Centre for Computational Finance and Economic Agents WP Working Paper Series. Hengxu Wang, John G. O Hara, Nick Constantinou

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1 Cenre for Compaional Finance and Economic Agens WP67-13 Working Paper Series Hengx Wang, John G. O Hara, Nick Consanino A pah-independen approach o inegraed variance nder he CEV model 13

2 A pah-independen approach o inegraed variance nder he CEV model Hengx Wang John G. O Hara Nick Consanino March 13 Absrac In his paper, a closed form pah-independen approximaion of he fair variance srike for a variance swap nder he consan elasiciy of variance (CEV) model is obained by applying he small disrbance asympoic expansion. The realized variance is sampled coninosly in a risk-neral marke environmen. Wih he applicaion of a Brownian bridge, we derive a heorem for he condiionally expeced prodc of a Brownian moion a wo differen imes for arbirary powers. This heorem enables s o provide a condiional Mone-Carlo scheme for simlaing he fair variance srike. Compared wih resls in he recen lierare, he mehod olined in or paper leads o a simplified approach for pricing variance swaps. The mehod may also be applied o oher more sophisicaed volailiy derivaives. JEL classificaion: C, C63, G13 Keywords CEV process, Realized variance, Small disrbance asympoic expansion, Brownian bridge, Condiional Mone-Carlo simlaion 1 Inrodcion There has been significan ineres in he valaion of variance swaps boh from he academic and praciioner commniies. The vale of a variance swap depends on he realized variance of he nderlying asse over is lifeime. Zh and Lian 19 presened a closed form formla for pricing a variance swap nder he Heson model 1 and hey defined he realized variance wih discree sampling imes. Demeerfi e al. 9 arged ha he fair variance srike (annalized variance of he nderlying asse accmlaed over he lifeime of he conrac) can be expressed in erms of a log-conrac, mariy and risk free ineres rae if he realized variance is sampled coninosly in a risk-neral marke environmen. The vale of a log-conrac is he disconed expecaion of he nderlying s log-rern (coninosly componded rae of rern) over he lifeime of he conrac. I can be obained by applying Mone-Carlo simlaion Cenre for Compaional Finance and Economic Agens, Universiy of Essex, Wivenhoe Park, Colcheser CO4 3SQ hwangd@essex.ac.k Corresponding ahor, johara@essex.ac.k Essex Bsiness School, Universiy of Essex, Wivenhoe Park, Colcheser, CO4 3SQ, ncons@essex.ac.k 1

3 o he disribion of he nderlying a mariy. The log-conrac made he fair variance srike pahindependen and simplifies he pricing of variance swaps whose nderlying is sricly non-zero. However, for hose variance swaps whose nderlying may possibly reach zero, he vale of a log-conrac is difficl o compe by sandard Mone-Carlo simlaion becase he naral logarihm of zero is negaive infiniy. As discssed by Davydov and Linesky 8, he CEV model allows he nderlying asse o reach zero when model parameers fall in a cerain range. Jordan and Tier 11 sed asympoic mehods o expand he probabiliy of he nderlying asse aaining zero and obained an approximaion of a log-conrac nder he CEV model. In his paper, he realized variance is defined as a inegral of he insananeos variance of he CEV process. We deermine a closed-form approximaion of he fair variance srike in erms of he iniial variance, ime o mariy, risk free ineres rae and model parameers nder he CEV model by applying he small disrbance asympoic expansion, of Kniomo and Takahashi 13, o he insananeos variance. Or mehod overcomes he compaion difficlies associaed wih he nderlying having vale zero. We show ha he approximaion is very accrae sing nmerical mehods. In addiion, corollary 4.4 provides a condiional Mone-Carlo simlaion scheme for comping fair variance srike of a variance swap by applying heorem 4.. Compared wih oher simlaion mehods which need o generae he enire pah of he nderlying asse o compe he realized variance, or approach aomaically simlaes he disribion of he realized variance a mariy. This new simlaion mehod is more ime efficien and also generaes less bias dring he simlaion procedre. The paper is srcred as follows. In secion, we give a sochasic differenial eqaion for insananeos variance of a CEV process by applying Iô s lemma. In secion 3, we derive he small disrbance asympoic expansion of he insananeos variance and provide a closed form approximaion of he fair variance srike. In secion 4, we give he closed form approximaion of condiionally expeced realized variance by sing a Brownian bridge and his approximaion can be implemened by a condiional Mone Carlo simlaion. Finally in secion 5, he fair variance srike of a variance swap is comped o demonsrae he correcness or mehods. Comparison wih a log-eler Mone-Carlo simlaion are also carried o o indicae ha or mehods are more efficien for comping he fair variance srike. CEV variance process In his secion we concenrae on he CEV process inrodced by Cox and Ross 7 ds = rs d + δs β+1 dw,, T. (1) Eqaion (1) gives he dynamics of he nderlying asse. In he above, S denoes he asse price a ime, T and S >, δ is a posiive consan and β is in he range 1, ), r is he consan riskless ineres rae, and W is a sandard Brownian moion nder he risk-neral measre Q. We chose he CEV process for hree reasons. Firs, he process is consisen wih one of well-known sylized facs ha volailiy changes are negaively correlaed wih asse rerns in sock markes for negaive β, i.e., local volailiy δs β is a decreasing fncion of S for β < and δ >. Second, as shown by Davydov and Linesky 8, he process is able o capre he observed volailiy skew for boh eqiy and index opions. Third, he process can be readily calibraed o he marke prices of Eropean opions sing closed form opion formlas provided in he papers of Schroder 16 as well as Davydov and Linesky 8. Cox and Ross 7 resriced β in he region β 1, whils Schroder 16 relaxed he range of β o

