Pricing index options in a multivariate Black & Scholes model

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1 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383

2 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, Introducton In ths paper, we consder the problem of prcng equty ndex optons (or basket optons) n a multvarate Black & Scholes settng. Although ths model su ers from some major drawbacks, t pays to consder the prcng of dervatves n the multvarate Black & Scholes model, because t s the most straghtforward multvarate extenson of the one dmensonal Black & Scholes model. Therefore, the multvarate Black & Scholes ndex opton prcng formula can be consdered as a benchmark prcng formula, smlar to the one-dmensonal Black & Scholes formula. A partcular applcaton where such a benchmark model plays a crucal role s the study of mpled correlaton; see e.g. Dhaene, Lnders and Schoutens (2013) and Tavn (2013). In ths paper, we derve approxmatons for the prce of an ndex opton usng the theory of comonotoncty. Comonotonc random varables are maxmal dependent: an ncrease n one component mples that all components must ncrease. In the multvarate Black & Scholes model, the ndex s a weghted sum of dependent lognormal random varables. The dstrbuton of ths ndex s not gven n closed form whch makes the prcng of ndex optons hghly unattractable. We transform the orgnal prcng problem to the prcng of an ndex opton wrtten on a mod ed ndex whch s a weghted sum of comonotonc lognormal random varables. By choosng ths mod ed ndex n an approprate way, one can derve upper and lower bounds for the ndex opton prce. It was proven n Chen et al. (2008) that the ndex opton prce of a comonotonc ndex can be decomposed n a lnear combnaton of vanlla opton prces for approprate choosen strke prces. The latter can be determned n closed form usng the Black & Scholes formula, whch results n an easy and fast algorthm to compute the upper and lower bounds. In a last step, we combne the upper and lower bound n an approxmate value. We prove that the approxmate ndex opton curve we obtan, can be nterpreted as an ndex opton curve under a synthetc stock market ndex, where the rst two moments of ths KU Leuven, Belgum, 1

3 mod ed ndex concde wth the rst two moments of the real ndex. Furthermore, we also derve the dstrbuton functon of ths synthetc ndex. The dea of valung an opton by replacng the real underlyng dstrbuton wth a more tractable dstrbuton was already proposed n Jarrow and Rudd (1982). The prcng of an ndex opton n a Black & Scholes context usng the theory of comonotoncty was also consdered n Deelstra et al. (2004). Here, the prce of an ndex opton s dvded n an exact part and a part for whch accurate upper and lower bounds can be determned by replacng the orgnal random varable by a convex ordered approxmaton. In Carmona and Durrleman (2006), the authors derve upper and lower bounds for ndex optons by expressng the ndex opton prce as an optmzaton problem. Both papers provde numercal examples to llustrate the accuracy of the approxmatons. In ths paper, we use the same numercal examples to show that the accuracy of our new approxmaton s comparable wth the exstng methods. There s a vaste lterature on approxmatng the prce of a basket or ndex opton. For example, Mlevsky and Posner (1998) propose to use the recprocal gamma dstrbuton to approxmate the prce of an ndex opton, whereas Hull and Whte (1993) and Rubnsten (1994) use a bnomal tree model. The prcng of basket optons usng Quas-Monte Carlo smulaton s dscussed n Joy et al. (1993). In order to prce ndex optons n a more realstc model, Xu and Zheng (2010) derve approxmatons for the ndex opton prce wthn a jump d uson model and McWllams (2011) derves approxmatons for the ndex opton prce n a stochastc delay model. The paper s organsed as follows. In Secton 2, we ntroduce the nancal market and the multvarate Black & Scholes model. Furthermore, we recaptulate the notons of convex order and comonotoncty. An analytcal formula for the approxmated prce of an ndex opton s derved n Secton 3. Numercal llstratons are gven n Secton 4. Fnally, Secton concludes 5 the paper. 2 Optons, convex order and comonotoncty 2.1 The nancal market We assume a nancal market where n d erent (dvdend or non-dvdend payng) stocks, labeled from 1 to n, are traded. Current tme s 0, whle the tme span under consderaton s T years. For each stock, ts random prce at tme t, 0 t T, s denoted by X (t). We denote the stochastc prce process of stock by fx (t) j 0 t T g. Hereafter, we wll always slently assume that each X (t) 0 and also that E [X 2 (t)] < 1: The market ndex s composed of a lnear combnaton of the n underlyng stocks. Denotng the prce of the ndex at tme t by S (t), 0 t T, we have that S (t) = w 1 X 1 (t) + w 2 X 2 (t) + : : : + w n X n (t) ; (1) where w ; = 1; 2; : : : ; n; are postve weghts that are xed up front. 2

