hal , version 4-27 May 2013

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1 Inernaional Journal of Theoreical Applied Finance c World Scienific Publishing Company Monooniciy of Prices in Heson Model hal , version 4-7 May 13 S. M. OULD ALY Universi Paris-Es, Laboraoire d Analyse e de Mahmaiques Appliques 5, boulevard Descares, Marne-la-Valle cedex, France sidi-mohamed.ouldaly@univ-mlv.fr Received 9 May 1 Revised 4 Ocober 1 In his aricle, we sudy he price monooniciy in he parameers of he Heson model for a conrac wih a convex pay-off funcion; in paricular we consider European pu opions. We show ha he price is increasing in he consan erm in he drif of he variance process decreasing in he coefficien of he linear erm in he drif of variance process. We also show ha he price is increasing in he correlaion for small values of he sock decreasing for he large values. Keywords: Heson model; Monooniciy; Pu opions; Maximum principle; Correlaion. 1. Inroducion The main aracion of he Black-Scholes model is he abiliy o express he price of European opions in erms of a volailiy parameer. Moreover, for convex payoffs, hese formulas are sricly increasing wih respec o he volailiy parameer, which can cover he risk associaed wih his parameer hrough he purchase or sale of opions. However, following he rejecion of deerminisic volailiy assumpion by empirical sudies, praciioners are increasingly convinced ha he bes way o model he dynamics of an underlying is o consider a model he process of insananeous variance is sochasic. In a general sochasic volailiy model, he variance process does no depend solely on is curren value. For example, under he Heson model, he variance process is given as he unique soluion of he following sochasic differenial equaion dv = a bv d + σ V dw, V = v. 1.1 The opions prices depend on he iniial value of he variance process v he parameers a, b, σ. These parameers are ofen calibraed o marke price of derivaives, so hey end o change heir values regularly. I is hen imporan o know he impac hey have on opion prices. The iniial value of he variance process has a posiive effec on prices of convex pay-off in a large class of sochasic volailiy models. See for example, Bergman e al. 1996, Hobson 1998, Janson Tysk Kijima. When 1

2 S. M. OULD ALY hal , version 4-7 May 13 he volailiy process is sochasic bu bounded beween wo values m M, El Karoui e al 1998 show ha he price of an opion is bounded beween he Black- Scholes prices wih volailiies m M. In [17], Romano Touzi show ha he derivaive of he value-funcion of an opion wih respec o he volailiy under models such as Hull Whie 1987 Sco 1987 has a consan sign, does no vanish before mauriy. Henderson 5 shows ha convex opion prices are decreasing in he marke price of volailiy risk. However, o our knowledge, he dependence of he European opion price on he correlaion parameer is no known in any sochasic volailiy model. In his aricle, we sudy he price monooniciy of European pu opions wih respec o he parameers v, a, b. We firs show ha he value funcion of pu price is a classical soluion of he Black-Scholes equaion. Then using a Maximum principle we show ha he price is increasing in he iniial value of he variance process as well as in he consan erm in he drif of he variance process decreasing in he coefficien of he linear erm in he drif of variance process. We also show ha he price is increasing in he correlaion for small values of he sock decreasing for he large values. This paper is organized as follows: In secion we recall some properies of he pu price in he Heson model. In secion 3 we sudy he monooniciy of he price wih respec o he parameers of he drif of he variance process. The secion 4 is devoed o he sudy of he monooniciy wih respec o he correlaion.. Preliminaries Under a complee filered probabiliy space Ω, F, F, P saisfying he usual condiions, we consider he Heson sochasic volailiy model for a price process S, defined by he following sochasic differenial equaions ds S = V dw 1, dv = a bv d + σ V dw, d W 1, W = d,.1 a, b, σ > < 1. Le S, V be he soluion of his equaion wih iniial value S = s, V = v. One can wrie S s as S = s exp Vs dws 1 1 V s ds. The process S is a local maringale; i is even a rue maringale, by Mijaović Urusov [14]. Thereby, using he Call-Pu pariy, all he resuls of his paper hold for Call opions. We consider an European pu opion on S wih srike K mauriy. Is curren price is given, for s, v R + R +, by P, s, v = E [K S + S = s; V = v]..3