4 β. In his paper, he range of β will be resriced in 1, ) for hree reasons. Firs, β > does no garanee he model o obey he sylized fac as indicaed in he previos paragraph. Second, he CEV model recovers he Black Scholes model for β =, which will no be concenraed on in his paper. Third, as sggesed by Lindsay and Brecher 14, he CEV process admis hree disinc ypes of solion wih respec o β <.5,.5 β < and β. We le β 1, ) becase his resricion allows wo ypes of solion wih respec o β <.5,.5 β < and more imporanly, i simplifies or nmerical experimens in secion 5 wiho loss of generaliy of he CEV model. In addiion, research based on he CEV process wih β in he Cox and Ross 7 range can also be fond in papers sch as 1, 6, 18 When.5 β <, bondary vale S = is aainable and absorbing 1. When 1 β <.5, he bondary S = is aainable, and can be absorbing or reflecing. Inrodcing an absorbing bondary is an appropriae choice becase i excldes any arbirage opporniy afer S aains. If a reflecing bondary is chosen for 1 β <.5, an invesor prchasing he asse when S his wold never lose money, since he asse price will always be posiive. By aking r >, 1 β < and δ > wih an absorbing bondary we have he insananeos variance of S given by Davydov and Linesky 8 as V = δ S β,, T, V >. () By Iô s lemma 17, we obain a sochasic differenial eqaion for V dv = β (β 1)V 3 + rv d + βv dw, (3) where is parameers are defined as in eqaion (1). From Davydov and Linesky 8, we also have he probabiliy of V given by Q(V ) = Γ( 1, ζ), (4) β where Γ(x, y) is he complemenary Gamma disribion fncion and ζ = r βv (1 e βr ) 1. 3 Small disrbance asympoic expansion of CEV variance process In his secion, we apply he small disrbance asympoic expansion firs proposed by Kniomo and Takahashi 13 for he CEV Variance process (3). The small disrbance asympoic expansion is closely relaed o he Taylor expansion. In order o implemen his echniqe, we inrodce a small parameer < ɛ 1 and a negaive variable β sch ha β = ɛ β. From eqaion (3), we have V = V + ɛ β (ɛ β 1)V + rv d + ɛ β V 3 dw, (5) and V ɛ= = V. The small disrbance asympoic expansion reqires s o consrc a Taylor expansion 1 Here absorbing means ha S remains a if i his. Here reflecing means ha vales of S may increase afer i his. 3

5 of V abo he poin ɛ = : V = V + ɛ V + ɛ ɛ=! V + ɛ3 ɛ= 3! 3 V 3 + ɛ4 ɛ= 4! 4 V 4 + O(ɛ 5 ). (6) ɛ= From eqaion (5), V depends on V, V and V 3. Eqaion (6) reqires s o compe he derivaive of V wih respec o ɛ p o he forh order, which means we also reqire higher order derivaives of V and V 3. The following lemma provides a general ieraion formla for he nh order derivaive of V m. Lemma 3.1. The nh order derivaive of V m n (V m ) n = mn β + mɛ β + mn β wih m and n N. is given by m(n 1) β n (V m+1 ) n + (mɛ β 1) n 1 (V m+1 ) n 1 + r n 1 (V m ) n 1 mn β n 1 (V m+1 ) n 1 (V m+ 1 ) n 1 n 1 + (mɛ β 1) n (V m+1 ) n + r n (V m ) n dw + mɛ β d d n (V m+ 1 ) n dw, (7) Proof. By Iô s lemma, we obain a sochasic differenial eqaion for V m given by V m = V m + mβ (mβ 1)V m+1 + rv m d + mβ V m+ 1 dw. Since β = ɛ β for < ɛ 1, we have V m = V m + mɛ β (mɛ β 1)V m+1 + rv m d + mɛ β V m+ 1 dw. (8) Lemma 3.1 is proved by indcion. We see he resl follows in he case n = 1, since V m = m β + m β (mɛ β 1)V m+1 V m+ 1 + rv m dw + mɛ β d + mɛ β V m+ 1 dw. m+1 m βv + (mɛ β m+1 V 1) + r V m d Sppose ha eqaion (7) holds for n = l. Then he derivaive of V m given by ( l (V m ) l ) = m(l + 1) β + mɛ β + m(l + 1) β wih respec o ɛ of order l + 1 is ml β l 1 (V m+1 ) l 1 + (mɛ β 1) l (V m+1 ) l + r l (V m ) l m(l + 1) β l (V m+1 ) l (V m+ 1 hence eqaion (7) also holds for n = l + 1. ) l d l + (mɛ β 1) l+1 (V m+1 ) l+1 + r l+1 (V m ) l+1 dw + mɛ β l+1 (V m+ 1 ) l+1 dw, Remark 3.. When n = 1, he derivaive of he fncion V m, V m+ 1 and V m+1 o he order n 1 in eqaion (7) are acally he fncions hemselves. Then by applying lemma 3.1 o eqaion (6), we have he small disrbance asympoic expansion of V. d 4

6 Lemma 3.3. The small disrbance asympoic expansion of V p o forh order is V = V + βg + β H + β 3 L + β 4 M + O(ɛ 5 ), (9) where G = V (r V ) + V 1 W, H = V (V 3rV + r ) V + 3V 1 (r V )W + 3V W V 1 (r + V ) W d, L = V ( V 3 + 4rV 14 3 r V r3 ) 3 + 3V 1 (5 4 V 4rV + 3r ) W V (3r V ) + 6V (r V )W 3V 3 W + 4V 3 W 3 + V (3r + V ) 3V (r + V )W + V 1 (V + rv r ) W d + V 1 (5 V rv 6r ) W s dsd V 3 W d W d, M = V (V 4 5rV r V 5r 3 V + 3 r4 ) 4 V (3V 8rV 14 3 r ) 3 V 1 (35 8 V rv + 49 r V 9r 3 ) 3 W + 6V ( 3 V 5rV + 4r ) W 3V 3 (4r 5 V ) W + V + 1V 3 (r V )W 3 6V W + 5V W 4 + V 3 (5V r) W d V V 1 (35 8 V rv + 1 r V + 9r 3 ) + 3V (3V rv 8r )W + 3V 3 (8r + 3V )W + V ( 5 V + 6rV 3r ) Wd + V 3 (r + V ) W d + 3V (V r) W d 4V (V + 4rV 3r ) W d + 3V ( 5 V 4rV + 6r )W V 1 (7V 3 9rV 16r V + 1r 3 ) V 1 (3V 3 1r V + 4r 3 ) Proof. From lemma 3.1, we have V = β + β W s dsd W W s dsd V 3 (4r + 13V ) s (ɛ β 1)V + rv d + ɛ β V 3 dw + ɛ β W d W s dsd W 3 d sw s dsd 6V 3 (r + V )W W a dadsd 3V W W d. W d W s dsd W d βv + (ɛ β 1) V + r V ) d V 3 dw (1) by aking m = n = 1. The second and he las erms on he righ hand side of eqaion (1) can be 5