4 A European call opton gves the buyer the rght to purchase a stock or an ndex at a prede ned prce at a prede ned tme. For example, a call opton wrtten on the ndex S wth maturty T and strke prce K has a pay-o at tme T of (S (T ) K) + : Its prce s denoted by C [K; T ] : A smlar de nton exsts for a put opton, whose prce s gven by P [K; T ] : 2.2 The multvarate Black & Scholes model Assume that the stock prces X (t) ; = 1; 2; : : : ; n; can be descrbed by the followng set of SDE s: dx (t) X (t) = dt + db (t) ; for = 1; 2; : : : ; n; (2) where B (t) = (B 1 (t) ; B 2 (t) ; : : : ; B n (t)) and fb (t) j t 0g s a standard n-dmensonal Brownan moton de ned on the ltered probablty space (; F; P). Ths probablty space s equped wth the ltraton (F t ) 0tT of F whch records the the past behavor of the multvarate Brownan moton. The ltered probablty space sats es the usual techncal condtons. The vector = ( 1 ; 2 ; : : : ; n ) contans the drft parameters of each stock. The Varance-Covarance matrx s de ned as ;2 1;n 2; ;n = C. A ; n;1 n;2 n 2 where ;j t = Cov [ B (t) ; j B j (t + s)] : (3) The correlaton ;j between the stocks and j s gven by ;j = Corr [ B (t) ; j B j (t + s)] (4) and we can wrte ;j = ;j j : We have that 2 R n and 2 R nn : The stock prce model descrbed above s called the multvarate Black & Scholes model. We assume that the matrx has full rank. It can be proven that n ths case the market s complete and free of arbtrage. Furthermore, there always exsts a unque equvalent martngale measure Q: If we replace each by r n (2) we obtan the set of SDE s descrbng the stock prce dynamcs under the rsk-neutral probablty measure Q. Here, r s the rsk-free rate, whch s assumed to be known at tme t = 0 and constant over tme. Under ths rsk-neutral prcng measure, the stock prces at tme T are followng a lognormal dstrbuton: ln X (T ) X (0) Q = N r T; 2 T ; for = 1; 2; : : : ; n; (5) where Q = denotes an equalty n dstrbuton under the Q-measure. For a detaled dscusson about condtons for completeness and no-arbtrage n the multvarate Black 3

5 & Scholes model, we refer to Dhaene, Kukush and Lnders (2013), Björk (1998) and the references theren. The current prce of any pay-o at tme T can be represented as the dscounted expectaton of ths pay-o. In ths prce-recpe, dscountng s performed usng r, whereas expectatons are taken wth respect to Q. The prce of a call opton wrtten on stock ; wth strke K and maturty T s denoted by C [K; T ] : The prce of a put opton wth the same specfatons s denoted by P [K; T ] : Call and put optons wrtten on stock can be expressed as dscounted expectatons: C [K; T ] = e rt E (X (T ) K) + ; (6) P [K; T ] = e rt E (K X (T )) + : (7) If the rsk-neutral dynamcs of the stock prce X (T ) can be descrbed by the lognormal dstrbuton (5), the opton prces C [K; T ] and P [K; T ] can be expressed as wth C [K; T ] = X (0) (d ;1 ) Ke rt (d ;2 ) ; (8) P [K; T ] = Ke rt ( d ;2 ) X (0) ( d ;1 ) ; (9) d ;1 = r T ln K X (0) p ; T d ;2 = d ;1 p T : Expressons (8) and (9) are the well-known Black & Scholes opton prcng formulae; see e.g. Black and Scholes (1973). In the remander of ths text, expectatons (dstrbutons) of functons of the random vector (X 1 (T ) ; : : : ; X n (T )) have to be understood as expectatons (dstrbutons) under the Q-measure. We wll often call them rsk-neutral expectatons (dstrbutons). Furthermore, the notatons F X (T ) and F S(T ) wll be used for the tme-0 cumulatve dstrbuton functons (cdf s) of X (T ) and S (T ) under Q. In order to avod unnecessary overloadng of the notatons, hereafter we wll omt the xed tme ndex T when no confuson s possble. For example, we wll wrte X ; C [K] and F X (x) for X (T ) ; C [K; T ] and F X (T ) (x), respectvely. 2.3 Convex order and comonotoncty In ths secton we summarze some de ntons and results concernng convex order, nverse dstrbutons and comonotoncty needed afterwards. A r.v. X s sad to precede a r.v. Y n convex order sense, notaton X cx Y, f ( E (X K)+ E (Y K)+ E (K X) + E (K Y )+ ; for all K 2 R: (10) 4

