An accurae analyical approximaion for he price of a European-syle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price average of he underlying asse. Due o he dependence srucure beween he prices of he underlying asse, no simple exac pricing formula exiss, no even in a Black-Scholes seing. In he recen lieraure, several approximaions and bounds for he price of his ype of opion are proposed. One of hese approximaions consiss of replacing he disribuion of he sochasic price average by an ad hoc disribuion (e.g. Lognormal or Inverse Gaussian) wih he same firs and second momen. In his paper we use a differen approach and combine a lower and upper bound ino a new analyical approximaion. This approximaion can be calculaed efficienly, urns ou o be very accurae and moreover, i has he correc firs and second momen. Since he approximaion is analyical, we can also calculae he corresponding hedging Greeks and consruc a replicaing sraegy. 1 Inroducion Consider a risky asse (a non-dividend paying sock) wih prices described by he sochasic process {A(), 0} and a risk-free coninuously compounded rae δ ha is consan hrough ime. In his secion all probabiliies and expecaions have o be considered as condiional on he informaion available 1 Corresponding auhor: K.U.Leuven, Dep. of Applied Economics, Naamsesraa 69, B-3000 Leuven, Belgium, email: david.vyncke@econ.kuleuven.ac.be 2 K.U.Leuven, Belgium, and Universiy of Amserdam, he Neherlands 1
a ime 0, i.e. he prices of he risky asse up o ime 0, unless oherwise saed. Noe ha in general, he condiional expecaion (wih respec o he physicalprobabiliy measure) of e δ A(), given he informaion available a ime 0, will differ from he curren price A(0). However, we will assume ha we can find a unique equivalen probabiliy measure Q such ha he discouned price process {e δ A(), 0} is a maringale under his equivalen probabiliy measure. This implies ha for any 0, he condiional expecaion (wih respec o he equivalen maringale measure) of e δ A(), given he informaion F 0 available a ime 0, will be equal o he curren price A(0). Denoing his condiionalexpecaion under he equivalen maringale measure by E [ Q e δ A() ],wehaveha E Q [ e δ A() F 0 ] = A(0), 0. (1) The noaion F A0 ()(x) will be used for he condiional probabiliy ha A() is smaller han or equal o x, under he equivalen maringale measure Q and given he informaion F 0 available a ime 0. Is inverse will be denoed (p). The exisence of an equivalen maringale measure is relaed o he absence of arbirage in he securiies marke, while uniqueness of he equivalen maringale measure is relaed o marke compleeness. Two models incorporaing such a unique equivalen maringale measure are he binomialree modelof Cox, Ross and Rubinsein (1979) and he geomeric Brownian moion model of Black and Scholes (1973). The exisence of he equivalen maringale measure allows one o reduce he pricing of opions on he risky asse o calculaing expeced values of he discouned pay-offs, no wih respec o he physicalprobabiliy measure, bu wih respec o he equivalen maringale measure, see e.g. Harrison and Kreps (1979) or Harrison and Pliska (1981). A reference in he acuarial lieraure is Gerber and Shiu (1996). A European call opion on he risky asse, wih exercise price K and exercise dae T generaes a pay-off (A(T ) K) + a ime T,hais,ifhe price of he risky asse a ime T exceeds he exercise price, he pay-off equals he difference; if no, he pay-off is zero. Noe he similariy beween such a pay-off and he paymen on a sop-loss reinsurance conrac. A ime his call opion will rade agains a price given by by F 1 A 0 () EC(K, T, ) =e δ(t ) E Q [(A (T ) K) + ] (2) A European-syle arihmeic Asian call opion wih exercise dae T, n averaging daes and exercise price K generaes a pay-off ( 1 n 1 n A T (T i) K ) + 2