3 Monooniciy of Prices in Heson Model 3 If we replace he pu pay-off by a funcion g C R such ha xg x g are bounded, hen Eksröm, Tysk 1 cf. [4] Theorem..3 show ha he funcion u, s, v := E [gs S = s, V = v].4 hal , version 4-7 May 13 is a classical soluion of he pricing equaion. In paricular, i saisfies u C 3 R + C 1,, R 3 + C 1,,1 R + R + R +. In addiion, a probabilisic represenaion of he derivaive of u wih respec o v is given as u, s, v = E v dŝs Ŝ s [ e bτ Ŝs τ u s τ, Ŝs τ, ˆV v τ dτ ],.5 Ŝs, ˆV v is he unique soluion saring from s, v o he sochasic differenial equaion = σd + ˆV v dw 1, d ˆV v = a + σ b ˆV v d + σ ˆV v dw..6 Obviously, he European pu pay-off does no saisfy he assumpions of his heorem. Neverheless, Proposiions of [4] which require only g o be coninuous bounded ensure ha P CR 3 + C 1,, R + 3 so ha Lϕ = ϕ + LP, s, v =,, s, v R + R + R +, P, s, v = K s +, s, v R + R +, a bv v + 1 s v.7 s + 1 σ v + σsv ϕ..8 v v s 3. Monooniciy wih respec o he parameers v, a b In his secion we sudy he monooniciy properies of he pu price wih respec o he parameers v, a b. We firs give an exension of he resul of [4] o he European pu pay-off. Theorem 3.1. In addiion o.7, we have P C 1,,1 R + R + R +. Furhermore, he derivaive of P wih respec o v is given by [ P ], s, v = E e bτ h τ, v Ŝs τ, ˆV τ v dτ, 3.1 Ŝs, ˆV v is he soluion saring from s, v o.6 h is defined on R + 3 by hτ, x, y = E y K N logx/k τ Vu dw u + 1 τ V udu τ V udu, τ V udu 3. N is he cumulaive disribuion funcion of he sard normal law.

4 4 S. M. OULD ALY Remark 3.1. Noe ha he funcion h, s, v is simply s ss P, s, v. As a direc consequence of his heorem, we have for any, s >, he funcion v P, s, v is increasing. Proof. Wriing S = s exp Vs dws + Vs dŵ s 1 V s ds, 3.3 hal , version 4-7 May 13 he Brownian moion Ŵ in independen from W, we have [ ] P, s, v = K E [Nd 1 ] s E e VudW u Vudu Nd, 3.4 d 1 = log s K Vu dw u + 1 V udu V udu d = d V u du. 3.6 We can wrie ss P, s, v, using his sochasic represenaion of P, as P s = E K/s v N logs/k Vu dw u + 1 V udu. Se V udu V udu 3.7 h, s, v = s P, s, v. 3.8 s The main purpose of he assumpion xg xg are bounded is o give a sochasic represenaion of he second derivaive of P wih respec o s o ensure ha i is coninuous bounded. Here we see ha we have a sochasic represenaion of ss P given by 3.7. Following he procedure of [4] cf Proposiion 4.1, 4., we only need o show ha he funcion [ ], s, v H, s, v := E e bτ h τ, Ŝs τ, ˆV τ v dτ 3.9 is coninuous on R + R + R + bounded by an inegrable rom variable. For his, we consider a sequence n, s n, v n, s, v show ha H n, s n, v n converges o H, s, v. As Ŝsn vn τ, ˆV τ converges o Ŝs τ, ˆV τ v in probabiliy, we only need o find an upper bound of H, s, v by an inegrable rom variable conclude by applying he dominaed convergence heorem.

5 Monooniciy of Prices in Heson Model 5 hal , version 4-7 May 13 To obain he desired upper bound, we firs noe ha for any x, y R τ we have hx, y, τ E y K π =: M τ, y. 3.1 τ V u du We can easily see ha for any y 1 y, we have M τ, y 1 M τ, y On he oher h, by he comparison heorem, we have I follows ha Then, ˆV v τ [ E hŝs τ, ˆV ] τ v, τ E [M τ, V v V v τ, a.s. 3.1 M τ, ˆV v τ M τ, V v τ, a.s τ ] = E E V v τ K π τ V u du = E v K π τ V udu The las line follows from he Markov propery of he process V. I follows ha [ ] E e bτ h v Ŝs τ, ˆV τ v, τdτ E v K π dτ τ V udu We have, by Dufresne [3], E v Moreover, for any v, we have 1 < +, τ > τ V udu lim I follows ha for any v, τ τ 3 Ev 1 = τ V udu E 1 dτ < τ V u v du The res of he proof of he Theorem is idenical o Proposiion 3.1 of [4] by using his upper bound. Thus, he funcion H is coninuous on R + R + R +.

