, x) xv ε (, y) B3 x y κ, (6.7) for any [, T ) where all conan B, B, B, B 3 and b, b, b, and κ are univeral, i.e. independen of ε. Lemma 6.3. Given ϕ(x) which aifie he condiion (6.), xv ε (, ) }, a a family of funcion of x indexed by ε, uniformly converge o xv (, ) on he compac e K ρ a ε end o for any fixed [, T ). Proof. The proof i imilar o ha of Theorem 5..5 in [. Indeed, becaue V ε ha up o econd order parial derivaive in x, by following he argumen in he proof of he Theorem 5..5 in [ we conclude ha x V ε converge o x V a ε end o, uniformly for all x K ρ. However, V ε (, ) i only wice differeniable wih repec o x. In order o obain he uniform convergence of he equence of econd parial derivaive xv ε}, we fir oberve from (6.7) ha xv ε} i a family of uniformly bounded and Hölder coninuou funcion of x on K ρ. I implie ha xv ε} i equi-coninuou. Then, here exi a ub-equence xv ε } which converge uniformly on K ρ. Togeher wih he convergence of x V ε } o x V, we conclude ha he xv ε } uniformly converge o xv for all x K ρ. 3

14 Since he limi xv i independen of he choice of ub-equence x V ε} converge uniformly o xv. Indeed, if here i a ub-equence converge o xv, hen according o he Arzela-Arcoli Theorem xv ε }, we claim ha xv ε } doe no xv ε } ha a uniformly xv ε }. Again, ogeher wih he convergence of x V ε } o x V, convergen ub-equence xv ε } ha o converge o xv, which i a conradicion wih he aumpion. Therefore, he lemma follow. Le S ε = x xv ε (, x) xv (, x), } ε ε K ρ. Noe ha S ε } ε a a family of e indexed by ε i non-increaing a ε decreae o. Define S := lim ε S ε. Lemma 6.4. Given ϕ(x) which aifie he condiion (6.), for any fixed [, T ) } S = x K ρ x V (, x) =. Proof. Noice ha if x K ρ uch ha xv (, x) =, hen x S ε ha } S x K ρ x V (, x) =. for all ε >. I implie On he oher hand, if xv (, x) > for any x K ρ, hen due o he uniform convergence of xv ε (, ) } here exi ε > uch ha xv ε (, x) > for all ε < ε. Hence, xv ε (, x) xv (, x) > for all ε < ε, i.e., x / S ε ε < ε. I i followed by x / S. Similarly, we can prove ha any x K ρ uch ha xv (, x) < doe no lie in S eiher. Therefore, we can claim ha } S x K ρ x V (, x) =. For any fixed [, T ), we ake S ε := he cloure of S ε for each ε. For hi equence of cloed, bounded and non-increaing e, we define i limi S := lim ε Sε. Due o he coninuiy of xv (, x) in x, i i rue ha S = S for any fixed [, T ). The following lemma ell u ha he ame relaionhip hold beween S ε and S ε. Lemma 6.5. Given ϕ(x) which aifie he condiion (6.) and any fixed [, T ), i i rue ha S ε = S ε. Proof. To prove he lemma, i uffice o how ha x S ε, eiher x S ε or x S. Indeed, if i i rue, hen ogeher wih he fac ha S ε S (by he monooniciy of S ε }) we can ee ha S ε = S ε, i.e. S ε i a cloed e. For any x S ε, according o he definiion of cloure, here exi a equence x n } S ε uch ha lim n x n = x. For each x n, here i a ε n ε uch ha xv εn (, x n ) xv (, x n ). Since < ε n ε for all n >, here exi a ε [, ε and a ub-equence ε n } uch ha lim n ε n = ε. 4

