International Journal of Mathematial Analysis and Appliations 2014; 1(3): 43-48 Published online August 20, 2014 (http://www.aasit.org/journal/ijmaa) A solution by stohasti iteration method for nonlinear Blak- Sholes equation with transation ost and volatile portfolio risk in Hilbert spae Bright O. Osu 1, Chidinma Olunkwa 2 1 Department of Mathematis, Abia State University, Uturu, Nigeria 2 Department of Mathematis Abia State University, Uturu, Nigeria Keywords BS Non-Linear Equation, Transation Cost Measure, Volatile Portfolio Risk Measure, LCP, Hilbert Spae Reeived: June 06, 2014 Revised: June 20, 2014 Aepted: June 21, 2014 Email address megaobrait@hotmail.om (B. O. Osu), ejihi24@yahoo.om (C. Olunkwa) Citation Bright O. Osu, Chidinma Olunkwa. A Solution by Stohasti Iteration Method for Nonlinear Blak-Sholes Equation with Transation Cost and Volatile Portfolio Risk in Hilbert Spae. International Journal of Mathematial Analysis and Appliations. Vol. 1, No. 3, 2014, pp. 43-48. Abstrat We introdue a stohasti iteration method for the solution of a non-linear Blak- Sholes equation whih inorporates both the transation ost and volatile portfolio risk measures. We first redue the equation to a linear omplementarity problem (LCP) and then propose an expliit steepest deent type stohasti method for the approximate solution of the LCP govern by a maximal monotone operator in Hilbert spae. This sheme is shown to onverge strongly to the non-zero solution of the LCP. 1. Introdution In a omplete finanial market without transation osts, the elebrated Blak- Sholes no-arbitrage argument [1] provides not only a rational option priing formula but also a hedging portfolio that repliates the ontingent laim. However, the Blak-Sholes hedging portfolio requires trading at all-time instants, and the total turnover of stok in the time interval [0, T] is infinite. Aordingly, when transations ost diretly proportional to trading- is inorporated in the Blak- Sholes model the resulting hedging portfolio is prohibitively expensive. It is therefore aeptable that in the ontinuous-timemodel with transation osts, there is no portfolio that an repliate the European alloption with finite transation osts. To proeed, the ondition under whih hedging an take plae has to be relaxed suh that the portfolio only dominates rather than repliates the value of the European all option at maturity. With this relaxation, there is always the trivial dominating hedging strategy of buying and holding one share of the stok on whih the all is written. From arbitrage priing theory, the prie of an option should not be greater than the smallest initial apital that an support a dominating portfolio. Interesting results have evolved from this line of approah to priing option without transation ost, however, in the presene of onstraints, in the presene of transation osts.soneret al [2] proved that the minimal hedging portfolio that dominates a European all option is the trivial one. In essene this suggests another way or tehnique to relaxing perfet hedging in models with transation osts. Leland [3] used a relaxation with the effet that his model allowed transations only
International Journal of Mathematial Analysis and Appliations 2014; 1(3): 43-48 44 at disrete times. By a formal δ - hedging argument, one an obtain a generalized option prie that is equal to a Blak- Sholes prie but with an adjusted volatility of the form; = 1, where > 0 is a onstant historial volatility, = is the Leland number and is time lag. Assuming that inventor s preferenes are haraterized by an exponential utility funtion, Barles and Soner[4] derived a nonlinear Blak- Sholes equation with volatility =,, given by = 1+"# $%& ',,, - (. - Where "( 3 2, for lose to the originand is a onstant. Market models with transation ost have been extensively dealt with (see for example Amster, et al [], Avellanda and Paras [6]). A solution in Sobolev spae whih implies a weak solution of the nonlinear Blak- Sholes equation has been obtained (see Osu and Olunkwa[7]). In a related paper,thesolution of a nonlinear Blak-Sholes equation with the Crank-Niholson sheme had also been obtained (see Mawah [8] and the referenes therein). The objetive of this paper is to further inorporate volatile portfolio risk and show that a stohasti iteration method for the solution of a non-linear Blak- Sholes equation whih inorporates both the transation ost and volatile portfolio risk measures exists and onverges strongly to the nonzero solution. 