6 6 S. M. OULD ALY Monooniciy wih respec o a b We now sudy he monooniciy properies of he pu price wih respec o he parameers a b. Noe ha he pahs of he variance process are increasing wih respec o a decreasing wih respec o b. This means ha increasing a generaes higher volailiy which will increase he Pu price. To verify his claim, we will le he pu price vary in erms of a b : We wrie [ P a,b, s, v = E K S a,b + S a,b = s; V a,b = v ], 3.19 hal , version 4-7 May 13 S a,b, V a,b is he unique soluion saring wih s, v of he sochasic differenial equaions = V a,b dw 1, ds a,b S a,b dv a,b = a bv a,b d + σ V a,b dw, d W 1, W = d. 3. The following maximum principle will be crucial for he proof of he main resul of his secion. The proof of his heorem can be found in he appendix. Theorem 3. Maximum Principle. For >, le µ = sup {µ > : ES µ < }. 3.1 Le L be he operaor defined by.8 ϕ C 1,, R + 3 CR 3 + so ha Suppose ϕ saisfies, M >, λ < µ : sup ϕ, s, v M λ. 3. τ, s M, v R Lϕ, s, v resp <,, s, v ], + [ R + R +, ϕ, s, v, s, v R + R Then ϕ resp ϕ > on R + 3. We esablish he monooniciy of P wih respec o a b in he following resul Proposiion 3.1. Le a > a 1 b 1 < b. We have P a1,b, s, v < P a,b, s, v, b,, s, v R P a,b1, s, v > P a,b, s, v, a σ,, s, v R Proof. For any a, b, le L a,b ϕ = rϕ ϕ + rs.. + a bv s v + 1 s v. s + 1 σ v. v + σsv. ϕ v s 3.6

7 Monooniciy of Prices in Heson Model 7 We can easily check ha L a,b P a,b P a1,b, s, v = a a 1 P a 1,b v,, s, v R + 3, P a,b P a1,b, s, v =, s, v R +. L a,b P a,b1 P a,b, s, v = b b 1 v P a,b 1 v,, s, v R + 3, P a,b1 P a,b, s, v =, s, v R hal , version 4-7 May 13 We have, by Theorem 3.1, he funcion v P a1,b v P a,b1 are posiive. Then, by Theorem 3., ha P a,b P a1,b > P a,b1 P a,b >. 4. Monooniciy wih respec o he correlaion This secion focuses on he monooniciy properies of he price of he European pu wih respec o he correlaion. Noe ha he mehod we used in he previous secion o esablish he monooniciy wih respec o v, a b can no be applied here. Indeed, he idea of his mehod was o differeniae.7 wih respec o he parameer considered obain a differenial sysem as Lu < on C u on C, which gives he sign of u by applying he maximum principle; while if we differeniae.7 wih respec o, we obain he sysem L P P P, s, v = σsv s v, s, v,, s, v ], T ] R + R +,, s, v =, s, v R + R As he sign of sv P is no necessarily consan, his does no allow us o deduce he sign of he derivaive of P wih respec o using he maximum principle. To analyze he impac of in he price P, we will sudy he sign of he derivaive of P wih respec o. This derivaive can be obained by differeniaing 3.4 wih respec o : P = E log K s e x N x Vu dw u + 1 I x Vu dw u + I dx, I udu 3 I 4. I := V udu. The sign P is no obvious, however he following figure shows ha here is a change of monooniciy depending on he value of he srike price. We see ha P is posiive for s < K = 1 negaive for for s > 1. In order o deermine if his change of monooniciy is unique, we will sudy in deails he sign of he derivaive of P wih respec o for s very large very small. For his we define he quaniies s, v = inf {s > : { s, v = sup s > : } P, s, v 4.3 } P, s, v. 4.4

