Loss of martingality in asset price models with lognormal stochastic volatility

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1 Loss of maringaliy in asse price models wih lognormal sochasic volailiy BJourdain July 7, 4 Absrac In his noe, we prove ha in asse price models wih lognormal sochasic volailiy, when he correlaion coefficien beween he Brownian moion driving he volailiy and he one driving he acualized asse price is posiive, his price is no a maringale Inroducion On a filered probabiliy space Ω, F, F, P, we consider he following risk-neural model for he acualized asse price X : dx = σ X ρ db + 1 ρ dw, X = x σ = e Y 1 dy = db + µd γy d, Y = y where B, W is a wo-dimensional F -Brownian moion, y, µ and γ belong o R, x and are posiive consans and he correlaion coefficien ρ beween he Brownian moion ρ B + 1 ρ W driving he asse price and he Brownian moion B driving he volailiy belongs o [ 1, 1] In his model, firs inroduced by Sco [7] p46, he volailiy σ is he exponenial of he Ornsein-Uhlenbeck process Y When he elasiciy coefficien γ is zero, σ evolves according o he Black-Scholes Sochasic Differenial Equaion dσ = σ db + µ + /d, and 1 is he model inroduced by Hull&Whie [3] and a special case of he very popular SABR model [1] Since X = x E e Ys ρ db s + 1 ρ dw s, where E sands for Dooleans-Dade exponenial, his process is a non-negaive F local maringale and herefore a super-maringale his leads o he following naural quesion : is X a ENPC-CERMICS, 6-8 av Blaise Pascal, Cié Descares, Champs sur Marne, Marne-la-Vallée Cedex, France and MAHFI projec - jourdain@cermicsenpcfr 1

2 rue maringale? he answer urns ou o be affirmaive if and only if he correlaion coefficien ρ is non-posiive see heorem 1 below In addiion, when ρ >, i is possible o check ha EX is decreasing see Proposiion 4 As a consequence, he Call-Pu pariy relaion does no hold whaever he mauriy and srike of he opions he exisence of such arbirage opporuniies invalidaes he model For ρ, X is a maringale and we invesigae he inegrabiliy of X δ for > and δ > 1 his quesion is imporan for numerical consideraions : for insance, inegrabiliy of X is necessary o ensure ha he convergence of a Mone-Carlo esimaor of EX K + is ruled by he cenral limi heorem Basically, we obain ha X δ is inegrable when δ < 1/1 ρ and ha EX δ is infinie when δ > 1/1 ρ 1 Sudy of maringale propery Our main resul is he following one : heorem 1 Process X is a maringale if and only if ρ Remark he fac ha X is no a maringale when ρ > is no so bad since in order o modelize he increase of volailiy in krach siuaions, ρ is generally chosen non-posiive For >, by Jensen inequaliy and since Y sds is a Gaussian random variable wih posiive variance, E 1 exp e Ys ds E exp e Ysds = + herefore we canno rely on Novikov crierion and corollaries see [5] p198 o prove ha X is a maringale in case ρ In conras, in he models proposed eiher in [7] p41, [8] where he sochasic volailiy σ in he firs line of 1 solves or in [4], [] where dσ = µ γσ d + db σ = Y for Y = y + one easily checks ha µ γy s ds + >, c >, sup E e c σ < + Ys db s wih µ, y As a consequence if 1 < and 1 c, hen by Jensen inequaliy, E e 1 σ d 1 1 E e 1 σ d < By [5] Corollary 514 p199, one concludes ha X is always a maringale in such models

3 In order o deal wih he general SABR model [1], le us consider X solving dx = e Y X β ρdb + 1 ρ dw, X = x wih β, 1 and Y like in 1 Inroducing τ n = inf{, X > n}, one has EX τ n = x + E τn x + x + E e Ys X β s ds e Ys/1 β 1 β EX s τn β ds E e Ys/1 β 1 β 1 + EX s τn ds Remarking ha s E e Ys/1 β is locally bounded, using Gronwall s Lemma, and leing n +, one obains ha E ds is locally bounded which ensures ha eys Xs β X is a maringale In conclusion, in he SABR model, he acualized asse price may fail o be a maringale only in he limi case β = 1 Proof : As X is a super-maringale, i is enough o check ha he non-increasing funcion EX /x is no consan if and only if ρ > Using he independence of W and B and he fac ha Y is adaped o he naural filraion of B, one obains EX = x E E ρ = x E E ρ 1 e Ys db s E E ρ e Ys dw s B s, s e Ys db s In case ρ =, one concludes ha EX = x for any posiive ime o deal wih he case ρ, we are firs going o use Girsanov heorem in order o be able o apply Exercice 1 p 354 [6] his way, EX /x urns ou o be equal o he probabiliy for he explosion ime of a well-chosen sochasic differenial equaion o be greaer han We will finally analyse wheher his explosion ime is finie wih posiive probabiliy hanks o Feller s es for explosions [5] p Le us inroduce he probabiliy measure Q such ha dq γy µ dp = E F According o Girsanov heorem, B = B + For any, y + B = y + B γ γy µ + γb s db s + γb s ds is a Brownian moion under Q y + B s ds + µ As rajecorial uniqueness holds for he Ornsein-Uhlenbeck Sochasic Differenial Equaion, by Yamada Waanabe heorem, he law of B, y + B under probabiliy measure Q is he 3

