VALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION

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1 Aca Universiais Mahiae Belii ser. Mahemaics, 16 21, Received: 15 June 29, Acceped: 2 February 21. VALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION TOMÁŠ BOKES Absrac. We exend he model for valuaion of American-syle of Asian opions inroduced by Hansen, Jørgensen 2 in 3] by including a nonrivial dividend rae q. We use he heory of condiioned expecaions o calculae he formula of he American-syle Asian floaing srike opion wih a general average of he underlying asse. We deermine an inegral equaion formula for he value of his ype of an opion wih coninuous geomeric average and approximae formula for he coninuous arihmeic average. 1. Inroducion Evoluion of rading sysems influences he developmen of he marke of financial derivaives. Firs, he simple derivaives as forwards and vanilla opions were used o hedge he risk of a porfolio. Progress in valuaion of hese simple financial insrumens pushed raders ino invening less predicable and more complex derivaives. Using financial derivaives wih more complicaed pay-offs brings ino aenion also new mahemaical problems. Asian opions belong o a group of pah-dependen opions, i.e. par of exoic opions. Here he pay-off depends on he spo value of he underlying during he whole or some pars of he life span of he opion. Asian opions depend on he arihmeic or geomeric average of he spo price of he underlying. Asian opions can be used as a ool for hedging he high volailiy of he price of asses or goods. The price of an underlying varies during he life span of he opion, he holder of he Asian opion can be secured for he case when he price jumps o he unpleasan region oo high for call holder or oo low for pu holder his loss will be reduced. Asian opions can be divided ino wo subgroups when considering he ype of heir pay-off funcion. The average srike Asian opion and he fixed srike Asian 2 Mahemaics Subjec Classificaion. 35K15, 35K55, 9A9, 91B28. Key words and phrases. opion pricing, maringales, exoic opions, Asian opions, American opion.

2 opion wih he pay-off funcion for he call opion 1 V T S, A = S A + and 2 V T S, A = A X +, respecively. 2. A probabilisic model for pricing of American-Syle of Asian opions In his secion we provide a formula for he valuaion of he early exercise boundary of an American-syle Asian opion paying nonrivial dividends. We follow he derivaion inroduced by Hansen, Jørgensen in 3]. Their formula for a floaing srike opion was derived using he heory of maringales and condiioned expeced values. We exend he formula o Asian opions on underlying paying non-zero dividend rae. This model is based on he sochasic behavior of he underlying in ime. I is assumed ha i is driven by sochasic process saisfying he following sochasic differenial equaion 3 ds = r qs d + σs dw Q on he ime inerval, T ], saring almos surely from he iniial price S >, where he consan parameer r > denoes he risk-free ineres rae, q is a dividend rae, σ is he volailiy of sock reurns and W Q is a sandard Brownian moion wih respec o he sandard risk-neural probabiliy measure Q. A soluion of equaion 3 corresponds o he geomeric Brownian moion 4 S = S e r q 1 2 σ2 +σw Q, for T. The bond risk-free marke is driven by he differenial equaion 5 db = rb d, wih B = 1, i.e. B = e r. As we have already menioned above we shall derive he value of an Americansyle Asian opion wih floaing srike. If we define he opimal sopping ime as T, he pay-off of he opion is se by +, 6 V T = ρs T A T where V is he value of he opion a ime, A is a coninuous average of he sock value during he inerval, ] and ρ = 1 for a call opion and ρ = 1 for a 18

