Ch 6. Opion Pricing When Volailiy is Non-Consan I. Volailiy Smile II. Opion Pricing When Volailiy is a Funcion of S and III. Opion Pricing Under Sochasic Volailiy Process I is convincingly believed ha he consan volailiy assumpion of he Black-Scholes model is rejeced by many empirical facs. Therefore, his chaper inroduces wo opion pricing models o deal wih he non-consan volailiy. The firs opion pricing model is proposed by Nelson and Ramaswamy (1990) o consider he volailiy of he underlying asse being a funcion of he ime and he price of he underlying asse. The second opion pricing model, developed by Richken and Trevor (1999), is o price opions when he underlying asse price follows he GARCH or sochasic volailiy processes. I. Volailiy Smile The phenomenon of volailiy smile shows he variaion of he Black-Scholes implied volailiy wih respec o he srike price. In oher words, he consan volailiy assumpion in he Black-Scholes is no so correc. Nex, I will show he volailiy smile for currency and equiy opions, respecively. Volailiy smile for currency opions Figure 6-1 Illusraion of he Volailiy Smile for Currency Opions Implied σ σ 6-1
I is inuiive ha boh implied volailiies of call or pu opions should be he same as long as hey share a common underlying asse. Thus, eiher he implied volailiies of calls or he implied volailiies of pus could be employed o explain he volailiy smile phenomenon. Figure 6- Explanaion of he Volailiy Smile for Currency Opions implied disribuion lognormal disribuion K K1 S 0 For a pu opion wih a low K1, he implied (rue) in-he-money probabiliy, i.e, prob( ST K1), is higher han he counerpar of he lognormal disribuion. Thus, K marke price > heoreical value (Noe ha no only OTM pu prices are higher han heir heoreical values, bu also he ITM call prices should be higher han heir heoreical values because hey share he same underlying asse.) implied S T For a call opion wih a high K, he implied (rue) in-he-money probabiliy, i.e, prob( ST K), is higher han he counerpar of he lognormal disribuion. Thus, K marke price > heoreical value (Noe ha no only OTM call prices are higher han heir heoreical values, bu also he ITM pu prices should be higher han heir heoreical values because hey share he same underlying asse.) implied Possible reasons for he volailiy smile of currency opions: (i) The volailiy of he underlying asse is never o be consan. (ii) The exisence of price jump causes exreme reurns and hus increase he probabiliies when S T is fairly high or low. Furhermore, for longer mauriy, he jumps are averaged ou and hus he jumps have weaker impacs on he probabiliies for S T being fairly high or low. So, he volailiy smile dies ou wih he increase of mauriy, which is consisen wih empirical sudies. (iii) The sochasic or GARCH process also can generae he probabiliy disribuion of he underlying asses wih faer ails. 6-
Volailiy smile for equiy opions (including index opions and individual sock opions): Figure 6-3 Illusraion of he Volailiy Smile for Equiy Opions Implied σ Figure 6-4 Explanaion of he Volailiy Smile for Equiy Opions implied disribuion lognormal disribuion K 1 K For a pu opion wih a low K1, he implied (rue) in-he-money probabiliy, i.e, prob( ST K1), is higher han he counerpar of he lognormal disribuion. Thus, K marke price > heoreical value (Noe ha no only OTM pu prices are higher han heir heoreical values, bu also he ITM call prices should be higher han heir heoreical values because hey share he same underlying asse.) implied S 0 S For a call opion wih a high K, he implied (rue) in-he-money probabiliy, i.e, prob( ST K), is lower han he counerpar of he lognormal disribuion. Thus, K marke price < heoreical value (Noe ha no only OTM call prices are lower han heir heoreical values, bu also he ITM call pu prices should be lower han heir heoreical values because hey share he same underlying asse.) implied 6-3
