Forward Price models and implied Volatility Term Structures

Transcription

1 Forward Price models and implied Volailiy Term Srucures Raquel M. Gaspar ISEG, Dep. ManagemenFinance, Universidade Tecnica de Lisboa, Rua Miguel Lupi 2, Lisboa, PORTUGAL February 26 Absrac This paper discusses he relaion beween forward price models (FPM) and he socalled implied volailiy erm srucure (VTS). We sar by considering he case of pure deerminisic forward price volailiies and suppose boh forward conracs and a-hemoney (ATM) opions, on a same underlying, are liquidly raded in he marke. We, hen, derive no arbirage condiions beween he funcional form of he ATM implied VTS and he funcional form of forward price volailiies. We conclude he firs par by characerizing a parameric family of ATM implied VTS ha is compaible wih a finie dimensional realizaion (FDR) of forward prices. Finally, we consider he possibiliy of sochasic forward price volailiies and derive a no arbirage drif condiion ha mus hold for he dynamics of ATM implied VTS. 1 Inroducion In his paper we sudy he connecion beween forward price models (FPM) and implied volailiy erm srucures (VTS). As far we know, his relaionship has no been exploied previously in he lieraure. The inspiraion comes from he so-called marke models on implied volailiy surfaces. In hese models liquidly raded plain vanilla opions and he underlying asse are modeled simulaneously and he ulimae goal is o find he arbirage free dynamics of he implied volailiy surface. Examples of such sudies are Dupire (1994), Schönbucher (1999), Skiadopoulos, Hodges, and Clelow (2), Haffner and Wallmeier (21), Brace, Goldys, Klebaner, and Womersley (21), Balland (22), Con and Fonseca (22) and Buehler (24). Unforunaely, up o dae, here is no good model for he dynamics of he enire implied volailiy surface, which is sill seen 1

2 as an ineresing open quesion. The difficulies do no have o do so much wih he fac ha we need o handle simulaneously boh several mauriies and several srike prices, bu raher resul from he messy procedure of invering he Black-Scholes formula. Here we ake a slighly differen approach and insead of modeling he underlying asse and all plain vanilla opions simulaneously, we ake all forward conracs and a-he-money (ATM) opion conracs on a same underlying as our basic asses, and model simulaneously forward prices and ATM opion prices. By choosing only ATM opions as our basic asses, we avoid having o deal wih several srike prices and focus on wha is know as he implied ATM volailiy erm srucure (VTS). The choice of only one srike price is a limiaion, bu i can be jusified by he fac ha hese are indeed he opions mos liquidly raded in he marke. We argue ha forward price erm srucures should conain informaion on expeced volailiies over fuure periods, and implied VTS should conain similar informaion. The goal is hen o formally characerize he link beween FPM and VTS. Ineresing quesions are he following. 1. Given a FPM, wha can we say abou he implied VTS of ATM opions? 2. Given a parameerized family of implied VTS curves, wha can we say abou he FPM? 3. If here is a link beween VTS and FPM, wha family of VTS is compaible wih a FPM ha admis a finie dimensional realizaion (FDR)? The firs quesion is he more absrac as i seeks he general connecion beween any VTS model and any FPM. The second quesion is quie specific, since i implicily assumes a deerminisic implied VTS. I erms of generaliy he hird quesion can be placed in beween he firs wo and can be seen as an applicaion of an earlier sudy on finie dimensional realizaions (FDR) of forward price erm srucure models (see Gaspar (25)). The main conribuions of he paper are: We give concree answers o all quesions in he special case where we assume he volailiy of he forward price is deerminisic. The resuls seem o be useful and insrucive, despie his assumpion. We show here is a no-arbirage relaionship beween FPM and VTS, and ha no all parameerizaions of implied VTS and compaible wih a FPM. Moreover, only very specific parameerizaions of VTS curves will be compaible wih FPM ha admi a FDR. We hen show ha if he volailiy of forward prices is no deerminisic, hen he implied VTS mus be sochasic. For his general case, we derive he a drif condiion for he dynamics of he implied VTS. The paper is organized as follows. In Secion 2 we presen he basic model seup for he case of deerminisic forward price volailiies. We derive he basic arbirage relaionship and give examples of parameerizaions of implied VTS curves ha do no admi a FPM. We hen characerize he parameerizaions ha are compaible wih FPM ha admi a FDR, and end up suggesing wo new classes of parameerized implied VTS ha have nice properies. 2