8 8 S. M. OULD ALY hal , version 4-7 May 13 Fig. 1. P for s [.4,.5] K = 1, v =.1, b = 3, σ =. =.5. Having s > resp s < + means ha P is posiive resp negaive for s small resp s large. We nex presen he main resul of his secion. Theorem 4.1. For any, v > ] 1, 1[, we have < s, v s, v < Proof. We use he resuls obained in [16], i is shown ha for R sufficienly large, we have ln P 1 I + Vu dwu 1 > R µ + R 4.6 ln P 1 I + Vu dwu 1 < R µ R, 4.7 wih µ + = inf {p >, T p = } > 1, µ = inf {p >, T p = } { p T p = sup >, E Q p } exp V u du < +, 4.8 under Q he process V saisfies he sochasic differenial equaion dv = a b σpv d + σ V dw Q. 4.9 We can easily see ha, for k sufficienly large, we have ln P, e k, v µ k, ln P, e k, v 1 e k µ + k 4.1 P, e k, v lim k + kp, e k, v = µ, lim k + P, e k, v kp, e k, v 1 e k = µ

9 Monooniciy of Prices in Heson Model 9 By he comparison heorem, he process V is increasing wih respec o under Q see also [15] for p > decreasing for p <. This means ha for p > 1, T p as a funcion of is decreasing for any p >, T p is increasing. On he oher h, p T p is decreasing near µ + p T p is increasing near µ. I follows ha µ + is decreasing wih respec o µ is increasing wih respec o. This means ha, for k sufficienly large, we have P, e k, v > 4.1 hal , version 4-7 May 13 Thus 4.5. P, e k, v < So far we confirmed ha < s s < +, which means ha he Pu price is increasing in he correlaion for small values of he sock price decreasing for large values. The quesion is wheher s = s, which means ha here is only one poin s, v so ha he derivaive of P wih respec o is posiive for s s negaive for s > s. All numerical experimens seem o confirm his inuiion. In he nex secions, we will show ha s = s for shor long mauriies Small-Time Asympoic Behavior We sudy here he monooniciy wih respec o he correlaion for shor mauriies. The main resul of his secion is he following Proposiion Proposiion 4.1. Consequenly, For any ] 1, 1[ any v R +, we have P lim sign, e x, v = sign x lim s, v = lim s, v = Proof. Le S, V be he unique soluion of.6 saring wih s, v. By Forde Jacquier cf [6], we have lim log EK S + = Λ log K, for s > K 4.16 s lim log ES K + = Λ log K, for s < K, 4.17 s Λ is he Fenchel-Legendre ransform of he funcion Λ defined by vp Λp = σ 1 co 1 σp, for p ]p, p + [, 1 Λp =, for p R\]p, p + [, 4.18

10 1 S. M. OULD ALY hal , version 4-7 May 13 wih p p + are given by 1 1 arcan p = 1 σ 1 < π π + arcan σ 1 = + 1 σ 1 >, π + arcan p + = 1 σ 1 < + π arcan σ 1 = + 1 σ 1 >. 4. The funcion Λ is given by p x is he unique soluion of Λ x = xp x Λp x, 4.1 x = Λ p x 4. Λ is given by Λ v p = σ co 1 σp + σvp csc 1 σp σ co 1 σp. 4.3 Le Σ x be he Black-Scholes implied volailiy, defined as he unique soluion of P BS, s, k, Σ = KN P, K e x, v = P BS, K e x, K; Σ x, 4.4 logs/k + Σ/ logs/k Σ/ sn. 4.5 Σ Σ By Theorem.4 of [6], we have lim Σ x x = Λ x. 4.6 Wriing P, s, v in erms of he Black-Schole implied volailiy as in 4.4 noing ha he dependence of he righ erm of 4.4 wih respec o is only hrough Σ using he fac ha he Black-Scholes pu price is is increasing wih respec o he implied volailiy, we see ha p, s, v Σ logk/s have he same monooniciy wih respec o he correlaion. Therefore sign P, s, v = sign Σ log K s. 4.7 The implied volailiy is differeniable wih respec o he correlaion. Moreover, using Lemma 4.1 below, we have Σ x lim = x Λ x Λ x Λ x. 4.8