Possible reasons for he volailiy smile of equiy opions: (i) The above volailiy smile paern for equiy opions is ure afer he crash of sock markes in Ocober of 1987. Before Ocober of 1987, he volailiy is almos consan for differen srike prices. Thus, Mark Rubinsein, a famous academic in he finance field and he invenor of he CRR binomial ree model, erms he reason underlying his volailiy paern as crashophobia, meaning invesors are afraid ha he sock markes crash again. (ii) Leverage effec: i is well known ha when a firm employs more leverage, he financial risk of he firm increases and hus he volailiy of he sock price increases o reflec his risk. Sock price, leverage raio, volailiy, which make he sock price furher decline. Sock price, leverage raio, volailiy, which make he sock price furher rise. (iii) The volailiy should follow an asymmeric GARCH or sochasic process o reflec he counercyclical variaion of he volailiy, i.e., when he sock price rises (declines), i volailiy decreases (increases). The relaionship beween he implied volailiy and he ime o mauriy can be expressed as a volailiy erm srucure. If boh he sirke price and he ime o mauriy are considered, a 3-D implied volailiy funcion can be derived, which is called he volailiy surface. By observing he volailiy surface, i can be found ha he volailiy smile is pronounced for shorer imes o mauriy, bu becomes minor for longer imes o mauriy. Opion pricing models for non-consan volailiies: (i) Volailiy is a funcion of K: Since he srike price K can be measured relaive o he sock price S, he volailiy o be a funcion of K implies he volailiy o be a funcion of S. Nelson and Ramaswamy (1990) deals wih he volailiy as a funcion of S and. (ii) Volailiy is a funcion of T : Since he curren ime poin and he ime o mauriy T always change by he same magniude bu in opposie direcions, he volailiy o be a funcion of T implies he volailiy o be a funcion of. Nelson and Ramaswamy (1990) deals wih he volailiy as a funcion of S and. (iii) Volailiy movemens conform he GARCH or sochasic variance processes: Richken and Trevor (1999) deals wih he GARCH or he sochasic variance process. 6-4
II. Opion Pricing When Volailiy is a Funcion of S and The seing of he sock price process and he binomial ree model in Nelson and Ramaswamy (1990), Simple Binomial Processes as Diffusion Approximaions in Financial Models, Review of Financial Sudies 3, pp. 393 430. The underlying asse price is assumed o be an Iô proces: ds = µ(s, )d + σ(s, )dz. (Noe ha boh he drif and he volailiy erms are funcions of S and and should be sochasic. In addiion, S can be any kind of underlying asse raher han only he sock price.) The binomial ree considered in Nelson and Ramaswamy (1990) is he model in Cox and Rubinsein (1985) as follows. Figure 6-5 Non-recombinaion of he binomial ree model when he volailiy is no a consan S S, q S S S S, S, S S q S S 1 q S q S S S S, S, S S S, S, S 1 q S S S S, 1 q S S S S, S, S If ( S, ), hese wo sock prices are he same, and he ree recombines S where q(s ) = E[S+ ] S + S + + S + = S+µ(S,) S +. S + + S + (I is obvious ha he non-consan σ(s, ) is he reason o make he binomial ree nonrecombined. As for µ(s, ), i only affecs he upward and downward branching probabiliies for each node.) 6-5