a T, ha is, if he average of he prices of he risky asse a he laes n daes before T is more han K, he pay-off equals he difference; if no, he pay-off is zero. Such opions proec he holder agains manipulaions of he asse price near he expiraion dae. The price of he Asian call opion a ime is given by [( ) ] n 1 AC(n, K, T, ) =e δ(t ) E Q 1 A (T i) K (3) n and he price of he Asian pu opion a ime equals [( ) AP(n, K, T, ) =e δ(t ) E Q K 1 A (T i) n Noe ha, due o he pu-call pariy, he price of an Asian pu opion can be easily derived from he price of an Asian call opion: ( ) ) e δ(t AP(n, K, T, ) =AC(n, K, T, )+ K E[A (T i)] n Hence, in he remainder we will only consider call opions. We can also assume ha T n +1>. Indeed, if a ime he averaging has already sared, i.e. T n +1, hen we know he prices A(T n + 1),...,A( ), where denoes he ineger par of. Sinceaime he prices A( +1),...,A(T ) are sill random, we can wrie AC(n, K, T, ) = e δ(t ) T n where EQ [( 1 T T 1 + + ]. A (T i) K ) = n n AC(n,K,T,) (4) n =min(t, n) and K = nk n 1 i=t A (T i). n Wih hese rescaled parameers he averaging has no ye sared since T n +1>. 3 + ]
Deermining he price of an Asian opion is no a rivialask, because in general we do no have an explici analyical expression for he disribuion of he average S = n 1 A (T i). One can use Mone-Carlo simulaion echniques o obain a numericalesimae of he price, see Kemna and Vors (1990) and Vazquez-Abad and Dufresne (1998), or one can numerically solve a parabolic parial differenial equaion, see Rogers and Shi (1995). Bu as boh approaches are raher ime consuming, i would be helpful o have an accurae, easily compuable analyical approximaion of his price. In Jacques (1996) an approximaion is obained by replacing he disribuion of he sum n 1 A(T i) by a Lognormalor an Inverse Gaussian disribuion. From he expression for he price of an arihmeic Asian call opion, we see ha he problem of pricing such opions urns ou o be equivalen o calculaing sop-loss premiums of a sum of dependen random variables. This means ha we can apply he resuls of Dhaene e al. (2002b,a) in order o find analyical lower and upper bounds for he price of Asian opions. By combining hese bounds a new approximaion arises. 2 Bounds and approximaions Assume ha a ime he averaging has no ye sared and hus A (T n + 1),...,A (T ) are random. The price AC(n, K, T, ) hen essenially consiss of a sop-loss premium of a sum of n dependen random variables. In Dhaene e al. (2002b,a) i is shown how o consruc upper and lower bounds for such sop-loss premiums by using he heory on comonoonic risks. In he acuarial lieraure i is common pracice o replace a random variable by a less aracive random variable which has a simpler srucure, making i easier o deermine is disribuion funcion, see e.g. Goovaers e al. (1990), Kaas e al. (1994) or Denui e al. (1999). Performing he compuaions (of premiums, reserves and so on) wih he less aracive random variable is a pruden sraegy. On he oher, considering more aracive random variables could help o give an idea abou he degree of overesimaion involved in replacing he original variable by he less aracive random variable. Of course, we have o clarify wha we mean by a less aracive random variable. Definiion 1. Consider wo random variables X and Y.ThenX is said o precede Y in he sop-loss order sense, noaion X sl Y,ifandonlyifX 4
has lower sop-loss premiums han Y : E[(X d) + ] E[(Y d) + ], <d<+. Hence, X sl Y means ha X has uniformly smaller upper ails han Y, whichinurnmeanshaapaymeny is indeed less aracive han a paymen X. Sop-loss order has a naural economic inerpreaion in erms of expeced uiliy. Indeed, i can be shown ha X sl Y if and only if E[u( X)] E[u( Y )] holds for all non-decreasing concave real funcions u for which he expecaions exis. This means ha any risk-averse decision maker would prefer o pay X insead of Y, which implies ha acing as if he obligaions X are replaced by Y indeed leads o conservaive or pruden decisions. The characerizaion of sop-loss order in erms of uiliy funcions is equivalen o E[v(X)] E[v(Y )] holding for all non-decreasing convex funcions v for which he expecaions exis. Therefore, sop-loss order is ofen called increasing convex order and denoed by icx. For more deails and properies of sop-loss order