A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL

Transcription

1 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL BRIAN EWALD*, WANWAN HUANG** Absrac. We propose a variance reducion mehod for Mone Carlo compuaion of opion prices in he conex of he Coupled Addiive-Muliplicaive Noise model. Four differen schemes are applied for he simulaion. The mehods selec conrol variaes which are maringales in order o reduce he variance of unbiased opion price esimaors. Numerical resuls for European call opions are presened o illusrae he effeciveness and robusness of his maringale conrol variae mehod. Keywords: Mone Carlo mehod; variance reducion;conrol variae mehod;cam model. Inroducion In financial mahemaics, sochasic differenial equaions (SDEs) play an imporan role as he seing for mos of he models used for pricing derivaives. The SDEs describe he evoluion of cerain financial variables, such as he sock price, volailiy of an asse, or ineres rae. The classic model which people usually apply for pricing European call opions is he Black-Scholes model, where he volailiy is assumed o be consan. Bu his assumpion is a limiaion of he sandard Black- Scholes model, which is proven by he so-called smile effec: ha implied volailiies of marke prices are no consan wih srike price and he ime o mauriy of he conrac. One way o ake his ino accoun is o rea volailiy as varying in ime as well. People have done pleny of research in a framework for pricing derivaives: for example, Fouque [], Hull and Whie [2]. Among hese works, mean reversion of he volailiy has been used o simplify he basic pricing and esimaion problems as well as reflec realiy o some exen, as volailiy doesn wander o arbirarily large or small values. Wha is volailiy? There are several noions of volailiy. Some of hem are model dependen, and ohers are daa dependen. *Assisan Professor of Mahemaics, he Florida Sae Universiy, Tallahassee, FL, ewald@mah.fsu.edu. **PhD. Candidae of Financial Mahemaics, he Florida Sae Universiy, Tallahassee, FL, whuang@mah.fsu.edu.

2 2 BRIAN EWALD*, WANWAN HUANG** Like menioned in Fouque s slides [3], realized volailiy, someimes referred o as he hisorical volailiy, measures one aspec of wha acually happened in he pas. The measuremen of he volailiy depends on he paricular siuaion. For example, one could look a he realized volailiy for he equiy marke in November of 28 by aking he sandard deviaion of he daily reurns wihin ha monh. One could also calculae he realized volailiy beween :AM and 2:PM of May 6, 28 by calculaing he sandard deviaion of one minue reurns. Here, le < < < N be a sequence of imes. Then () T N Σ 2 s ds N N i= (log S i log S i ) 2 i i, where Σ s is he realized volailiy, and S i is he sock price a ime i. In conras o realized volailiy, implied volailiy, as explained by Beckers [4] and Mayhew [5], refers o he marke s assessmen of fuure volailiy under he assumpion ha he dynamics can be modelled by a Black-Scholes model. I is an esimaion of he volailiy of a sock as implied by he price of an opion on ha sock, as follows. Given an observed European call opion price C obs for a conrac wih srike price K and expiraion dae T, he implied volailiy I is defined o be he value of he volailiy parameer ha mus go ino he Black-Scholes formula o mach his price: (2) C BS (, x; K, T ; I) = C obs. Model dependen volailiy has wo main caegories: local volailiy and sochasic volailiy. People usually se a lognormal model for he asse price X : (3) dx = µx d + σx dw. One popular way o modify he lognormal model is o suppose ha volailiy is a deerminisic posiive funcion of ime and sock price: σ = σ(, X ). This is called he local volailiy. The sochasic differenial equaion modeling he sock price is hen (4) dx = µx d + σ(, X )X dw. In sochasic volailiy models, he value σ, called he volailiy process, is allowed o vary sochasically. I does no have o be an Iô process: i can be a jump process, a Markov chain, ec. I should be posiive, as i is a volailiy. Unlike he local volailiy, he sochasic volailiy process need no be perfecly correlaed wih he Brownian moion, W, in he asse price model: (5) dx = X (µ d + σ dw ).

3 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL3 In oher words, he sochasic volailiy is a funcion of some process Y, where Y conains an addiional source of randomness: (6) σ = f(y ). 2. Sochasic Volailiy Models 2.. Sandard Models. As menioned in he las secion, ypically, sochasic volailiy is aken o be a funcion of a sochasic process (Y ) in general (bu we consider only he Iô process case here), ha is σ = f(y ). The process (Y ) saisfies a sochasic differenial equaion driven by a second Brownian moion. The desired models should make he volailiy posiive. An imporan feaure ofen applied in he sochasic volailiy models is mean reversion. The definiion for he erm mean reversion is a linear pull-back erm in he drif of he volailiy process iself, or in he drif of some (underlying) process of which volailiy is a funcion. I also refers o he characerisic ime i akes for a process o ge back o he mean level of is invarian disribuion. The sochasic differenial equaion for (Y ) inroduces a new Brownian moion Z (7) dy = α(m Y ) d + β(, Y ) dz. Here he parameers are α and m. α is he rae of he mean reversion and m is he long-run mean level of Y. Y will approach m wih speed α, on average. The simples mean-revering model is an Ornsein-Uhlenbeck (OU) process, which is defined as a soluion of equaion (7) where β(, Y ) = β is consan. Also noice ha he second Brownian moion (Z ) is ypically correlaed wih he Brownian moion (W ) from he asse price equaion (5). ρ [, ] is he insananeous correlaion coefficien defined by equaion (8). ρ is ofen found o be negaive because of he leverage effec beween sock price and volailiy shocks. I s ofen convenien o wrie i like equaion (9), where (Z ) is a sandard Brownian moion independen from (W ): (8) dw dz = ρ d wih (9) Z = ρw + ρ 2 Z. Besides he Ornsein-Uhlenbeck process, here are some oher common meanrevering processes. The Feller or Cox-Ingersoll-Ross (CIR) process is anoher common one: () dy = κ(m Y ) d + ν Y dz. The popular Heson model [6] is based on he CIR process wih f(y ) = Y.