Main idea of Nelson and Ramaswamy (1990): Suppose is a funcion of S and, and is wice differeniable wih respec o S and once differeniable wih respec o. According o he Iô s Lemma: d(s, ) = (µ(s, ) (S,) S + 1 σ (S, ) (S,) S Deliberaely se he volailiy erm o be 1: + (S,) )d + (σ(s, ) (S,) S )dz. (S,) S σ(s, ) = 1 (S, ) = 1 dk (a mapping beween and S) S σ(k,) Three-sep process o build a recombined ree for S: (i) Build he binomial ree for firs. Since he volailiy erm of is a consan and equal o 1, he binomial ree for recombines (see Figure 6-6). (ii) For each node, ransform he value of o he corresponding value of S such ha S(, ) = {S : (S, ) = }. (iii) The resuling binomial ree for S will recombine, and he upward and downward probabiliy for each node can be derived via q(s ) = E[S+ ] S + S + + S + Figure 6-6 = S+µ(S,) S +. S + + S + S( ) S( ) S( ) S( ) Afer he ransformaion of deriving he S-ree, he risk neural upward probabiliies for each node can be calculaed via S( ) S( ) ES [ ] S qs ( ) S S S ( S, ) S = S S 6-6
Oher implemenaion issues: (i) σ(s, ) canno be zero, oherwise he value of will approach infiniy. (Since S is a sochasic process, σ(s, ) should no be zero. However, when he value of S is very small, someimes σ(s, ) is very close o 0, e.g., he CIR ineres rae process.) (ii) should be small enough o ensure ha 0 < q < 1 and hus guaranee he binomial ree of S o model he process S() properly. (iii) There is no consrain for he negaive value of. However, for he underlying asse price S, i should be nonnegaive. (If you need o avoid negaive S or someimes he ransformaion fucnion S(, ) only acceps posiive values of, e.g., S(, ) = ln(), he simples modificaion you can ry is o le he volailiy erm of o be a number far smaller han 1 and hus resric he variaion of. More specifically, inead of seing (S,) S for example, (S,) S σ(s, ) = 0.01.) σ(s, ) = 1, you can consider, 6-7
Several examples for Nelson and Ramaswamy (1990): Example 1: Consan elasiciy of variance (CEV) sock price process: ds = µsd + σs γ dz, 0 < γ 1 The { feaure of CEV process: S > 1 S γ < S (higher S, lower volailiy han he lognormal disribuion) S < 1 S γ > S (exermely lower S, higher volailiy han he lognormal disribuion) (S, ) = σ 1 S K γ dk = S1 γ σ(1 γ) If γ = 1, ds = µsd + σsdz, (S, ) = 1 σ ln S. [σ(1 γ)] 1 1 γ if > 0 S(, ) = 0 o/w S S(, ) S + + S( + J+, + ) S + S( J, + ) q = S+µ S + S + + S + q if 0 q 1 q = 0 if q < 0 1 if q > 1 J + and J are inroduced o allow muliple jumps such ha in mos cases, he E[S + ] is inbeween wo following branches and hus he upward probabiliy q is in [0, 1]. The rules o decide J + and J are as follows. he smalles, odd, posiive, ineger j s.. J + S( + j, + ) S(, ) µ(s, ) if < = L 1 if L To preven he explosive growh of he number of nodes on he binomial ree, he upward muliple jumps are allowed o reach he nodes already on he binomial ree only when < L. In addiion, he incremen of he underlying asse price of he upward branch should be higher han he expeced growh of he underlying asse price. (For L, if he upward muliple jumps are allowed, new nodes should be generaed, which could resul in he unexpeced grow of he binomial ree.) 6-8