15 We ake he correponding ub-equence x n } which converge o x by aumpion. Then, lim n xv εn (, x n ) xv (, x n ) = xv ε (, x ) xv (, x ). (6.8) Indeed, x V ε n (, x n ) xv (, x n ) xv ε (, x ) xv (, x ) x V ε n (, x n ) xv (, x n ) xv ε n (, x ) xv (, x n ) + x V ε n (, x ) xv (, x n ) xv ε n (, x ) xv (, x ) + xv ε n (, x ) xv (, x ) xv ε (, x ) xv (, x ) xv (, x n )B 3 x n x κ + xv ε n (, x )B 3 x n x κ + xv (, x ) x V ε n (, x ) xv ε (, x ) (6.9) where he la inequaliy i due o Hölder coninuiy of xv ε for all ε. Then, due o he coninuiy of xv (, ), xv (, x n ) converge o xv (, x ) a n end o. Baed on he fac of he uniform boundedne of xv ε (, ) } on K ρ and convergence of xv ε n (, x ) } o xv ε (, x ), i i clear ha all hree erm in (6.9) converge o. Therefore, (6.8) follow. We dicu wo poible cae for he value of ε.. Cae ε > : We alo know ha ε ε, ince ε n ε. According o he definiion of S ε, x S ε.. Cae ε = : From he fac (6.8), ( xv (, x ) ). Therefore, x V (, x ) =, i.e. x S due o he lemma 6.5. Therefore, he lemma i proved. Noe ha any x S i a zero poin of xv a ime. Le U(, x) := xv (, x). We hall conider zero e of U(, x) for any fixed [, T ). Wih he aumpion (6.) on ϕ, V ha up o fourh derivaive wih repec o x. Therefore, we can derive he equaion for U(, x) from (4.) a follow U + σ U + σx x U + σ x xu =, U(T ) = ϕ. Le x := log y, Ũ(, y) := U(, x). Then, Ũ(, x) olve he following PDE Ũ + 3 σ y Ũ + σ yũ + σ Ũ =. Noice ha all coefficien in he above equaion are conan. Therefore, Theorem B in [ and he remark below i are applicable o Ũ. They direcly implie ha he ize of he zero e of Ũ } Z = y Ũ(, y) =, i non-increaing a variable goe from T o. Noing ha he change of variable x = log y i a one-o-one mapping, we can conclude he following propoiion. Propoiion 6.3. If ϕ aifie (6.), hen xv ha a mo he ame number of zero poin a ϕ doe for any fixed. A hi poin, we can conclude ha if ϕ ha a finie number of zero poin hen he g d,ε will be vanihing a ε decreae o. However, o achieve he goal ha expecaion of compac par goe o, we ill need o how ha he law of variable X,ε doe no give a poiive probabiliy o any ingle poin for any fixed [, T. 5

16 Recall he main Theorem. in [4 ha for every (, T ), he marginal law of M = α dw doe no weigh poin, where (α ) i any progreively meaurable proce uch ha < σ α σ + ε. The following lemma i a imple exenion o he above reul. Lemma 6.6. Le (X,ε ) olve he SDE (5.). For any (, T, X,ε doe no weigh poin. Proof. Due o he ranformaion applied in he ecion 5., we only need o prove he claim for he proce (Y,ε ) which olve on he probabiliy pace (Ω, F, Q). Le ( ξ = exp Define a meaure Q on F T by dy,ε = (α,ε ) 8 ( α,ε) d + α,ε dw, d + d Q = ξ T dq. According o Giranov heorem, under he meaure Q W = ) α,ε dw, for T. α,ε d + W i a Brownian moion and (Y,ε ) ha he following dynamic dy,ε = α,ε d W. Noe ha he wor cae cenario volailiy proce for χ, (α,ε ) i a adaped and bounded proce. According o he Novikov condiion, (ξ ) i a maringale, and herefore Q and Q are wo equivalen meaure. From he Theorem. in [4, we learn ha he law of Y,ε doe no weigh poin under he meaure Q, for any [, T ). Due o equivalence beween Q and Q, we can claim ha doe no weigh poin under he meaure Q. Therefore, he lemma follow. Y,ε For given [, T [, we can no direcly ue he coninuiy of a probabiliy meaure o claim ha lim ε Q x X,ε S [ ε = Qx X, S, becaue boh he proce (X,ε ) and he e S ε depend on ε. Therefore, we define a capaciy from he law of a family of random variable X,ε } ε a follow for any A B(R). c(a) := up Q x (X,ε A), ε [, Propoiion 6.4. X,ε converge weakly o X for any >. Proof. I i a direc implicaion of he lemma 3.. If ( ϕ ha a finie number of zero poin, hen due o he lemma 6.6 Q x X,ε S ) = for any ε >. Thi fac direcly lead o c( S ) =. Due o he weak convergence of X,ε }, he family of law of X,ε } i weakly compac. 6