4 same as he law of B, Y under probabiliy measure P As a consequence, EX ] = E [E ρ expy + B s d x B s = E [E ρ ] = E [E bb s db s expy + B s db s exp ρ γy µ expy + B s E γy µ + γb s ds + γb s db s ] where bz = ρ expy + z + µ γy γz Le us briefly recall he link made in [6] Exercice 1 p354 beween he las expecaion and he probabiliy for he explosion ime of he Sochasic Differenial Equaion Z = B + 3 bz s ds 4 o be greaer han Since funcion b is locally Lipschiz coninuous, for any n N, here exiss a bounded and globally Lipschiz coninuous funcion b n which coincides wih b on inerval [ n, n] We denoe by Z n he soluion of he equaion similar o 4 wih b replaced by b n and inroduce τ n = inf{ : Z n > n} hen Z = n 1 {τn 1 <τ n}z n convenion : τ = solves 4 on ime-inerval [, τ where τ = lim n + τ n Le us also define σ n = inf{ : B > n} By Girsanov heorem and since b n coincides wih b on inerval [ n, n], Pτ n > = E 1 {σn>}e b n B s db s = E 1 {σn>}e Leing n + hen using 3, we conclude ha Pτ > = E E bb s db s = EX x bb s db s In order o analyse he explosion ime τ of he sochasic differenial equaion 4 hanks o Feller s es for explosions, we inroduce consans ρ = ρe y / and η = µ γy / so ha he drif coefficien of his equaion wries bz = ρ expz/ + η/ γz Noice ha he sign of ρ is he same as he one of ρ Funcion z pz = e ρ exp γx ηx ρe x dx 5 4

5 is a scale funcion According o [5] heorem 59 p348, Pτ = + = 1 is equivalen o v+ = v = + where funcion v is given by vz = = z z p x p y dydx exp γx ηx ρe x exp γy + ηy + ρe y dydx 6 o conclude he proof, we are going o check ha v+ < + if ρ > and ha v+ = v = + if ρ < Case ρ > ie ρ > : le > be such ha y, γy + η + ρe y > By inegraion by pars, one has for x, exp γy + ηy + ρe y dy = exp γx + ηx + ρe x γx + η + ρe x exp γx 1 + η + ρe γ + η + ρe + ρey γ exp γy + ηy + ρe y γy + η + ρe y dy One may choose large enough o ensure ha y, ρ e y γ 1 γy + η + ρey hen for any x, exp γy + ηy + ρe y dy exp γx + ηx + ρe x γx + η + ρe x and one easily concludes ha v+ < + Case ρ < ie ρ < : hen p+ = + which implies v+ = + according o [5] Problem 57 p348 If γ >, hen p = and herefore v = + I only remains o check ha v = + in case γ Since for non posiive y, exp ρe y belongs o [exp ρ, 1, v = + is equivalen o w = where wz = z If γ =, hen η = µ/ and expγx ηx exp γy + ηydydx wz = { z if µ = µ z + µ e µz/ 1 if µ which ensures w = + If γ <, seing < η/γ, one obains by inegraion by pars ha for x exp γy + ηydy = exp γx + ηx η γx exp γx + ηx η γx exp γ + η η γ exp γ + η η γ γ exp γy + ηy η γy dy 5

6 Hence for any x, one has exp γy + ηydy exp γx + ηx η γx + C, where he consan C does no depend on x One deduces ha w = + Remark 3 he argumen given a he end of he previous proof o check ha v = + in case ρ < also leads o he same conclusion in case ρ > When he correlaion coefficien ρ is posiive, he non-increasing and non-negaive funcion EX is no consan I is naural o wonder wheher his funcion is decreasing and wheher i ends o as + he nex proposiion answers boh quesions : Proposiion 4 Assume ha ρ > hen EX is decreasing In addiion, EX ends o as ends o + if and only if eiher γ > or γ = and µ Remark 5 As a consequence, when ρ >,, K >, EX K + EK X + < x K ie he Call-Pu pariy relaion does no hold Proof : Le us firs deal wih he limi of EX = x Pτ > as ends o + One easily checks ha he scale funcion pz defined by 5 saisfies p = if and only if eiher γ > or γ = and η Because η = µ γy /, he laer condiion is equivalen o γ = and µ Since v+ is finie according o he proof of heorem 1 and v = + according o Remark 3, by [5] Proposiion 53 p35, one concludes ha Pτ < + = 1 and equivalenly lim + EX = if and only if γ > or γ = and µ Le us now check ha EX is decreasing As we need o emphasize he dependence on he iniial condiions, we denoe X x,y, Y y he soluion of 1 One has, x >, y R, EX x,y = x EX 1,y Le us firs check ha for any posiive, he se A = {y R : EX 1,y < 1} has posiive Lebesgue measure Indeed if > is such ha he Lebesgue measure of A is zero, remarking ha by he Markov propery, EX 1,y = E EX 1,y F = E EX x,y x,y=x 1,y,Y y = E X 1,y EX 1,y y=y y, and ha since he law of Y y is absoluely coninuous wih respec o he Lebesgue measure PY y A =, we obain EX 1,y = EX1,y herefore A = A By inducion, for any n N, A n = A and for y R \ A, X 1,y is a maringale, which conradics heorem 1 Le now s < Again by he Markov Propery, EX 1,y = E X 1,y +s/ E X s/ 1,y y=y y +s/ 6