4 4 BRIAN EWALD*, WANWAN HUANG** To revise he lognormaliy assumpion (3) of Black-Scholes [7], he Consan Elasiciy of Variance (CEV) model [8] is also focused on by researchers. The CEV model is in he form: () dx = µx d + σx θ 2 dw, ha is, (2) dx X = µ d + σ X θ 2 dw. The reurn variance wih respec o price X, ν(x, ) = σ 2 X θ 2, has he relaionship (3) which implies ha dν(x, )/dx ν(x, )/X = θ 2, (4) dν(x, )/ν(x, ) = (θ 2)dX /X. The quaniy θ 2 is called elasiciy of reurn. In paricular, if θ = 2, hen he elasiciy is zero and he sock price is lognormally disribued as in he Black-Scholes model. If θ =, hen he elasiciy is. This is he model proposed by Cox and Ross. The CEV model has been exploied a lo recenly, for example, Anderson [9] and Lord []. As menioned in [], he asse price process (X ) and he variance process (Y ) evolve according o he following SDEs: (5) dx = µx d + λ Y X β dw, (6) dy = κ(m Y ) d + ωy α dz. Here he process is specified under he risk-neural probabiliy measure. The parameer µ is he risk neural drif of he asse price, κ is he speed of mean-reversion of he variance, m > is he invarian average variance, ω is he so-called volailiy of variance, and λ is a scaling consan. Finally, as explained above, W and Z are correlaed Brownian moions, wih insananeous correlaion coefficien ρ. β is resriced o lie in (, ] and α o be posiive. The popular Heson model is he special case when α = and β = CAM Model. The coupled addiive-muliplicaive noise (CAM) model was inroduced by Sardeshmukh and Sura in heir papers [] and [2]. In [], hey found a link beween he skewness and kurosis of daily sea surface emperaure (SST) variaions. If he sandard deviaion of SST anomalies T a a paricular

5 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL5 poin on he ocean s surface is denoed by σ, he skewness (skew) and kurosis (kur) become (7) skew T 3 σ 3 and kur T 4 σ 4 3. Skewness is a measure of asymmery of a probabiliy densiy funcion. If he lef ail is heavier han he righ ail, he probabiliy densiy funcion has negaive skewness. If he reverse is rue, i has posiive skewness. If he probabiliy densiy funcion is symmeric, like he Gaussian, i has zero skewness. Kurosis (or more accuraely, excess kurosis, since he kurosis of 3 for a Gaussian disribuion is subraced) measures he excess probabiliy (faness) in he ails, where excess is defined in relaion o a Gaussian disribuion. The kur-skew relaionship was gained from he scaerplo of emprically calculaed kurosis vs skewness of he ime series of all high-resoluion observaional daa poins a mos locaions around he globe. The scaerplo evinced a lower parabolic bound on kurosis in heir daase: kur (3/2) skew 2. All of he daa poins lay above his parabola, and his is evidenly a very srong consrain on he non-gaussian characer of he SST variabiliy. From hese observaions, a deailed dynamical explanaion was provided. They inroduced a univariae linear model wih muliplicaive noise o capure he observed non-gaussianiy of SST anomalies over almos all he globe: (8) T = λt φf T + F + R + φf T. Here T is he SST anomalies, λ and φ are locally consan parameers, F and R are rapidly varying forcing erms. They assumed ha he rapidly varying erms F and R can be approximaed as independen, zero mean Gaussian whie noise processes, under which (8) becomes an SDE for SST anomalies T. They also derived an analyical equaion from (8) o explain he kurosis-skewness relaionship shown in he scaerplo figure, and hey finally concluded ha he CAM model is applicable for anomalous SST variabiliy. Empirical plos of skewness vs kurosis for log volailiy of commodiy or sock prices also exhibis he parabolic lower bound. So he log volailiy of a commodiy also has a non-gaussian disribuion. In order o capure his non-gaussian behavior, we propose o model sochasic volailiy by a CAM model. So we apply his new model for pricing he European call opion in his paper. If we make he volailiy of opion price σ(y ) = exp(y ), he log volailiy is jus he diffusion process Y. Based on he asse price model (5) and (6), we consider he CAM model for he diffusion process Y in his way: (9) dy = α(m Y )d + βdẑ() + γy dẑ(2).

6 6 BRIAN EWALD*, WANWAN HUANG** This is he so-called CAM process. This SDE is developed from he simples Ornsein-Uhlenbeck (OU) process, and i has a mean-revering drif erm. A second source of randomness Ẑ(2) is added o he equaion besides Ẑ(). And he hree whie noises W, Ẑ(), Ẑ(2) are correlaed. We can use he coefficiens of correlaion ρ, ρ 2, ρ 3 and hree independen whie noises W, Z (), Z (2) o represen he correlaions: (2) W = W, (2) Ẑ () = ρ W + and (22) Ẑ (2) = ρ 2 W + ρ 3 ρ ρ 2 ρ 2 Z () + ρ 2 Z () ρ 2 2 (ρ 3 ρ ρ 2 ) 2 Z (2) ρ 2. Besides is mean-revering propery, we apply he CAM model for pricing he European call opion because of is several advanages. Firs, i s analyically racable in some ways. We can ask his quesion: when does he momen EY n say bounded as? We can solve he ordinary differenial equaion for EY n o find a relaionship beween he parameers which can ensure ha a paricular momen says bounded for all he ime. This relaionship urns ou o be (n ) (23) α γ 2. 2 For example, in order for fifh momens of he saionary disribuion o exis, we would need ha (24) α 2γ 2. The proof of (23) and (24) will be shown in he appendix A. 3. Opion Pricing using CAM Model 3.. Differen Numerical Mone Carlo Schemes Black-Scholes Formula. The price of he European call opion for a nondividend paying underlying sock in erms of he Black-Scholes parameers is C(, S) = SN(d ) Ke r(t ) N(d 2 ), d = log( S σ2 ) + (r + )(T ) K 2 σ, T d 2 = log( S K σ2 ) + (r )(T ) 2 σ, T