17 Lemma 6.7. If ϕ aifie he condiion (6.) and ϕ ha a finie number of zero poin, hen [ lim Q x X,ε S ε ε = for any (, T ). Proof. Fir, we oberve ha Q x ( X,ε S ε ) c( Sε ). Noice ha } Sε i a equence of decreaing cloed e and converge o S a ε goe o. Becaue of he weak compacne of he law of X,ε } and Lemma 8 in [7, i can be een ha c( S ) ε c( S ) =. Then, he lemma i rue. Theorem 6.3. If ϕ aifie (6.) and ϕ ha a finie number of zero poin, hen here exi ε > uch ha E x [ i < δ for any fixed ρ >, any fixed poin (, x) (, T R and δ > a long a ε < ε. Proof. Recall from (6.5) ha E x [i = E x g d,ε (, X,ε )I Kρ (X,ε )σ (X,ε ) xv (, X,ε )d Noe ha g d,ε (, x) can only ake hree poible value:,, }. Indeed, g d,ε (, x) if = xv ε (, x) xv (, x) <, if xv ε (, x) xv (, x). Therefore, due o (6.6) i follow ha E x [ i = E x g d,ε (, X,ε ) I Kρ(δ) (X,ε )σ (X,ε ) x V (, X,ε ) d σm [ E x T (X,ε σm M8 ( + x m+4 ) Due o he lemma 6.7, i follow ha T ) 4 ( + X,ε m ) d a ε. Therefore, he lemma i concluded. / Q x (X,ε S ε )d /. Q x (X,ε S ε )d, Now, we are ready o claim he main reul in hi ecion. Q x (X,ε. S ε )d Theorem 6.4. If ϕ aifie (6.) and ϕ ha a finie number of zero poin, hen lim ε I = for any fixed (, x) [, T ) R. Proof. Noe ha I E x [ i + E x [ ii. For any δ >, due o he propoiion 6. here exi ρ (, x, δ) > uch ha E x [ ii < δ for all ρ > ρ (, x, δ). By he heorem 6.3, for he given ρ (, x, δ) and δ here exi ε (, x, ρ (, x, δ)) uch ha E x [ i < δ for any ε < ε (, x, ρ (, x, δ)). Therefore, he heorem follow. / 7

18 From he proof, we eenially derived ha ᾱ = σ + εḡ i a good approximaion of he wor cae cenario volailiy α,ε ; ee heir definiion in he ecion 5. Togeher wih he properie of he law of he ae price proce in he wor cae cenario, we proved he main heorem ha Z ε (, x) goe o a ε, for any (, x) [, T ) R. Therefore, we can conruc an approximaion of V ε of order o(ε): V + εv. The performance of hi approximaion procedure i udied numerically in he nex ecion. 7 Numerical reul In hi ecion, we will work on a non-rivial example: a ymmeric European buerfly wih he following payoff funcion ϕ(x) = (x 9) + ( x) + + (x ) + (7.) which i neiher convex nor concave; ee he Figure. Payoff: ϕ Ae Price Figure : The payoff funcion of a ymmeric European buerfly Even hough he payoff funcion (7.) doe no aify he condiion of he main heorem 4., we can conider a regularizaion ϕ of ϕ, which aifie he condiion of he Theorem 4.. Moreover, ϕ can be choen ufficienly cloe o ϕ uch ha V (ϕ) + εv (ϕ) [V ( ϕ) + εv ( ϕ) ε. Tha i, he approximaion of ϕ: V ( ϕ) + εv ( ϕ) i a good proxy for ha of ϕ. Indeed, we numerically compue he wor cae cenario price V ε (ϕ), by he cheme provided in [5. I i proved by Barle [4 ha he numerical oluion from ha cheme i locally uniformly convergen o V ε, he unique vicoiy oluion of (4.), a he cheme become finer. We alo compare he numerically compued wor cae cenario price wih i approximaion: V (ϕ) + εv (ϕ), where V (ϕ) i given by he Black-Schle formula and V (ϕ) i numerically compued by a imple difference cheme according o he equaion (4.3). Becaue he cheme for compuing V ε (ϕ) ue Newon ieraion in each ime ep o deal wih he nonlineariy, our approximaion i compued a lo more efficienly. For viual comparion of he wor cae cenario price wih correponding approximaion, we how complee numerical reul for a very mall ε =.6, a mall ε =., and a ε =.5 8