2. The Model Transation osts as well as the volatile portfolio risk depend on the time lag between two onseutive transations. Minimizing their sum yields the optimal length of the hedge interval time lag. This leads to a fully nonlinear paraboli PDE. If transation osts are taken into aount perfet repliation of the ontingent laim is no longer possible. Modeling the short rate / = / by a solution to a one fator stohasti differential equation. 0 = 12,0 +2,03, (1.1) where1,0 represent a trend or drift of the proess and, represents volatility part of the proess, the risk adjusted Blak-Sholes equation an be viewed as an equation with a variable volatility oeffiient + 4, 1 1 6' 4 +/2 / = 0, ( 1.2) where 2, depends on a solution = 2, and ' 6 1 = 3 7, sine 2, = 1 1, 6. Inorporating both transation osts and risk arising from a volatile portfolio into equation (1.2) we have the hange in the value of portfolio to beome. + 2, 2 4 +/ / = / % +/ 9: where / % = <.. is the transation osts measure, / 9= = >? Γ Δ is the volatile portfolio risk measure and Γ = 4. Minimizing the total risk with respet to the time lag yields min / % +/ 9: = - 7 ' 6 E 6. They hange in the value of the portfolio after minimizing the total risk with respet to time lag is given as +.&FG H I4,6 4 +/ / - 7 ' 7 ' 6 6 E 6 = 0. (1.3) LetJ, = - E 6, and if we assume that there is no round trip transation ost,ie if we say there is no transation ost by making J, = 0 then equation (1.3) beomes +.&FG H I4,6 4 +/ / = 0. (1.4) Equation (1.4) above is one form of Blak-Sholes equation that inorporates both transation osts and the volatile portfolio risk measures. We are interested in the Blak-Sholes equation of the form below that inorporates both the transation osts measure and the volatile portfolio measure. That is a fully nonlinear paraboli equation of the form I +.. K 2 L1±3 7 ', - 24 N., -O 4 N +/2 4 N /N = 0, 2 > 0, 0,Q. (1.) Note: (i) if R=0 or C=0, equation (1.) redues to the lassial Blak-Sholes equation. (ii) Minus sign indiates Bid option prie [9]. Denote RS = 1 1S.., -'S1 = 3 7 ', -,T = Q and ( = ln, V ( > 2 > 0 with S(,Q = 4 N2,, then equation (1.) an be transformed into a quasilinear equation of the form X S = Y RS+ X RS+/ Y SQ 0,Q, ( >, (1.6)
4 Bright O. Osu and Chidinma Olunkwa: A Solution by Stohasti Iteration Method for Nonlinear Blak-Sholes Equation with Transation Cost and Volatile Portfolio Risk in Hilbert Spae with boundary onditions: S,Q = S,Q = 0. For S \ [ S]h,_`,` = %,h = b we have (1.6) a K beoming; d R. S \&. [&. + /'S \&. d [&. e1+ d R. S \&. [&. / d d R. S \&. [ fs \ [ d R. S \&. \ [ S [&. = S \&. [ + RS \&. d [ RS \&.g [&. (1.7) For ] = +1,, 1 and _ = 1,,i,S K \ = 0 = S K \ and S j j = S( [,0. In matrix X form, we have ks [ \ +l = 0. (1.8) Equation (1.8) in a linear omplementarity problem (LCP) whih is finding n-dimensional S > K suh that with ks +l 0,S 0,S ks +l = 0, (1.9) / K r q k = p1 / o s' ab ab s 0 0 ab + ab s s 0 0 0 1 / + ab abs s / K r z y, s' ab ab s x where s = R S \&. [&.,s = R S \&. [ and ab } l = { 0 ~. ab Interest on LCP stems from the fat that many important Mathematial problems an be formulated as LCP (Cottle et al., [10]). This problem has been extensively studied by many authors inluding Murty[11]. We formulate (1.9) into an equivalent minimization problem: mins % ks +Sl: 0,kS +l 0. (1.10) We observe that from (1.10) that JS = S % ks +S % l has zero as a feasible minimize. Thus the problem redues to that of searhing for the global minimizer: Sƒ> K :S 0,l +ks 0,S % ks +l = 0. But S = 0,JS = 0. This suggests a reformation of the problem as a searh for = S ƒ> K :S > 0:JS = 0, (1.11) where JS = G G = ks +l. In this study stohasti gradient type reursive sequene is suggest: S \. = S \ ˆ\0 \, (1.12) where 0 [ is the estimate of JS = ks +l and ˆ\Š is a sequene of positive salars to be speified. The proedure is a way of stohastially loating the set (1.11) when it exists. The iteration method desribed in this study differs from most iterative methods mainly in the way the searh diretion at eah iteration and the starting point of the searh algorithm are estimated to determine the optimum diretion and provide maximum