6 If X and Y are two r.v. s such that X cx Y; then E [X] = E [Y ] ; but Y has heaver (upper and lower) tals than X: The usual nverse F 1 X wth nf ; = +1, by conventon. of the cdf F X of a r.v. X s de ned by F 1 X (p) = nf fx 2 R j F X (x) pg ; p 2 [0; 1] ; (11) The weghted sum S s de ned by S = w X ; (12) where w > 0: Assume that the margnal stop-loss premums E (X K) + can be determned for any K. Even f we have full nformaton about the margnal dstrbutons, calculatng the stop-loss premum E (S K) + s not straghtforward as t requres nformaton about the dependence among the margnals. Specfyng ths dependence structure can be done by choosng an approprate copula, but the correspondng dstrbuton of S s n most stuatons unknown or wll be too cumbersome to work wth. The random vector (X 1 ; : : : ; X n ) s sad to be comonotonc f (X 1 ; : : : ; X n ) d = F 1 X 1 (U) ; : : : ; F 1 X n (U) ; (13) where U s a unform (0; 1) r.v. and = d denotes equalty n dstrbuton. If S s a sum of comonotonc random varables, the stop-loss premum E (S K) + can be decomposed n stop-loss premums of the margnals wth approprate chosen retentons. We state ths result n Theorem 1. For a proof of ths theorem, we refer to Kaas et al. (2000). Theorem 1 plays a crucal role n Secton 3, where we search for an accurate prcng formula for the ndex opton prces C [K; T ] and P [K; T ] n a multvarate Black & Scholes model. Theorem 1 (Decomposton formula) Consder a comonotonc random vector (X 1 ; X 2 ; : : : ; X n ) and denote the weghted sum by S. Assume that F S s contnuous and strctly ncreasng. For K 2 F 1+ S (0); F 1 S (1), the stop-loss premum E (S K) + can be decomposed nto a lnear combnaton of stop-loss premums of the margnals nvolved: where E (S K) + = w E (X K ) + ; (14) K = F 1 X (F S (K)) ; = 1; : : : ; n; (15) and F S (K) sats es the followng relaton: X w K = K: (16) 5

7 In case the cdf F S s not contnuous and strctly ncreasng, a smlar decomposton formula (14) can be proven for the stop-loss premum E (S K) + of a comonotonc sum S, but now the expresson for the strke prce K wll be slghtly d erent. A su cent condton for F S to be strctly ncreasng and contnuous s that the margnal cdfs F X are strctly ncreasng and contnuous. Ths condton s especally met when dealng wth lognormal r.v. s. Furthermore, for approprate choosen K ; the decompston formula (14) remans to hold when K =2 F 1+ S (0); F 1 S (1) ; see.e.g Dhaene et al. (2002a) and Chen et al. (2013). For an extensve overvew of the theory of comonotoncty, ncludng proofs of the results mentoned n ths subsecton, we refer to Dhaene et al. (2002a). Fnancal and actuaral applcatons of the concept of comonotoncty are descrbed n Dhaene et al. (2002b). An updated overvew of applcatons of comonotoncty can be found n Deelstra et al. (2011). 3 Convex approxmatons for ndex optons In ths secton we derve the approxmate ndex opton prces C [K] and P [K] for C [K] and P [K] ; respectvely. Furthermore, we show that the curves C and P can be consdered as ndex opton curves wrtten on a synthetc market ndex S; whch serves as an approxmate ndex for the real ndex S; see Theorem 11. The approxmate ndex opton prce C [K] s a lnear combnaton of the upper bound C c [K] and the lower bound C l [K] ; where the nterpolaton weght s chosen such that Var[S] = Var S : Smlarly, P [K] s a lnear combnaton of the upper bound P c [K] and the lower bound P l [K] ; where we use the same nterpolaton weghts. Note that usng only the upper or lower bound s not desrable as ths wll lead to a consstent over or under estmaton of the real ndex opton prce. 3.1 Upper bound In ths subsecton, we replace the real sum S by the random sum S c ; whch s de ned as S c = w 1 F 1 X 1 (U) + : : : + w n F 1 X n (U) : (17) The ndex S c s called the comonotonc stock market ndex and t s, lke the ndex S; a weghted average of the margnals X 1 ; X 2 ; : : : X n ; but the dependence structure s assumed to be comonotonc. In Kaas et al. (2000) t s proven that the comonotonc sum S c s a convex upper bound for the sum S : S cx S c : (18) Consder the pay-o s (S c K) + and (K S c ) + at tme T: These pay-o s can be nterpreted as pay-o s of an ndex call and put opton wrtten on a stock market ndex that can be descrbed by S c : Note, however, that these optons are not traded actvely 6

8 and ts prces cannot be observed n the market, because the stock market ndex S s n general not equal to the comonotonc stock market ndex S c. If we denote the theoretcal prces of these synthetc ndex optons by C c [K] and P c [K] ; we can determne them as: C c [K] = e rt E (S c K) + ; (19) P c [K] = e rt E (K S c ) + : For the comonotonc ndex opton prces, we can prove the put-call party C c [K] = P c [K] e rt K + e rt E [S] : (20) From expresson (17) we nd that the components of S c are all non-decreasng functons of the same r.v. U: Therefore, we can nterprete the ndex S c as a worst case scenaro. All stocks composng the ndex wll go smultaneously up or smultaneously down. As a result, the prce of an ndex opton wrtten on S c s an upper bound for the real ndex opton prce. Theorem 2 The ndex call and put opton prces C [K] and P [K] are constraned from above by C c [K] and P c [K] ; respectvely: C [K] C c [K] ; for all K 0; P [K] P c [K] ; for all K 0: Proof. Ths s a drect consequence of the convex order relaton (18) and the characterzaton of convex order; see (10). Theorem 2 holds regardless the assumpton about the margnal dstrbutons F X : In Chen et al. (2008) and Hobson et al. (2005), model-free upper bounds for ndex optons are derved usng Theorem 2 together wth Theorem 1. Furthermore, t s shown that there exsts an optmal statc super-replcatng strategy for an ndex opton, whch conssts n buyng a lnear combnaton of vanlla optons. Addtonal detals and computatonal ssues are gven n Chen et al. (2013) and Lnders et al. (2012). In ths secton, we spec ed the margnal dstrbutons to be lognormally dstrbuted. In ths specal case, we can determne S c explctly n terms of the margnal volatltes, the rsk-free rate r and the cdf of a standard normal dstrbuton. Theorem 3 (A closed form expresson for S c :) Consder a market where the assets follow the multvarate Black & Scholes model (2). The comonotonc market ndex S c s gven by the followng expresson: S c Q = w X (0) exp r 2 p T + T 1 (U) ; (21) 2 where denotes the cdf of a standard normal random varable and U denotes a r.v. whch s unformly dstrbuted on the unt nterval. Its varance s gven by Var [S c ] = w w j X (0) X j (0) e 2rT e j T 1 : (22) j=1 7