in a general conex, see Shaked and Shanhikumar (1994) or Kaas e al. (1994), where sochasic orders are considered in an acuarialconex. If X sl Y,henalsoE[X] E[Y ],andiisinuiivelyclearhahe bes approximaions arise in he borderline case where E[X] = E[Y ]. This leads o he so-called convex order. Definiion 2. Consider wo random variables X and Y. Then X is said o precede Y in he convex order sense, noaion X cx Y,ifandonlyif E[X] =E[Y ] and E[(X d) + ] E[(Y d) + ], <d<+ From E[(X d) + ] E[(d X) + ]=E[X] d, we find X cx Y { E[X] =E[Y ], E[(d X) + ] E[(d Y ) + ], <d<+. Noe ha parialinegraion leads o E[(d X) + ]= d F X (x) dx, which means ha E[(d X) + ] can be inerpreed as he weigh of a lower ailof X. We have seen ha sop-loss order enails uniformly heavier upper 5
ails. The addiional condiion of equal means implies ha convex order also leads o uniformly heavier lower ails. I can be proven ha X cx Y if and only E[v(X)] E[v(Y )] for all convex funcions v, provided he expecaions exis. This explains he name convex order. Noe ha when characerizing sop-loss order, he convex funcions v are addiionally required o be non-decreasing. Hence, sop-loss order is weaker: more pairs of random variables are ordered. We also find ha X cx Y if and only E[X] =E[Y ]ande[u( X)] E[u( Y )] for all non-decreasing concave funcions u, provided he expecaions exis. Hence, in a uiliy conex, convex order represens he common preferences of all risk-averse decision makers beween random variables wih equal mean. In case X cx Y, he upper ails as well as he lower ails of Y eclipse he corresponding ails of X, which means ha exreme values are more likely o occur for Y han for X. This observaion also implies ha X cx Y is equivalen o X cx Y. Hence, he inerpreaion of he random variables as paymens or as incomes is irrelevan for he convex order. As he funcion v defined by v(x) =x 2 is a convex funcion, i follows immediaely ha X cx Y implies Var[X] Var[Y ]. The reverse implicaion does no hold in general. For he problem a hand, we have he following resul. Theorem 1. Consider a sum S of random variables X 1,X 2,...,X n define and S c = F 1 X 1 (U)+F 1 X 2 (U)+...+ F 1 X n (U) (5) S u = F 1 1 1 X 1 Λ (U)+FX 2 Λ (U)+...+ FX n Λ (U) (6) S l = E[X 1 Λ] + E[X 2 Λ] +...+E[X n Λ] (7) wih U Uniform(0,1) and wih Z an arbirary random variable, independen of U. The following relaions hen hold: E[(S l ) + ] E[(S ) + ] E[(S u ) + ] E[(S c ) + ], (8) for all real, ande[s l ]=E[S] =E[S u ]=E[S c ]. These lower and upper bounds can be considered as approximaions for he disribuion of a sum S of random variables. On he oher hand, any convex combinaion of he sop-loss premiums of he lower bound S l and he upper bounds S c or S u oo could serve as an approximaion for he sop-loss 6
premium of S. Since he bounds S l and S c have he same mean as S, any random variable S m defined by is sop-loss premiums E[(S m ) + ]=z E[(S l ) + ]+(1 z)e[(s c ) + ], 0 z 1, will also have he same mean as S. By aking he (righ-hand) derivaive we find F S m(x) =zf S l(x)+(1 z)f S c(x), 0 z 1, so he disribuion funcion of he approximaion can be calculaed fairly easily. By choosing he opimal weigh z, wewans m o be as close as possible o S. A naural opimaliy crierion could be o choose z such ha E[(S m ) + ] E[(S ) + ]d =0. Since S, S c, S l and S m all have he same mean, he relaion (see Kaas e al. (2001)) Var[X] =2 (E[(X ) + ] (E[X] ) + )d implies ha he opimal weigh in his case equals z = Var[Sc ] Var[S] Var[S c ] Var[S l ]. (9) Noe ha his choice doesn depend on he reenion. Choosing z as in (9), we have ha Var[S m ]=Var[S] so he opimalapproximaion S m can also be seen o be a momens based approximaion. As an alernaive one could consider he improved upper bound S u and define a second approximaion as follows E[(S m2 ) + ]=z E[(S l ) + ]+(1 z)e[(s u ) + ], wih z =(Var[S u ] Var[S])/(Var[S u ] Var[S l ]). For a comparison of hese approximaions we refer o Secion 4 where i is shown ha S m and S m2 yield almos he same resuls. This echnique could also be used in oher acuarial problems concerning sums of (dependen) random variables. Such a sum appears for insance when considering discouned paymens relaed o a single policy or a porfolio a differen fuure poins in ime, i.e. when combining he (acuarial) echnical risk wih he (financial) invesmen risk, see Dhaene e al. (2002b,a). 7