3 In Secion 3 we deal wih sochasic implied VTS and derive a non-arbirage drif condiion. In Secion 4 summarized he resuls and poins ou direcions for fuure research. 2 Deerminisic forward price volailiies Since we wan o alk abou forward prices, we consider a sochasic ineres rae seing. The resuls will only hold for fuures prices whenever he wo prices are he same. The oher key assumpion is on he basic asses exisen on he marke. Besides zero-coupon bonds, hese are forward conracs and ATM call opions on he same underlying. Assumpion 2.1. We have sochasic ineres raes and we assume ha here exiss a liquid marke for bond prices of all mauriies. Furhermore, here exiss a liquid marke for boh forward conracs and ATM opion conracs wrien on he same underlying. In wha follows he resuls would go hrough for any fixed srike price K. We choose o consider ATM opions because hose are ypically he mos liquid opions in he marke. 2.1 Forward price model We sar by assuming deerminisic forward price volailiies. Then, using he fac ha forward prices are maringales under he T -forward measure we can define heir dynamics. Assumpion 2.2. We consider forward price models (FPM) ha can be described by he following dynamics, under he T -forward measure, df (, T )=f(, T )γ(, T )dw T (1) where W T is a muli-dimensional Q T -Wiener process and we ake he volailiy of he forward prices, γ(, T ) o be a row vecor of deerminisic funcions of running ime and he mauriy T. 2.2 Implied Volailiy Term Srucure We now define wha exacly we mean by ATM implied VTS Definiion 2.3. Le us denoe by c (, T, K) he observed call prices in he marke. For each K and T, he implied volailiy, σ (, T, K) is defined by he following closed-form formula ln d 1 = ( f(,t ) K c (, T, K) =p(, T )[f(, T )N(d 1 ) KN(d 2 )] (2) where N( ) is he cumulaive disribuion of a sandard normal, and ) (σ (, T, K)) 2 (T ) σ (, T, K) T d 2 = d 1 σ (, T, K) T, and p(, T ), f(, T ) are, respecively, he price he zero-coupon bonds and he forward price, wih he same mauriy as he opion conrac. 3

4 If, for each T we choose K = f(, T ), he marke call price for each mauriy is denoed c (, T ), and equaion (2) simplifies o [ ( 1 c (, T )=p(, T )f(, T ) 2N 2 σ (, T ) ) ] T 1. (3) The mapping σ : T R +, for σ in (3), is called he implied a-he-money (ATM) volailiy erm srucure (VTS). We noe ha equaion (2) is an adapaion of he Black-Scholes formula, frequenly used by raders in he marke o correc from he effec of sochasic ineres raes. Here see (2) simply as par of he definiion of ATM implied VTS. Since we consider K = f(, T ), sricly speaking, we are no dealing wih ATM opions bu almos ATM. We use of he ATM expression boh because i is also a marke sandard and because mos opions on exchanges are conracually defined in erms of he forward price wih he same mauriy as he opion. 2.3 Arbirage relaionship beween FPM and implied VTS We now go on wih he analysis and use he forward price dynamics in (1) o explicily compue he arbirage free price of call opions on same underlying. The main inuiion comes from c(,k,t) = E Q [e ] T r sds max [S T, ] = p(, T )E T max S T }{{} f(t,t),. In he nex lemma we derive he opion pricing formula using forward prices. Lemma 2.4. Suppose Assumpions hold. The price of a call opion wrien on S, wih mauriy T, and srike price K is given by c(, T, S, K) =p(, T )[f(, T )N(d 1 ) KN(d 2 )] (4) where N( ) is he cumulaive disribuion of a sandard normal, and d 1 = ln f(,t ) K Σ(, T ) Σ(, T ) d 2 = d 1 Σ(, T ). (5) where Σ(, T )= In paricular, if for each mauriy we choose K = f(, T ) we have [ ( ) ] 1 c(, T )=p(, T )f(, T ) 2N Σ(, T ) 1 2 γ(s, T ) 2. (6). (7) 4