11 Monooniciy of Prices in Heson Model 11 Le s consider he derivaive of Λ x wih respec o, for x R. This derivaive is given by Λ x = p x x Λ p x Λ p x = Λ p x as Λ p x = x [ vp x 1 coθ x θ x csc θ x ] + 1 = σ, 4.9 coθ x hal , version 4-7 May 13 θ x := 1 σp x. 4.3 Using Lemma 4. below, which ensures ha, for any x R, we have coθ x θ x csc θ x + 1 >, 4.31 sign Λ x = sign vp x. 4.3 On he oher h, as p x has he same sign as x, we deduce ha for sufficienly small, we have Lemma 4.1. sign P = sign Σ x = sign Λ x = sign log K s. For any x, we have Σ x lim = x Λ x Λ x Λ x Lemma 4.. For any [ 1, 1] x R, we have coθ x θ x csc θ x + 1 > Large-Time Asympoic Behavior I is known ha for long mauriies he implied volailiy curve in a sochasic volailiy model flaens, so i does no depend on he srike. Under Heson model, Forde e al [7] showed ha under he assumpion b σ > he implied volailiy can be wrien as Σ x = 8V + a 1 x/ + o, 4.35 V a 1 are given below. The main resul of his secion is he following resul

12 1 S. M. OULD ALY Proposiion 4.. For any ] 1, 1[ such ha b σ > for any v >, we have lim + s, v = lim + s, v = Proof. We will use he noaions of [7]. Under he assumpion b σ >, we have, for any p ]p, p + [, hal , version 4-7 May 13 V p = lim 1 log E [exp px x ] = a b σp d ip, 4.37 σ d ip = b σp + σ p p 4.38 p ± := b + σ ± σ + 4b 4bσ Le s consider he funcion p : R ]p, p + [ defined by 1/ σ b + a + xσ σ +4b 4bσ p x x := σ +xaσ+a σ, for x R. 4.4 For sufficienly large x R, we have cf. [7] 1 S ES S e x + = 1 + A π exp 1 p x V 1 + O1/, 4.41 V is he Fenchel-Legendre ransform of V defined by V x := sup {px V p, p ]p, p + [ } 4.4 A is he funcion defined in a neighborhood of by 1 Up x Ax = V p x p x1 p x, 4.43 Up := d ip b σp + d ip a σ exp v a V p Similarly, he Black-Scholes implied volailiy can be wrien as cf. [7], Theorem 3. Σ x = 8V + a 1 x/ + o, 4.45 a 1 x = 8 log A V + 4p 1x In paricular, for any x R, we have lim + Σ x = 8V. 4.47

13 Monooniciy of Prices in Heson Model 13 Now using 4.45 we show, in a similar way as Lemma 4.1, ha We have V Σ x lim + = 8 V = V p + x V p p = V p = a σ σp + σp b σp b σp + σ p p hal , version 4-7 May 13 = a p 1 p b σp + σ p p The firs wo lines follow from he fac ha V p =. For =, we have V = a b 1 + >. 4.5 = σ b + σ /4 Lemma 4.3 below ensures ha he funcion ϕ defined in 4.55 is increasing. Noe ha, for any [ 1, 1], we have ϕ = p As ϕ = 1, we deduce ha for any, ϕ 1/ has he same sign as. This means ha Therefore, we have I follows ha for any x R, V ϕ 1/ >. 4.5 >, [ 1, 1] Σ x lim > I follows ha for sufficienly large, he pu price is increasing wih respec o he correlaion. Thus Lemma 4.3. The funcion ϕ defined by ] 1, 1[ ϕ := σ b + σ + 4b 4bσ σ 4.55 is increasing.