7 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL7 and d 2 = d σ T. Here N( ) is he cumulaive disribuion funcion of he sandard normal disribuion. The ime o mauriy is T, he sock price is S, he srike price is K, he risk free rae is r, and he volailiy of reurns of he underlying asse is σ Euler Scheme. Here our model is (25) dx = µx d + σ X dw wih he CAM model diffusion process (26) dy = α(m Y ) d + β dẑ() + γy dẑ(2), and σ = exp(y ). And apply (2), (2) and (22), we can rewrie he diffusion process Y in he form of (27) ( ) ( ) X µx d = d Y α(m Y ) ( ) exp(y )X dw + βρ + γy ρ 2 β ρ 2 ρ + γy 3 ρ ρ 2 γy ρ 2 ρ 2 2 (ρ 3 ρ ρ 2 ) 2 dz (), ρ 2 dz (2) where W, Z (), and Z (2) are hree independen sandard Brownian Moions. We use some simple noaions here: a = µx, b = exp(y )X, a = α(m Y ), b = βρ + γy ρ 2, b 2 = β ρ 2 ρ + γy 3 ρ ρ 2, and b 3 = γy ρ 2 ρ 2 2 (ρ 3 ρ ρ 2 ) 2. As in ρ 2 [], using he risk-neural heory, here is an equivalen maringale measure P under which he discouned sock price X = e r X is a maringale. And we can compue he European call opion price wih ime-t payoff H using he formula (28) C = E {e r(t ) H F } for all T, when here is no arbirage opporuniy. Thus C is a possible price for he European call opion. We can ry o consruc he equivalen maringale measures now. Like wha Fouque s group did, we absorb he drif erm of X in is maringale erm by seing (29) W = W + (µ r) exp(y s ) ds. Any shif of he second and he hird independen Brownian moions of he form (3) Z (j) = Z (j) + θ (j) s ds (j =, 2)

8 8 BRIAN EWALD*, WANWAN HUANG** will no change he drif of X. By he muliple dimensional Girsanov s heorem from [3], (W ) and (Z (j) ) are independen sandard Brownian moions under a measure P (θ() θ (2)) defined by ( exp T ((θ 2 s) 2 + (θ () s ) 2 + (θ s (2) dp (θ() θ (2) ) = dp ) 2 )ds T θ sdw s T θ() s dz s () θ = (µ r) exp(y ). ) T θ(2) s dz s (2), Here (θ (j) ) are any adaped (and suiably regular) processes. We assume ha he newly defined measure P (θ() θ (2)) is well-defined, so ha f is bounded away from zero and (θ (j) ) are bounded. Then, under his new risk-neural measure, he SDEs (25) and (26) become (3) ( ) ( ) X rx d = d Y α(m Y ) Φ(, x, y) ( exp(y )X + βρ + γy ρ 2 β ρ 2 ρ + γy 3 ρ ρ 2 γy ρ 2 ρ 2 2 (ρ 3 ρ ρ 2 ) 2 ρ 2 where (32) ( Φ(, x, y) = (βρ + γy ρ 2 ) (µ r) + e Y +γy ρ 2 2 (ρ 3 ρ ρ 2 ) 2 β ρ 2 ρ + γy 3 ρ ρ 2 ρ 2 θ (2). ρ 2 ) ) θ () dw dz () dz (2) And he hree Brownian moions W, Z () and Z (2) under he new measure are independen. The funcion Φ(, x, y), as explained in [], is relaed o he risk premium facor from he second and he hird sources of he randomness ha drive he volailiy. For he Mone Carlo compuaion of he derivaive prices, i is used o rea Φ(, x, y) = for simplificaion as in [6], and his won affec he compuaion resuls. We will have he ime inervals equal o each oher, so T = N where T is he ime o mauriy of he opion wih = and N is he number of ime seps. As saed in [4], here are general srong and weak Iô-Taylor approximaions. For srong approximaions, he sochasic process X saisfies he convergence condiion [E[(X X δ ) 2 ]] 2 O( α ). For weak schemes, any funcion, f, of X should saisfy he convergence condiion E f(x ) f(x δ ) O( β ) provided f and enough of is parial derivaives have polynomial growh. Here X δ is he numerical discreizaion of X, and α, β are orders of he schemes. Since he payoff funcion of he European,

9 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL9 call opion is jus a simple funcion of he sock price a mauriy X T, he weak scheme here is sufficien for pricing he opion. The Euler scheme corresponds o he runcaed Iô-Taylor expansion which conains only he ordinary ime inegral and he simple Iô inegral. We shall see from a general convergence resul for weak Taylor approximaions, as saed in Theorem 4.5. of chaper 4 of [4], ha he Euler approximaion has order of weak convergence., if amongs oher assumpions, aa, bb, a, b, b 2 and b 3 are four imes coninuously differeniable. This means ha he Euler scheme is he order. weak Taylor approximaion. So for he SDEs (33) dx = a (, X ) d + b (, X ) dw and (34) dy = a(, Y ) d + b (, Y ) dw + b 2 (, Y ) dz () + b 3 (, Y ) dz (2), he Euler scheme has he form (35) X + = X + a (X ) + b (X ) W and (36) Y + = Y + a(y ) + b (Y ) W + b 2 (Y ) Z () + b 3 (Y ) Z (2), wih iniial value X = x and Y = y, where wih j =, 2. = n+ n, W = W n+ W n and Z (j) = Z (j) n+ Z (j) n Simplified Weak Euler Scheme. From [4], for weak convergence we only need o approximae he measure induced by he Iô process Y, so we can replace he Gaussian incremens W and Z (j) (i) in (36) by oher random variables Z (i =, 2, 3) wih similar momen properies. We can hus obain a simpler scheme by choosing more easily generaed noise incremens. This leads o he simplified weak Euler scheme () (2) (3) (37) Y n+ = Y n + a(y n ) + b (Y n ) Z + b 2 (Y n ) Z + b 3 (Y n ) Z, (i) where he Z for i =, 2,..., m (here m = 3) mus be independen measurable random variables wih momens saisfying he convergence condiion: (38) E( (i) Z ) + E(( (i) Z ) 3 (i) ) + E(( Z ) 2 ) C 2