5 Proof. As saed c(, T, K) =p(, T )E T [max [f(t,t) K, ]]. Le us denoe y(, T, K) =E T [max [f(t,t) K, ]]. Then we have y(, T, K) = { E T [f(t,t) f(t,t) K] kqt (f(t,t) K) }. Using he fac ha he volailiy of he forward price is deerminisic we have, under he Q T forward measure, ( ln f(t,t) F N ln f(, T ) 1 ) T γ 2 (s, T )ds, γ 2 (s, T )ds. 2 Thus, Q T (f(t,t) K) =Q T (ln f(t,t) ln K) =Q T z ln K ln f(, T )+ 1 T 2 γ 2 (s, T )ds γ 2 (s, T )ds Where z is a sandard normal random variable under Q T. γ 2 (s,t )ds To compue E T [f(t,t) f(t,t) K] we noe ha f(t,t) K z ln K ln f(,t )+ 1 2 hus E T [f(t,t) f(t,t) K] = f(, T )e 1 T 2 γ 2 (s,t )ds+ T γ 2 (s,t )dsz ϕ(z)dz. z γ 2 (s,t )ds The resul follows from expanding he exponen and using he properies of he Normal disribuion. By simple comparison of equaions (3) and (7), we have he following resul. Proposiion 2.5. Suppose Assumpions 2.1 and 2.2 hold. Then, he following relaion mus hold beween he volailiy of forward prices γ in (1) and he ATM implied VTS from (3). σ γ(s, T ) 2 (, T )=. (8) T We noe ha he same expression holds for all possible fixed srike price K, as he same resul can also be obained by comparing (2) and (4). The immediae conclusion one can draw from (8) is ha, a leas in he deerminisic volailiy case, when choosing he forward price volailiies, one deermines no only he enire forward price erm srucure, bu also he implied VTS for he srikes assumed o be liquidly raded in he marke. In he case of his paper, hese are he ATM opions. This connecion can be exremely useful from an applied poin of view, especially because i ells us ha one can use informaion from exremely liquid marke opions o decide on he FPM. Many financial insiuions have forward conracs rading desks and mus decide, on a daily basis, which prices o pracice. In order words, need o choose a FPM. The same insiuion, possibly in a differen deparmen, ypically rades also opions on he same underlying. Then, i is common pracice o choose some parameerized family of VTS, fi i o marke daa from opion markes, and use i o price oher opions or more exoic srucures. If he link beween he wo conracs is ignored, hen i is very likely ha a financial insiuion is quoing prices ha allow for arbirage opporuniies. In order no o inroduce arbirage here are wo opions., 5

6 Given a parameerized family of VTS curves, characerize he FPMs ha are compaible wih hem. Given a FPM, choose a parameerized family of VTS ha is compaible wih i. Proposiion 2.5 old us how o proceed in he second case. The nex example should make is applicabiliy clear. Example 2.6. Suppose forward prices are driven by a one-dimensional Wiener process and ha a(t ) γ(, T )=be hen [ γ(s, T ) 2 ds = γ 2 (s, T )ds = b 2 e 2a(T s) ds = b2 1 e 2a(T )], 2a and he ATM implied VTS becomes σ b (, T )= 2 [ ] 1 e 2a(T ). (9) 2a(T ) In he firs case, we are given a parameerized family of VTS curves and we would like o arrive o a FPM. Many sudies and praciioners books propose several alernaive parameerizaions o implied VTS. Examples can be found in Rebonao (1999), Balland (22) and Haffner (24). In pracice, here are a limied number of mauriies liquidly raded on he marke, and raders use he proposed parameerizaions o inerpolae he volailiies implied a hose mauriies and ge a smooh implied VTS for all possible mauriies. The nex proposiion gives us he main resul. Proposiion 2.7. Suppose Assumpions 2.1 and 2.2 hold. Suppose, furhermore, ha we are given a parameerized family of implied VTS curves, σ (, T ) such ha (??) for all T raded in he marke, and σ : R 2 + R + is differeniable w.r.. o. Then, he following relaion hold beween he ATM implied VTS σ (, T ) and he forward price volailiy γ(, T ): γ(, T ) = (σ (, T )) 2 2 σ (, T )[T ]. (1) Moreover, o have a well defined FPM we mus have σ (, T ) < σ (, T ) 2(T ). (11) Proof. From he definiion of Σ(, T ) in (6) i follows γ(, T ) = Σ (, T ), and o have a well defined forward price volailiy 1, we need Σ (, T ) <. 1 This indeed makes sense since, for a fixed mauriy T, an increase in running ime is equivalen o a decrease in ime o mauriy and hus on he non-annualized variance over [, T ]. 6

