Rev Deriv Res (207) 20: 3 DOI 0.007/s47-06-922-2 On he mulipliciy of opion prices under CEV wih posiive elasiciy of variance Dirk Veesraeen Published online: 4 April 206 The Auhor(s) 206. This aricle is published wih open access a Springerlink.com Absrac The discouned sock price under he Consan Elasiciy of Variance model is no a maringale when he elasiciy of variance is posiive. Two expressions for he European call price hen arise, namely he price for which pu-call pariy holds and he price ha represens he lowes cos of replicaing he call opion s payoffs. The greeks of European pu and call prices are derived and i is shown ha he greeks of he risk-neural call can subsanially differ from sandard resuls. For insance, he relaion beween he call price and variance may become non-monoonic. Such unfamiliar behavior hen migh yield opion-based ess for he poenial presence of a bubble in he underlying sock price. Keywords Bubbles Consan elasiciy of variance Opion pricing Pu-call pariy Risk-neural valuaion JEL Classificaion G3 Inroducion The elasiciy of he variance of reurns o he sock price under Geomeric Brownian Moion (GBM) is zero and reurns hus are assumed o have consan variance. This assumpion was challenged in, for insance, Black (976) and became he main I am very graeful o he anonymous referee whose consrucive commens allowed me o remove erroneous claims and o grealy enhance he focus of he paper. The usual disclaimer applies. B Dirk Veesraeen d.j.m.veesraeen@uva.nl Amserdam School of Economics, Universiy of Amserdam, Roeerssraa, 08WB Amserdam, The Neherlands 23
2 D. Veesraeen argumen behind he developmen of he Consan Elasiciy of Variance (CEV) opion pricing model in Cox (975). The discouned sock price is a maringale under he CEV process wih negaive or zero elasiciy, bu is no a maringale for posiive values of he elasiciy parameer as was firs noed in Emanuel and MacBeh (982) and Lewis (2000). Cox and Hobson (2005) and Heson e al. (2007) inerpreed his loss of he maringale propery under posiive elasiciy of variance as evidence for he presence of a bubble in he underlying sock price. As a resul, here are (a leas) wo candidae prices for he call opion, namely he price for which he pu-call pariy holds and he price ha represens he lowes cos of replicaing he payoff of he call (see Heson e al. 2007). This paper calculaes he greeks of European pu and call opions under CEV wih posiive elasiciy of variance and inerpres hem as poenial indicaors for he presence of bubbles in he underlying. I is shown ha he greeks for he risk-neural call can reveal behavior ha is in sharp conras wih he sandard resuls as suggesed earlier in Cox and Hobson (2005), Eksröm and Tysk (2009) and Pal and Proer (200). For insance, risk-neural call prices can be non-monoonic funcions of he remaining ime o mauriy in which increasing ime iniially increases he call s price bu afer some poin sars o depress he value of he call. Deecing such paern hen migh poin o he presence of a bubble in he sock price. Indeed, he bubble may in he shor run inflae furher and hus sep up opion prices, bu he likelihood of is collapse grows for longer ime horizons causing he call price hen o decrease in he ime o mauriy. Addiional ess could be based on a similar non-monoonic relaion beween he risk-neural call price and he variance of he underlying as well as he poenial concave relaion beween call and sock prices. The remainder of his paper is organized as follows. Secion 2 discusses he loss of he maringale propery for he discouned sock price under CEV wih posiive elasiciy of variance. Secion 3 calculaes he European call and pu prices. Secion 4 discusses he sensiiviies of he opion prices o heir underlying parameers and inerpres heir behavior as poenial indicaors of bubbles in sock prices. Secion 5 concludes. 2 The loss of he maringale propery under CEV wih posiive elasiciy of variance The sock price is assumed o follow a CEV diffusion process wih risk-neuralized drif rs and diffusion coefficien σ 2 S 2α, respecively ds = rsd + σ S α dw, () wihr being he posiive, consan risk-free ineres rae. The elasiciy of he variance of reurns o he sock price equals α and GBM hus emerges as he limi for α =. This paper focuses on posiive elasiciy of variance, i.e. α>, and for breviy he acronym CEV from now on refers o CEV wih posiive elasiciy of variance. 23