14 14 S. M. OULD ALY 5. Appendix Appendix A. Proof of Theorem 3. Suppose, s, v R + 3 so ha ϕ, s, v <. Consider S s, V v he unique soluion of he sochasic differenial equaions hal , version 4-7 May 13 ds s S s dv v = V v dw 1, = a bv v d + σ V v = v. S s = s, V v Le s define he F-sopping imes τ = inf dw, d W 1, W = d, { u [, ] : ϕ u, S s u, V v u } ϕ, s, v { [ ] c } 1 τ n = inf u [, ] : Su s Vu v n, n. A.1 A. A.3 We have Pτ < = 1. Applying he Iô formula o he process ϕ u, Su, s Vu v u beween τ τ n, we have ϕ τ τ n, Sτ τ s n, Vτ τ v n = ϕ, s, v + σ + As S V are in ], n], we have τ τn τ τn τ τn τ τn S s u V v u s ϕ u, S s u, V v u dw 1 u + V v u v ϕ u, S s u, V v u dw τ Lϕ u, S s u, V v u du. A.4 ϕ, s, v = E Lϕ u, Su, s Vu v du + E [ ϕ τ τ n, Sτ τ s n, Vτ τ v n ] E [ ϕ τ τ n, Sτ τ s n, Vτ τ v n ] ϕ, s, v P τ τ n + E [ ϕ τ n, S s τ n, V v τ ] n 1 τ> τn. A.5 Wriing {τ > τ n } = {sup u τ V u n} {sup S u n}, u τ A.6 we have E ϕ τn, S s τ n, V v τ n 1 τ> τn n λ P sup V u n + P sup S u n. A.7 u u Now using Doob s maringale inequaliy, we have P sup S u n u µ ES +λ n µ +λ = n λ P sup S u n u µ ES +λ n µ λ n. A.8

15 Monooniciy of Prices in Heson Model 15 Similarly, applying Doob s maringale inequaliy o he maringale e b V a b aking ino accoun he fac ha hal , version 4-7 May 13 we obain Therefore EV p <, p >, lim n nλ P sup V u n u lim n E ϕ τ n, S s τ n, V v τ n 1 τ> τn =. A.9 =. A.1 A.11 This means ha ϕ, s, v ϕ, s, v. A.1 Hence he conradicion ϕ, s, v is supposed o be negaive. Thus ϕ. Now assume Lϕ <. Le s ake, s, v wih >. Applying he Iô formula o he process ϕ u, Su, s Vu v u beween τ n, we have ϕ τ n, S s τn, V v τn = ϕ, s, v + τn Su s V v u s ϕ u, Su, s Vu v dwu 1 + τn σ V v u v ϕ u, Su, s Vu v dwτ + We ge, by he same way as before, Thus ϕ, s, v >. ϕ, s, v E τn Lϕ u, S s u, V v u du. Lϕ u, S s u, V v u du >. A.13 A.14 Appendix B. Proof of Lemma 4.1: The pu price P is given, in erms of he Black-Scholes implied volailiy, by logs/k + P, s, v = KN Σ logs/k sn Σ. B.1 Σ Σ Differeniaing his expression on boh sides wih respec o, we can wrie P as P logs/k +, s, v = KN Σ Σ logs/k. Σ B. On he oher h, by 4.6, we know ha lim Σ x x = Λ x. B.3

16 16 S. M. OULD ALY Moreover he funcion x / Λ x is C 1 on ] 1, 1[. We claim ha Σ is bounded near. This is equivalen o say ha P, s, v K N logs/k + Σ / is bounded. B.4 Σ hal , version 4-7 May 13 Wriing P = E K s exp we can wrie P as [ P, s, v = E Vu dw 1 u + Vu dw 1 u + Applying he Hölder inequaliy, wih p > 1, we have P, s, v [ E S Vu dw 1 u + PK S can be wrien as PK S = P K = N S logs/k + Σ Σ On he oher h, for any y >, we have N y 1 y Vu dwu I /, B.5 + Vu dw u S 1 K S ]. B.6 p] 1/p Vu dwu [PK S ] p 1 B.7 p, logs/k + +KN Σ Σ Σ K. B.8 exp y / π. B.9 I follows ha for any s > K sufficienly small, we have logs/k + N Σ Σ logs/k + Σ logs/k N Σ. B.1 Σ Σ Then, for s > K, here exiss a consan M > such ha, for sufficienly small, we have PK S M logs/k + N Σ. B.11 Σ I follows ha P, s, v N logs/k+ M [E Y p ] 1/p Σ Σ [ N logs/k + ] 1 Σ p, Σ B.1