7 ignored since < ɛ 1, and from eqaion (6), we have V V when ɛ. So we have V = β ɛ= V + rv d + β V 3 dw. (11) = βv (r V ) + V 1 W Defining G = V (r V ) + V 1 W, we have ɛ V ɛ= = βg. By by repeaing his procedre we can obain he oher erms wih higher order derivaives in eqaion (6). See appendix A for a deailed proof. The vale of a volailiy derivaive depends on is nderlying s annalized realized variance over is lifeime. For example, he vale of a fair variance srike of a variance swap, P, a ime = is defined as P := E Q V (,T ), (1) T where V (,T ) := 1 T V d, T is he mariy, V is he insananeos variance of he nderlying asse and E Q represens expecaion nder risk-neral measre Q. V (,T ) depends on he vale of V for, T and hence i is a pah-dependen random variable, which makes i compaionally expensive o evalae by Mone-Carlo simlaion, since he vale of V reqires o be comped for every elemen in a discree ime inerval from ime o T. However, lemma 3.3 enables s o obain a closed form approximaion for he fair variance srike, which is pah-independen, by sbsiing eqaion (9) ino (1). Lemma 3.4. The fair variance srike of a variance swap is given by E Q V (,T ) = V + βg + β H + β 3 L + β 4 M + O(ɛ 5 ), (13) where G = 1 T T EQ Gd = 1 V (r V )T, H L M = 1 T T EQ Hd = 1 T T EQ Ld = 1 T T EQ Md = 1 3 V (V 3rV + r )T + V T, = V 1 4 ( V 3 + 4rV 14 3 r V r3 )T 3 + V (3r 3 V )T, = V 1 5 (V 4 5rV r V 5r 3 V + 3 r4 )T 4 + V ( 31 1 V 79 1 rv r )T V T. Proof. Sbsiing eqaion (9) ino (1) gives E Q 1 T ( V (,T ) = T EQ V + βg + β H + β 3 L + β 4 M + O(ɛ 5 ) ) d = V + β T T EQ Gd T + β4 T EQ Md T T + β T EQ Hd + β3 T EQ Ld + O(ɛ 5 ). 6

8 Then he lemma is proved by sbsiing G, H, L and M of eqaion (9) ino he above formla. 4 Expecaion of annalized inegraed variance In his secion we apply he resls from secion 3 o obain an efficien Mone-Carlo mehod o compe he fair variance srike by ransforming he vale of V (,T ) o a pah-independen random variable. This mehod allows s o simlae he condiional disribion of he realized variance. In essence, he fair variance may be calclaed by he mehods olined in secion 3, however, for hose more sophisicaed volailiy derivaives whose payoff is no linear on he realized variance, for example, opions on realized variance, a closed-form solion may be difficl o be obained. Using an appropriae adjsmen, he Mone-Carlo mehod provided in his secion is able o simlae he disribion of he realized variance and hence can be applied o evalae oher more sophisicaed volailiy derivaives. I is well known ha EV (,T ) = EEV (,T ) F (14) and by Jensen s ineqaliy we have V arev (,T ) F V arv (,T ) holding for any condiion F. This implies ha simlaing he condiional expecaion E Q V (,T ) F is more efficien han simlaing V (,T ) iself. So in his secion we provide a framework o compe he condiional expecaion of he annalized inegraed variance given a Brownian moion wih fixed end poin a ime T, sch ha W T = b. The pah dependen random variable V (,T ) is ransformed o a pah independen random variable E Q V (,T ) W T = b, via a Brownian bridge. This is one of he key resls of or paper. A Brownian bridge is a coninos sochasic process whose probabiliy disribion is he disribion of a Brownian moion wih a fixed iniial and end poin. Nex we give one of is proposiions. Proposiion 4.1. (See Karazas and Sherve1) Le W, T be a Brownian moion, hen a Brownian bridge b T X = + (T ) dw T ; for < T, b; for = T, is Gassian wih almos srely coninos pahs, wih mean µ = EX = b T and variance σ = V arx = T for a consan b. Lemma 3.3 expanded V ino a fncion of Brownian moion. So wih his lemma, he calclaion of E Q V (,T ) W T = b involves he compaion of some condiional expecaion sch as E Q W W T = b, E Q W W W T = b, E Q W W W T = b and so on. We provide a heorem o compe hese condiional expecaions in a more general form, E Q W m W n W T = b, for T. For simpliciy, E Q W T = b is denoed by E Q b. Theorem 4.. Le W for, T be a Brownian moion. Then we have he condiional expecaion E Q b W m W n = m!n! j m l k n δ ν k+j ( l + j j ) (T ) j k (T ) k+l ( ) k l n k b m+n j k (m j)!(n k)!( k l l+j j l j l )!( )! m+ T m+n k (15), (16) 7