9 Proof. The margnal rsk-neutral dstrbutons are gven by (5). If we combne ths expresson wth Theorem 1 n Dhaene et al. (2002a), the nverse cdf F 1 X s gven by F 1 2 p X (p) = X (0) exp r T + T 1 (p) : (23) 2 Takng nto account the de nton of S c n (17) proves (21). We wrte the varance Var[S c ] as We have that h Cov F 1 Var [S c ] = X (U) ; F 1 X j j=1 (U) h w w j Cov F 1 h Note that f 2 R; then E e 1 (U) = e 2 2 and X (U) ; F 1 X j (U) : = X (0) X j (0) e 2rT 2( j)t h Cov e p T 1 (U) ; e j h Cov e p T 1 (U) ; e p j T (U) 1 = e 1 2( 2+2 j)t e j T for each par ; j = 1; 2; : : : n; whch proves (22). p T 1 (U) : 1 ; In the followng theorem, we prove that the upper bound C c [K] for the ndex call opton and the upper bound P c [K] for the ndex put opton can be expressed n terms of vanlla call and put opton prces on the components of S: Theorem 4 The prces C c [K] and P c [K] of the ndex optons wth pay-o at tme T gven by (S c K) + and (K S c ) +, respectvely, can be expressed as follows: C c [K] = P c [K] = w C [K ] ; (24) w P [K ] ; (25) where K = X (0) exp r and F S c (K) s determned usng the relaton 2 p T + T 1 (F S c (K)) ; (26) 2 w K = K: (27) 8

10 Proof. The sum S c s a sum of comonotonc r.v. s. Furthermore, the margnal cdf s F X are strctly ncreasng and contnuous for all = 1; 2; : : : ; n: It can be proven that F S c s also contnuous and strctly ncreasng; see e.g. Dhaene et al. (2002a). From (21), t follows that F 1+ S (0) ; F 1 c S (1) = ( 1; +1) : Combnng (19) and (6) wth Theorem 1 c results n (24). The put-call party (20) proves expresson (25). The choce (26) for K follows from (23) and (15). The rght hand sde of (24) s a lnear combnaton of vanlla call optons and the rght hand sde of (25) s a lnear combnaton of vanlla put optons. Usng the Black & Scholes opton prcng formula we can nd an analytcal expresson for the prces C c [K] and P c [K] : Theorem 5 The prces C c [K] and P c [K] of the ndex optons wth pay-o at tme T gven by (S c K) + and (K S c ) +, respectvely, can be expressed as follows: C c [K] = P c [K] = w X (0) (d ;1 ) K e rt (d ;2 ) ; (28) w K e rt ( d ;2 ) X (0) ( d ;1 ) (29) where K s de ned n (26) of Theorem 4 and d ;1 = ln X (0) K d ;2 = d ;1 p T : + r T p ; T Proof. Usng expressons (8) and (9) n Theorem 4 proves the result. 3.2 Lower bound In ths subsecton we replace the market ndex S by the condtonal sum S l ; whch s de ned as follows: S l = w 1 E [X 1 j ] + : : : + w n E [X n j ] ; where = ln X X (0) ; (30) for > 0: In Kaas et al. (2000) t s proven that the sum S l s a convex lower bound for the sum S : S l cx S: (31) 9