10 BRIAN EWALD*, WANWAN HUANG** (i) for some consan C, and his is from [4]. A very simple example of such Z (37) are wo-poin disribued random variables wih ( (i) P Z = ± ) = Order 2. Weak Taylor Scheme. Firs, we need o inroduce he sochasic Taylor expansions. From [4] we denoe he following noaions. The muli-index α = (j, j 2,..., j l ) is a row vecor wih j i {,,..., m} for i {, 2,..., l} and m =, 2, 3,.... The lengh of α is l = l(α) {, 2,... }. The vecor ν denoes he muli-index of lengh zero, which means l(ν) =. In addiion, he number n(α) denoes he number of componens of a muli-index α which are equal o. We denoe he se of all muli-indices by M = {(j, j 2,..., j l ) : j i {,,..., m}, i {,..., l}} {ν}, for l =, 2, 3,.... For adaped righ coninuous sochasic processes f(), we can define cerain funcion spaces H α. The firs such is he oaliy of all such processes, which is H ν. I conains all he f wih f() being almos surely finie, for each. The second space, H (), is he subspace of H ν consising of hose f wih (39) f(s) ds < almos surely, for every. And he hird space, H (j) wih j, is he subspace of H ν consising of hose f wih (4) f(s) 2 ds < almos surely, for every. Le ρ and τ be wo sopping imes wih ρ(ω) τ(ω) T, w.p.. Then he muliple Iô inegral I α [f( )] ρ,τ is defined by f(τ) : l = τ (4) I α [f( )] ρ,τ := ρ I α [f( )] ρ,s ds : l and j l = τ I ρ α [f( )] ρ,s dw j l s : l and j l, and α denoes α wih is las componen j l removed. Now H α wih α M and lengh l(α) >, is considered recursively by (42) I α [f( )], H (jl ) almos surely, for every. To define he Iô Taylor expansion we also need o learn he Iô coefficien funcions. There are wo ypes of differenial operaors relaed o a SDE. These are, for in

11 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL he general SDE (43) du = a(, u ) d + b(, u ) dw, we have (44) L = + k and a k (45) L j = k u + b kj b lj 2 k 2 u k u l b kj j,k,l u. k For each α = (j,..., j l ), we define L α = L j L j l, and fα = L α = L α f, f ν = f. The Iô-Taylor expansion is considered for he Iô process m (46) X = X + a(s, X s ) ds + b j (s, X s ) dws j, wih T, he equivalen, inegral form of SDE above. Using he previous conens in his secion, le s define a hierarchical se A M as a nonempy se of muliindices such ha sup α A l(α) <, and α A whenever α ν is in A. The remainder se B(A) consiss of all hose α no in A such ha α is in A. Finally, we ge he Iô Taylor expansion of he funcion f applied o a soluion X of (46): (47) f(τ, X τ ) = I α [f α (ρ, X ρ )] ρ,τ + I α [f α (, X. )] ρ,τ, α A j= α B(A) and for γ =, 2,..., we denoe by A γ he hierarchical se consising of all as of lengh a mos γ, and we call he Iô Taylor expansion wih A = A γ he (weak) Iô Taylor expansion o order γ. A weak Taylor scheme is simply he srong Taylor-expansion wih high order erms runcaed. To see he derivaion, refer o [4]. Now we can consider he order 2. weak Taylor scheme, which is obained by adding all of he double sochasic inegrals from he Iô-Taylor expansion (47) o he Euler scheme. Applying he Iô Taylor expansion (47) in he case d = 2, m = 3 for f y (or x), we obain he following for he CAM model: (48) Y + = Y + a + b W + b 2 Z () + b 3 Z (2) + L b I (,) +L 2 b I (2,) + L 3 b I (3,) + L b 2 I (,2) + L 2 b 2 I (2,2) + L 3 b 2 I (3,2) +L b 3 I (,3) + L 2 b 3 I (2,3) + L 3 b 3 I (3,3) + L b I (,) + L b 2 I (,2) +L b 3 I (,3) + L ai (,) + L 2 ai (2,) + L 3 ai (3,) + 2 L a 2 + R 2 (),

12 2 BRIAN EWALD*, WANWAN HUANG** (49) X + = X + a + b W + L b I (,) + L 2 b I (2,) + L 3 b I (3,) +L b I (,) + L a I (,) + L 2 a I (2,) + L 3 a I (3,) + 2 L a 2 + R 2 (), where R 2 () and R 2 () are remainders. The differenial operaors here are (5) L = + rx X + α(m Y ) Y + 2 e2y X 2 +e Y X (βρ + γy ρ 2 ) 2 X Y + (β ρ 2 ρ 2 + γy 3 ρ ρ 2 2 X γ2 Y 2 ( ρ 2 2 (ρ 3 ρ ρ 2 ) 2 ρ 2 ) 2 Y 2 (5) L = e Y X X + (βρ + γy ρ 2 ) Y, (52) L 2 = and ( (53) L 3 = γy β ) ρ 2 ρ 3 ρ ρ 2 + γy, ρ 2 Y ρ 2 2 (ρ 3 ρ ρ 2 ) 2 ρ (βρ + γy ρ 2 ) 2 2 X 2, Y. ρ 2 ) 2 Y 2 We have he muliple Iô inegrals involving differen componens of he Wiener process, which is no easy o generae in realiy. Under weak convergence, we can sill (i) use some Z o replace W and Z (j), use (i) Z 2 o replace I (,i) and I (i,). The las ype of muliple inegrals I (i,i 2 ) can be replaced by ( Z (i ) 2 Z (i 2) + V i,i 2 ). (i) Here he Z for i =, 2, 3 are independen random variables saisfying he momen condiions explained in [4] and hree-poin disribued wih (i) (54) P ( Z = ± 3 ) = (i), P ( Z = ) = And he independen variables V i,i 2 are in a wo-poin disribuion wih (55) P (V i,i 2 ) = 2 for i 2 =,..., i, (56) V i,i 2 = and (57) V i2,i = V i,i 2 for i 2 = i +,..., m and i +,..., m. Finally, he convergence of his weak order 2. scheme was proved by [4].