7 I follows from (8) ha Σ (, T )=2σ (, T ) σ (T )[T ] 1 2 [T 1 ] 2 (σ (, T )) 2. and he equivalen condiion in erms of σ (, T ) is hus 2σ (, T ) σ (T )[T ] 1 2 [T ] 1 2 (σ (, T )) 2 < σ (, T ) < σ (, T ) 2(T ). Finally, rewriing (8) we have and deriving w.r.. we ge (σ (, T )) 2 [T ] = γ(s, T ) 2. 2σ (, T ) σ (, T )[T ] (σ (, T )) 2 = γ(, T ) 2. Remark 2.8. If we are dealing wih ime homogeneous funcions σ, ypically hey depend on ime o mauriy x = T and are differeniable w.r.. x. Then we can rewrie (1) and (11) in erms of x, using σ = σ x. From (11) we see ha here are parameerized families of curves ha are no compaible wih any FPM which is an ineresing resul. We now give an exemplify how o choose a FPM if we choose curves from he Nelson-Siegel family for fiing VTS. Firs, we derive he necessary condiion on he parameer values, o guaranee exisence of a FPM. Then, we propose one concree FPM The Nelson-Singel Family Consider we choose he following parameric implied VTS where x = T. Then using (11), we mus have σ (z,x) =z 1 + z 2 e z4x + z 3 e z4x. (12) z 4 1 2x + z 1 2xβ(z,x). where β(z,x) =(z 2 + z 3 )e z4x. Moreover, i follows from (1) ha [ γ(x) 2 =[z 1 + β(z,x)] 2 2z 4 x z 1 β(z,x)+(β(z,x)) 2]. (13) A FPM compaible wih (12) mus hen obey (13), and since he expression is quie messy, he bes ake a FPM driven by a one-dimensional W wih [ γ(x) = [z 1 + β(z,x)] 2 2z 4 x z 1 β(z,x)+(β(z,x)) 2]. 7

8 2.3.2 Oher Popular parameerizaions Oher popular parameerizaions, proposed in several exs, are o inerpolae he observed implied volailiies by square roo in ime or logarihmic rules. The logarihmic rule is, for insance, proposed in Haffner (24, page 87). The implied volailiy is ypically defined by σ (x) =z 1 + z 2 w(x) where we again use x = T and from (1) we have Popular choices for he funcion w are: γ(x) 2 =[z 1 + z 2 w(x)] 2 +2z 2 (x)[z 1 + z 2 w(x)] w x w 1 (T ) = ln(1+(t )) (14) w 2 (T ) = ɛ +(T ) ɛ R + (15) 1 w 3 (T ) = ɛ R +. (16) ɛ +(T ) In he case (14) we have γ(, T ) 2 =[z 1 + z 2 ln (1 + (T ))] 2 1 2z 2 (T )[z 1 + z 2 ln (1 + (T ))] 1+(T ), in he case (15) we have [ ] 2 ] γ(, T ) 2 1 = z 1 + z 2 ɛ +(T ) +2z2 (T ) [z 1 + z 2 ɛ +(T )) 2 ɛ +(T ), and for (16) [ ] 2 γ(, T ) = z 1 + z 2 +2z 2 (T )[z 1 + z 2 w(t )]. ɛ +(T ) 2(ɛ +(T )) 3 2 As in he case of he Nelsson-Singel family, he above expressions are quie messy and FPMs compaible wih hem will also have quie cumbersome volailiy expressions. Besides, possible racabiliy problems, his may have serious consequences in erms of forward prices erm srucures and heir properies. We can argue ha given, he simpliciy of he condiion (8) as compared o (1), he bes sraegy is o choose a FPM and use (8) o find he parameerized family of curves compaible wih i. This also avoids he problem of non-exisence of he FPM. Ineresingly, his is exacly he inverse of wha is common pracice in he marke. 2.4 Special case of a FPM ha admis a FDR We now characerize he parameerized family of VTS curves ha is compaible wih a FPM ha admis a FDR. If he marke agens would like o fi simulaneously he forward price erm srucure and he ATM implied VTS his is he only family of funcions ha allows hem o do i in a compaible way. 8

