Term Structure of Prices of Asian Options


 Magnus Osborne
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1 Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy Nojihigashi, Kusasu, Shiga , Japan {akahori, rp4992, yasuomi, Absrac In he presen paper, we will sudy he pricing of Asian opions. The main conribuion is he model consrucion; our model is compaible wih marke convenions, can be calibraed o observable marke daa, and compuaionally racable, while i is 2facor and he ineres rae is sochasic. The model akes i ino accoun ha he ineres rae risks can influence he prices of he opions in he long run. Thus our model can be used o analyse he erm srucure, from shor o long erm, of he prices of he opions 1 Inroducion 1.1 Backgrounds In 1998, he Financial Accouning Sandards Board (FASB) issued Saemen No.133, Accouning for Derivaive Insrumens and Hedging Aciviies[3], which is known as FAS133. Since hen, pricing Derivaives is geing more and more imporan because he FAS133 requires a firm o be aware of he risk exposure of Derivaives in is porfolio. When a Derivaive is raded Over The Couner (OTC), however, pricing becomes difficul since he marke does no quoe he price. This is ofen he case wih Exoic Derivaives such as knockou opions or Asian (average) opions. To evaluae he fair price of an OTC Derivaive, one hen needs o rely on a mahemaical model which inerpolaes he price of he Derivaive from marke observables. The model should be
2 compaible wih marke convenions, capable of calibraion o marke quoes, compuaionally racable. To consruc such a model deserves a careful sudy from a pracical poin of view. 1.2 The problem In he presen paper, we will sudy he pricing of Asian opions, a ypical example of Exoic opions. The payoff of he Asian call opion we will sudy is 1 N max S Tk K, N, (1) k=1 where S denoe he price a ime of an asse, T 1 < < T N are he daes for aking average, and K is he srike price. We le T( T N ) be he mauriy dae, and he fair price a ime (< T) of he opion will be denoed π(, T), a funcion of T; we consider he cases where T 1,..., T N is compleely deermined by he choice of T. By a sandard argumen of mahemaical finance [2, e.g.], he price should be π (T, K) = P T ET max 1 N S Tk K, N F, (2) where P T is he price a ime of a zero coupon bond which maures a T, he (condiional) expecaion E T is aken in erms of a probabiliy measure ha makes S/P T a maringale (which is usually called forward measure and will be denoed P T ) wih respec o he given filraion {F }. In he nex secion, we specify he sochasic dynamics of S and P T o mee he requiremens of he las paragraph of secion 1.1. We sress ha k=1 he ineres raes are sochasic in our model. This means ha we ake he ineres rae risks ino accoun. This is new in he conex of pricing of exoics, and is why we use he word erm srucure, which is usually associaed wih ineres rae. Then in secion 3, we will discuss how o calibrae he model o marke observables, and in secion 4, a numerical sudy will be presened using MoneCarlo mehod. Remark. The pricing of Asian opions has been inensively sudied, especially by M. Yor and his coworkers (see [6] and references herein). They 2
3 derived explici formulas for Asian opions, which may be useful in our cases. Bu in his paper our sress is on modelling, no in numerical analysis. So we ook an easy way o use MoneCarlo mehod. 2 The Model 2.1 Marke convenion In he conex of opion pricing, he marke convenion usually means he convenional consensus ha he acual price of a plain opion follows he BlackScholes formula (see e.g.[4]). In he sandard BlackScholes model, he ineres rae is nonsochasic, while we wan o consruc a model including ineresrae risks. We can overcome his dilemma by using he forward measure P T menioned in he previous secion; he price a ime of a plain call opion mauring a ime T is P T ET [max(s T K, ) F ] (3) where r is he (consan) ineres rae. To read his as a BlackScholes formula, log S T S is Gaussian under P T (4) is required. Our goal is o price he Asian opions, so we impose a lile sronger condiion: log S is Gaussian under P T for all (, T) (5) 2.2 Ineres rae risks aken ino accoun The convenion discussed above implicily require ha he model should be based on Brownian moions. In he presen paper, we use wo independen Brownian moion (W, B) as he risk facors; we use a 2facor model. The firs Brownian moion W is mean o represen he inheren risk of he asse, while W represens he ineres rae risk. The sae price densiy is hen, in is fully general form, { D = P exp f (s, ) db s 1 } f (s, ) 2 ds (6) 2 3