9 where m, n are non-negaive inegers, < T, and δ ν is a Kronecker dela fncion sch ha δ ν = {, if ν 1, if ν = for ν = k + j and he floor fncion x = max{p x p N} for any x. l + j k l, Proof. Since W for, T is a Brownian moion, E Q b W m W n = E Q X m X n. Take Y, = dw s T s for T. Then Y, is a Gassian random variable wih mean µ = ( ) E Q Y, =, and variance σ, = E Q dw s ( ) T s = E Q 1 T s ds = (T )(T ). From eqaion (15), we have ( ) m ( ) n b b E Q X m X n = E Q T + (T )Y, T + (T )Y, (T ) j (T ) k m j n k ( b = m!n! j!k!(m j)!(n k)! T = m!n! = m!n! = m!n! j m k n j m k n j m l k n j m l k n (T ) j (T ) k m j n k j!k!(m j)!(n k)! ) m+n j k E Q Y j, (Y, + Y, ) k ( ) m+n j k b E Q Y j T, l k (T ) j (T ) k m j n k ( ) m+n j k b E Q Y l+j, j!l!(m j)!(n k)!(k l)! T Y, k l (T ) j (T ) k m j n k ( b j!l!(m j)!(n k)!(k l)! T According o Miller and Childers 15, we have for i = k l k l ν = i + h, sch ha ) m+n j k E Q Y l+j, k! l!(k l)! Y l,y, k l E Q Y, k l. (17) E Q ( ) k l Y, k l = δi k l (k l)! ( k l )!, (18) (T )(T ) ( ) l+j E Q Y l+j, = δ h l+j (l + j)! ( l+j )!, (19) T (T ), h = l+j l+j. δ is a Kronecker dela fncion, where {i, h, ν} and δ = {, if 1, if =. Then we have δ i δ h = δ i+h = δ ν and he heorem is proved by sbsiing eqaions (18) and (19) ino eqaion (17). 8

10 Corollary 4.3. Theorem 4. holds for some special vales of m and n, sch as when m = n = 1, or when n =, where ν = j j. E Q b W m = m! E Q b W W = b + T T, () j m δ ν j b m+j (T ) j m j (m j)!( j )!, (1) T m j Theorem 4. allows s o compe he vale of E Q b V (,T ), given by he following corollary. Corollary 4.4. The condiional expecaion of he annalized realized variance over he lifeime of a variance swap is given by E Q b V (,T ) = V + βg b + β H b + β 3 L b + β 4 M b + O(ɛ 5 ), () where G b = V 1 (r V )T + V 1 b, H b = V 1 3 (V 3rV + r )T V 1 (1r 7V )bt + V b, L 1 b = V 4 ( V 3 + 4rV 14 3 r V r3 )T 3 + V 1 (59 48 V 37 1 rv r )bt V (r V )T + V ( 11 4 r 41 4 V )b T 1 6 V 3 bt + V 3 b3, M 1 b = V 5 (V 4 5rV r V 5r 3 V + 3 r4 )T 4 + V 1 ( V rv r V r3 )bt 3 + V ( 7 48 V + 15 rv r )T 3 + V ( V 43 6 rv r )b T + V 3 ( 1 3 V 71 6 r)bt V T + V 3 (16 5 r 53 V )b 3 T 11 4 V b T + V b 4. Proof. See Appendix B. Remark 4.5. Given eqaion (14), we have E Q V (,T ) = E E Q Q b V (,T ). Then lemma 3.4 may also be proved sing corollary 4.4 combined wih he fac ha E Q W T = E Q W 3 T =, EQ W T = T and E Q W 4 T = 3T. Sbsiing hese vales ino eqaion () gives he vale of E Q V (,T ) direcly. 5 Nmerical experimens In his secion we will perform a nmerical sdy of he approximaion given by he corollary 4.4 and lemma 3.4. Secion 5.1 gives he log-eler Mone Carlo scheme o simlae he insananeos variance and compe E Q V (,T ). We also illsrae why his mehod may no be siable for some model parameers. Secion 5. compes E Q V (,T ) by he small disrbance asympoic expansion as described in lemma 3.4 and corollary

11 5.1 Log-Eler Mone-Carlo scheme for he CEV variance process In he lierare, some mehods are sed o simlae he nderlying price, S, of he CEV process (1). For example, Lindsay and Brecher 14 provided a qasi Mone-Carlo mehod o simlae he nderlying price from a non-cenral Chi sqare disribion. Chen e.al. 5 also sggesed sing a momen-mached qadraic Gassian approximaion mehod and a direc inversion scheme o simlae he nderlying price S. However, since or aim is primarily o obain some benchmark vales of he fair variance srike, we will no focs or aenion on he se of hese echniqes. In or Mone-Carlo simlaions, we have employed he sandard log-eler scheme for he CEV variance process (3). Given eqaion (3), we have a sochasic differenial eqaion of ln V by Iô s lemma, sch ha d ln V = β(r V )d + β V dw, for β < and, T. Then applying he firs-order Taylor approximaion scheme for ln V gives: ln V n+1 = ln V n + β(r V n )h + β V n Z h + O(h ), wih Z a sandard normal random variable, n = n N T for n =, 1,,, N and h = T N. By aking exponenials on boh side, we have V n+1 = V n exp β(r V n )h + β V n hz + O(h ), (3) and V n+1 > for all V n >, i.e., V n > if V >. However, his nmerical scheme becomes nsable, i.e., V n = δ S β n when S n for some model parameers. Figre 1: Sample pahs of V by he log-eler Mone-Carlo simlaion for differen vales of β and iniial volailiies σ = V. r =.1, T = 1. Model parameers are shown on he op of each plo. The vale of Q(V ) represens he probabiliy of insananeos variance V reaching infiniy wih corresponding model parameers. 1