13 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL A Sochasic Adams-Bashforh Scheme. The Sochasic Adams-Bashforh (SAB) scheme can be represened in several versions. The simples one is for he ordinary differenial equaion φ = F (φ): (58) φ n+ = φ n + 2 [3F (φ n) F (φ n )], and is of order 2. The paper [5] lised a sochasic analog of he previous one wih srong convergence. The version we applied here is a weak convergen form. This is derived from he order 2. Iô-Taylor expansion which is U + = U + j bj W j + a + j,k Lj b k I (j,k) (59) + k L b k I (,k) + j Lj ai (j,) + 2 L a 2 + R 2 () = U + a + 2 L a 2 + M (), where each coefficien is evaluaed a he poin (, U ), and each sochasic inegral is from o +, =. We can also apply he Iô-Taylor expansion for he coefficien a in orders and : (6) a( +, U + ) = a + L a + N (), wheren () = j Lj a W j + R (), and (6) L a( +, U + ) = L a + P (), where P () = R (). We combine hese resuls o yield, for any η and θ, (62) U + = U + [ηa( +, U + ) + ( η)a] +( 2 η)[θl a( +, U + ) + ( θ)l a] 2 η N () ( 2 η)θ 2 P () + M (). So, if = n, = 2, η = θ =, and wriing U n for U n, (63) U n+2 = U n + 2a( n, U n ) + 2L a( n, U n ) 2 + M 2 ( n ), and if = n, =, η = 3, and θ =, 2 (64) (65) Hence, U n+ = U n 3 2 a( n+, U n+ ) a( n, U n ) +2L a( n, U n ) N ( n ) + M ( n ). U n+2 = U n+ + (U n+2 U n ) (U n+ U n ) = U n+ + [ 3 2 a( n+, U n+ ) 2 a( n, U n )] 3 2 N ( n ) + (M 2 ( n ) M ( n )).

14 4 BRIAN EWALD*, WANWAN HUANG** So, we will consider he following version of a SAB scheme: [ 3 (66) Y n+2 = Y n+ + 2 a( n+, Y n+ ) ] 2 a( n, Y n ) + B n+ ( n+, Y n+ ), in which (67) B n+ (, x) = j b j (, x) W j + j L b j (, x)i (,j) + j L j a(, x)i (j,) + j,k L j b k (, x)i (j,k), where he random inervals are evaluaed over he inerval from n+ o n+2. This was proved o be convergen by [5]. The exac scheme for CAM model is: (68) (69) Y +2 = Y + + [ 3 a( +, Y 2 + ) a(, Y 2 ) ] + b W + b 2 Z () +b 3 Z (2) + L b I (,) + L b 2 I (,2) + L b 3 I (,3) + L ai (,) + L 2 ai (2,) +L 3 ai (3,) + L b I (,) + L b 2 I (,2) + L b 3 I (,3) + L 2 b I (2,) + L 2 b 2 I (2,2) +L 2 b 3 I (2,3) + L 3 b I (3,) + L 3 b 2 I (3,2) + L 3 b 3 I (3,3), X +2 = X + + [ 3 a 2 ( +, X + ) a 2 (, X ) ] + b W + L b I (,) +L a I (,) + L 2 a I (2,) + L 3 a I (3,) + L b I (,) + L 2 b I (2,) + L 3 b I (3,). Here W and Z (j) (i) can also be replaced by Z, and he muliple Iô inegrals can also be replaced by he simple hree-poin disribued random variables as menioned before. 4. The Maringale Conrol Variae Mehod for Opion Pricing Under CAM Model Under a risk-neural pricing probabiliy P paramerized by he combined volailiy risk relaed erms Λ (y) and Λ 2 (y), we consider he following CAM model: (7) dx = rx d + σ X dw, σ = f(y ), (7) dy = where [ ε c (Y ) g (Y ) Λ (Y ) g ] 2(Y ) Λ 2 (Y ) d + g (Y ) ε ε ε dẑ() + g 2(Y ) Y ε dẑ(2), (72) Ẑ () (73) Ẑ (2) = ρ 2 W + ρ 3 ρ ρ 2 Z () ρ 2 + = ρ W + ρ 2 Z (), ρ 2 2 (ρ 3 ρ ρ 2 ) 2 Z (2) ρ 2.

15 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL5 Here X is he underlying asse price process wih a consan risk-free ineres rae r as explained before. And c (Y ) = (m y), α =, β = ν 2 ε ε, γ = χ ε, g (y) = ν 2, g 2 (y) = χ. Given he CAM model, he price of a plain European opion wih he inegrable payoff funcion H and expiry T is given by (74) P ε (, x, y) = E,x,y{e r(t ) H(X T )}, where E,x,y denoes he expecaion wih respec o P condiioned on he curren saes X = x, Y = y. A basic Mone Carlo simulaion esimaes he opion price P (, S, Y ) a ime by N (75) N i= e rt H(X (i) T ), where N is he oal number of independen sample pahs and X (i) T denoes he i-h simulaed sock price a ime T. Assuming ha he European opion price P (, X, Y ) is smooh enough, we apply Iô s lemma o is discouned price e r P, and hen inegrae from ime o he mauriy T. The following maringale represenaion is obained (76) P ε (, X, Y ) = e rt H(X T ) M (P ) ε M (P ) ε M 2 (P ), where cenered maringales are defined by (77) M (P ) = (78) M (P ) = (79) M 2 (P ) = T T T ε rs P e x σ X dw, ε rs P e y ν 2 dẑ(), ε rs P e x χy dẑ(2). The maringales play he role of perfec conrol variaes for Mone Carlo simulaions and heir inegrands would be he perfec Dela hedges if P were known and volailiy facors were raded. Like menioned in [7], neiher P nor is gradien a any ime s T are in any analyic form even hough all he parameers of he model have been calibraed as we suppose here. We can approximae he opion price and subsiue for P in he maringales above and sill reain maringale properies. The approximaion of he Black-Scholes ype is derived in[8] for coninuous payoffs: (8) P ε (, x, y) P BS (, x; σ).