9 Proposiion 2.9. A family of parameerized implied VTS curves is compaible wih a FPM ha admi a FDR, only if we have he following represenaion (σ (, T )) 2 2 σ N (, T )[T ] = δi 2 (, T ) (17) where for all iδ i is a quasi-exponenial (QE) funcion 2. Moreover he minimum N for which (17) holds give us he minimal dimension of he Wiener process W in he forward price dynamics (equaion (1)). Proof. The firs par of he proposiion follows immediaely from (8) and from Proposiion 5.1. in (Gaspar 25). The second par follows from he fac ha γ 2 = i γ2 i where differen funcions γ i are associaed wih differen funcions γ i. An ineresing remark is ha none of he common parameerizaions sudied above is compaible wih a FPM ha admis a FDR. Nex we propose new parameerizaions which are compaible wih relaively simple FPM ha admi a FDR. i=1 2.5 New families of parameerized implied VTS Based on all he above resuls and using he inuiion form he previous forward price sudies we can now propose new parameerized families of implied VTS. Boh involve a maximum of 4 parameers, like he Nelson-Singel family, wih he advanage ha do no inroduce arbirage opporuniies and are compaible wih exisen of a FDR for forward prices. Obviously, many ohers could have be suggesed, now ha we know he exac condiion we need o saisfy o guaranee boh absence of arbirage an exisence of FDR of forward prices. The logic in he derivaion of boh expressions is o sar from easy QE funcions for he volailiy of forward prices. Here we simply look a forward price volailiies of he ype γ i (x) = e aix p i (k, x) i i where x = T, i is a finie ineger, a i R and p i (k, x) are a polynomial of degree k in x for all i. If we wan o limi our selves o 4 parameers, we can pick wo possible parameerizaions, simply by (I) γ i (x) =e ax b + cxe dx, (18) i 2 A quasi-exponenial (or QE) funcion is by definiion any funcion of he form f(x) = u e λux + j e α jx [p j (x) cos(w j x)+q j (x) sin(w j x)] where λ u,α j,w j are real numbers, whereas p j and q j are real polynomials. 9

10 (II) γ i (x) =e dx (a + bx + cx 2 ). (19) i To compleely specify he FPM we should also choose he dimension of he Wiener process and decompose i γ i in specific funcions γ i. When we allow for several random sources in he FPM he family of parameerized implied VTS has nicer formulas, bu one could derive parameerizaions compaible wih FPM driven by one-dimensional Wiener process. The nex resul is useful resul for he following compuaions. Resul 2.1. e au u n du = n! a n+1 e au i = n n! (n i)!a i+1 xn i. (I): We consider he case of a 2-dimensional W : Thus, γ 1 (x) =e ax b γ 2 (x) =cxe dx. (σ (x)) 2 x = γ(u) 2 du = b 2 e 2au du + c 2 u 2 e 2du du = b2 2a = b2 σ (x) = [ 1 e 2ax ] + c { 2 1 2d e 2dx x 2 1 4d 2 e 2dx 2x 1 ( ) 2a b2 2a e 2ax + c2 c 2 4d 3 e 2dx 4d 3 + c2 2d 2 x + c2 2d x2 ( ) b 2 2ax b2 2ax e 2ax + c2 c 4d 3 x 2 e 2dx 4d 3 x + c2 2d 2 + c2 2d x } 8d 3 2(e 2dx 1) (II): We consider he case of a 3-dimensional W : γ 1 (x) =e dx a γ 2 (x) =bxe dx γ 3 (x) =cx 2 e dx. 1