4 where f (, ) is an adaped process defined for every. Under he NoArbirage hypohesis, he bond prices are P u = E[D F u ]/D u ( = P u P u exp { f (s, ) f (s, u)} db s 1 u ) { f (s, ) 2 f (s, u) 2 } ds, 2 (7) and he process {D S } mus be a maringale (see e.g. [1]). As a consequence, a fully general expression of S is as follows: { S = D 1 S D exp 1 2 σ(s) dw s + λ(s) db s } ( σ(s) 2 + λ(s) 2 ) ds, (8) where σ, λ are adaped processes. 2.3 Specificaion of he model The forward measure is defined via is densiy on F T wih respec o he physical measure P; dp T dp = D T /E[D T ], (9) or dp T dp = E[D T F ]/E[D T ] = P T D /E[D T ]. (1) F Then, by MaruyamaGirsanov Theorem B T := B + f (s, T) ds (11) 4
5 is Brownian moion under P T (see e.g. [5]). Since we have S = 1 { S u P exp σ(s) dw s u u {λ(s) f (s, )} db s } { σ(s) 2 + λ(s) 2 f (s, ) 2 } ds u (subsiuing (11)) = 1 { P exp σ(s) dw s + u u 1 2 u u { σ(s) 2 + λ(s) f (s, T) 2 f (s, T) f (s, ) 2 } ds } u {λ(s) f (s, )} db T s (12) Consequenly, o be consisen wih (5), σ, λ, f mus be nonrandom. (13) If his is he case, hen under he forward measure we have log S S ( u N log P u 1 2 u u { σ(s) 2 + λ(s) f (s, T) 2 f (s, T) f (s, ) 2 } ds, { σ(s) 2 + λ(s) f (s, ) 2} ) ds. (14) 3 Calibraions In his secion we will presen a calibraion scheme for he model we gave in he previous secion. 3.1 A Convenion Firsly, we inroduce an approximaion which faciliae he calibraion scheme. For k =, 1,, N 1, se X k := log S T k S Tk 1. 5
6 Here we le T 1 =. By (12), we have X k = log P T k T k 1 { Tk + σ(s) dw s + T k Tk Tk T k 1 {λ(s) f (s, T k )} db T s { σ(s) 2 + λ(s) f (s, T k ) 2 T k 1 } f (s, T) f (s, T k ) 2 } ds, (15) so ha S Tk = S exp k k X j = S j= 1 j= P T j T j 1 where Y k = log P T k T k 1 + X k. Noe ha Y, Y 1,..., Y N 1 are muually independen. e k j= Y j, By he approximaion of k 1 j= P T j T j 1 1 P T k, (16) he value of he Asian opion in quesion is given by he following muliple inegral: π (T, K) P T 1 max N N 1 k= + S P T k + (2π) N/2 e 1 2 Nk=1 x 2 k e k j= (α j x j+1 +β j ) K, dx 1 dx N. (17) Here we le α k be he sandard variaion of Y k, and β k be he mean of Y k. More precisely, and α k = ( Tk β k = 1 2 { σ(s) 2 + λ(s) f (s, T k )) 2} ) 1/2 ds (18) T k 1 Tk { σ(s) 2 + λ(s) f (s, T k ) 2 T k 1 (19) f (s, T) f (s, T k ) 2 } ds. 6
7 3.2 Parameer esimaion To ge a value of he inegral (17), we need o esimae he followings., T, K, N : No problem! S /P T k is (heoriically) equal o he foward price, which is usully quoed in he marke, P T α k is quoed in he marke as ineres rae, β k (k =, 1,..., N 1); see he followings. (As a maer of course, he acual marke quoaions are he discree daa, and so we need o inerpolae hem o ge coninuous daa.) Following he marke convenion we menioned above, we can observe he volailiy of he asse price S : v(, T) = T { σ(s) 2 + λ(s) f (s, T) 2} ds (2) from he price of he plain opion mauring a T. Noe ha a ime we can observe v(s, T) for s and T. We have α 2 k = v(t k 1, T k ) = v(, T k ) Tk 1 { σ(s) 2 + λ(s) f (s, T k )) 2} ds, and he hird erm can be seen as he implied volailiy of he plain opion whose selemen dae is T k 1 and he paymen dae is T k. Inuiively, a leas when T is enough large, he difference of he wo dae does no effec he opion price so much. So, we claim ha he following convenion is appropriae: Tk 1 { σ(s) 2 + λ(s) f (s, T k )) 2} ds v(, T k 1 ) (21) Consequenly, we can esimae he value of α k by α k v(, T k ) v(, T k 1 ). (22) 7