11 Consder the pay-o s (S l K) + and (K S l ) + whch have to be pad at tme T: These pay-o s can be nterpreted as the pay-o s of an ndex call and put opton wrtten on a stock market ndex whch can be descrbed by S l : The stock market ndex S wll d er from S l ; whch makes t mpossble to nvest n the ndex S l. We denote by C l [K] and P l [K] the prces of the synthethc ndex call and put opton wrtten on S l : The theoretcal prce of these optons are gven by h C l [K] = e rt E S l K ; (32) + h P l [K] = e rt E K S l : + We can prove the put-call party for these opton prces: C l [K] = P l [K] e rt K + e rt E [S] : (33) Theorem 6 The ndex call and put opton prces C [K] and P [K] are constraned from below by C l [K] and P l [K] ; respectvely: C l [K] C [K] ; for all K 0; P l [K] P [K] ; for all K 0: Proof. Ths s a drect consequence of the convex order relaton (31). If the margnals are lognormal dstrbuted, an analytcal expresson for S l n terms of the r.v. ; the rsk-free rate r, the maturty T and the margnal volatltes can be derved. Theorem 7 Consder a market where the assets follow the multvarate Black & Scholes model (2). The convex lower bound S l can be expressed as follows: S l Q = w X (0) exp r h In ths formula, r = Corr ln X ; X and (0) The varance s gven by Q = r 2 p 2 r2 T + r T 1 (U) : (34) 2 p T + T B (1) : (35) 2 Var S l = w X (0) w j X j (0) e 2rT j=1 e r r j j T 1 : (36) 10

12 Proof. Note that B (T ) = d p T N (0; 1) ; for = 1; 2; : : : ; n: From (5), we nd that the rsk-neutral dynamcs of the logreturns are gven by Q ln = r T + B (T ) : X X (0) Combnng ths expresson wth (30), we nd that (35) holds. Furthermore, has a normal dstrbuton wth mean T and varance 2 T: Remember that for a bvarate normal dstrbuton (X; Y ) wth = Corr[X; Y ] ; we have that X j Y has agan a normal dstrbuton wth mean: s Var [X] E [X j Y ] = E [X] + (Y E [Y ]) ; (37) Var [Y ] and varance Var[X] (1 dstrbuton wth mean E ln and varance 2 ) : Usng expresson (37), we nd that ln X X (0) X X (0) j E[] Fnally, the equalty p Var[] = r! 2 p E [] T + r T p ; 2 Var [] Var ln X X (0) j = 2 T 1 r 2 : d = 1 (U) proves (34). The proof of (36) follows the same lnes as the proof of (22). j has a normal Remark 8 (Calculaton of r ) The varance of and X j are both nvolved n the calculaton of r : The varance of s denoted by 2 T. Usng (4) and (35), we nd that 2 = j ;j j : Pluggng the varance 2 n the formula for the correlaton r results n r = Corr ln X X (0) ; h Cov ln X ; X (0) = r h Var ln X X Var [] (0) j< Gven that ;j 0; all correlatons r are postve. (38) (39) = T P n j=1 jcov [ B (1) ; j B j (1)] T P n j=1 = j ;j j : (40) 11

13 The sum S l s a comonotonc sum f all r are non-negatve. In Deelstra et al. (2004) t s proven that there always exsts ; = 1; 2; : : : ; n n (30) such that S l s a comonotonc sum. However, for sake of smplcty, we make the followng assumpton: O Assumpton : ;j > 0 for all ; j = 1; 2; : : : ; n: (41) Under ths assumpton, the market ndex S l can be expressed as a sum of n comonotonc lognormal random varables V : S l Q = w V ; n Q where V = X (0) exp r p 2 2 r 2 T + r T 1 (U) o can be consdered as an adjusted stock prce process for stock : Each r.v. V s an ncreasng functon of the same r.v. U: Under the adjusted prce process fv (t) j t 0g, the prce of stock at tme T s agan lognormal dstrbuted ln V V (0) Q = N r 2 2 r2 T; r 2 2 T : (42) The followng theorem states that the prces C l [K] and P l [K] can be expressed as a weghted sum of n vanlla opton prces, wrtten on the adjusted stock prces V : Theorem 9 Consder a market where the assets follow the multvarate Black & Scholes model (2), where ;j 0; for all ; j: The prces C l [K] and P l [K] of the ndex optons wth pay-o at tme T gven by S l K and K Sl, respectvely, can be expressed + + as follows: C l [K] = w e rt E (V K ) + ; (43) where K P l [K] = = X (0) exp r and F S l (K) s determned usng w e rt E (K V ) + ; (44) 2 p 2 r2 T + r T 1 (F S l (K)) ; w K = K: Proof. The assumpton (41) assures that S l s a sum of the comonotonc r.v. s V 1 ; V 2 ; : : : ; V n : Each V has a lognormal dstrbuton and from Theorem 1 n Dhaene et al. (2002a), we nd that F 1 V (p) = X (0) exp r 12 r 2 2 p T + r T 1 (p) : (45)