11 Thus, (σ (x)) 2 x = γ(u) 2 du = a 2 e 2du du + b 2 u 2 e 2du du + c 2 u 4 e 2du du [ ] [ 1 1 = a 2 2d [1 e 2dx ] + b 2 4d 3 e 2dx σ (x) = +c 2 [ 15 8d 5 e 2dx ( ) a 2 2d + v e 2dx ( 1 d x d 2 x + 1 4d 3 )] ( 1 2d x d 2 x d 3 x d 4 x d 5 x d 5 5 v i x i i= )] where v = b2 4d c2 16d 6, v 1 = b2 2d c2 4d 5, v 2 = b2 d + 15c2 2d 4, v 3 = 5c2 4d 3 v 4 = 5c2 2d 2, v 5 = c2 2d 3 Beyond deerminisic volailiies In his secion we discuss he implicaions of relaxing Assumpion 2.2. Using arbirage argumens we are able o derive general arbirage condiions. On he one hand, we show ha if he forward price volailiies are no deerminisic, assuming a deerminisic implied VTS inroduces arbirage in he model. On he oher hand, when considering sochasic implied VTS, arbirage allows for he free choice of he volailiy of volailiy bu no he drif. The drif under he forward measure Q T will depend on boh he volailiy of volailiy and on he volailiy of he forward price. We impose a new assumpion ha replaces he previous Assumpion 2.2 on he forward price dynamics and adds he implied volailiy dynamics. Assumpion 3.1. We consider forward price models (FPM) ha can be described by he following dynamics, under he T -forward measure, df (, T )=f(, T )γ(, T )dw T (2) where W T is a mulidimensional Q T -Wiener process and we ake he volailiy of he forward prices, γ(, T ) o be a row-vecor of adaped process. Furhermore, for each fixed T,K R + we assume he following Q T -dynamics for he implied volailiies as defined in (2), dσ (, T, K) =u(, T )d + ν(, T )dw T (21) where W T is w.l.o.g. assume o be he same as in (2). u, ν are adaped processes. Since he volailiy of forward prices is no longer deerminisic, equaion (8) is also no longer he correc arbirage free condiion. We sill have c(, T, K) =p(, T ) E T [max f(t,t) K, ] }{{} y(,t,k) 11

12 and we know ha y(, T, K) isaq T -maringale. Also, by Definiion 2.3, we have a closed-form soluion for y(, T, K) in erms of wo sochasic variables f(, T ) and σ (, T, K). We recall here ha closed-form expression ln d 1 = ( f(,t ) K y(, T, K) =f(, T )[N(d 1 ) N(d 2 )] (22) where N( ) is he cumulaive disribuion of a sandard normal, and ) (σ (, T, K)) 2 (T ) σ (, T, K) T d 2 = d 1 σ (, T, K) T. (23) Using boh hese fac we can derive he new arbirage condiion. This ime we obain a drif condiion on he dynamics of he implied VTS. Proposiion 3.2. Suppose Assumpions 2.1 and 3.1 hold. Then, he following drif condiion mus hold for he ATM implied VTS u(, T )= σ (, T ) 2(T ) 1 γ(, T ) 2 2 σ (, T )(T ) σ (, T ) ν(, T ) γ(, T )ν (, T ) (24) Proof. We sar by nohing ha for y = f [N(d 1 ) N(d 2 )], wih d 1 = ln( f K )+ 1 2 (σ ) 2 (T ) σ, T d 2 = d 1 σ (, T, K) T, and using he properies of he normal disribuion as well as he relaions beween d 1 and d 2 we have y = fϕ(d 1)σ 2 T y σ = f T ϕ(d 1 ) y f = N(d 2 y 1) f 2 = ϕ(d 1 ) fσ T 2 y σ = fϕ(d 1) T 2 y 2 σ d 1 d 2 f σ = 1 ϕ(d 1 ) σ d 2. wih ϕ(x) = 1 2π e x2 2. Applying he Iô s lemma o (22) we have and for simpliciy omiing (, T, K) or(, T ) in all erms and using x = T we have under Q T dy = y + y f df y 2 f 2 (df )2 + σ f dσ y 2 σ 2 (dσ ) y f σ = { fϕ(d 1)σ 2 + f xϕ(d 1 )u + 1 ϕ(d 1 ) x 2 fσ x f 2 γ fϕ(d 1 ) x 2 σ d 1 d 2 ν 2 + 1ϕ(d } 1) σ d 2 fγν d + { + N(d 1 )fγ(, T )+ fϕ(d 1) } x σ d 1 d 2 dw. Since y is a Q T -maringale we mus have he drif above equal do zero: fϕ(d 1)σ 2 x + f xϕ(d 1 )u + 1 ϕ(d 1 ) 2 fσ x f 2 γ fϕ(d 1 ) x σ d 1 d 2 ν 2 + 1ϕ(d 1) σ d 2 fγν =. 12