8 3.3 Convenions for he implied volailiy of ineres rae To esimae β k = 1 2 α2 k + Tk we use he implied volailiy δ(, T 1, T 2 ) := T k 1 { f (s, T) f (s, T k )} 2 ds, (23) T1 of he ineres rae of he period [T 1, T 2 ]: { f (s, T 2 ) f (s, T 1 )} 2 ds. (24) L(T 1, T 2 ) = P T 2 T 1. (25) The daa of (24) are implied by he prices of he plain opion on L i.e. he caple, which is no OTC. The problem is ha he usual marke convenion assumes ha L is lognormally disribued while in our model under (13), L+1 is lognormal random variable under P T. The difference is no ignorable, since he laer allows he negaive ineres raes, while he former does no. We overcome his hardship by inroducing anoher convenion. We assume ha he marke convenion is doing a momen maching. Namely (under he hypohesis of (13)) he marke assumes ha for each L(T 1, T 2 ) e R on F where R is a Gaussian random variable such ha and E[e R ] = E T 2 [L(T 1, T 2 ) F ] (26) E[e 2R ] = E T 2 [ L(T 1, T 2 ) 2 F ]. (27) In oher words, he marke convenion under (13) is jus E T [max(l(t 1, T 2 ) K, ) F ] E[max(e R K, )]. (28) The new convenion implies ha on F, e R = 1 + PT 1 ecx 1 2 c2 (29) for an X N(, 1) and a consan c c(, T 1, T 2 ). In paricular, c 2 is he (convenional) implied volailiy. 8 P T 2
9 On he oher hand, since we have we can insis ha E T [ L(T 1, T 2 ) 2 F ] = 1 2 PT 1 + exp PT 1 P T 2 2 { T1 P T 2 { T1 exp { f (s, T 2 ) f (s, T 1 )} 2 ds } { f (s, T 2 ) f (s, T 1 )} 2 ds 1 + T1 { f (s, T 2 ) f (s, T 1 )} 2 ds, }, provided ha T 2 T 1 is small enough. If his is he case, we have { T1 } { f (s, T 2 ) f (s, T 1 )} 2 ds 1 + PT 2 P T 1 2 c 2 (, T 1, T 2 ). (3) (31) (32) Hence our convenion here is ( ) 2 L (T 1, T 2 ) δ(, T 1, T 2 ) = c(, T 1, T 2 ) 2 (33) 1 + L (T 1, T 2 ) where L (T 1, T 2 ) = 1 + PT 1 4 Numerical Analysis P T 2. (34) The acual daa is no coninuous bu discree. Mos common daa is monhly, like Fig 1. One can ge daily daa, or even shorer, bu anyway he/she mus inerpolae he daa o some exen. In his secion we exhibi he resuls of several numerical calculaions using he monhly daa Fig 1 o illusrae how our model works. We se he period of aking average is one monh, aking he average of daily daa; T k T k 1 = 1 (day) and N = 2 (day) = 1 (monh). Looking a he inegral (17), we need a daily erm srucure of Forward prices, volailiies of he asse, spo ineres raes, and he volailiies of forward raes of a fixed lengh. The sample daa (1) is missing he las of he 9
10 Figure 1: Sample Daa 1
11 four, So we se hem zero for simpliciy. I has firs hree of he four, which we inerpolae o ge daily daa as follows: Suppose ha we are given Forward prices F T[i] and volailiies V T[i] for i N. Here T[i] N and T[i] T[i 1] = 2 (days = 1 monh). We se F and V for T[i 1] < < T[i], N as and F[i] F = F[i 1]( F[i 1] ) T[i] T[i] T[i 1], T[i] V = V[i 1] + (V[i] V[i 1]) T[i] T[i 1]. And we se S for T[i 1] < T[i] as T[i 1] S = F exp (a[h]n h + b[h]), where N h are muually independen Gaussian random variables, and T[i 1] a[] = V T[i 1] Y b[] = 1 2 V T[i 1] T[i 1] Y 1 a[h] = V T[i 1]+h Y b[h] = 1 2 V 1 T[i 1]+h Y. h= Here we have se Y = 24; he number of days in a year. We used a sandard and simples MoneCarlo mehod, using RAND as quasirandom number generaor. The firs example Fig 2 shows he prices of Asian Pu and Call when T = 4 (days). he inersecion poin indicaes he PuCall pariy for Asian opion. The second one Fig 3 shows he prices for a lile longer erm Asian opions. The las one Fig 4 shows a erm srucure of prices of Asian opions. I is parallel o ha of plain opions. This is because we have se he volailiy of ineres raes as zero. 11
12 Figure 2: Asian opion pu and call price wih T=4 5 Conclusions As have been shown, we have successfully consruced a proper model for pricing Asian opions and describing heir erm srucure. A he same ime we have inroduced a calibraion scheme which acually works quie well. References [1] D. Duffie: Dynamic Asse Pricing Theory, Princeon Universiy Press, [2] R. J.Ellio, and P.E. Kopp: Mahemaics of Financial Markes, Springer,
13 Figure 3: Asian opion pu and call price wih T=46 [3] Financial Accouning Sandards Board: he URL of is web pages: hp://www.fasb.org/ [4] M. Musiela, and M. Rukowski Maringale Mehods in Financial Modelling, Springer, [5] B. Oksendal: Sochasic Differenial Equaions; an inroducion wih applicaions, 5h ediion, Springer, [6] M. Yor: Exponenial funcionals of Brownian moion and relaed processes, Springer Finance. SpringerVerlag, Berlin,
14 Figure 4: Term srucure of Asian opions 14
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