17 Monooniciy of Prices in Heson Model 17 Y = exp Vu dw 1 u + Se x = logs/k. For small, Vu dw u Vu dw 1 u + Vu dwu I /. B.13 hal , version 4-7 May 13 We choose p so ha For his paricular p, we have x + Σ Σ x Σ. p = p = c. B.14 B.15 [ N logs/k + ] 1 Σ p M 3 exp log x M 4. Σ c Σ B.16 We nex show ha [E Y p ] 1/p is bounded for close o. For his, we use he usual inequaliy We ge Y 1 = exp 1 y e y + e y, y R. B.17 Y Y 1 + Y, B.18 Vu dw 1 u + + Vu dw u I / B.19 Y = exp 1 + Vu dw 1 u + Vu dw u I /. B. Boh Y 1 Y can be wrien as Y i = exp α i Vu dwu 1 + β i Vu dwu 1 I, i = 1,. B.1 In paricular, we have E Y p i = E exp α i p = E exp α i p Vu dwu 1 + β i p Vu dwu p I Vu dwu 1 + β i p p I. B.

18 18 S. M. OULD ALY By [16] he fac ha we have, for p sufficienly large, Vu dw 1 u = V v a + bi /σ, B.3 E Y p i = exp α i pv + a/σ + aϕ + vψ, B.4 hal , version 4-7 May 13 wih I follows ha ψ = b λ i σ + pσ b σ an g, p, ϕ = b σ + log cos g, p log cos g, p σ B.5 λ i g, p = pσ b λ i + arcan 1pσ b λ i pσ b, λ i 1p = α i p/σ λ i p = β i p p B.6 + α i bp/σ. B.7 [E Y p i ]1/p = exp α i v + a/σ + a ϕ p + v ψ. B.8 p In paricular, for p = p = c/, we have, for sufficienly small, Similarly, we have a ϕ p + v ψ p g, p cβ iσ vβ ic σ + arcan α i β i. B.9 cβi σ an + arcan α i. B.3 β i Noe ha he coefficien c in B.15 was chosen so ha π/ < cβ iσ We finally have, for i = 1,, lim [E Y p vβi c i ]1/p = exp σ + arcan α i β i < π/, for i = 1,. B.31 cβi σ an + arcan α i < +. B.3 β i I follows ha, using B.1 B.16, he claim B.4 is verified. We proceed similarly for s < K, by using he call price insead of he pu price.

19 Monooniciy of Prices in Heson Model 19 Appendix C. Proof of Lemma 4.: Le s se ηx = coθ x θ x csc θ x + 1 C.1 hal , version 4-7 May 13 ηx = ϕθ x, ϕ is defined by cosθ ϕθ = sinθ 1 θ sin + 1. C. θ For any x R, p x [p, p + ], we have θ := 1 σ p θ x 1 σ p + =: θ, x R. C.3 So we only need o show ha ϕ is posiive on [θ, θ]. We can easily see ha ϕ is C 1 on [ θ, θ ] \{}, is derivaive is given by ϕ θ = A simple sudy of he sign of he funcion sinθ + θ cosθ sin 3. C.4 θ θ + cosθ sinθ, C.5 shows ha i reaches is maximum on [ θ, θ ] a his maximum is equal o <. We deduce ha We only have wo possible siuaions: ϕ θ, θ [ θ, θ ]. C.6 Case > : In his case, we have θ = π + arcan e θ = arcan. C.7 On he oher h, he funcion ϕ is decreasing on [ θ, θ ]. In paricular, we have, for any θ [ θ, θ ], ϕθ ϕ θ = 1 arcan = 1 + arcan. We do he following change of variables y = = y. C.8

20 S. M. OULD ALY We obain ϕθ ϕ θ = 1 y + y y arcan y >, C.9 hal , version 4-7 May 13 as he minimum of he funcion y +y y arcan y is reached a he 3+ poin y = 17 is.78. Case < : In his case, we have θ = arcan e θ = π + arcan. C.1 The funcion ϕ is increasing on [ θ, θ ]. In paricular, we have, for any θ [ θ, θ], ϕθ ϕθ = arcan 1 = 1 + arcan. C.11 We do he following change of variables z = = z C.1 Thus ϕθ ϕθ = 1 z +z z arcan z >. Appendix D. Proof of Lemma 4.3 The funcion ϕ is defined on [ 1, 1], by for ] 1, 1[, ϕ = σ b + σ + 4b 4bσ σ, D.1 ϕ 1 = σ + 4b 4b σ, ϕ1 = 4σ + b 4b σ. D. The funcion ϕ is C 1 on [ 1, 1], is derivaive is given, by ϕ = b + σ + 4b 4bσ σ for ] 1, 1[, ϕ 1 = b σ σ + b 3 + σ σ + b, ϕ 1 = bσ σ σ + 4b 4bσ + p, D.3 b σ b σ 3 + σ. D.4 b σ