3 Applicaion in a Black & Scholes seing In he model of Black and Scholes (1973), he price of he risky asse is described by a sochasic process {A(), 0} following a geomeric Brownian moion wih consan drif µ and consan volailiy σ: da() A() = µd + σdb(), 0, (10) wih iniialvalue A(0) > 0 and where{b(), 0} is a sandard Brownian moion. Under he equivalen maringale measure Q, he price process {A(), 0} sill follows a geomeric Brownian moion wih he same volailiy bu wih drif equalo he coninuously compounded risk-free ineres rae δ: da() A() = δd + σdb(), 0, (11) wih iniialvalue A(0) and where {B(), 0} is a sandard Brownian moion in he Q-dynamics. Hence, under he equivalen maringale measure, we have ha ) (δ A 0 () =A(0) e σ2 2 +σb(), 0. (12) This implies ha under he equivalen maringale measure, ( he random ) variables A 0() are lognormally disribued wih parameers δ σ2 and σ 2 A(0) 2 respecively: ] σ2 (δ F A0 ()(x) =Prob [A(0)e 2 )+ σφ 1 (U) x, (13) where U is uniformly disribued on he inerval (0, 1). Using he famous Black and Scholes (1973) pricing formula for European call opions, we find EC(K, T, ) = e δ(t ) E Q [ (A (T ) K) + ] where d 1 and d 2 are given by = A()Φ(d 1 ) Ke δ(t ) Φ(d 2 ), (14) d 1 = ln(a()/k)+(δ + σ2 /2)(T ) σ T (15) 8
and d 2 = d 1 σ T. (16) Wihin he Black & Scholes model, no closed form expression is available for he price of an arihmeic Asian call opion. Therefore, we will derive bounds and approximaions for he price of such opions. Because of (4) we will only consider he case ha he averaging has no ye sared. To avoid lenghy formulas, we will use T i as a shorhand for T i. Comonoonic upper bound From (5) we find he following comonoonic upper bound for he price of an Asian call opion: ) e δ(t AC(n, K, T, ) E [ ] (S c nk) n + = A() e δi Φ [σ T i Φ ( 1 F S (nk) )] c n e δ(t ) K ( 1 F S c (nk) ), (17) which holds for any K > 0. Noe ha his upper bound corresponds o he opimal linear combinaion of he prices of European call opions as menioned in Dhaene e al. (2002a). The remaining problem is how o calculae F S c (nk). This quaniy follows from ( F 1 A (T i) FS c (nk)) = nk, or, equivalenly, from (12) we find ha F S c (nk) followsfrom A() exp ) [(δ σ2 T i + σ T i Φ ( 1 F S c 2 (nk) )] = nk. Lower bound For he lower bounds for AC(n, K, T, ) we consider he condiioning random variable Λ defined by Λ= e j=0 (δ σ2 2 9 ) T j B (T j). (18)
From (12) we find ha, in he Q-dynamics, A (T i) =A() e (δ σ2 2 ) T i +σb (T i). (19) So, Λ is a linear ransformaion of a firs order approximaion o n 1 A (T i). The variance of Λ is given by σλ 2 = e j=0 k=0 (δ σ2 2 ) (T j +T k ) min(t j,t k ). (20) We have ha (B (T n +1),B (T n +2),...,B (T )) has a mulivariae normaldisribuion. This implies ha given Λ = λ, he random variable B (T i) is normally disribued wih mean r Ti i σ Λ λ and variance T i (1 ri 2) where We find ( r i = Cov(B n 1 (T i), Λ) j=0 = e σ Λ Ti S l n 1 ) T j min(t i,t j ). (21) σ Λ Ti δ σ2 2 [ n 1 ] d = E Q A (T i) Λ d = A() ) (δ e σ2 2 r2 i T i +σ r i Ti Φ 1 (U) (22) where U is uniformly disribued on he uni inerval. From his expression, we see ha S l is a comonoonic sum of lognormal random variables. Hence, we find he following lower bound for he price of he Asian call opion: ) [ e δ(t (S ] l AC(n, K, T, ) E nk) + = A() n n e δ(t ) K ( )] e δi Φ [σr i Ti Φ 1 F S l (nk) ( ) 1 F S l (nk) which holds for any K>0. In his case, F S l (nk) followsfrom ) A() exp [(δ σ2 ( 2 r2 i T i + σr i Ti Φ 1 F S l (nk)) ] = nk. 10 (23)