13 Dividing all erms by fϕ(d 1 ) x we ge σ 2x + u + 1 γ 2 2 σ x + 1 d 1 d 2 ν 2 2 σ + d 2γν σ x = and he resul follows from solving w.r.. u and using he fac ha in he ATM case we have d 2 = d 1 = 1 2 σ x and he resul follows. Remark 3.3. For he paricular case when he implied VTS is deerminisic (1) follows from (24). To see his noe ha when σ is a smooh deerminisic funcion we have ν(, T )=for all (, T ) and u(, T )= σ (,T ). Taking hese wo facs ino accoun (24) reduces o (1), σ (, T ) σ = 2(T ) 1 γ(, T ) 2 2 σ (, T )(T ) σ (, T )= γ(s, T ) 2 T and implies deerminisic implied VTS are only compaible wih FPM wih deerminisic volailiies. 4 Conclusion In his paper we sudied he arbirage link beween FPM and implied VTS when boh forward and opion conracs are wrien on a same underlying. We showed ha assuming deerminisic implied VTS, σ, is compaible wih FPM wih deerminisic volailiies, γ, and derived he arbirage funcional dependence beween σ and γ. Mos parameerizaions of implied VTS used in pracice lead o cumbersome FDR ha, in paricular do no allow for FDR of forward prices. We propose wo alernaive parameerizaions ha are compaible wih nice FPM. We hen go one sep furher a sudy he connecion beween VTS and FPM when he implied VTS σ is allowed o be a sochasic process. For his case we derive a no-arbirage drif condiion for σ. A deeper analysis of his general siuaion is lef for fuure research. References Balland, P. (22). Deerminisic implied volailiy models. Quaniaive Finance 2 (1), Brace, A., B. Goldys, F. Klebaner, and R. Womersley (21). Marke model of sochasic implied volailiy wih applicaions o he BGM model. Working paper. Deparmen of Saisics, Universiy of new Souh Wales. Buehler, H. (24). Consisen variance curve models. Deusche Bank AG London working paper. Con, R. and J. Fonseca (22). Dynamics of implied volailiy surfaces. Quaniaive Finance 2 (1), Dupire, B. (1994). Pricing wih a smile. Risk 7,

14 Gaspar, R. M. (25). Finie dimensional markovian realizaions for forward price erm srucure models. In M. Grossinho, A. Shyriaev, M. Esquível, and P. Oliveira (Eds.), Sochasic Finance, Chaper 1. Spring verlag. Haffner, R. (24). Sochasic Implied Volailiy. Springer Verlag. Berlin Heidelberg New York. Haffner, R. and M. Wallmeier (21). The dynamics of DAX implied volailies. Quarerly Inernaional Journal of Finance 1, Rebonao, R. (1999). Volailiy and Correlaion in he pricing of equiy, FX and ineres rae opions. Wiley. Schönbucher, P. (1999). A marke model for sochasic implied volailiy. Philosophical Transacions of Royal Sociey 357, Skiadopoulos, G., S. Hodges, and L. Clelow (2). Dynamics of he S&P5 implied volailiy surface. Review of Derivaives Research 3,