On he mulipliciy of opion prices under CEV wih... 3 The ransiion probabiliy densiy funcion p S T, T ; S, is derived in Emanuel and MacBeh (982) as ( p S T, T ; S, =2 (α ) k 2( α) wih k = z z 4α T r σ 2 ( α)(exp 2r ( α)(t ) ) > 0, z = ks 2( α) exp 2r ( α)(t ), ) exp z z T I ( ) 2 z z T, 2(α ) (2) z T = ks 2( α) T, I q = he modified Bessel funcion of he firs kind of order q. The disribuional relaions in Schroder (989) allow o specify he following wo inegral expressions ha faciliae calculaion of he below resuls and J ;β,β 2 = β 2 β Q J 2;β,β 2 = p S T, T ; S, ds T = Q 2kβ 2( α), 2 + α, 2z β 2 β S T p S T, T ; S, ds T = S exp r (T ) Q 2z, { Q 2z, α, 2kβ2( α) 2 2kβ 2( α) 2, 2 + α, 2z α, 2kβ2( α) }, where Q a, b, c is he complemenary non-cenral chi-square disribuion evaluaed a a wih b degrees of freedom and non-cenraliy parameer c. 2 The condiional expecaion for he CEV process (), E S T, T ; S,, hen is E S T, T ; S, = + 0 S T p S T, T ; S, ds T The expression for he ransiion probabiliy densiy funcion in (7) inheson e al. (2007) conains a misprin as he erm (xz α) 4 α should be (xz 4α) 4( α). Also, he numeraor in he firs fracion in he erm u on p. 367 should be r raher han 2r. 2 The complemenary chi-square disribuion arises if he non-cenraliy parameer equals 0. See Zelen and Severo (964) for more deail on he (non-cenral) chi-square disribuion. 23
4 D. Veesraeen E S T, T ; S, = J 2;0,+ { E S T, T ; S, = S exp r (T ) Q 2z, E S T, T ; S, = S exp r (T ) { Q } α, + Q 2z, α, 0 } 2z,, (3) α, 0 given ha Q a, b, + = as can be inferred from definiion 26.4.25 in Zelen and Severo (964). The condiional expecaion (3) hus falls below S exp r (T ) since Q 2z, α, 0 is a disribuion. As a resul, he discouned sock price is no a maringale. 3 Cox and Hobson (2005) and Heson e al. (2007) inerpreed his loss of he maringale propery as evidence ha he sock price has a bubble and as a resul is expeced o decrease in he fuure. 3 European call and pu opion prices The risk-neural value of a European pu opion wih exercise price K and ime o mauriy T, P, is given by P = exp r (T ) K 0 K S T p S T, T ; S, ds T P = exp r (T ) { } K J ;0,K J 2;0,K { ( P = exp r (T ) K Q 2kK 2( α), 2 + α, 2z Q +, 2 + )} α, 2z { ( exp r (T ) S exp r (T ) Q 2z, α, + )} Q 2z, α, 2kK2( α) P = K exp r (T ) Q 2kK 2( α), 2 + α, 2z ) S ( Q 2z, α, 2kK2( α), (4) which corresponds wih he pu price ha is derived in Hull (2006). 3 Noe ha resul 26.4. in Zelen and Severo (964) implies ha Q 2z, α, 0 = 0forα =. The discouned sock price hen is a maringale in line wih he fac ha α = yields GBM. 23
On he mulipliciy of opion prices under CEV wih... 5 The risk-neural expeced discouned value of he payoff o he European call opion, in shor he risk-neural call price C,is C = exp r (T ) + K S T K p S T, T ; S, ds T C = exp r (T ) { } J 2;K,+ K J ;K,+ { C = exp r (T ) S exp r (T ) )} Q 2z, α, 0 { ( exp r (T ) K Q 0, 2 + Q C = S (Q 2z, 2kK 2( α), 2 + K exp r (T ) α, 2z α, 2kK2( α) Q ( Q ( Q 2z, α, 2z )} 2z, α, 2kK2( α) ) α, 0 2kK 2( α), 2 + α, 2z ). (5) The call price (5) corresponds wih he price G 2 (S, ) ha was obained in Heson e al. (2007). Emanuel and MacBeh (982) repored a second expression for he call price, denoed by C EM, ha can be obained via he pu-call pariy. Plugging he risk-neural pu price (4) ino he pu-call pariy yields C EM C EM = P + S K exp r (T ) = S Q 2z, α, 2kK2( α) ( K exp r (T ) Q 2kK 2( α), 2 + ) α, 2z. (6) This price is documened in, amongs ohers, Schroder (989), Davydov and Linesky (200), Hull (2006) and is specified as G (S, ) in Heson e al. (2007). The relaion beween he call prices (5) and (6)is C EM C = S Q 2z, α, 0. (7) 23