3 pu opion. We may consider eiher he coninuous arihmeic average 7 A = 1 or he coninuous geomeric average 8 ln A = 1 or he weighed arihmeic average 9 A = 1 S u du, ln S u du a us u du, where he kernel funcion a. wih he propery aζ dζ < is usually defined as as = e λs for λ >. 3. Valuaion We recall ha derivaion of he more simple ype opion was inroduced in 3]. According o Hansen and Jørgensen, American-syle coningen claims can be priced by he condiioned expecaions approach. The opion prices are evaluaed by considering all possible sopping imes in he inerval, T ] 1 V, S, A = ess sup s T,T ] E Q e rs ρs s A s + S = S, A = A where T,T ] denoes he se of all sopping imes in he inerval, T ] and E Q X] = E Q X F ] is he condiioned expecaion wih informaion of ime he informaion is represened by he filraion F of he σ-algebra F, where he Brownian moion is suppored. To simplify he formula we change he probabiliy measure by he maringale 11 η = e r q S = e 1 2 σ 2 +σw Q S he new probabiliy measure Q is defined by 12 dq = η T dq. According o Girsanov s heorem, he process 13 W Q = W Q σ is a sandard Brownian moion wih respec o he measure Q. The value of he sock under his measure is defined by 14 S = S e r q+ 1 2 σ2 +σw Q. 19 ],

4 All asses priced under his measure are Q-maringales when discouned by he sock price. According o his fac, we can reduce he dimension of sochasic variables. We inroduce a variable ξ = A S and so we can derive V, S, A = ess sup E Q e rs + ] S ρs s A s = S, A = A s T,T ] = ess sup E Q s T,T ] = ess sup E Q s T,T ] = ess sup E Q s T,T ] = ess sup E Q s T,T ] = ess sup e qs S E Q s T,T ] η e rs + S ρs s A s = S, A = A] η T e r s + e S r q ρs s A s E Qe r qt s S T + e e q rs r qs S ρs s A s e qs S ρ 1 A s S s ] ] S = S, A = A ] S = S, A = A + ] S = S, A = A S s + ] S ρ1 ξ s = S, A = A. The las expression can be rewrien in erms of he new variable ξ = A S as follows: q V, S, A + ] 15 Ṽ, ξ = e = e qt E Q ρ1 ξ T S, where T = inf{s, T ] ξ s = ξs } and he funcion ξ describes he early exercise boundary. The sopping region S and coninuaion region C for he call and pu opions are defined by S call = C pu = { T, ξ < ξ }, C call = S pu = { T, ξ < ξ < }. Now we solve he problem wih one sochasic variable formulaed in 15. In wha follows, we generalize he resul by Hansen, Jørgensen 2 from 3] for he case of a nonrivial dividend rae q. Theorem 3.1. The value of he floaing srike Asian opion on sock underlying wih dividend rae q is given by 18 Ṽ, ξ = ṽ, ξ + ẽ, ξ, where 19 ṽ, ξ E Q + ] e qt ρ1 ξ T 2

5 and 2 ẽ, ξ E Q T ρe qu dau ξ u 1 S u, ξ u A u ] r qξu 1 du, wih average given by he funcion A and sopping region S. Here he funcion 1 S is he indicaor funcion of he se S, ρ ses he call opion by he value 1 and he pu opion by he value 1. In he proof of Theorem 3.1 we will use he following lemma. Lemma 3.2. The auxiliary variable ξ = A S differenial equaion: saisfies he following sochasic da 21 dξ = ξ r qξ d σξ dw Q. A A Proof of Lemma 3.2. We express he differenial dξ = d S as dξ = 1 S da A S 2 ds + A S 3 ds 2 = ξ da A r qξ d σξ dw Q, and he proof of lemma follows. Noice ha, when comparing o he original expression wih a zero dividend rae, q =, he only difference is ha he parameer r is replaced by r q. The value of da A depends on he mehod of averaging of he underlying used in he valuaion. The expression for he arihmeic averaging has form da a 22 A a = 1 1 ξ a 1 d. As far as, he geomeric average is concerned, we have da g 23 A g = 1 ln ξg d and for he weighed arihmeic averaging da wa 24 A wa = 1 a + a u Su S du ξ wa 1 d, where a is he derivaive of he funcion a. The las equaion is unusable in is general form. Neverhless, if we se as = e λs, i becomes da wa 25 A wa = 1 1 ξ wa 1 + λ d. 21