7 Since he law of Y y +s/ is equivalen o he Lebesgue measure, P Y y +s/ A s/ > One deduces ha EX 1,y < EX 1,y +s/ As he righ-hand-side is no greaer han EX 1,y s, one concludes ha EX 1,y is decreasing Inegrabiliy of X δ for δ > 1 Proposiion 6 Le > and δ > 1 If ρ = hen EX δ = + If ρ <, hen EX δ < + if and only if one of he following condiions is saisfied : δ < 1/1 ρ δ = 1/1 ρ and γ > δ = 1/1 ρ, γ = and µ + Proof : Le us compue X o he power δ wih δ > 1 : X δ = x δ exp δ ρ e Ys db s + 1 δ1 ρ 1 e Ys ds E δ 1 ρ herefore, reasoning like in, one obains [ EX δ = x δ E exp δ ρ e Ys db s + 1 δ1 ρ 1 e Ys dw s ] e Ys ds 7 In case ρ =, by Jensen inequaliy and since Y sds is a Gaussian variable wih posiive variance, δδ 1 EX δ = xδ [exp E ] [ δδ 1 e Ys ds x δ E exp e Le us now deal wih he case ρ < According o 1 and Iô s formula, e Y e y = e Ys db s + µ + / γy s e Ys ds Ysds ] = + Insering in 7 he expression of eys db s obained from his formula, one obains ρ EX δ = xδ [exp E δ ey e y ρ + γy s µ / + 1 ] δ1 ρ 1e Ys e Ys ds Under any of he hree condiions saed in he Proposiion, funcion ρ y R γy µ / + 1 δ1 ρ 1e y e y is bounded from above by a finie consan C As a consequence, ρ δ ey e y ρ + γy s µ / + 1 δ1 ρ 1e Ys e Ys ds δc ρe y / 7

8 and for any >, sup EX δ xδ exp δc ρey / [, ] Le us now suppose ha none of he hree condiions saed in he Proposiion is saisfied hen here is a posiive consan ε such ha funcion ρ y R γy µ / + 1 δ1 ρ 1e y e y εe y is bounded from below by a finie consan As a consequence here is a posiive consan C such as [ ρ ] EX δ CE exp δ ey + ε e Ys ds By Jensen inequaliy, EX δ CE [exp δ ρ ey + εe 1 Ysds] [ = CE exp δρe Y / E exp δεe 1 Since he covariance marix of he Gaussian vecor Y, Y sds is non-degenerae, E exp δεe 1 Ysds Y = + almos surely and one concludes ha EX δ = + Ysds ] Y Remark 7 For ρ >, when one of he following condiion is saisfied δ > 1/1 ρ δ = 1/1 ρ and γ > δ = 1/1 ρ and γ = and µ +, hen funcion y R ρ γy µ / + 1 δ1 ρ 1e y e y is bounded from below herefore EX δ CEexpδρeY / = + when > Bu i does no seem easy o analyse wheher EX δ is finie when none of he previous condiions holds References [1] PS Hagan, D Kumar, AS Lesniewski, and DE Woodward Managing Smile Risk Preprin, [] SL Heson A Closed-Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions he Review of Financial Sudies, 6:37 343, 1993 [3] J Hull and A Whie he Pricing of Opions on Asses wih Sochasic Volailiies he Journal of Finance, XLII:81 3, june 1987 [4] J Hull and A Whie An analysis of he bias in opion pricing caused by a sochasic volailiy Advances in Fuures and Opions Research, 3:9 61,

9 [5] I Karazas and SE Shreve Brownian Moion and Sochasic Calculus Springer-Verlag, 1988 [6] D Revuz and M Yor Coninuous maringales and Brownian moion Springer-Verlag, 1991 [7] L Sco Opion pricing when he variance changes randomly : heory, esimaion and an applicaion Journal of Financial and Quaniaive Analysis, : , 1987 [8] EM Sein and JC Sein Sock Price Disribuions wih Sochasic Volailiy: An Analyic Approach he Review of Financial Sudies, 44:77 75,