14 The sum S l s a sum of comonotonc r.v. s. Furthermore, the margnals F V are strctly ncreasng and contnuous for all = 1; 2; : : : ; n: So the cdf F S l s also contnuous and strctly ncreasng. From (34), f follows that F 1+ (0) ; F 1 (1) = ( 1; +1) : Fnally, S l S l combnng (32) and (6) wth Theorem 1 proves (43). Applyng the put-call party (33) proves (44). Theorem 10 Consder a market where the assets follow the multvarate Black & Scholes model (2), where ;j 0; for all ; j: The prces C l [K] and P l [K] of the ndex optons wth pay-o at tme T gven by S l K and K Sl, respectvely, can be expressed + + as follows: where the K C l [K] = P l [K] = w X (0) (d ;1 ) K e rt (d ;2 ) ; (46) w K e rt ( d ;2 ) X (0) ( d ;1 ) ; (47) s de ned as n Theorem 9 and d ;1 = ln X (0) K d ;2 = d ;1 r p T : + r r 2 T p ; r T Proof. The margnals V ; = 1; 2; : : : ; n have a lognormal dstrbuton; see (42). Combnng ths observaton wth (8) and (9) results n a closed form expresson for e rt E (V K ) + and e rt E (K V ) + for = 1; 2; : : : ; n: Usng these expressons n (43) and (44) proves the result On the choce of In Cheung et al. (2013) t s proven that for a su cently nce convex functon 1 u, we have that E u S l E [u (S)] : Moreover, f u s strctly convex E u S l = E [u (S)] s equvalent wth S l = d S: Therefore, t s reasonable to take such that E u S l s as close as possble to E [u (S)] ; for a partcular strctly convex functon u: Here, we choose u to be equal to u (x) = (x E [S]) 2 : Then E [u (S)] = Var[S] : Followng the deas of Vandu el et al. (2005), we approxmate the varance of S l as follows: E u S l = Var S l w X (0) w j X j (0) e 2rT r r j j T: (48) j=1 1 A functon u s su cently nce f has an absolutely contnuous dervatve u 0. 13

15 h Remember that r = Corr ln X ; X and Var[] = 2 (0) T: Then we can wrte h p Cov ln X ; X (0) r T = p : T Usng ths relaton, the rght hand sde of (48) becomes j=1 Cov 2rT w X (0) w j X j (0) e h ln X X (0) ; Cov h ln X j X j (0) ; 2 T : whch can be wrtten as h Pn 2 Cov w X (0) e rt ln X ; X (0) 2 T : Fnally, we can approxmate the varance as follows: E u S l = Var " #! S l Corr w X (0) e rt ln X 2 X (0) ; (49) " # Var w X (0) e rt ln X : X (0) If E u S l reaches ts maxmal value, E u S l s as close as possble to E [u (S)] : The rght hand sde of (49) s maxmal f we take such that " # Corr w X (0) e rt ln X X (0) ; = 1: So we nd that a globally optmal choce for s = w X (0) e rt ln X X (0) ; hence = w X (0) e rt ; for = 1; 2; : : : ; n: (50) 3.3 Moments based approxmaton The upper and lower bounds derved n Theorems 5 and 10 can be combned n one approxmate value for the prces C [K] and P [K] ; whch we wll denote by C [K] and 14

16 P [K] ; respectvely: Ths approxmaton wll be a lnear combnaton of the convex upper and lower bound, usng a factor z 2 [0; 1] : C [K] = zc l [K] + (1 z) C c [K] ; for all K 0; (51) P [K] = zp l [K] + (1 z) P c [K] ; for all K 0: (52) The non-ncreasng convex curve C and the non-decreasng convex curve P can be nterpreted as opton curves of a synthetc stock market ndex S: The ndex S s not traded n the market, but the theoretcal prce of an ndex opton on S s gven by h C [K] = e rt E S K + h P [K] = e rt E K S + The cdf F S can be expressed n terms of the cdf s F S l and F S c: ; for all K 0; (53) ; for all K 0: (54) Theorem 11 Consder a market where the assets follow the multvarate Black & Scholes model (2), where ;j 0; for all ; j: Assume that the prces for call and put optons wrtten on the ndex S are gven by (51) and (52), respectvely. Then we have that the cdf F S of S s gven by F S (x) = zf S l (x) + (1 z) F S c (x) ; for all x 2 R: Proof. The curve C s a call opton curve wrtten on S; so (53) must hold. Then the cdf F S s fully determned by the opton curve C va the relaton F S (x) = 1 + e rt C 0 [x+] ; where C 0 [x+] denotes the rght dervatve of C n x; see e.g. Breeden and Ltzenberger (1978). Applyng ths relaton n (51) proves (53). Relaton (54) follows from the put-call party. D erent values for z wll lead to d erent opton curves C and P and as a result also to d erent dstrbutons for S. The value for z n (51) s determned such that E u S = E [u (S)] : The latter equalty cannot be used to conclude that S d = S because the r.v. s S and S are not convex ordered. Theorem 12 Consder a market where the assets follow the multvarate Black & Scholes model (2), where ;j 0; for all ; j: If n (51) and (52), z s gven by then E u S = E [u (S)] : z = E [u (Sc )] E [u (S)] E [u (S c )] E [u (S l )] ; (55) 15