6 Proof of Theorem 3.1. We follow he proof of he original heorem including necessary modificaions relaed o he form of averaging and he fac ha q. Firs, we suppose ha, ξ belongs o he coninuaion region C. The opion is held and so we use Iô s lemma o calculae he differenial dṽ = Ṽ ξ dξ Ṽ 2 ξ 2 dξ2 + Ṽ d = ξ Ṽ da ξ A + r qξ Ṽ ξ σ2 ξ 2 2 Ṽ ξ 2 = σξ Ṽ ξ dw Q, + Ṽ ] d σξ Ṽ ξ dw Q where he las equaliy holds rue, because Ṽ is Q-maringale. Now we suppose ha, ξ belongs o he sopping region S. The value of he opion is defined by Ṽ, ξ = ρe q 1 ξ. So he differenial dṽ has form dṽ = ρqe q 1 ξd ρe q dξ = ρe q ξ da A + ρe q rξ qd + ρe q σξdw Q. For boh regions we have an equaion 26 dṽ, ξ = ρe q da 1 S, ξ ξ A rξ qd + dm Q, where M Q is a Q-maringale. Inegraing 26 from o T and aking expecaion we have E Q Ṽ T, ξt ] T Ṽ, ξ = E Q ρe qu dau ξ u 1 S u, ξ u r q ] du A u ξ u T ] + E Q dmu Q, }{{} = Ṽ, ξ = + ] E Q e qt ρ1 ξ T }{{} =ṽ,ξ T + E Q ρe qu dau ξ u 1 S ξ u r q ] du. A u ξ }{{ u } =ẽ,ξ his complees he proof of Theorem

7 Conclusions In his paper we exended he Hansen and Jørgensen s formula for valuaion of he floaing srike American-syle Asian opion by assuming a non-zero dividend rae q. The heory of he maringales and condiioned expeced values was used in he calculaion of an inegral equaion for he posiion of he early exercise boundary. We also presen he formula for he weighed arihmeic average wih ime dependen weighs. The presened formula can be used in he comparison of he value of he early exercise boundary o he projeced SOR mehod for Asian opion due Kwok, Dai in 1] as well as inegral ransformaion mehod described in 7]. The numerical experimens and asympoic analysis of he early exercise boundary will be he subjec of he forhcoming paper being prepared. Acknowledgmens The work has been suppored by bilaeral Slovak-Bulgarian projec APVV SK- BG References 1] M. Dai and Y. K. Kwok, Characerizaion of opimal sopping regions of American Asian and lookback opions, Mahemaical Finance, Vol. 16, No. 1 January 26, p ] J. N. Dewynne, S. D. Howison, I. Rupf and P. Wilmo, Some mahemaical resuls in he pricing of American opions, European Journal of Applied Mahemaics 1993, Vol. 4, p ] A. T. Hansen and P. L. Jørgensen, Analyical Valuaion of American-syle Asian Opions, Managemen Science 468 2, p ] J. C. Hull, Opions Fuures and Oher Derivaive Securiies, 2 nd Ediion, Prenice Hall, New Jersey, ] Y. K. Kwok, Mahemaical Models of Financial Derivaives, 2 nd Ediion, Springer Berlin Heildelberg, 28. 6] D. Revuz and M. Yor, Coninuous Maringales and Brownian moion, Springer-Verlag, Berlin, 25. 7] D. Ševčovič, Transformaion mehods for evaluaing approximaions o he opimal exercise boundary for linear and nonlinear Black-Scholes equaions, Nonlinear Models in Mahemaical Finance New Research Trends in Opion Pricing, Mahias Ehrhard ed., Nova Science Publishers, Inc., 28, p ] J. M. Seel, Sochasic Calculus and Financial Applicaions, Springer-Verlag, New York, 21. 9] P. Wilmo, S. Howison and J. Dewynne, The Mahemaics of Financial Derivaives, Cambridge Universiy Press, New York, Deparmen of Applied Mahemaics and Saisics, Faculy of Mahemaics, Physics and Informaics, Comenius Universiy, Braislava, Slovak Republic address: bokes@fmph.uniba.sk 23

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