12 Figre (1) shows sample pahs of {V },T generaed by eqaion (3) wih differen iniial volailiies and vale of β. Each sbplo conains 1 sample pahs and hose in he lef colmn shows pahs wih iniial volailiy σ = V = 5%, β =.3 and β =.9 from op o he boom respecively. Plos in he righ colmn also indicaes pahs wih iniial volailiy σ = 5%, β =.3 and β =.9 from op o he boom respecively. I is observed ha he nmber of nsable pahs increases wih β for boh high and low iniial volailiies σ. By applying he log-eler scheme, we obain he sqare roo of he fair variance srikes and he sandard deviaions of his nmerical scheme shown in able 1. The sqare roo of he fair variance srike is comped for differen iniial volailiies and vales of β. We choose nine vales of iniial volailiy eqally spaced in inerval 1%, 5% and en vales of β eqally spaced in inerval 1,.1 3. We se ime seps a 5 and generae 16 1 sample pahs by sing a Sobol seqence. I is clear ha here are several infiniy a he pper righ corner of he able. This is de o he resl shown by eqaion (4) ha he probabiliy of V increases wih iniial variance V. This behavior cold also be seen in able, which gives he probabiliies of V before mariy for differen vales of he iniial volailiy and β. When his probabiliy is large enogh o ensre one sample pah can reach infiniy, he average realized variance will be infiniy. For example, wih β =.9 and iniial volailiy increasing from 5% o 3%, he probabiliy of V increased from.4% o.135%. When iniial volailiy is 5%, here migh be.4% ( 16 1), i.e.,.6 pahs reaching infiniy before T. In oher words, he realized variance for every sample pah does no reach infiniy since he insananeos variance a every sep before mariy is less han infiniy, so does he expeced realized variance. B when he iniial volailiy increase o 3% wih he same vale of β, he increasing probabiliy of V makes.135% ( 16 1), approximaely 9 sample pahs o reach infiniy. As a resl, he expeced realized variance diverges. I follows ha he log-eler scheme is no always sable when comping he fair variance srike of a variance swap nder he CEV model. Sandard deviaions are shown in he brackes as percenage nmbers. We can see ha his vale increases wih iniial volailiy σ and β. There are 16 1 sample pahs generaed by he Sobol seqence ensring he sandard deviaion less han.3%. Using a comper wih a 3.33GHz Inel Core Do CPU and 4GB memory, i akes seconds o compe all he vales in his able,.7817 seconds for each vale. 5. Small disrbance asympoic expansion of he CEV variance process As indicaed in secion 4, small disrbance asympoic expansion of he CEV variance process gives s wo disinc mehods o price he fair variance srike of a variance swap. By applying lemma 3.4, he fair variance srike is obained explicily as a deerminisic fncion of he iniial volailiy, ineres rae and mariy. Table 3 shows he sqare roos of fair variance srikes, as percenages, wih differen vales of parameers. Iniial volailiy σ and β are sampled in a similar way as in he log-eler simlaion. The vales shown in he brackes are he absole vale of relaive difference beween he sqare roos of he fair variance srike calclaed by he log-eler simlaion of eqaion (3) and he closed form approximaion of lemma13 respecively. These vales are also expressed in percenage. I is ineresing o poin o ha by sing he small disrbance asympoic expansion we solve he problem of he aberraion of infiniies for he fair variance srike in or calclaion, when he probabiliy of V increases. Alhogh hese differences increase wih he iniial volailiy σ and β, hey are 3 As menioned in secion, he CEV model recovers he Black Scholes model when β =. We ignore his well known problem in his paper. 11

13 β Iniial Volailiies (%) (4.1) (14.6) (38.6) (119.9) (3.7) (13.1) (33.7) (76.9) (3.3) (11.6) (9.3) (63.4) (161.9) (.9) (1.1) (5.) (53.) (1.8) (.5) (8.7) (1.4) (44.) (81.8) (145.7) (.1) (7.3) (17.7) (35.9) (65.) (11.) (18.7) (1.7) (5.8) (14.1) (8.3) (5.4) (83.3) (13.7) (198.8) (99.5) (1.3) (4.4) (1.6) (1.) (37.) (6.1) (9.4) (136.1) (194.6) (.9) (3.) (7.1) (13.9) (4.3) (39.) (59.3) (85.9) (1.) (.4) (1.5) (3.6) (7.) (1.1) (19.3) (9.) (41.6) (57.5) Table 1: Sqare roo of fair variance srikes (%) obained by log-eler Mone Carlo scheme of eqaion (3) wih β 1,.1, iniial volailiies σ.1,.5, r = 1% and T = 1. Vales shown in brackes are he sandard deviaions ( 1 6 ) of his scheme wih respec o he corresponding model parameers. β Iniial Volailiies (%) Table : Probabiliy (%) of V wih β 1,.1, iniial volailiies σ.1,.5, r = 1% and T = 1. 1

14 β Iniial Volailiies (%) (.31) (.758) (.3784) (.785) (.3) (.56) (.351) (.5131) (.159) (.396) (.858) (.398) (.7351) (.99) (.59) (.561) (.3155) (.454) (.6) (.157) (.34) (.671) (.3337) (.4799) (.1) (.83) (.183) (.357) (.67) (.361) (.4437) (.) (.8) (.77) (.159) (.97) (.58) (.93) (.3637) (.4938) (.1993) (.1994) (.17) (.47) (.98) (.176) (.3) (.498) (.89) (.1984) (.1986) (.1985) (.1995) (.7) (.7) (.53) (.93) (.155) (.1975) (.1978) (.1979) (.198) (.1981) (.198) (.1984) (.1987) (.1991) Table 3: Sqare roo of he fair variance srikes (%) obained by he closed form approximaion of lemma 3.4, when iniial volailiies σ.1,.5, β 1,.1, r = 1% and T = 1. Vales shown in brackes are he relaive differences (%) beween he fair variance srikes obained by his closed form approximaion and he log-eler scheme wih respec o he corresponding model parameers. Symbol is sed for hose variance swaps whose fair variance srike is in able 1. β Iniial Volailiies (%) (.35) (.14) (.3) (.74) (1.41) (.677) (4.869) (8.56) (14.61) (.31) (.111) (.84) (.611) (1.194) (.1) (3.89) (6.666) (11.111) (.8) (.99) (.49) (.56) (1.5) (1.8) (3.97) (5.158) (8.377) (.5) (.87) (.16) (.449) (.839) (1.466) (.45) (3.965) (6.6) (.) (.75) (.184) (.377) (.691) (1.181) (1.9) (3.4) (4.638) (.18) (.63) (.153) (.39) (.557) (.934) (1.485) (.76) (3.393) (.15) (.51) (.1) (.44) (.435) (.716) (1.117) (1.673) (.431) (.11) (.38) (.9) (.18) (.3) (.51) (.799) (1.174) (1.67) (.8) (.6) (.61) (.11) (.11) (.34) (.515) (.746) (1.43) (.4) (.13) (.31) (.61) (.15) (.168) (.5) (.361) (.499) Table 4: Sqare roos of he fair variance srikes (%) and heir sandard deviaions obained by he condiional Mone-Carlo simlaion of corollary 4.4, when iniial volailiies σ.1,.5, β 1,.1, r = 1% and T = 1. Vales shown in brackes are he sandard deviaions ( 1 6 ) of his scheme wih respec o he corresponding model parameers. 13