16 6 BRIAN EWALD*, WANWAN HUANG** We denoe by P BS (, x; σ) he soluion of he Black-Scholes parial differenial equaion wih he erminal condiion P BS (T, x) = H(x). The average volailiy σ is defined by (8) σ = exp( m). Noe ha he approximae opion price P BS (, x; σ) is independen of he variable y. A maringale variae esimaor is formulaed as N (82) [e rt H(X (i) T N ) M(i) (P BS )], where M (P BS ) = i= T e rs P BS x (x, X s; σ)f(y s )X s dw s. 5. Variance Analysis of Maringale Conrol Variaes For he sake of simpliciy, we firs assume ha he insan correlaion coefficiens, ρ, ρ 2 and ρ 3 in (7), (72) and (73), are zero. From (76), he variance of he conrolled payoff (83) e rt H(S T ) M (P BS ) is simply he sum of quadraic variaions of maringales: (84) V ar(e rt H(S T ) M (P BS )) = E,,x,y{ T e 2rs ( P P BS x x )2 (s, S s, Y s )f 2 (Y s )Ss 2 ds + T ε e 2rs ( P y )2 2ν 2 ds + T ε e 2rs ( P y )2 χ 2 Ys 2 ds}. Theorem.. Under he assumpions made above and he payoff funcion H being coninuous piecewise smooh as a call (or a pu), for any fixed iniial sae (,x,y), here exiss a consan C > such ha for ε, V ar(e rt H(S T M (P BS )) Cε. The proof of Theorem. is given in he Appendix B. The proof is from he similar procedure given in Fouque s paper [7]. 6. Numerical Resuls The numerical experimens are implemened o illusrae ha he maringale conrol variae mehod is efficien and robus for European opion problems under CAM model wih is relevan parameers and iniial values specified in Table and Table 2.

17 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL7 Table. Parameers used in he CAM model. r m β ρ ρ 2 ρ 3 Φ f(y) exp(y) Table 2. Iniial condiions and call opion parameers. $X Y $K T years Compared o plain Mone Carlo simulaions, significan variance reducion raios for European opions are obained. These resuls confirm he robusness of our mehod based on maringale conrol variaes consruced as in dela hedging sraegies. The effeciveness of our mehod depends on opion price approximaions o he pricing problem considered. Resuls of variance reducion under he four differen schemes are illusraed in Table 3 Table 6 wih various parameers α and γ. The ime sep size for all he schemes is = 3 and he number of realizaions is N =,. Figure Figure 4 presen sampled European opion prices wih respec o he number of realizaions. The dash line correponds o basic Mone Carlo simulaions, while he do line corresponds o he same Mone Carlo simulaions using he maringale conrol variae M (P BS ). Figure 5 Figure 8 presen sandard deviaion of simulaed opion prices wih respec o he number of realizaions. The dash line corresponds o basic Mone Carlo simulaions, while he do line corresponds o he same Mone Carlo simulaions using he maringale conrol variae mehod. The resuls confirm ha he sandard deviaion under conrol variae mehod converges faser. 7. Conclusion In his paper we have presened he applicaion of a Coupled Addiive-Muliplicaive Noise model in opion pricing. We have focused our aenion on four differen schemes: Euler scheme, simplified weak Euler scheme, order 2. weak Taylor scheme and SAB scheme. The effeciveness of he four schemes is presened. A maringale conrol variae mehod is proposed o price European opions by Mone Carlo simulaions. The size of he variance reducion by his generic conrol variae mehod has been characerized by a heoreical variance analysis. We also obain he significan variance reducion raio by comparing o he resuls from plain Mone Carlo simulaions. The resuls confirm he pracical applicaion of he CAM model and he robusness of he maringale conrol variaes mehod consruced as in dela hedging sraegies.

18 8 BRIAN EWALD*, WANWAN HUANG** Figure. Implici Euler scheme: Mone Carlo simulaions under implici Euler scheme for a European call opion price when α = 3 and γ =. Sampled prices are obained along he number of realizaions. Figure 2. Implici Euler scheme wih coin flips

19 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL9 Figure 3. Weak order 2. scheme Figure 4. SAB scheme

20 2 BRIAN EWALD*, WANWAN HUANG** Figure 5. Implici Euler scheme: Mone Carlo simulaions under implici Euler scheme for a European call opion price when α = 3 and γ =. Sanderd error are obained along he number of realizaions. Figure 6. Implici Euler scheme wih coin flips

21 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL2 Table 3. Implici Euler Scheme: Comparison of sandard errors wih various α and γ. The noaion Sd BMC sands for he sandard error esimaed from basic Implici Euler Scheme Mone Carlo simulaions, and Sd MCV he sandard error from he same Mone Carlo simulaions bu using he maringale conrol variae. Numbers wihin parenhesis in he hird and fourh columns are sample means esimaed from he wo Mone Carlo mehods, respecively. The fifh column records he variance reducion raio, which is calculaed by (Sd BMC /Sd MCV ) 2. α γ Sd BMC Sd MCV Variance Reducion Raio.3..37(2.7965).2(2.8989) (2.295).94(2.374) (9.5385).893(9.6697) 2.5 Table 4. Implici Euler Scheme wih coin flips α γ Sd BMC Sd MCV Variance Reducion Raio (2.2).895(2.86) (2.68).843(2.3546) (2.6).786(9.687) 3.25 Table 5. Weak order 2. scheme α γ Sd BMC Sd MCV Variance Reducion Raio.3..39(2.882).27(2.95) (2.2783).962(2.459) (9.7554).85(9.6829) 2.89 Table 6. SAB Scheme α γ Sd BMC Sd MCV Variance Reducion Raio.3..38(2.82).27(2.943) (2.2724).962(2.497) (9.7457).852(9.6734) 2.88 Appendix A. When does EY n say bounded as? For n =, he soluion of he CAM process (9) is explicily given in erms of is (assumed known) saring value y by (85) Y = y + α(m Y s ) ds + β dẑ() + γy s dw (2).