17 Proof. In Cheung et al. (2013), t s shown that E u S E [u (S)] can be expressed as E u S E [u (S)] = Z E[S] 0 u 00 (K) P [K] Insertng (51) n ths expresson results n P [K] dk+ Z +1 E[S] u 00 (K) C [K] C [K] dk: E u S E [u (S)] = + Z E[S] 0 Z +1 E[S] u 00 (K) zp l [K] + (1 z) P c [K] P [K] dk u 00 (K) zc l [K] + (1 z) C c [K] C [K] dk = E [u (S c )] E [u (S)] z E [u (S c )] E u S l ; from whch we nd that E u S E [u (S)] = 0 f z s gven by (55). Throughout ths paper, we wll use the choce u (x) = (x E [S]) 2. Theorem 12 states that f we take z = Var [Sc ] Var [S] Var [S c ] Var [S l ] ; (56) then the ndex opton surface s approxmated such that Var S = Var[S]. In the sequel of the paper we wll use C [K] and P [K] as approxmatons for the prces of ndex call and put optons: C [K] C [K] ; for all K 0; P [K] P [K] ; for all K 0: Convex approxmatons for sums of dependent lognormal r.v. s proved to be successful n earler lterature; see e.g. Vandu el et al. (2005), Dhaene et al. (2005) and Van Weert (2011). The dea of combnng an upper and lower bound n an approxmate opton value was proposed n Vyncke et al. (2004) for the prcng of an Asan opton. 4 Numercal llustraton The e cency of the comonotonc approxmatons C [K] and P [K] for the opton prces C [K] and P [K] s dscussed n ths secton wth the help of numercal llustratons. We rst consder the bvarate case, so n = 2. The correlaton between the stocks s denoted by and S (T ) = w 1 X 1 (T ) + w 2 X 2 (T ) : The nterest rate r s set to 5%. An example wth equal weghts and another example wth unequal weghts wll be nvestgated. In each stuaton, we compare the approxmate opton prces wth the correspondng Monte Carlo estmates, where 10 6 smulated values are used. We determne opton prces for the maturtes T = 1 and T = 3: Note that strke prces are expressed n terms of forward moneyness. A basket strke prce K has forward moneyness equal to K : We assume that E[S] the current prces of the non-dvdend payng stocks are gven by X 1 (0) = X 2 (0) = 100 and the weghts are equal, w 1 = w 2 = 0:5: The results are lsted n Table 1. 16

18 Table 2 gves the numercal values for the stuaton where the weghts and the ntal stock prces are d erent. Here, we have that X 1 (0) = 130 and X 2 (0) = 70. Furthermore, w 1 = 0:3 and w 2 = 0:7: Both tables show that the approxmate values are very close to the smulated values. The two stuatons consdered n Table 1 and 2 are also handled n Deelstra et al. (2004). In ths paper, the authors dscuss varous approxmatons for the prce of an arthmetc basket opton, whch are also based on comonotonc approxmatons. We also consder the prcng of an ndex opton, where the ndex S s composed of n = 50 stocks. For smplcty, we take r = 0%; T = 1 and X (0) = 100; w = 1=50; for = 1; 2; : : : ; 50: Ths partcular stuaton was also consdered n Carmona and Durrleman (2006). The performance of the approxmatons C [K] s compared wth the Monte- Carlo smulaton C mc [K] and lsted n Table 3. Tables 1-3 show that the approxmaton C [K] s close to the smulated opton prce C mc [K] : Indeed, the error " K s de ned as C[K] " [K] = 1 and s always less than 1%: C mc [K] 17

19 Table 1: Approxmatons and smulatons: r = 5%; w 1 = w 2 = 0:5; X 1 (0) = X 2 (0) = 100 K T C mc [K] C[K] [K] 10%OTM % % % % % % % % ATM % % % % % % % % 10% ITM % % % % % % % % 18

20 Table 2: Approxmatons and smulatons: r = 5%; w 1 = 0:3; w 2 = 0:7; X 1 (0) = 130; X 2 (0) = 70 K T C mc [K] C[K] [K] 10%OTM % % % % % % % % ATM % % % % % % % % 10%ITM % % % % % % % % 19

21 Table 3: Approxmatons and smulatons: r = 0%; n = 50; T = 1 year; w = 1=50; X(0) = 100 = 10% = 20% = 30% K C mc [K] C[K] [K] C mc [K] C[K] [K] C mc [K] C[K] [K] % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 20