When he number of averaging daes n equals 1, he Asian call opion reduces o a European call opion. I is sraighforward o prove ha in his case he upper and he lower bounds (17) and (23) for he price of he Asian opion boh reduce o he Black & Scholes formula for he price of he European call opion. Improved upper bound By Theorem 1 we can also consruc a smaller upper bound for AC(n, K, T, ). We choose Λ = B(T ) since hen he dependence srucure of he erms in S u is almos comonoonic, see Vanmaele e al. (2002). This yields wih AC(n, K, T, ) E [ (S u nk) ] + = nk(1 F S u (nk)) + 1 0 ( e σr i Ti Φ 1 (v) Φ σ e δ(t ) n E [ (S u nk) + ) ] A()exp [(δ σ2 2 r2 i T i 1 r 2 i Ti Φ 1 (F S u V =v (nk)) ] ) dv. where he correlaion coefficiens r i are given by r i = Cov(B (T i), Λ) σ Λ Ti = T i T. The condiionaldisribuion F S u V =v(nk) followsfrom A() σ2 (δ e 2 )T i+σr i Ti Φ 1 (v)+σ 1 ri 2 Ti Φ 1 (F S u V =v(nk) = nk. Momens based approximaions Severalauhors propose o replace he unknown disribuion of S by an ad hoc disribuion wih he correc firs wo momens. The quesion remains which disribuion one should use. For reasonable values of he parameers, Levy (1992) subsaniaes he lognormal disribuion as an approximaion for he disribuion of a sum of lognormal random variables. Jacques (1996) 11
Approximaions minus lower bound -0.0010 0.0 0.0005 0.0015 80 90 100 110 120 Srike price Figure 1: Price of an Asian opion wih T = 120, n =30andσ =0.2 according o he lognormal approximaion (black line) and he inverse gaussian approximaion (grey line) minus he lower bound (23). Negaive values show ha he approximaions perform worse han he lower bound. concludes ha an Inverse Gaussian approximaion gives prices comparable o hose given by he lognormal approximaion when he parameers are chosen in he same range as in Levy (1992). Alhough hese approximaions appear o be very accurae, hey have wo srucuraldisadvanages. Firs, for some values of he parameers, he approximaions urn ou o be smaller han our heoreical lower bound, see Figure 1. Moreover, if a differen process is used o modelhe sock prices, he approximaions will no be valid anymore. By using he momens based approximaion from Secion 2 hese drawbacks can be avoided. A firs approximaion can be obained by combining he lower bound S l and he comonoonic upper bound S c. For he variance of S l and S c we find Var[S l ]=A 2 () j=0 ( ) e δ(t i+t j ) e σ2 r i r j Ti T j 1 and Var[S c ]=A 2 () j=0 ( e δ(t i+t j ) e σ2 ) T i T j 1. 12
As shown in Secion 2, he random variable S m = zs l +(1 z)s c will have he correc variance Var[S] =A 2 () j=0 e δ(t i+t j ) ( ) e σ2 min(t i,t j ) 1 if we choose z = Var[Sc ] Var[S] Var[S c ] Var[S l ]. (24) Replacing he comonoonic upper bound S c by he improved upper bound S u yields a second approximaion S m 2 = z u S l +(1 z u )S u.inhiscase,he weigh z u is given by z = Var[Su ] Var[S] Var[S u ] Var[S l ] where ( Var[S u ]=A 2 () e δ(t i+t j ) e σ2 (r i r j + (1 ri 2)(1 r2)) ) j T i T j 1. j=0 In he following secion, we will show ha he approximaions S m and S m 2 give almos equal prices. Hence, we propose o use S m, since he compuaion of S m 2 involves much more calculaions. 4 Numerical illusraion In his secion we numerically illusrae he bounds and approximaions for he price of an Asian opion in a Black & Scholes seing, as obained in he previous secion. We consider a ime uni of one day and se =0. The parameers ha were used o generae he resuls given in Tables 1, 2 and 3 are he same as in Jacques (1996): an iniialsock price A(0) = 100, a riskfree ineres rae of 9% per year, hree values (0.2, 0.3 and 0.4) for he yearly volailiy, and five values (80, 90, 100, 110 and 120) for he exercise price K. Noe ha he risk-free force of ineres per day is given by δ =ln(1.09)/365, while he daily volailiy σ is obained by dividing he yearly volailiy by 365. In Table 1 we compare he bounds and approximaions wih Mone Carlo esimaes (based on 10000 pahs each) in case T = 120 and n = 30. Noe 13