6 D. Veesraeen Hence, C EM exceeds C. 4 The call price C EM hus has a bubble given ha i exceeds he cos of a sraegy ha replicaes he payoffs of he call opion (see Heson e al. 2007 for more deail). 4 The greeks of European pu and call opion prices I is possible o calculae he greeks for he above opion prices for general values of α as definiion 26.4.25 in Zelen and Severo (964) expresses he non-cenral chi-square disribuion as an infinie sum of chi-square disribuions. The required derivaives hen be compued via relaion 26.4.9 in Zelen and Severo (964) and relaion 6.5.25 in Davis (964). Bu, he resuling expressions become oo elaborae for a meaningful analysis. However, opion prices and heir greeks can be expressed in (surprisingly) compac and easy-o-inerpre form for paricular values of α. Forα = 2, he densiy 2 funcion (2) can be simplified via he ideniy I (x) = sinh x (see Polyanin 2 π x 2002) ino 5 p α=2 S T, T ; S, = S k exp r (T ) exp 2 ST 3 {exp π { 2kS T The condiional expecaion hen is { 2kS T 2 + } } 2 2kS exp r (T ). } 2 2kS exp r (T ) E α=2 S T, T ; S, = S exp r (T ) { 2 q }, where q = 2kS exp r (T ) and q = 2π sandard normal disribuion funcion. The risk-neural pu price, P α=2, emerges as P α=2 q exp 2 x2 dx is he = K exp r (T ) σ S exp 2r (T ) {φ q 2 φ q 3 } 2r + K exp r (T ) { q 2 + q 3 } S { q 3 q 2 }, (8) 4 Boh expressions for he call price saisfy he valuaion parial differenial equaion of Black and Scholes (973) and is boundary condiions as discussed in Heson e al. (2007) andeksröm and Tysk (2009). Heson e al. (2007) and Eksröm and Tysk (2009) furhermore argued ha he muliples of he difference (7) also saisfy he valuaion equaion such ha acually an infinie number of soluions arises. 5 The ideniies in Polyanin (2002) andschroder (989) can be combined o yield similar simplificaions for various oher values of α. Cox and Hobson (2005) andeksröm and Tysk (2009) also focused on he case of α = 2 bu wih he added resricion of r = 0 in order o express he sock price as he reciprocal of he radial par of a 3-dimensional Brownian moion. 23
On he mulipliciy of opion prices under CEV wih... 7 where q 2 = 2k { S exp r ( T ) K }, q 3 = 2k { S exp r ( T ) + K } and φ q = exp 2π 2 q2. The risk-neural value of he call opion, C α=2, is given by C α=2 = S {2 q q 2 q 3 } K exp r (T ) { q 3 q 2 } K exp r (T ) σ S exp 2r (T ) {φ q 3 φ q 2 } (9) 2r and he relaion beween he risk-neural pu and call prices, i.e. he risk-neural pu-call pariy, is C α=2 = P α=2 + S { 2 q } K exp r (T ). (0) The call price of Emanuel and MacBeh (982), C EM,α=2, can be obained by plugging ino he pu-call pariy P α=2 C EM,α=2 = P α=2 + S K exp r (T ) () C EM,α=2 = S {2 q 2 q 3 } K exp r (T ) { q 3 q 2 } K exp r (T ) σ S exp 2r (T ) {φ q 3 φ q 2 }. 2r (2) The call price of Emanuel and MacBeh (982) exceeds he risk-neural call price given ha C EM,α=2 = C α=2 + 2S q (3) 4. Dela The dela of he risk-neural pu price (8) can be calculaed as P α=2 S = q 2 q 3 + K 2k {φ q 2 φ q 3 }. I is no possible o analyically show ha he dela of he pu has he familiar negaive value. However, an exensive grid search confirmed ha he dela is always negaive. 6 The dela of he pu ogeher wih he risk-neural pu-call pariy (0) hen specifies 6 The grid search used values for S beween 0.0 and 0, K ranged beween 0.0 and 5, r moved beween 0.02 and 0., σ beween 0.5and 0.45and T ranged from0. o 5. The oal number of cases ha was evaluaed was a,048,576. 23
8 D. Veesraeen he dela of he risk-neural call as C α=2 S =2 q q 2 q 3 2S 2k exp r (T ) φ q + K {φ q 2 φ q 3 }. 2k The grid search showed ha he dela of he risk-neural call always had he familiar posiive sign. The dela for he call price of Emanuel and MacBeh (982) can mos easily be obained via he pu-call pariy () as C EM,α=2 S = q 2 q 3 + + K 2k {φ q 2 φ q 3 } > 0. The laer expression is always posiive given ha he sum of he firs hree erms on he righ-hand side is posiive and his also holds for he remaining erm. Boh call prices increase in he value of he underlying sock price. However, he dela of he call price of Emanuel and MacBeh (982) exceeds he dela of he riskneural call price as, for insance, relaion (3) implies C EM,α=2 S Cα=2 S = 2 q + 2S 2k exp r (T ) φ q > 0. 4.2 Gamma The risk-neural pu price (8) is characerized by he familiar convexiy in he sock price given ha 2 P α=2 S 2 = S 3 2kK exp 2r (T ) {φ q2 φ q 3 } > 0. Combining he laer resul wih he pu-call pariy () implies ha he call price of Emanuel and MacBeh (982) is a convex funcion of he sock price 2 C EM,α=2 S 2 = 2 P α=2 S 2 > 0. The gamma for he risk-neural call price follows from he risk-neural pu-call pariy (0) as 2 C α=2 S 2 = 2 P α=2 S 2 2 5 2 S 4 k 3 2 exp 3r (T ) φ q. The gamma of he risk-neural call hus falls below ha of he pu in view of k > 0. Moreover, he gamma will be negaive for K = 0 in view of he zero value of he 23