22 5 Concluson Ths paper handles the problem of prcng ndex optons. The ndex opton prce s n uenced by the dstrbuton of the ndvdual components and the dependence structure, whch makes t a hard task to derve closed form solutons. We assume that the rsk-neutral dynamcs of the stocks can be descrbed by a multvarate Black & Scholes model. In ths smple stock prce model, the vanlla optons can be prced usng the celebrated Black & Scholes opton prcng formula, but the prce C [K] of an ndex opton s not gven n an analytcal formula. We derve a closed form approxmaton, whch s based on convex approxmatons for a sum S of dependent lognormal random varables. Comparng our new approxmate opton prcng formula wth Monte Carlo smulatons shows that the approxmate values are close to the smulated values. Furthermore, we also show that the approxmate ndex opton curve C can be nterpreted as the ndex opton curve under the approxmate ndex S; where the cdf of S has a more attractve form than the cdf of the orgnal ndex S: Acknowledgement: Danël Lnders acknowledge the nancal support of the Onderzoeksfonds KU Leuven (GOA/12/002/TBA: Management of Fnancal and Actuaral Rsks: Modelng, Regulaton, Incentves and Market E ects) and the support of the AXA Research Fund (Measurng and managng herd behavor rsk n stock markets). Furthermore, the author would lke to thank Jan Dhaene and Ben Stassen (KU Leuven, Belgum) for useful dscussons. References Björk, T. (1998), Arbtrage theory n contnuous tme, Oxford Unversty Press. pp Black, F. and Scholes, M. (1973), The prcng of optons and corporate labltes, The Journal of Poltcal Economy 81(3), pp Breeden, D. T. and Ltzenberger, R. H. (1978), Prces of state-contngent clams mplct n opton prces, Journal of Busness 51(4), Carmona, R. and Durrleman, V. (2006), Generalzng the black-scholes formula to multvarate contngent clams, Journal of Computatonal Fnance 9, Chen, X., Deelstra, G., Dhaene, J. and Lnders, Danel annd Vanmaele, M. (2013), On an optmzaton problem related to statc super-replcatng strateges, Research report AFI, FEB, KU Leuven. Chen, X., Deelstra, G., Dhaene, J. and Vanmaele, M. (2008), Statc super-replcatng strateges for a class of exotc optons, Insurance: Mathematcs & Economcs 42(3), Cheung, K. C., Dhaene, J., Kukush, A. and Lnders, D. (2013), Ordered random vectors and equalty n dstrbuton, Scandnavan Actuaral Journal. 21

23 Deelstra, G., Dhaene, J. and Vanmaele, M. (2011), An overvew of comonotoncty and ts applcatons n nance and nsurance, n B. Oksendal and G. Nunno, eds, Advanced Mathematcal Methods for Fnance, Sprnger Berln Hedelberg, pp Deelstra, G., Lnev, J. and Vanmaele, M. (2004), Prcng of arthmetc basket optons by condtonng, Insurance: Mathematcs and Economcs 34(1), Dhaene, J., Denut, M., Goovaerts, M., Kaas, R. and Vyncke, D. (2002a), The concept of comonotoncty n actuaral scence and nance: theory, Insurance: Mathematcs & Economcs 31(1), Dhaene, J., Denut, M., Goovaerts, M., Kaas, R. and Vyncke, D. (2002b), The concept of comonotoncty n actuaral scence and nance: applcatons, Insurance: Mathematcs & Economcs 31(2), Dhaene, J., Kukush, A. and Lnders, D. (2013), The multvarate black & scholes market: condtons for completeness and no-arbtrage, Theory of Probablty and Mathematcal Statstcs (88), Dhaene, J., Lnders, D. and Schoutens, W. (2013), A new mpled correlaton ndex, Workng papers, KU Leuven, FBE. Dhaene, J., Vandu el, S., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2005), Comonotonc approxmatons for optmal portfolo selecton problems, Journal of Rsk and Insurance 72, Hobson, D., Laurence, P. and Wang, T. (2005), Statc-arbtrage upper bounds for the prces of basket optons, Quanttatve Fnance 5(4), Hull, J. and Whte, S. (1993), E cent procedures for valung european and amercan path-dependent optons, The Journal of Dervatves 1(1), Jarrow, R. and Rudd, A. (1982), Approxmate opton valuaton for arbtrary stochastc processes, Journal of Fnancal Economcs 10(3), Joy, C., Boyle, P. P. and Tan, K. S. (1993), Quas-Monte Carlo methods n numercal nance, Management Scence 4, Kaas, R., Dhaene, J. and Goovaerts, M. J. (2000), Upper and lower bounds for sums of random varables, Insurance: Mathematcs and Economcs 27(2), Lnders, D., Dhaene, J., Hounnon, H. and Vanmaele, M. (2012), Index optons: a modelfree approach, Research report a feb, Leuven: KU Leuven - Faculty of Busness and Economcs. McWllams, N. (2011), Opton prcng technques understochastc delay models, PhD thess, Unversty of Ednburgh. Mlevsky, M. and Posner, S. (1998), A closed-form approxmaton for valung basket optons, The Journal of Dervatves 5(4),

24 Rubnsten, M. (1994), Impled bnomal trees, The Journal of Fnance 49(3), Tavn, B. (2013), Hedgng dependence rsk wth spread optons va the power frank and power student t copulas, Techncal report, Unversté Pars I Panthéon-Sorbonne. Avalable at SSRN: Van Weert, K. (2011), Optmal portfolo selecton: the comonotonc approach, PhD thess, Faculty of Busness and Economcs (FBE), Leuven. Vandu el, S., Hoedemakers, T. and Dhaene, J. (2005), Comparng approxmatons for rsk measures of sums of non-ndependent lognormal random varables, North Amercan Actuaral Journal 9(4), Vyncke, D., Goovaerts, M. and Dhaene, J. (2004), An accurate analytcal approxmaton for the prce of a European-style arthmetc Asan opton, Fnance 25, Xu, G. and Zheng, H. (2010), Basket optons valuaton for a local volatlty jump d uson model wth the asymptotc expanson method, Insurance: Mathematcs and Economcs 47(3),

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