15 sill in fairly low level, ha all of hem are less han 1%. We se he symbol o denoe when a vale may no be rerned, becase he resl of log-eler scheme is infiniy. The alernaive approach is o apply corollary 4.4 by implemening a condiional Mone-Carlo simlaion. The vale of he condiional expeced realized variance only depends on he iniial volailiy, ineres rae, mariy and a sample of Brownian moion W T, which is a pah independen random variable and is sraighforwardly simlaed by qasi-mone Carlo mehod. The firs sep is o generae samples of a Brownian moion W T via a Sobol seqence. Nex, we compe he vale of E Q b V (,T ) by aking b as a sample of W T and sbsie b o eqaion (). Then he vale of fair variance srike is he mean of E Q b V (,T ) comped by repeaing he second sep for all samples of W T. Table 4 shows he sqare roo of fair variance srikes in percen by applying his condiional Mone- Carlo simlaion wih differen vales of parameers. The model parameers σ and β are he same as hose sed in able 1. We generae 16 1 elemens in a Sobol seqence o simlae he disribion of E Q b V (,T ) for all he parameer vales, and also compe is he sandard deviaions, which are shown in he brackes as a percenage vale. As is shown, he fair variance srikes in his able are vary close o he vales in able 3. The sandard deviaion also increases wih iniial volailiy σ and β, which agrees wih he rend shown in able 1. However, he level of sandard derivaions generaed by he small disrbance asympoic expansion is approximaely 1 imes smaller han by he log-eler simlaion when hey boh se 16 1 pahs. In addiion, he condiional Mone-Carlo simlaion wih small disrbance asympoic expansion is shown o be very fas in or case. I akes.815 seconds o compe all he vales in able 4, i.e.,.31 seconds for each of hem, an approximae improvemen of a facor of 9. This mehod also has he poenial o be applied o price more complicaed volailiy derivaives, sch as corridor variance swaps and opion on realized variance. See Carr and Madan, Carr and Lewis 3, 4 for a deailed discssion. 6 Conclsion In his paper we apply he small disrbance asympoic expansion o he CEV variance process and compe he fair variance srike of a variance swap by giving a pah-independen closed from approximaion. A condiional Mone Carlo simlaion is also applied o obain he fair variance srike by deriving a heorem for he condiionally expeced prodc of a Brownian moion a wo differen imes wih arbirary powers. The resls presened in he paper have hree main advanages. Firs, he fair variance srike is expressed as a deerminisic fncion of ineres rae, expiry ime and iniial volailiy and can be easily implemened nmerically. Second, he condiional expecaion of he realized variance only depends on ineres rae, expiry ime, iniial volailiy and a Brownian moion a expiry, which can be applied by a condiional Mone-Carlo simlaion scheme. This mehod is compared wih he firs order log-eler scheme. The resls demonsrae ha he speed and accracy of or condiional Mone-Carlo simlaion is improved significanly compared wih he firs order log-eler scheme. Finally, ignoring he pahological case when he variance goes o infiniy, we gained reasonable and accrae resls. References 1 S. Beckers. The consan elasiciy of variance model and is implicaion for oprion pricing. Jornal of Finance, 35: ,

16 F. Black and M. Scholes. The pricing of opions and corporae liabiliies. The Jornal of Poliical Economy, 81(3): , P. Carr and K. Lewis. Corridor variance swaps. Risk, 17():67 7, 4. 4 P. Carr and D Madan. Towards a heory of volailiy rading. Working paper, Morgan Sanley, B. Chen, C. Ooserlee, and H. Weide. A low-bias simlaion scheme for he SABR sochasic volailiy model. Inernaional Jornal of Theoreical and Applied Finance, 15(): , 1. 6 S. Chng, P. Shih, and W. Tsai. Saic hedging and pricing American knock-in p opion. Jornal of Banking & Finance, 37:191 5, J. Cox and S. Ross. The valaion of opions for alernaive sochasic processes. Jornal of Financial Economics, 3(1-): , D. Davydov and V. Linesky. Pricing and hedging pah-dependen opions nder he CEV process. Managemen Science, 47(7): , 1. 9 K. Demeerfi, E. Derman, M. Kamal, and J. Zo. A gide o volailiy and variance swaps. Jornal of Derivaives, S. L. Heson. A closed-form solion for opions wih sochasic volailiy wih applicaions o bond and crrency opions. Review of Financial Sdies, 6():37 343, R. Jordan and C. Tier. The variance swap conrac nder he CEV process. Inernaional Jornal of Theoreical and Applied Finance, 1(5):79 743, 9. 1 I. Karazas and S. Shreve. Brownian moion and sochasic calcls. Springer, New York, N. Kniomo and A. Takahashi. The asympoic expansion approach o he valaion of ineres rae coningen claims. Mahemaical Finance, 11: , A. Lindsay and D. Brecher. Simlaion of he CEV process and he local maringale propery. Mahemaics and Compers in Simlaion, 8: , S. Miller and D. Childers. Probabiliy and Random Processes: Wih Applicaions o Signal Processing and Commnicaions - Insrcor s manal. Academic Press, Oxford, M. Schroder. Comping he consan elasiciy of variance opion pricing formla. Jornal of Finance, 44(1):11 19, S. Shreve. Sochasic calcls for finance II : coninos-ime models. Springer, New York, G. Sorwar and K. Dowd. Esimaing financial risk measres for opions. Jornal of Banking & Finance, 34: , S. Zh and G Lian. A closed-form exac solion for pricing variance swaps wih sochasic volailiy. Mahemaical Finance, 1():33 56,