σ K LB MB MB2 LN IG IUB UB MC (s.e. 10 4 ) 0.2 80 21.9212 21.9212 21.9212 21.9213 21.9208 21.9246 21.9269 21.9213 (2.51) 90 12.6768 12.6768 12.6768 12.6771 12.6767 12.7038 12.7204 12.6764 (2.36) 100 5.4609 5.4609 5.4609 5.4611 5.4629 5.5200 5.5557 5.4616 (2.31) 110 1.6252 1.6252 1.6252 1.6250 1.6259 1.6762 1.7072 1.6250 (1.64) 120 0.3317 0.3317 0.3317 0.3315 0.3307 0.3536 0.3673 0.3319 (1.15) 0.3 80 22.2332 22.2332 22.2332 22.2336 22.2313 22.2571 22.2720 22.2340 (5.69) 90 13.8521 13.8521 13.8521 13.8528 13.8544 13.9137 13.9512 13.8519 (5.48) 100 7.4787 7.4788 7.4788 7.4791 7.4855 7.5686 7.6229 7.4800 (5.36) 110 3.4826 3.4827 3.4827 3.4825 3.4875 3.5690 3.6214 3.4829 (4.60) 120 1.4125 1.4126 1.4126 1.4121 1.4123 1.4733 1.5105 1.4124 (3.52) 0.4 80 22.9646 22.9646 22.9646 22.9658 22.9638 23.0190 23.0525 22.9665 (10.05) 90 15.3589 15.3589 15.3589 15.3602 15.3682 15.4539 15.5115 15.3600 (9.82) 100 9.5113 9.5114 9.5114 9.5121 9.5277 9.6315 9.7041 9.5118 (9.05) 110 5.4794 5.4795 5.4795 5.4795 5.4936 5.5994 5.6720 5.4794 (8.33) 120 2.9608 2.9609 2.9609 2.9603 2.9666 3.0611 3.1222 2.9614 (7.36) Table 1: Upper (UB, IUB) and lower bounds (LB) for he price of an Asian opion a =0wihT = 120 and n = 30, compared o he Mone Carlo esimaes (MC) wih heir sandard error imes 10000 (s.e. 10 4 )andhe momens based approximaions (MB, MB2, LN, IG) ha he random pahs are based on aniheic variables and ha we use he geomeric average as a conrolvariae in order o reduce he variance of he Mone Carlo esimae. Also noe ha we generaed differen pahs for each value of σ and K. For each esimae we compued he sandard error. As is well-known, he (asympoic) 95% confidence inerval is given by he esimae plus or minus 1.96 imes he sandard error. On he oher hand, he range beween he lower bound and he (improved) upper bound conains he exac price wih cerainy. Despie he quie large number of pahs (and consequenly a long compuing ime) and he variance reducion echniques used, he Mone Carlo esimae (MC) violaes he lower bound (LB) 4 imes ou of 15. This migh indicae ha he lower bound is very close o he real price. The momens based approximaions all give similar prices, bu he lognormal approximaion (LN) appears o violae he lower bound for opions ha are far ou-ofhe-money. Also he inverse gaussian approximaion (IG) violaes he lower bound, no only for ou-of-he-money opions bu for in-he-money opions oo. Alhough he comonoonic upper bound (UB) and he improved upper bound (IUB) give quie differen prices, he corresponding momens based approximaions (MB, MB2) are almos equal. In Table 2 we use he same parameers as in Table 1 bu we change he expiraion ime o T = 60. In his case, he Mone Carlo esimae 14
σ K LB MB MB2 LN IG IUB UB MC (s.e. 10 4 ) 0.2 80 20.7841 20.7841 20.7841 20.7841 20.7841 20.7843 20.7845 20.7841 (2.43) 90 11.0273 11.0273 11.0273 11.0277 11.0275 11.0470 11.0599 11.0276 (2.43) 100 3.2013 3.2013 3.2013 3.2016 3.2021 3.2903 3.3443 3.2013 (2.15) 110 0.3373 0.3373 0.3373 0.3367 0.3366 0.3805 0.4080 0.3372 (1.33) 120 0.0116 0.0116 0.0116 0.0115 0.0114 0.0156 0.0185 0.0117 (0.55) 0.3 80 20.8122 20.8123 20.8123 20.8126 20.8122 20.8208 20.8268 20.8115 (5.44) 90 11.4929 11.4929 11.4929 11.4944 11.4942 11.5599 11.6017 11.4931 (5.40) 100 4.5063 4.5063 4.5063 4.5070 4.5086 4.6406 4.7221 4.5051 (4.55) 110 1.1516 1.1517 1.1517 1.1505 1.1508 1.2515 1.3134 1.1522 (3.81) 120 0.1915 0.1915 0.1915 0.1906 0.1898 0.2269 0.2503 0.1912 (1.97) 0.4 80 20.9708 20.9708 20.9708 20.9724 20.9709 21.0072 21.0309 20.9716 (9.57) 90 12.2468 12.2469 12.2469 12.2498 12.2505 12.3655 12.4384 12.2482 (9.47) 100 5.8157 5.8159 5.8159 5.8171 5.8210 5.9952 6.1038 5.8155 (8.49) 110 2.2082 2.2083 2.2083 2.2067 2.2088 2.3630 2.4582 2.2091 (7.63) 120 0.6783 0.6783 0.6783 0.6761 0.6750 0.7663 0.8223 0.6777 (4.96) Table 2: Upper (UB, IUB) and lower bounds (LB) for he price of an Asian opion a =0wihT =60andn = 30, compared o he Mone Carlo esimaes (MC) wih heir sandard error imes 10000 (s.e. 10 4 )andhe momens based approximaions (MB, MB2, LN, IG) violaes he lower bound 8 imes ou of 15. So again, he lower bound mus be very close o he real price. For he lognormal and he inverse gaussian approximaion, we see a similar paern as in he previous case. The momens based approximaions are again almos equal and very close