22 22 BRIAN EWALD*, WANWAN HUANG** Figure 7. Weak order 2. scheme Figure 8. SAB scheme Taking expecaions for his soluion (85) will give (86) EY = Ey + E α(m Y s ) ds,

23 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL23 so (87) hen EY d = E[α(m Y )] (88) EY = m + Ce α. The condiion for EY say bounded is α. For n = 2, he soluion of he sochasic differenial equaion dy 2 = 2Y dy + 2 (2)dY dy (89) = 2Y [α(m Y ) d + β dẑ() + γy dẑ() ] + β 2 d + γ 2 Y 2 d + 2βγY ρ 2 d is (9) = [2αY (m Y ) + β 2 + γ 2 Y 2 + 2βγY ρ 2 ] d + 2βY dẑ() + 2γY 2 Y 2 = y 2 + [(γ2 2α)Ys 2 + (2αm + 2βρ 2 γ)y s + β 2 ] ds + 2βY s dẑ() + 2γY 2 s dẑ(2). Taking expecaions on boh sides of his soluion we will ge (9) EY 2 so = Ey 2 + E [(γ 2 2α)Y 2 s + (2αm + 2βρ 2 γ)y s + β 2 ] ds, dey 2 = E[(γ 2 2α)Y d 2 + (2αm + 2βρ 2 γ)y + β 2 ] (92). = (γ 2 2α)E[Y 2 ] + (2αm + 2βρ 2 γ)e[y ] + β 2. The soluion for his ordinary differenial equaion is dẑ(2) (93) EY 2 = 2αm2 + 2βρ 2 γm + β 2 + (2αm + 2βρ 2γ)C e α + C e (2α γ2 ). 2α γ 2 α γ 2 The condiion for he momen o say bounded as is α γ2. 2 For n = 3, he soluion of he sochasic differenial equaion (94) dy 3 = 3Y 2 dy + (6Y 2 )dy dy = 3Y 2 [α(m Y ) d + β dẑ() + γy dẑ(2) ] + 3Y [β 2 d + γ 2 Y 2 d + 2βρ 2 γy d] = [(3γ 2 3α)Y 3 + (3mα + 6βρ 2 γ)y 2 + 3β 2 Y ] d + 3βY 2 dẑ() + 3γY 3 dẑ(2)

24 24 BRIAN EWALD*, WANWAN HUANG** is (95) Y 3 = y 3 + [(3γ2 3α)Ys 3 + (3mα + 6βρ 2 γ)ys 2 + 3β 2 Y s ] ds + 3βY 2 s dẑ() + 3γY 3 s dẑ(2). Taking expecaions on boh sides of he soluion gives (96) EY 3 = Ey 3 + E So he corresponding ordinary differenial equaion is (97) dey 3 d [ ] (3γ 2 3α)Ys 3 + (3mα + 6βρ 2 γ)ys 2 + 3β 2 Y s ds. = 3(γ 2 α)e[y 3 ] + 3(mα + 2βρ 2 γ)e[y 2 ] + 3β 2 E[Y ]. From he soluion of he ordinary differenial equaion, he condiion for no blowing up is α γ 2. Similarly as before, when n = 4, (98) dey 4 d = (6γ 2 4α)EY 4 + (4mα + 2βρ 2 γ)ey 3 + 6β 2 EY 2. From he soluion of his ordinary differenial equaion, he condiion for no blowing up is α 3 2 γ2. Similarly, we can conclude ha for any posiive inerger n, he condiion under which he nh momen of Y does no blow up is α (n ) γ 2. In our simulaion, we 2 wan n = 5, so we use he relaionship α 2γ 2. Appendix B. Derivaion of he accuracy of he variance analysis In order o prove Theorem., we need he following hree lemmas. Lemma A.. Under he assumpions of Theorem., for any fixed iniial sae (,x,y), here exiss a posiive consan C > such ha for ε, one has E,,x,y { T ( P e 2rs ε x P BS x Proof: By Cauchy-Schwarz inequaliy ) 2 (s, S s, Y s )f 2 (Y s )S 2 s ds } C ε. (99) E,,x,y{ T e 2rs ( P ε P BS x x )2 (s, S s, Y s )f 2 (Y s )Ss 2 ds} E { T ( P ε P BS x x )4 (s, S s, Y s )} ds T E {f 4 (Y s )(e rs S s ) 4 } ds.

25 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL25 The second facor on he righ hand side is bounded by T T () E {f 4 (Y s )(e rs S s ) 4 } ds C 2 f E {(e rs S s ) 4 } ds for some consan C f, as he volailiy funcion f is bounded. Using he noaion σ = f(y ) as in (7), and if W = W for simpliciy, one has and herefore e rs S s = S e s σudwu 2 s σ2 u du, E {(e rs S s ) 4 } = SE 4 {e σ s σ2 udu e s 4σudWu 2 C f S4 E {e s 4σudWu s 2 6σ2 udu } = C f S4, s 6σ2 u du } where we have used again he boundness of f and he maringale propery. Combined wih () we obain T () E {f 4 (Y s )(e rs S s ) 4 }ds C 2, for some posiive consan C 2. In order o sudy he firs facor on he righ hand side of he inequaliy (99), we have o conrol he dela approximaion, P ε P BS, as opposed o he opion x x price approximaion, P ε P BS, sudied in [8] for European opions, or in [] for digial-ype opions. By pahwise differeniaion (see [9] for insance), he chain rule can be applied and we obain { } P ε (, S, Y ) = E e r(t ) S T I {ST >K} S S S, Y. A ime =, (2) e rt S T = e T σdw T 2 σ2 d S gives an exponenial maringale, and herefore one can consruc a P -equivalen probabiliy measure P by Girsanov Theorem. As a resul, he dela P ε S (, S, Y ) has a probabilisic represenaion under he new measure P corresponding o he digial-ype opion P ε (, S, Y ) = S Ẽ{I {S T >K} S, Y }, where he dynamics of S become ds = (r + f 2 (Y ))S d + σ S d W,

26 26 BRIAN EWALD*, WANWAN HUANG** wih W being a sandard Brownian moion under P. The dynamics of Y remain he same because we have assumed here zero correlaion beween Brownian moions. The one can apply he accuracy resul in [] for digial opions o claim ha Ẽ{I {S T >K} S, Y } Ē{I { S T >K} S = S } C 3 (Y ) ε, where he consan C 3 may depend on Y, and he homogenized sock price S saisfies d S = (r + σ 2 ) S d + σ S d W wih W being a sandard Brownian moion []. In fac, he homogenized approximaion Ē{I { S T >K} S } is a probabilisic represenaion of he homogenized dela, P BS x. Consequenly, we obain he accuracy resul for dela approximaion: ( P ε x P ) BS (, S, Y ) x C 3(Y ) ε. The exisence of momens of Y ensures he exisence of he fourh momen of C 3 (Y ), and herefore he firs facor on he righ hand side of (99) is bounded by { T ( (3) E P ε x P ) 4 BS (s, S s, Y s ) ds} C 4 ε x for some posiive consan C 4. From (99), (3) and (), we conclude ha { T ( P E e 2rs ε x P ) } 2 BS (s, S s, Y s )f 2 (Y s )Ss 2 ds C ε x for some consan C. Lemma A.2. Under he assumpions of Theorem. for any fixed iniial sae (, x, y), here exiss ε a posiive consan C such ha for ε, one has T ( ) P e 2rs ε 2 (s, S s, Y s )g 2 y (Y s ) ds Cε 2. Proof: Condiioning on he pah of he volailiy process and by ieraive expecaions, he price of a European opion can be expressed as (4) P ε (, x, y) = E,x,y{E {e r(t ) (S T K) + σ s, s T }} = E,x,y{P BS (, x; K, T ; σ2 )}, where he realized variance is denoed by σ 2 : (5) σ 2 = T T f(y s ) 2 ds.