17 Appendix A Proof of lemma 3.3. Proof. By lemma 3.1, we have V 3 V 3 4 V 4 = β + ɛ β = 3 β + ɛ β = 4 β + ɛ β βv + (ɛ β 1) V V + r d + 4 β 4 β V + (ɛ β 1) (V ) + r V 4 β V + (ɛ β 1) (V ) + r V 6 β (V ) + (ɛ β 1) 3 (V ) + r 3 V 3 3 V 3 dw d + ɛ β d + 6 β 6 β (V ) + (ɛ β 1) 3 (V ) + r 3 V β 3 (V ) + (ɛ β 1) 4 (V ) + r 4 V d + ɛ β d + 8 β (V 3 ) dw ; (V 3 ) dw d + ɛ β 3 (V 3 dw ; and 3 ) ) 3 (V 3 dw 3 4 (V 3 ) 4 dw. Taking ɛ gives s he following expressions. V 3 V 3 4 V 4 = β ɛ= = 3 β ɛ= = 4 β ɛ= βv V 4 β V 6 β (V ) + rg ɛ= d + 4 β (V ) ɛ= + rh ɛ= 3 (V ) ɛ= 3 + rl ɛ= V 3 d + 6 β dw, (4) ɛ= (V 3 ) dw, (5) ɛ= 3 (V 3 ) dw. (6) 3 ɛ= d + 8 β The derivaives of V 3 and V needs o be comped p o he hird order. They will be comped in hree seps. Sep 1: he firs derivaive V m = m βv m ɛ= Sep : he second derivaive (V 3 ) = 6 β ɛ= (V ) = 6 β V 3 = 4 β ɛ= + 1 β V (r V ) + V 1 W for m. (7) 3 βv 5 1 V 5 + r V 3 d + 6 β V dw ɛ= ɛ= ɛ= 3 V 5 V(r 1 V) 5 ( V 3 W + 3r (r 1 ) V) + V 1 W d (r V ) + V 1 W dw. (8) 4 βv 3 V 3 + r V ɛ= 4V 3V (r V ) 6V 3 d + 8 β ɛ= ɛ= V 5 dw ɛ= = 4 β ( ) V W + 4r (r V ) + V 1 W d + 4 β V 5 (r 1 V) + V 1 W dw. (9) (V 5 ) = 1 β 5 βv 7 1 V 7 + r V 5 d + 1 β V 3 dw ɛ= ɛ= ɛ= 16

18 (V 3 ) = 1 β V 5 5 = 6 β ɛ= + 3 β V 3 = 1 β V 3 V 7 V(r 1 V) 7 ( V 3 W + 5r (r 1 ) V) + V 1 W d (r V ) + V 1 W dw. (3) + r V 3 ɛ= d + 1 β V 7 dw ɛ= ɛ= ( ) 3V V (r V ) 4V 3 W + 3r (r V ) + V 1 W d 6 βv 4 V β V 7 By Iô s lemma, we have W (31) may be wrien as (V 3 ) = 3 β V 3 ɛ= + 4V W V 1 (r + V ) (V ) = 4 β V ɛ= (r 1 V) + V 1 W dw. (31) = WdW + and W = Wd + ( 5 4 V 4rV + 3r ) V + 4V 1 (r V )W dw. So eqaion (8) o eqaion W d. (3) ( 3 V 5rV + 4r ) V + 5V 1 (r V )W + 5V W V 1 (r + V ) (V 5 ) = 5 β V 5 ( 7 4 V 6rV + 5r ) V + 6V 1 (r V )W ɛ= + 6V W V 1 (r + V ) (V 3 ) = 6 β V 3 ɛ= Sep 3: he hird derivaive 3 (V 3 ) = 9 3 β ɛ= 3 (V 3 ) W d. (33) W d. (34) (V 7rV + 6r ) V + 7V 1 (r V )W + 7V W V 1 (r + V ) = 9 β 3 V 3 = 6 β ɛ= W d. (35) 3 β V 5 1 (V 5 3 ) + r (V ) ɛ= ɛ= ɛ= 15 V((r V) + V 1 W ) 5 ( V ( 7 4 V + 6V 1 (r V )W + 6V W V 1 (r + V ) V + 4V 1 (r V )W + 4V W V 1 (r + V ) + 36 β 3 V V 1 (r + V ) = 1 β 3 V d + 9 β (V ) 6rV + 5r ) V ) ( W sds + 3r ) W sds d ( 3 V 5rV + 4r ) V + 5V 1 (r V )W + 5V W W sds dw ɛ= ( 5 4 V 4rV + 3r ) dw. (36) 8 β V 3 (V 3 ) ɛ= + r (V ) ɛ= d + 1 β ɛ= ( ) ( 1V (r V ) + V 1 W 3V (V + 7V 1 (r V )W + 7V W V 1 (r + V ) V + 5V 1 (r V )W + 5V W V 1 (r + V ) (V 5 ) dw ɛ= 7rV + 6r ) V ) ( W sds + 4r ) W sds d ( 3 V 5rV + 4r ) 17