o he lower bound. In Table 3 we change he expiraion ime back o T = 120 bu we reduce he number of averaging days o n = 10. Wih hese parameers, he Mone Carlo esimae violaes he lower bound 13 imes ou of 15. The firs 4 columns (LB, MB, MB2, LN) are almos equal while he inverse gaussian approximaion appears o underesimae he price of in-he-money opions and ou-of-he-money opions. Comparing he resuls in Tables 1 and 3, we see ha he comonoonic upper bound performs beer for he opion wih n = 10 han for he opions wih n = 30. This illusraes he fac ha he dependency srucure of he A(T i) is more comonoonic-like if all poins in ime T i are close o each oher. To assess he overall accuracy of he bounds and approximaions, we assume ha he Mone Carlo esimae gives he exac price and calculae he oal absolue difference of all 45 cases. As can be seen from Table 4, he momens based approximaions MB and MB2 boh have he smalles error, closely followed by he lower bound. The lognormal approximaion 15
σ K LB MB MB2 LN IG IUB UB MC (s.e. 10 4 ) 0.2 80 22.1712 22.1712 22.1712 22.1712 22.1706 22.1724 22.1735 22.1712 (0.85) 90 13.0085 13.0085 13.0085 13.0085 13.0081 13.0162 13.0232 13.0083 (0.81) 100 5.8630 5.8630 5.8630 5.8630 5.8651 5.8791 5.8934 5.8629 (0.75) 110 1.9169 1.9169 1.9169 1.9168 1.9181 1.9313 1.9442 1.9168 (0.59) 120 0.4534 0.4534 0.4534 0.4533 0.4525 0.4603 0.4665 0.4533 (0.33) 0.3 80 22.5656 22.5657 22.5657 22.5657 22.5631 22.5729 22.5795 22.5656 (1.89) 90 14.3149 14.3149 14.3149 14.3150 14.3172 14.3321 14.3475 14.3148 (1.84) 100 8.0101 8.0101 8.0101 8.0101 8.0178 8.0346 8.0563 8.0099 (1.72) 110 3.9475 3.9475 3.9475 3.9475 3.9540 3.9715 3.9928 3.9474 (1.37) 120 1.7297 1.7297 1.7297 1.7297 1.7307 1.7474 1.7633 1.7297 (1.14) 0.4 80 23.4194 23.4194 23.4194 23.4195 23.4181 23.4351 23.4493 23.4194 (3.43) 90 15.9549 15.9549 15.9549 15.9550 15.9654 15.9811 16.0045 15.9554 (3.33) 100 10.1735 10.1735 10.1735 10.1736 10.1925 10.2062 10.2354 10.1733 (3.02) 110 6.1019 6.1019 6.1019 6.1019 6.1196 6.1349 6.1643 6.1018 (2.80) 120 3.4683 3.4683 3.4683 3.4682 3.4775 3.4966 3.5220 3.4682 (2.41) Table 3: Upper (UB, IUB) and lower bounds (LB) for he price of an Asian opion a =0wihT = 120 and n = 10, compared o he Mone Carlo esimaes (MC) wih heir sandard error imes 10000 (s.e. 10 4 )andhe momens based approximaions (MB, MB2, LN, IG) also performs quie well, while he inverse gaussian approximaion yields a oal absolue error which is 10 imes bigger. In comparison o he lower bound, he upper bounds IUB and UB perform really bad, so we sugges o use hem only o consruc he momens based approximaions. 5 Replicaing porfolio Since he bounds and approximaions of Secion 3 have an analyical form, we can explicily calculae he so-called Greeks for hese approximaions. The Greeks are quaniies represening he marke sensiiviies of he opions as each Greek measures a differen aspec of he risk in an opion posiion. Through undersanding and managing hese Greeks, marke raders can manage heir risks appropriaely. In his secion, we will focus on he dela of he opion since his quaniy allows us o consruc a dynamical replicaing porfolio. The dela of an opion is defined as he rae of change of he opion price wih respec o he price of he underlying asse, i.e. (n, K, T, ) = AC(n, K, T, ). (25) A() 16
Approximaion Toalabsolue error Momens based MB 0.0174745 Momens based MB2 0.0174745 Lower bound LB 0.0176689 LognormalLN 0.0245574 Inverse Gaussian IG 0.1702414 Improved upper bound IUB 2.3196765 Comonoonic upper bound UB 3.8251189 Table 4: Toal absolue error of he bounds and approximaions in comparison o he Mone Carlo esimae. The dynamical replicaing porfolio consising of (n, K, T, ) shares of sock and AC(n, K, T, ) (n, K, T, )A() Ω(n, K, T, ) =. A(0)e δ shares of he bond, will reproduce he value of he opion a mauriy. This can be seen as follows: If he sock price drops wih a uni amoun, hen we would lose (n, K, T, ) on our porfolio. On he oher hand, if we inves all our money in opions and he underlying sock drops wih a uni