21 Monooniciy of Prices in Heson Model 1 We will show ha ϕ >, for any ] 1, 1[. We firs noe ha ϕ has he same sign as h = 1 + σ b + σ + 4b 4bσ 1 + bσ σ + 4b 4bσ. D.5 On he oher h, h can be wrien as h = α + α + β + γ γ/ α + β + γ, D.6 hal , version 4-7 May 13 α = σ b, β = σ e γ = 4bσ. D I follows ha h has he same sign as he quaniy λ α, β, γ Γ = α α + β + γ + α + β + γ/, Γ = { α, β, γ : α + β + γ }. D.8 D.9 Noe ha if α γ, hen h. To sudy he sign in he general case, we consider he derivaive of λ wih respec o γ. I is given by We discuss four cases γ λα, β, γ = α + α + β + γ. D.1 Case α γ : On Γ {α, γ }, we have λα, β, γ Γ = α α + β + γ + α + β + γ/. D.11 Case α γ : On Γ {α, γ }, we have λα, β, γ Γ = α α + β + γ + α + β + γ γ/. D.1 Case α γ : In his case, he minimum of λ on Γ {α, γ } is reached a γ = β his minimum is equal o β /. Case α γ : In his case, for any β, he funcion γ λα, β, γ is increasing on [, + [. In paricular, we have λα, β, γ λα, β, = α α + β + α + β. D.13 In all cases, we have λα, β, γ >, α, β, γ Γ. D.14 Thus, ϕ >, [ 1, 1].

22 S. M. OULD ALY Acknowledgmens I would like o hank Professor Damien Lamberon for many useful discussions anonymous referee who read he firs version helped me improve he presenaion. hal , version 4-7 May 13 References [1] Y. Z. Bergman, B. D. Grundy Z. Wiener, General Properies of Opion Prices, Journal of Finance, American Finance Associaion [] R. Con, Empirical properies of asse reurns: sylized facs saisical issues, Quaniaive Finance [3] D. Dufresne, The inegraed square-roo process. Research Paper no. 9, Cener for Acuarial Sudies, Universiy of Melbourne, 1. [4] E. Eksröm J. Tysk, The Black Scholes equaion in sochasic volailiy models, J. Mah. Anal. Appl [5] N. El Karoui, M. Jeanblanc-Picqu, S. E. Shreve, Robusness of he Black Scholes formula, Mah. Finance8 1998, [6] M. Forde A. Jaquier, Small-ime asympoic for implied volailiy under he Heson model, Inernaional Journal of Theoreical Applied Finance, [7] M. Forde, A. Jacquier A. Mijaović, Asympoic formulae for implied volailiy under he Heson model, Proceedings of he Royal Sociey A, : [8] V. Henderson, Analyical Comparisons of Opion prices in Sochasic Volailiy Models, Mahemaical Finance [9] D. Hobson, Volailiy Misspecificaion, Opion Pricing Superreplicaion via Coupling, Ann. Appl. Probab [1] D. Hobson, Comparison resuls for sochasic volailiy models via coupling, Finance Sochasics [11] J. Hull A. Whie, The Pricing of Opions on Asses wih Sochasic Volailiies, J. Finance [1] S. Janson J. Tysk, Volailiy ime Properies of Opion Prices, Ann. Appl. Probab [13] M. Kijima, Monooniciy convexiy of opion prices revisied, Mah. Finance [14] A. Mijaović M. Urusov, On he maringale propery of cerain local maringales, Acceped in Probabiliy Theory Relaed Fields 1. [15] S. M. Ould Aly, Parameer sensiiviy of CIR process, Forhcoming in Elecronic Communicaions in Probabiliy. 13. [16] S. M. Ould Aly, From he momen explosions o he asympoic behavior of he cumulaive disribuion, preprin 13. [17] M. Romano N. Touzi, Coningen claims marke compleeness in a sochasic volailiy model, Mahemaical Finance [18] L.O. Sco, Opion pricing when he variance changes romly : heory, esimaion, an applicaion, J. Finan. Quan. Anal