19 + 6 β 3 V 5 V 1 (r + V ) ( 7 4 V 6rV + 5r ) V + 6V 1 (r V )W + 6V W W sds dw. (37) Again, by Iô s lemma, we have W 3 = 3 W dw + 3 Wd, W = W d + WdW + 1, W = Wd + dw and s WsdsdW = W Wd W d. Afer sbsiing hese eqaions o eqaion (36) and (37), we derive ha 3 (V 3 ) = 9 3 β 3 V 3 ɛ= 3 (V 3 ) ( 35 4 V rv 49 6 r V + 3r 3 ) 3 + 4V 1 ( 3 V 5rV + 4r ) W V (4r 5 V) + 1V (r V )W 4V 3 W + 3 V 3 W 3 + V (8r + 3V ) + V 1 (3V rv 8r ) + V 1 ( 5 V + rv 6r ) = 1 β 3 V ɛ= W d W d 4V (r + V )W W d W sdsd V 3 W d. (38) ( V 3 + 9rV 38 3 r V r3 ) 3 + 5V 1 ( 7 4 V 6rV + 5r ) W V (5r 3V ) + 15V (r V )W 5V 3 W + 1V 3 W 3 + V (5r + V ) + V 1 ( 7 4 V rv 5r ) + V 1 (3V + rv 8r ) W d W d 5V (r + V )W W d W sdsd V 3 W d. (39) Now, we are able o derive H, L and M from eqaions (4) o (6). Firs, sbsiing eqaion (11) and (7) o eqaion (4) derives V = β ( V (V V ɛ= + 3V 1 ( (r V ) + V 1 W ) + r (r V ) + V 1 W ) dw ( (r V ) + V 1 W ) )d = β V (V 3rV + r ) V + 3V 1 (r V )W + 3V W V 1 (r + V ) Second, sbsiing eqaions (7), (3), (33) and (4) ino eqaion (5) gives s ha 3 V 3 = 3 β ( 3 V 8V ɛ= V + 5V W V 1 (r + V ) (r V ) + V 1 W ) 4V ( ( 3 V 5rV + 4r ) + 5V 1 (r V )W + 3V 1 (r V )W V 1 (r + V ) ) ( W sds + 4r (V 3rV + r ) V + 3V W V + 4V 1 (r V )W + 4V W V 1 (r + V ) ) W sds d + 18 β 3 V 3 W sds dw ( 5 4 V 4rV + 3r ) = 6 β 3 V ( V 3 + 4rV 14 3 r V r3 ) 3 + 3V 1 ( 5 4 V 4rV + 3r ) W V (3r V ) + 6V (r V )W 3V 3 W + 4V 3 W 3 + V (3r + V ) + V 1 ( 5 V rv 6r ) W d + V 1 (V + rv r ) W d. (4) Wd 3V (r + V )W W d W sdsd V 3 W d. (41) 18

20 Nex, we derive 4 V ɛ=, b we firsly give he following eqaions by applying Iô s lemma. 4 W 4 = 4 W 3 = 3 W = 3 W = s W 3 dw + 6 (W 3 + 3W )d + 3 W d + W d + W d, (4) Ws dsdw = W Wd W dw, (43) 3 dw, (44) W dw , (45) sw sdsdw = W W d W adadsdw = W W W sdsdw = 1 ( W W 3 d, (46) W d, (47) W sdsd W W sdsd, (48) ) W d Wd 3 W sdsd. (49) Then sbsiing eqaions (33), (38), (39), (41) and (4) o (49) ino eqaion (6), we obain 4 V 4 = 4 β 4 V (V 4 5rV ɛ= 3 r V 5r 3 V + 3 r4 ) 4 V 1 ( 35 8 V rv + 49 r V 9r 3 ) 3 W V (3V 8rV 14 3 r ) 3 + 6V ( 3 V 5rV + 4r ) W 3V 3 (4r 5 V) W + V + 1V 3 (r V )W 3 6V W + 5V W 4 + V 3 (5V r) + V 3 (r + V ) + 3V (V r) W sdsd V 1 ( 35 8 V rv Wd V 3 (4r + 13V ) V 1 (7V 3 9rV 16r V + 1r 3 ) V 1 (3V 3 1r V + 4r 3 ) + 3V (3V rv 8r )W + 3V ( 5 V 4rV + 6r )W s W d V + 1 r V + 9r 3 ) W d W 3 d 4V (V + 4rV 3r ) sw sdsd + V ( 5 V + 6rV 3r ) W adadsd + 3V 3 (8r + 3V )W Wd W d W s dsd W W sdsd W d 6V 3 (r + V )W W d W sdsd 3V W W d. (5) Finally, H, L and M can be derived by sbsiing eqaions (4), (41) and (5) ino eqaion (6). Appendix B Proof of lemma 4.4. Proof. From lemma (3.3), we have he expecaion of inegraed variance wih condiion {W T = b} in he form E b V (,T ) = 1 T E b = 1 T E b T T = V + β T E b V d (V + βg + β H + β 3 L + β 4 M + O(ɛ 5 ))d T T T T Gd + β T E b Hd + β3 T E b Ld + β4 T E b Md + O(ɛ 5 ). (51) 19

21 Then by corollary 4.4, we derive he condiional expecaion E b T Gd as follows: T T E b Gd = V (r V )E b d + V 3 T 1 E b W d = V T (r V)T + V 1. b (5) T Likewise, by repeaing his calclaion, we have he E b Hd T, E b Ld T, and E b Md as T 1 E b Hd = V T 3 (V 3rV + r )T V 1 (1r 7V )bt + V b. (53) T 1 E b Ld = V T 4 ( V 3 + 4rV 14 3 r V r3 )T 3 + V 1 ( V rv r )bt V(r V)T + V 11 ( 4 r 41 4 V)b T 1 6 V 3 bt + V 3 b 3. (54) T 1 E b Md = V T 5 (V 4 5rV r V 5r 3 V + 3 r4 )T 4 + V 1 ( V rv r V r3 )bt 3 + V 7 ( 48 V + 3 rv r )T 3 + V 131 ( 48 V rv r )b T + V 3 ( 1 71 V 3 6 r)bt V T + V 3 ( 16 5 r 53 V)b3 T 11 4 V b T + V b 4. (55) Then he lemma is proved by sbsiing eqaion (5) o (55) ino eqaion (51).