amoun, hen we would also lose (n, K, T, ). Since we can calculae he dela of he approximaions of Secion 3 explicily, we can also use i o assess he qualiy of he approximaions. In order o do ha, we will consruc he corresponding hedging porfolio and check how well he porfolio replicaes he value of he opion along a random rajecory of he sock price. For he comonoonic upper bound (17) we find c (n, K, T, ) = 1 n ( e δi Φ σ ) T i Φ 1 (F S c (nk)) if T n +1 >.ThecaseT n +1 is essenially he same because of (4) bu one has o be carefulwhen =. ThenalsoK is a funcion of A() and we pick up an exra erm in he differeniaion. Analogously, for he lower bound (23) we find l (n, K, T, ) = 1 n ( ) e δi Φ σr i Ti Φ 1 (F S l (nk)). 17
price of replicaing porfolio (MB) 15 20 25 0 20 40 60 80 100 120 day Figure 2: Momens based approximaion for he price of an Asian opion (black line) along a randomly generaed pah of he sock price, almos compleely eclipsing he price of is replicaing porfolio (grey line). The inrinsic value of he opion is indicaed by a doed line. Since z in (24) is independen of A(), he dela of he momens based approximaion S m equals m (n, K, T, ) =z l (n, K, T, )+(1 z) c (n, K, T, ). We will use his value as an approximaion for he real dela (25) and consruc he corresponding replicaing porfolio. Noe ha in heory he replicaing porfolio ough o be updaed coninuously, bu in pracice i will only be updaed a discree imes. In our numerical example we will updae he porfolio 24 imes a day. Using a higher updaing frequency doesn seem o have a significan influence on he resuls. Figure 2 shows he hedging porfolio along a randomly generaed pah of he sock price. The parameers are chosen as in Jacques (1996): an iniial sock price A(0) = 100, a srike price K = 90, a risk-free ineres rae δ =9% per year and a yearly volailiy σ = 20%. The number of days unilmauriy is se o T = 120 and he number of averaging days equals n = 30. Recall ha for pricing purposes we have o replace he drif parameer µ wih he risk-free rae of reurn δ. In he presen seing however, we consider physical pahs of he sock price process and hence we have o use he realrae of 18
reurn µ. We choose his parameer o be 0.15, significanly larger han he risk-free rae of reurn. The price of he opion is also ploed in Figure 2, bu i is almos indisinguishable from he hedging porfolio. So, he hedging porfolio replicaes he price of he opion along he pah wih very high precision. We also calculae he inrinsic value of he opion based on he n-periods moving average. This number indicaes how much he opion is worh assuming ha mauriy is aained a ha ime. For <n, we need values of he sock price a negaive imes o compue he moving average and we assume ha A() =A(0) if <0. The smooh curve in Figure 2 is he inrinsic value a he curren ime. I can be seen ha he hedging porfolio reproduces he inrinsic value of he opion a mauriy very well. References F. Black and M. Scholes (1973), The pricing of opions and corporae liabiliies, Journal of Poliical Economy, 81, 637 659. J. Cox, S. Ross and M. Rubinsein (1979), Opion pricing: A simplified approach, Journal of Financial Economics, 7, 229 263. M. Denui, F. De Vijlder and C. Lefèvre (1999), Exrema wih respec o s-convex orderings in momen spaces: A generalsoluion, Insurance: Mahemaics and Economics, 24, 201 217. J. Dhaene, M. Denui, M. Goovaers, R. Kaas and D. Vyncke (2002a), The concep of comonooniciy in acuarialscience and finance: Applicaions, Insurance: Mahemaics and Economics, 31(2), 133 161. J. Dhaene, M. Denui, M. Goovaers, R. Kaas and D. Vyncke (2002b), The concep of comonooniciy in acuarialscience and finance: Theory, Insurance: Mahemaics and Economics, 31(1), 3 33. H. Gerber and E. Shiu (1996), Acuarialbridges o dynamic hedging and opion pricing, Insurance: Mahemaics and Economics, 18, 183 218. M. Goovaers, R. Kaas, A. Van Heerwaarden and T. Bauwelinckx (1990), Effecive Acuarial Mehods, volume 3 of Insurance Series, Norh-Holland, Amserdam. 19
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