27 A MARTINGALE CONTROL VARIATE METHOD FOR OPTION PRICING WITH CAM MODEL27 Taking a pahwise derivaive for P ε wih respec o he fas varying variable y, we deduce by he chain rule { P ε (6) y (, x, y) = P BS E,x,y σ (, x; K, T ; σ 2 (y) } σ2. y Inside of he expecaion he firs derivaive, known as Vega, P BS σ = xe d2 /2 T, 2π wih d = log(x/k)+(r+ 2 σ2 )(T ) σ, is uniformly bounded in σ. Using he chain rule one T obains (7) σ2 y = (T ) σ2 T [ f y (Y s) Y ] s f(y s ) ds. y In order o conrol he growh rae of Ys we consider is dynamics: y ( ) [ ] d Ys (8) = ds y ε + ν 2 Λ ε y (Y s) + χ Λ 2 ε y (Y Y s s) y wih he iniial condiion Y =. y Rescaling he sysem (8) by defing Ỹsε = Ysε, we deduce d ds ( Ỹ ε s y ) = Ỹ s ε y + ε ( ν 2 Λ (Ỹs y ε ) + χ Λ 2 (Ỹs y ε ) ) ε Ỹs y. The funcions Λ and Λ 2 are defined according o he rescaling and hey are smooh and bounded as Λ s. By a classical sabiliy resul [2], we obain Ys y < C 5 e (s )/ε for some consan C 5. Applying hese esimaes o (7) and by he smooh boundedness of f, we obain σ2 Cε y for some C, and consequenly a similar bound for P ε (, x, y) in (6). Finally, as y g = ν 2, Lemma A.2 follows. Lemma A.3. Under he assumpions of Theorem., for any fixed iniial sae (, x, y), here exiss a posiive consan C such ha for ε, one has T ( ) P e 2rs ε 2 g 2 y 2(Y s ) ds C ε

28 28 BRIAN EWALD*, WANWAN HUANG** wih g 2 (Y ) = µy. Proof: The proof is similar o Lemma A.2. From he bounds in Lemma A., A.2 and A.3, we deduce Theorem.. References [] J. P. Fouque, G. Papanicolaou, & K. R. Sircar, Derivaives in Financial Markes wih Sochasic Volailiy, firs ediion, Cambridge (2). [2] J. Hull & A. Whie, The Pricing of Opiions on Asses wih Sochasic Volailiies, he Journal of Finance, Vol. XL, No. 2, June 987. [3] J. P. Fouque, Sochasic Volailiy Modeling, 28 Daiwa Lecure Series, Kyoo Universiy, Kyoo. [4] S. Beckers, Sandard Deviaions Implied in Opion Prices as Predicors of Fuure Sock Price Variabiliy, Journal of Banking and Finance, Vol. 5, Issue 3: , Sepember 98. [5] S. Mayhew, Implied Volailiy, Financial Analyss Journal, Vol. 5, No. 4: 8-2, July/Augus 995. [6] S. L. Heson, A Closed-Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions, he Review of Financial Sudies, Vol. 6, No. 2: , 993. [7] B. Fisher & M. S. Scholes, The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, Vol. 8, No. 3: , 972. [8] S. Beckers, The Consan Elasiciy of Variance Model and Is Implicaions for Opion Pricing, he Journal of Finance, Vol. XXXV, No. 3, June 98. [9] L.B.G. Anderson and V.V. Pierbarg, Momen Explosions in Sochasic Volailiy Models, Finance and Sochasics, Vol., No. : 29-5, 27. [] R. Lord, R. Koekkoek & D. V. Dijk, A comparison of Biased Simulaion Schemes for Sochasic Volailiy Models, Quaniaive Finance, Vol., No. 2: 77-94, February 2. [] P. Sura & P. D. Sardeshmukh, A Global View of Non-Gaussian SST Variabiliy, Journal of Physical Oceanography, vol. 38: , March 28. [2] P. D. Sardeshmukh & P. Sura, Reconciling Non-Gaussian Climae Saisics wih Linear Dynamics, Journal of Climae, vol. 22: 93-27, March 29. [3] S. E. Shreve, Sochasic Calculus for Finance II Coninuous-Time Models, Springer, 24. [4] P. E. Kloeden, E. Plaen, Numerical Soluion of Sochasic Differenial Equaions, Applicaions of Mahemaics 23, Springer-Verlag, Berlin, 992. [5] B. D. Ewald, R. Temam, Numerical Analysis of Sochasic Schemes in Geophysics, SIAM J. NUMER. ANAL., Vol. 42. No. 6 pp , 25. [6] J. P. Fouque, T. A. Tullie, Variance reducion for Mone Carlo simulaion in a sochasic volailiy environmen, Quaniaive Finance, Vol. 2 (22) [7] J. P. Fouque&C. Han A Maringale Conrol Variae Mehod for Opion Pricing wih Sochasic Volailiy. [8] J. P. Fouque, G. Papanicolaou, K. R. Sircar, &K. SolnaMuliscale Sochasic Volailiy Asympoics, SIAM Journal on Muliscale Modeling and Simulaion 2(), 23 (22-42). [9] P. Glasserman, Mone Carlo Mehods in Financial Engineering, Springer Verlag, 23. [2] R. Bellman, Sabiliy Theory of Differenial Equaions, McGraw-Hill, 953.