On he mulipliciy of opion prices under CEV wih... 9 0.7 0.9 call price 0.35 call price 0.45 0 0 0 4 8 0 0.5 0.3 sock price vola liy (a) K = 5, r = 0.03, T- = 0.75 and =0.2 (b) K = 5, S = 5, r = 0.03 and T- = 0.75 3.26 0.8 call price 3.2 call price 0.4 3.4 0 0 0. 0.2 0 0.5 ineres rae me o mauriy (c) K = 0. 5, S = 5, T- = 0.75 and =0.2 (d) K = 5, S = 5, r = 0.03 and = 0.2 Fig. Sensiiviy of he risk-neural call price o is underlying parameers for α =2 gamma of he pu and he posiive value of k. The limiing case of K = 0 hus produces a concave relaion beween he risk-neural call price and he sock price as noed in Cox and Hobson (2005) and Eksröm and Tysk (2009). However, larger values for K may creae more inricae paerns in which for insance he familiar convex relaion urns ino a concave relaionship for larger sock prices. Panel (a) in Fig. illusraes his for he parameer values K = 5, r = 0.03, T = 0.75 and σ = 0.2. This unfamiliar behavior of he gamma of he risk-neural call could be of ineres for he idenificaion of bubbles. In fac, deecing concaviy in he call opion price migh indicae ha he underlying sock price has a bubble. Furhermore, observing a larger gamma for he pu han for an idenical call migh also poin o he presence of a bubble in he sock price. 4.3 Vega The sandard resul of a posiive relaion beween opion prices and σ is presen for he risk-neural pu price as well as for he call price of Emanuel and MacBeh (982) 23
0 D. Veesraeen since P α=2 σ = KS σ 2k {φ q 2 φ q 3 } > 0 and C EM,α=2 σ = Pα=2 σ > 0. The vega of C α=2 hen can be obained via he risk-neural pu-call pariy (0) as C α=2 σ = Pα=2 σ 2 2 3 k 2 exp r (T ) φ q. σ The vega of he risk-neural call hus falls below he vega of he idenical pu. Also, a sricly negaive vega for he risk-neural call arises for K = 0. However, he relaion beween C α=2 and σ acually can grow non-monoonic as is illusraed in Panel (b) of Fig. for he parameer values S = 5, K = 5, r = 0.03 and T = 0.75. This non-monoonic relaion beween he risk-neural call price and σ can be inerpreed agains he background of he presence of a bubble in he underlying sock price. Iniially, a rise in σ for low values of he laer may furher inflae he bubble and hus boos he call price. However, afer some criical level of σ is reached, furher rises in σ (sharply) increase he likelihood ha he bubble will burs hrough which he call price mus decrease. Eksröm and Tysk (2009) used an advanced reamen of parial differenial equaions o show ha convexiy (concaviy) of he call in he sock price auomaically implies a posiive (negaive) relaion beween he call opion price and volailiy. This direc link is confirmed for he presen CEV process given ha he vega of he above hree prices can always be obained by muliplying he corresponding gamma wih he posiive erm S4 exp 2r (T ). 2kσ The analysis of he vega hus may offer an addiional opion-based es for he poenial presence of a bubble in he underlying sock price. Such bubble migh be presen if he call price decreases in response o an increase in volailiy or increases o a smaller degree when compared wih he reacion in he price of an idenical pu. 4.4 Rho The rho of he risk-neural pu price is given by P α=2 r = KS ( ) 2kσ 2 (T ) exp 2r (T ) {φ q 2 2 3 k 2 3 φ q 2 } r + K (T ) exp r (T ) { q 3 q 2 }. 23