he smalles, odd, posiive, ineger j s.. J S(, ) S( j, + ) µ(s, ) = or S( j, + ) = 0 (S should be nonnegaive) The decremen of he underlying asse price of he downward branch should be smaller han he expeced growh (may be negaive) of he underlying asse price. However, he smalles underlying asse price which can be reached is 0 becuase S is nonnegaive. Example : ds = µ(s, )d + σ(s, )dz, where µ(s, ) = µs and σ(s, ) = σs σ S S = 1 = σ 1 S S (S) = σ 1 ln S S = eσ Therefore, for d(s) = ( µ σ 1 σ)d + 1 dz, -ree can be buil firs. Nex, S-ree can be derived hrough he ransformaion S = e σ. In fac, he resuling S-ree is exacly indenical o he CRR binomial ree. 6-9
Example 3: ds = µ(s, )d + σ()sdz, where σ() is a sepwise funcion of defined as follows. Suppose he whole period is pariioned ino 3 subperiods, and σ() = σ 1 0 1 σ 1 < σ 3 < T (The sep funcion of σ is useful o discribe he differen sages of a growing firm, i.e., seed, growing, and maured sages of a firm.) Nelson and Ramaswamy (1990) (S, ) = 1 σ 1 ln S S = e σ1 0 1 1 σ ln S S = e σ 1 < 1 σ 3 ln S S = e σ3 < T (I is worh noing ha for he 3 subperiods, he volailiy erms of he prcoess are all equal o 1. Therefore, he -ree is idenical o he one in Figure 6-7 regardless of differen subperiods. For differen subperiods, only he ransformaion funcions lised above are no he same.) Figure 6-7 3 3 4 4 5 3 3 5 6 4 4 6 S e 1 x S e x S e 3 x 1 T 0 1 3 6-10
III. Opion Pricing under Sochasic Volailiy Process Richken and Trevor (1999), Pricing Opion under Generalized GARCH and Sochasic Volailiy, Journal of Finance 54, pp. 377 40. Main idea: (i) A rinomial ree framework for he log sock price is considered. (ii) Unlike he sandard rinomial ree framework inroduced in Chaper 4, a grid srucure of log sock price is consruced and hus he upward and downward ick changes are fixed iniially. (iii) The mos imporna originaliy of his paper: The change of each branching log sock price implies an innovaion of he sandard normally disribued variable in he Wiener process. In addiion, he innovaion is employed o updae he condiional variance process. (iv) Insead of recording all possible values of variances on each node, only several seleced represenaive variances are recorded on each node. During he backward inducion, he linear inerpolaion mehod is employed o find he corresponding opion values for he missing variances. General seings: (i) Nonlinear asymmeric GARCH (NGARCH) ln( S+1 S ) = r + λ h 1 h + h υ +1 h +1 = β 0 + β 1 h + β h (υ +1 C) (The above wo equaions represen he NGARCH process under he physical measure, in which h represens he condiional variance process, λ is he marke price of risk of S, β 0, β 1, β, and C are consans, and υ +1 follows he sandard normal disribuion) Under he risk-neural measure, he corresponding NGARCH process is as follows. ln( S+1 S ) = (r 1 h ) + h ε +1 h +1 = β 0 + β 1 h + β h (ε +1 C ) where C = C + λ, and ε +1 follows he sandard normal disribuion. Change variable o y by defining y = ln(s ) E [y +1 ] = y + r 1h and var(y +1 ) = h 6-11
(ii) For each, which is assumed o be 1 day in Richken and Trevor (1999), (i) n + 1 branches are employed o span he normal disribuion for y +1, e.g., 3 branches are used if n = 1. Fuhermore, he verical spacing parameer beween node is defined as γ n γ n n h0. The illusraion of he case of n = 1 is shown in Figure 6-8. (ii) The value of h +1 is updaed a he end of each. Figure 6-8 n n defined as h0 n y n n 1 (iii) η muliple-sized jumps are allowed such ha he probabiliies of he upward, middle, and downward branches are guaraneed o be in [0, 1]. The rule o decide η for any variance h is as follows. η 1 < h γ η, where γ h 0 6-1