rate of derease of JS. Definition 1: Let be a real Hilbert spae with inner produt (.,.) and norm. If Q is a mapping with domain ŽQin, then Q is said to be monotone if.,s. S 0 S.,S ŽQ,. QS., QS (1.13) We shall be interested in an important lass of monotone operator whih onsists of the gradient of onvex funtions:let J be a onvex lower semi ontinuous funtion from into,+. We assume that J + and let ŽJ = S :JS<+ be the effetive domain J. For S ŽJ, the set JS = :J JS, S ŽJS is alled the sub differential of J at. The set JS is losed and onvex. We assume that is a real separable Hilbert spae with inner produt (.,.) and norm. A random vetor in is a measurable mapping defined on a probability spae Ω,I, and taking values in. If,N are random vetors in and is fixed vetor in, then,,n,, are real-valued random variables in the usual sense. Let V denote the expetation operator. If V <, then V is defined by the requirement V, =,V. Definition 2: Let Q š S \ Š be a omplete orthonormal basis of assoiated with the data points S.,S,,S.ThenœJ is linear least square estimable in terms of some disrete funtion values omputed from data point S.,S,,S if the data points are suitably hosen suh that: Q š S \,Q $ S \ Š = Ÿ 0 ]J` / ]J` = / g (1.14)
International Journal of Mathematial Analysis and Appliations 2014; 1(3): 43-48 46 If is dimensional Eulidean spae a onvenient basis for J, onsidered in (Okoroafor and Ekere,[12]) with onrete examples, is the set \ Š in > K satisfying [\ = 0,] = 1,2,,, +2 < < KK.. \žj,. [\ = 1 This yields the same result. For the onvex funtion J with ŽJ = S :JS < Let œj be a single valued seletion of J. For every and œj J, thetaylor theorem implies that J +N J = N,œJ +. N,N +0 N, (1.1) for N. Where 0 indiates terms whih an be ignored in the limit and is the seond derivative of J if it exists. Remark 1: Where the seond derivative of J does not exist in any sense, we onsider the Taylor theorem of the form J +N J = œj,n +0 N, (1.16) where, 0. indiates terms whih an be ignored relative to N in the limit and ignore all seond onditions sine they have no influene on the onvergene analysis of the method as we shall see in the sequel. For ompleteness assume the seond derivative exists in some sense.let S \ = J S +QS \ ' JS,S ŽJfor a fixed ` and _ = 1,2,,. Definition 3: The non-observable random errors of approximation on the data points S.,S,,S, is the sequene of random variables S \ Š satisfying V S \ = 0 for eah _ and V S \ S \ = [\ where 0 < <. A onvenient basis for the funtion (1.1) is the omplete orthonormal basis Q $ in, so that the approximation funtion is given by: S \ = JS,Q $ S \ +. S,Q 4 S \ + S \ (1.17) whih is identifiable with (1.1). The disrete funtion values S \, for eah _, are real valued independent observable random variables performed on \ whose distribution is that of S \. If at the point S, for eah `, the data points S.,S,,S suitably hosen so that S \ = S +Q $ S \. (1.18) Then; Theorem 1 A strong approximation of œj at S that is onsistent is the random vetor 0 = Q $ S \,Q $ &. Q $ S \, S \ (1.19) Whih is the least square approximation omputed from different data points S.,S,,S. Proof: Assume then So that and Moreover, 0 = Q $ S \,Q $ &. Q $ S \, S \ 1 [\ = 1 V0 = 1 Q $ S \,V S \ = œj. V 0 œj = V 0 VœJ = V Q $ S \, S \ V 0 œjs = 0. V 0 œj = V Q $ S \, S \,Q $ S \, S \ =. Hene V 0 œj 0 as. 3. Getting the Domain of Attration Let R «N0 be partitioned into exlusive segments, S,j = 1,2,,t,n < 2 «. Let F be hosen randomly in S, suh that ff > 0, _ Let P = PF = αbe the probability that F = α so that Put P = ¹ so that, º ¹» ¹ P 0, ž. P = 1. (1.20) ¹ ¹ F¼ = ž. F P = ž. º. (1.21) ¹» ¹ It is shown in (Okoroafor and Osu,[13]) that if F½ = F¼ ρd,ρ > 0, (1.22)