On he mulipliciy of opion prices under CEV wih... Combining he laer resul wih he pu-call pariy () allows o express he rho of he call price of Emanuel and MacBeh (982) as C EM,α=2 r = Pα=2 r + K (T ) exp r (T ). A grid search showed ha he familiar posiive (negaive) rho for calls (pus) also holds for hese wo opion prices. The rho for he risk-neural call is given by C α=2 r = C EM,α=2 r + 2k exp r (T ) r ( ) 2kσ 2 (T ) φ q. The laer expression ypically reurns a posiive value. However, i can urn negaive for opions ha are deep in-he-money as is illusraed in Panel (c) of Fig. where K = 0.5, S = 5,σ = 0.2 and T = 0.75. 4.5 Thea The expression for he sensiiviy of he risk-neural pu o he remaining ime unil mauriy, hea, is Pα=2 (T ) = σ 2 k 2 KS 2 2 exp 2r (T ) {φ q 3 φ q 2 } rk exp r (T ) { q 3 q 2 }, and his derivaive, as in he sandard case of GBM, may be posiive or negaive. The pu-call pariy () hen specifies he hea of he call price of Emanuel and MacBeh (982) as EM,α=2 C (T ) Pα=2 = rk exp r (T ). (T ) The erm rk exp r (T ) guaranees ha he hea of he call of Emanuel and MacBeh (982) is negaive given ha he erms φ q 2 φ q 3 and q 3 q 2 boh are posiive. The hea for he risk-neural call is Cα=2 (T ) EM,α=2 = C (T ) + 2 3 2 k 2 exp r (T ) (kσ 2 exp 2r (T ) + r ) φ q. Adding he second erm on he righ-hand side hen acually can generae a posiive value for he hea of he risk-neural call. The possibiliy of encounering a posiive 23
2 D. Veesraeen hea was discussed in Pal and Proer (200) wihin he conex of he inverse Bessel process. Noe ha he relaionship beween C α=2 and T can be non-monoonic as is illusraed in Panel (d) of Fig. for K = 5, S = 5, r = 0.03 and σ = 0.2. Such sequence of an iniial rise of he call price in he remaining ime unil mauriy followed by a decline migh be compaible wih he presence of a bubble in he underlying. In fac, he bubble may grow furher in he shor run causing he call o increase in value. However, longer ime horizons make i more likely ha he bubble will collapse which hen negaively impacs upon he price of he call. The above findings for he hea of he call may be useful wihin he search for bubbles in he underlying sock on wo accouns. Firs, a negaive link beween European call prices and he remaining ime unil mauriy migh poin o he presence of a bubble in he sock price as noed earlier. Second, a negaive relaion beween he European call price and is ime o mauriy immediaely implies ha he early-exercise feaure of he idenical American call opion will now have posiive value even when no dividends are paid (Cox and Hobson 2005; Pal and Proer 200). Hence, observing prices for American calls on socks ha pay no dividends being in excess of prices of oherwise idenical European call opions migh likewise signal he presence of a bubble in he underlying. 5 Conclusions The Consan Elasiciy of Variance (CEV) model allows for a more elaborae specificaion of he volailiy of reurns. Opion pricing under he CEV process wih posiive elasiciy of variance, however, is complicaed by he fac ha he discouned sock price no longer is a maringale as was firs noed in Emanuel and MacBeh (982) and Lewis (2000). This loss of he maringale propery under posiive elasiciy of variance was inerpreed as evidence of sock-price bubbles in Cox and Hobson (2005) and Heson e al. (2007).Heson e al. (2007) proceeded by showing ha such bubbles generae (a leas) wo candidae prices for he call opion, namely he price for which pu-call pariy holds and he price ha represens he lowes cos of replicaing he opion s payoffs. This paper calculaes and illusraes he greeks of European pu and call opions under CEV wih posiive elasiciy of variance and discusses heir poenial role in esing for he presence of bubbles in he underlying sock price. I was found ha he greeks of he risk-neural call may exhibi behavior ha deviaes considerably from he sandard resuls. For insance, call prices may become non-monoonic in he variance of he underlying sock. Deecing such paerns hen migh indicae he presence of a bubble in he underlying as low levels of variance may furher inflae he bubble and sep up opion prices whereas he likelihood of a collapse in he bubble and he opion price sharply increases for larger dispersion. Open Access This aricle is disribued under he erms of he Creaive Commons Aribuion 4.0 Inernaional License (hp://creaivecommons.org/licenses/by/4.0/), which permis unresriced use, disribuion, and reproducion in any medium, provided you give appropriae credi o he original auhor(s) and he source, provide a link o he Creaive Commons license, and indicae if changes were made. 23
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