(iv) For y +1, n + 1 branches span he corresponding normal disribuion, i.e., y +1 = y + θ η γ n, θ = 0, ±1, ±,..., ±n Therefore, each possible value of y +1 implies a realized value of ε +1 hrough ε +1 = θ η γn (r h ) h, and hus leads o an updae of h +1 as follows. h +1 = β 0 + β 1 h + β h [ε +1 C ]. (v) How o decide he probabiliy for each branch? (1) Pariion (1 day in Richken and Trevor (1999)) ino n inervals, and he rinomial ree model is employed o model he movemens of y for each inerval. () Therefore, if n =, here are n + 1 = 5 branches for he period of. Noe ha alhough n inervals are considered in he period of, we do no consider he inermediae value of y during, and insead we use direcly he n + 1 branches for each. (3) For each of he ougoing n + 1 branches of y, is probabiliy equals he sum of condiional probabiliies of n-inerval pahs saring from y and reaching ha node. The deails o decide P (θ) Prob(y +1 = y + θηγ n ), for θ = 0, ±1, ±,..., ±n, are as follows. P (θ) = ( ) n Pu ju Pm jm P j d j u j m j d d j u,j m,j d s.. n = j u + j m + j d where P u = θ = j u j d h η γ + P m = 1 h η γ P d = h η γ (r h ) 1 n η γ (r h ) 1 n η γ (Noe ha here are ypos in he formuale for P u, P m, and P d in Richken and Trevor (1999). To correc heir formulae for P u, P m, and P d, you need o replace γ n wih γ and hen you can derive he above equaions.) 6-13
Illusraion of he n-subperiod pahs o reach y +1 = y + θ η γ n for differen θ. If n = 1, i is no necessary o make pariions for and he Richken and Trevor s model reduces o a rinomial ree model since for each, n + 1 = 3 branches are considered. For θ = 1, (j u, j m, j d ) = (1, 0, 0) For θ = 0, (j u, j m, j d ) = (0, 1, 0) For θ = 1, (j u, j m, j d ) = (0, 0, 1) If n =, he Richken and Trevor s model becomes a penanomial ree model since for each, n + 1 = 5 branches are considered. For θ =, (j u, j m, j d ) = (, 0, 0) For θ = 1, (j u, j m, j d ) = (1, 1, 0) For θ = 0, (j u, j m, j d ) = (0,, 0) and (j u, j m, j d ) = (1, 0, 1) For θ = 1, (j u, j m, j d ) = (0, 1, 1) For θ =, (j u, j m, j d ) = (0, 0, ) (vi) Forward inducion process o build he sock price ree and derive he possible variances reaching each node. Noe ha here could be so many condiional variances for each node because differen pahs reaching ha node generaes differen variances. Since he oal number of pahs is a leas 3 N, where T/N =, in he case of n = 1, we can infer ha he number of pahs grows exponenially. I is infeasible o record all condiional variances reaching each node due o he availabiliy of memory space in a PC and he concern of he efficiency problem. The soluion proposed by Richken and Trevor (1999): For each node, record only he maximum and minimum condiional variances among all condiional variances generaed by he pahs reaching ha node. In addiion, M inerpolaed represenaive condiional variances are equally-spaced placed from he maximum o minimum condiional variances. The able of represenaive condiional variances are consruced as follows. h(i, j, k) = M k M 1 h max(i, j) + k 1 M 1 h min(i, j), for k = 1,..., M, where h max (i, j) and h min (i, j) denoe he maximum and minimum condiional variances reaching node(i, j). Based on hese M inerpolaed represenaive condiional variances, he condiional variances h are updaed in he nex period of. When M approaches infiniy, he error caused by he above approximaion can be ignored. 6-14