47 Bright O. Osu and Chidinma Olunkwa: A Solution by Stohasti Iteration Method for Nonlinear Blak-Sholes Equation with Transation Cost and Volatile Portfolio Risk in Hilbert Spae where0 is as in(1.19), thenff½ = min ff :F s. It follows that the segment S Á where F½ S Á ontains x > 0 for whih ff is minimum and hene we have φv ż S Á so that if 0 is the attrator of the point x¼ and φ0 φv È = then 0 ½ = or else 0 = ½ with global domain of attration "0 = " ¼. Where V = F R «:F > 0:JF = 0 (1.23) is a way of stohastially solving problem (1.11). Thus we have Lemma 1: suppose that V Ê ϕ. thus there exists a neighborhood NV Ê D f of V Ê suh that for any initial guess F½ φv È, the non-negative minimizer V Ê is obtained as the limit of iteratively onstruted sequene F ž. generated form F½ by F. = F ρ d. Then with F½ as our starting point we searh for the minimizer of J as follows: starting at S½ as in Eq. (1.22). A. Compute the 0 as in Eq. (1.19) B. Compute the orresponding ˆ as speified below C. Compute F. = F ρ d. Has the proess onverged? i.e., S \. S \ <, > 0 if yes, then S \. = S \ if no return to A. Here we prove the strong onvergene of the sequene to the solution of (1.22) Theorem 1: Let ˆ\be a real sequene suh that I. ˆj = 1,0 < ˆ\ < 1 _ > 1 II. \žj ˆ\ = III. žj ˆ\ < Then the sequene S \ \žj generated by S½ " ½ ŽJ and defined iteratively by F. = F ρ d.remain in ŽJ and onverges strongly to ½. Proof: Let l \ = ˆ\ 0 \ J \ Then l \ is a sequene of independent random variable and from (1.18) Vl \ = 0 for eah _. Notiing that the sequene of partial sums \, \ = l \, is a Martingale. Therefore, V \ = Vl \ = Ó &. ˆ\. And = ˆ\ VÒ 0 \ J \ Ò Vl \ <, 2]Ô ˆ\ < Hene by a version of Martingale onvergene theorem (Whittle, [1]), we have So that lim = l \ < lim = 0 \ J \ = 0 \ ˆ\ Notiing that in (1.22), A is positive definite so that J is onvex and hene J is monotone. But an earlier result in theory of monotone operators, due to (Chidume, [1]), shows that the sequene S \ Š generated by S j ŽJ and defined iteratively by: F. = F ρ d. remain in ŽJ and onverges strongly to S :JS = 0. It follows from this result that our sequene onverges strongly to if 0. 4. Conlusion Transation osts as well as the volatile portfolio risk depend on the time-lag between two onseutive transations. Minimizing their sum yields the optimal length of the hedge interval - time-lag, whih leads to a fully nonlinear paraboli Blak-Sholes PDE.We have used an impliit finite differene approximation and transform this PDE to the linear omplementarity problem. We then onstruted a steepest deent type stohasti sequene in a separable Hilbert spae and show strong onvergene to the solution of the LCP when exit. MSC: 6M; 6C0; 6D30; 98B28. Referenes [1] F. Blakand M. Sholes. The priing of options and orporate liabilities.journal of politial eonomy 81(1973), 637-69. [2] H.M. Soner, S.E. Shreve, and J. Cvitani. There is no nontrivial hedging portfolio foroption priing with transation osts. [3] H. E. Leland. Option priing and repliation with transations osts.the journal of finane, vol. 40, No..(198), pp.1283-1301. [4] G. Barles and H. M. Soner.Option priing with transation osts and nonlinear Blak-Sholes equation. Finane Stohast. 2 (1998),369-397. [] P. Amster, C. G. Averbuj, M.C. Mariani and D. Rial. A Blak-Sholes option model with transation osts. Journal of Mathematial Analysis and Appliations. 303(2) (200), 688-69. [6] M.Avellaneda,A. Levy and A. Paras. Priing and Hedging derivative seurities in markets and unertain Volatilities.Applied Mathematial Finane,2(199),73-88. [7] B. O. Osu and C.Olunkwa (2014). The Weak solution of Blak-Shole s option priing model with transation ost. Applied Mathematis and Sienes: An International Journal (MathSJ ), Vol. 1, No. 1: 43-. [8] B. Mawah. Option priing with transation osts and anonlinear Blak-Sholes equation. Department of Mathematis U.U.D.M. Projet Report 2007:18.Uppsala University. [9] D. ˇSevˇoviˇ. Analytial and numerial methods for priing finanial derivatives. Letures at Masaryk University, 2011.
International Journal of Mathematial Analysis and Appliations 2014; 1(3): 43-48 48 [10] R. W. Cottle, J. S. Pang and R. E. Stone.The linear omplementarity problem.aademi press, San Diego, 1992. [11] K. G. Murty. Linear omplementarity, linear and nonlinear programming.sigma series in applied Mathematis.HeldermannVerlag Berlin, Germany, 3(1988). [12] A. C. Okoroafor and A. E. Ekere.A Stohasti Approximation for the Attrator of a Dynamial System. In Diretions in Mathematis (G. O. S. Ekhagwere and O. O. Ugbeboreds). Asso. Books, (1999) 131-141. [13] A. C.Okoroaforand B. O. Osu.The Solution by Stohasti Iteration of an Evolution Equation in Hilbert Spae. Asian Journal of Mathematial and Statistis 1(2) (2008), 126-131. [14] P. Whittle. Probability, John Wiley and sons, (1976). [1] C. E. Chidume. The iterative solution of nonlinear equation of the monotone type in Banah spaes. Bull. Aust. Math. So. 42 (1990) 2-31.