19 which i no o mall. Throughou all he experimen, we e σ =.5 and T =.5. When ε =.5, he upper bound of he volailiy inerval i σ + ε =., which i /3 larger han he bae volailiy level σ =.5. In oher word, even if ε =.5 i mall, 5% volailiy i ignifican. From Figure, we noe ha he wor cae cenario price are higher han he Black-Schole price. Tha i, we need exra cah o uper-replicae he opion when facing he model ambiguiy. I alo can be noed ha he fir order correced price capure he main feaure of he wor cae cenario price V ε for differen value of ε. Cae: ε =.6 Price 3 4 Price Ae Price 5 5 Ae Price Cae: ε =. Price 3 4 Price Ae Price 5 5 Ae Price Cae: ε =.5 Price Price Ae Price 5 5 Ae Price Figure : The black curve repreen he wor cae cenario price and he red curve repreen he Black-Schole price; he blue curve repreen he difference beween he wor cae cenario price and i Black-Schole price and he green curve i ε V ; all curve are ploed again ae price. To ee he rend of he error of our approximaion a ε increae, we chooe 8 equally paced value from o.5 for ε. For each ε, we compue he error of he approximaion, which i defined by error(ε) = up V ε (, x) V (, x) εv (, x). (7.) x 9

20 Error: V ε (V +εv) ε = σ σ Figure 3: Error for differen value of ε; A hown in he Figure 3 he error increae a ε become larger, o he econd order approximaion will be needed o improve accuracy for large value of ε. However, here i no an abrup change in he range of ε we chooe. 8 Concluion In hi paper, we have udied he aympoic behavior of he wor cae cenario opion price a he degree of model ambiguiy vanihe. Thi udy no only help u underand how a linear expecaion urn ino a ublinear expecaion, bu alo give u an approximaion procedure of wor cae cenario opion price when ε i mall. From he numerical reul, we ee ha he approximaion procedure work well even when he upper bound volailiy i /3 larger han he lower bound. Noe ha he wor cae cenario price i ofen compued o evaluae he rik in a porfolio. Our approximaion procedure improve he efficiency of hi evaluaion, becaue i avoid he Newon ieraion which i employed in he cheme for V ε. Moreover, he wor cae cenario price V ε ha o be re-compued for a new value of ε. However, he equaion (4.) and (4.3) for V and V are independen of ε, o he approximaion only require u o compue V and V once for all mall value of ε. Reference [ S. Angenen. The zero e of a oluion of a parabolic equaion. J.reine angew. Mah., 39:79 96, 988. [ P. Arzner, F. Delbaen, J.-M. Eber, and D. Heah. Coheren meaure of rik. Mahemaical Finance, 9:3 8, Jul 999. [3 M. Avellaneda, A. Levy, and A. Para. Pricing and hedging derivaive ecuriie in marke wih uncerain volailiie. Appl. Mah. Finance, :73 88, 995. [4 G. Barle. Convergence of numerical cheme for degenerae parabolic equaion ariing in finance heory. Numerical mehod in finance, 997.

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