Figure 6-9 Laice Model of Richken and Trevor (1999) Over Three Days. ln S S 6.9601 13.48 13.48 1053.74 6.9496 104.76 6.939 1.9 1.9 11.71 11.71 11.71 1031.90 6.987 1.8 10.57 101.16 0.0105 6.918 0.0105 6.9078 0.0105 6.8973 6.8868 h 0 =*10-5 10.13 variance beween 0 and 1 10.53 10.57 10.53 10.13 1.7 13.46 10.13 10.60 9.77 11.70 10.1 1.7 10.51 10 100 989.59 979.8 Among 3 's of his node, h wo of hem are wih =, and he oher is wih =1. Therefore, here are 5 branches for his node. h 's 1h 1 1 3 h 's 1h 1 1 h 's 1 1.7 η= 6.8763 13.44 969.08 11.395 0 1 3 Day η=1 This figures shows he firs hree days of he firs phase of he laice for an NGARCH model wih parameers r = 0, λ = 0, β 0 = 6.575 10 6, β 1 = 0.90, β = 0.04, and C = 0. The grid of values for he logarihmic price of he underlying, y = ln S is deermined by aking inervals of size γ = h 0 = 0.0105 around he log of he iniial price S 0 = 1000. In his example, n=1, giving hree possible pahs from each node for a given condiional variance. Each node is represened by a box conaining wo numbers. The op (boom) number is he maximum (minimum) condiional variance (muliplied by 10 5 ) of all pahs reaching ha node. In his example M = 3, so for each node, one addiional represenaive condiional variance is insered bween he maximum and minimum condiional variances. All hese hree variances deermine wheher he successor nodes are one or more unis of γ apar on he gard. The formulae o updae he condiional variance are as follows. h +1 = β 0 + β 1 h + β h (ε +1 C ), h 0 = 01096, where ε +1 = j η γn (r h h ), C = C + λ 6-15
Backward inducion process for opion pricing: Sep 1. Decide he payoff of each condiional variance of each erminal node. Since he condiional variance is indepden of he opion value, he payoffs of differen condiional variances on each node are he same. See Figure 6-10. Sep. For every condiional variance h(i, j, k) on node(i, j) for i = N 1, N,..., 0, (i) Find he evoluions of he condiional variance on he nex ime sep o be h nex (θ) = β 0 + β 1 h(i, j, k) + β h(i, j, k) ( θηγ n (r h(i, j, k)/) h(i, j, k) c ), for θ = 0, ±1, ±,..., ±n. (ii) Suppose ha h nex (θ) is inside he range [h(i+1, j+θη, k θ ), h(i+1, j+θη, k θ 1)]. By he linear inerpolaion mehod, he opion value C θ for he condiional variance h nex (θ) can be approximaed as C θ = w θ C(i + 1, j + θη, k θ ) + (1 w θ )C(i + 1, j + θη, k θ 1), where w θ = (h(i + 1, j + θη, k θ 1) h nex (θ))/(h(i + 1, j + θη, k θ 1) h(i + 1, j + θη, k θ )). (iii) The coninuaion value for each h(i, j, k) is C(i, j, k) = e r n θ= n P (θ)c θ. If he feaure of early exercise is aken ino accoun, aking vanilla call opions as examples, he opion value corresponding o h(i, j, k) becomes max(c(i, j, k), e y(i,j) K). Sep 3. Repea Sep for all h(i, j, k) s backward over he laice model, he value of C(0, 0, 1) will be he GARCH opion price derived by Richken and Trevor (1999). See Figure 6-10 for a numerical example of he above backward inducion process. 6-16
Figure 6-10 Laice for Pricing Three-Period A-The-Money Call Opion. h 13.48 13.48 c S 53.74 53.74 1053.74 53.74 104.76 h 1.9 1.9 c 31.90 31.90 31.90 11.71 11.71 31.90 31.90 31.90 1031.90 1.8 10.57 1.16 1.16 1.16 101.16 h c 1.0 1.0 1.0 10.57 10.53 13.46 10.13 10 h c 6.63 6.63 6.63 5.5 5.5 5.5 10.13.6 5.04 4.84 10.60 9.77 100 1.30 1.30 1.30 1.7 1.46 1.35 11.70 10.1 989.59 1.7 10.51 979.8 13.44 969.08 This figures shows he valuaion of a hree-period a-he-money European call opion. Each node is represened by a box conaining five mumbers. The op (boom) number in he firs column is he maximum (minimum) variance (muliplied by 10 5 ) of all pahs reaching ha node, as shown in Figure 6-9. As for he second column, he op number is he opion value corresponding o he maximum variance, he boom number is he value corresponding o he minimum variance, and he middle number corresponds o he midpoin variance. 6-17