A Solution to a Non Linear Black Schole s Equation with Transaction Cost and Volatile Portfolio Risk in Sobolev Space

Size: px

Start display at page:

Download "A Solution to a Non Linear Black Schole s Equation with Transaction Cost and Volatile Portfolio Risk in Sobolev Space"

Claribel Hopkins
6 years ago
Views:

1 Applied Mathematics 4, 4(): 4-46 DOI:.593/j.am.44. A Solution to a Non Linear Black Schole s Equation with Transaction Cost and Volatile Portfolio Risk in Sobolev Space Bright O. Osu,*, Chidinma Olunkwa Department of Mathematic, Abia State University, Uturu, Nigeria Abstract This paper considers a weak solution of non-linear Black-Scholes equation with transaction cost and portfolio risk in Sobolev space. We first obtain a weak formulation and obtain its solution in Sobolev space that is characterized by the Fourier transform. Keywords Sobolev space, Non Linear Black-Scholes equation, Transaction cost, Portfolio risk. Introduction Price in different markets tends to converge and this is due to the effect of arbitrage. The concept of arbitrage on some levels is analogous to the principle of the invisible hand of the market. Black and Scholes ([], []) derived a singular equation that has moved the frontier of financial world to hitherto unimaginable length. In their derivation of this equation some basic assumptions were made: one such assumption is that there are no transaction costs in hedging. Since then there have been several and successful remodeling of the Black Scholes equation that takes into account transaction costs. Accordingly, when transactions cost directly proportional to trading- is incorporated in the Black-Scholes model the resulting hedging portfolio is prohibitively expensive. It is therefore acceptable that in the continuous-time model with transaction costs, there is no portfolio that can replicate the European call option transaction costs. To precede, the condition under which hedging can take place has to be relaxed such that the portfolio only dominates rather than replicates the value of the European call option at maturity. With this relaxation, there is always the trivial dominating hedging strategy of buying and holding one share of the stock on which the call is written. From arbitrage pricing theory, the price of an option should not be greater than the smallest initial capital that can support a dominating portfolio. Interesting results have evolved from this line of approach to pricing option without transaction cost, however, in the presence of constraints, in the presence of transaction costs, Soner et al * Corresponding author: megaobrait@yahoo.com (Bright O. Osu) Published online at Copyright 4 Scientific & Academic Publishing. All Rights Reserved [3] proved that the minimal hedging portfolio that dominates a European call option is the trivial one. In essence this suggests another way or technique to relaxing perfect hedging in models with transaction costs. Leland [4] used a relaxation with the effect that his model allowed transactions only at discrete times. By a formal δ - hedging argument, one can obtain a generalized option price that is equal to a Black- Scholes price but with an adjusted volatility of the form; σσ = σσ LLLLLLLLLL( SS VV), where σσ > is a constant historical volatility, LLLL = CC ππ σσ tt is the Leland number and tt is time lag. Assuming that inventor s preferences are characterized by an exponential utility function, Barles and Soner [5] derived a nonlinear Black- Scholes equation with volatility σσ = σσ( SS VV, SS, tt) given by σσ = σσ + φφaa ee rr(tt tt) SS SS VV, 3 XX 3 where φφ(xx) 3 for close to the origin and σσ is a constant. Market models with transaction cost have been extensively dealt with (see for example Amster, et al [6], Avellanda and Paras [7]). A solution in Sobolev by stochastic iteration method for nonlinear Black-Scholes equation with transaction cost and volatile portfolio risk measure in Hilbert space had obtain (see Osu and Olunkwa [8]). In a related paper, the solution of a nonlinear Black-Scholes equation with the Crank-Nicholson scheme had also been obtained (see Mawah [9] and the references therein). The objective of this paper is to further incorporate volatile portfolio risk and show that solution in Sobolev space subject to some boundary conditions is possible. We obtain weak formulation which leads to a weak solution.

2 4 Bright O. Osu et al.: A Solution to a Non Linear Black Schole s Equation with Transaction Cost and Volatile Portfolio Risk in Sobolev Space. The Model Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval time lag (for numerical example, see references in [8]). This leads to a fully nonlinear parabolic PDE. If transaction costs are taken into account perfect replication of the contingent claim is no longer possible. Modeling the short rate rr = rr(tt) by a solution to a one factor stochastic differential equation, dddd = μμ(ss, tt)dddd + σσ(ss, tt)dddd, (.) where μμ(ss, tt)dddd represent a trend or drift of the process and σσ(ss, tt) represents volatility part of the process, the risk adjusted Black-Scholes equation can be viewed as an equation with a variable volatility coefficient tt VV + σσ (ss, tt) SS μμ(ss SS VV) 3 ss VV +rrrr SS VV rrrr =, (.) where σσ (ss, tt) depends on a solution VV = VV(ss, tt) and μμ = 3 CC RR 3, since ππ σσ (ss, tt) = σσ ( μμ(ss SS VV(SS, tt)) 3. Incorporating both transaction costs and risk arising from a volatile portfolio into equation (.) we have the change in the value of portfolio to become. tt VV + σσ (ss,tt) SS ss VV + rrrr SS VV rrrr = (rr TTTT + rr VVVV )SS, (.3) where rr TTTT = CC Γ σσss is the transaction costs measure, ππ tt rr VVVV = RRσσ4 SS Γ Δtt is the volatile portfolio risk measure and Γ = ss VV. Minimizing the total risk with respect to the time lag tt yields; min tt (rr TTTT + rr VVVV ) = 3 RR CC 3 σσ SS 4 ππ SS VV 3. For simplicity of solution and without loss of generality, we choose the minimized risk as with {mmmmmm tt (rr TTTT + rr VVVV )} 3 = AAss ss vv, (.4a) AA = ( 3 )3 CC RR σσ 3. (.4b) ππ They change in the value of the portfolio after minimizing the total risk with respect to time lag is given as tt VV + σσ ( μμ(ss ss vv(ss, tt)) 3 SS ss VV +rrrr SS VV rrrr = AAss ss vv, (.5) which can also be written as + σσ (` μμ(ss SS VV(SS, tt)) 3ss VV SS +rrrr rrrr = VV AAss SS. The left hand of equation.5 is the usually Black-Sholes formula. Setting SS = ee xx, VV(xx, yy) = uu(ee xx, tt)aaaaaa h(ee xx ) = gg(xx) we have equation (.5) becoming; + σσ (` μμ(ss SS UU(SS, tt)) 3ss uu + rr xx rrrr = AA uu xx, which implies; + σσ (` μμ(ss SS uu(ee xx, xx)) 3 uu xx +rr rrrr = AA( uu xx ). (.6) Let kk = σσ (` μμ(ss SS uu(ee xx, xx)) 3, then equation(.6) reduces to + uu kk( xx ) + rr rrrr = AA( uu xx ), (.7) or + kk UU xx + (rr kk) rrrr = AA UU, (.8) xx which is equivalent to that in (Mariama []). We further assume that there is no accumulated interest on the portfolio. Hence rr = and the new portfolio becomes kk uu + kk xx = AA uu xx or kk uu xx + kk = AA uu (.9) xx with the initial condition UU(xx, ) = max( ee xx, ) AA = AA. Our interest in this paper is to show that a solution of the equation kk uu xx + kk = AA FF, uu xx (, TT) (.) uu(xx, ) = uu (xx) oooo xx (.) is possible in Sobolev space. 3. The Sobolev Space We are considering functions for which all the derivatives, in distribution sense, belongs to LL. Let R dd and let uu be a function of LL, it can be identified to a distribution on as a function of LL llllll ( ), also denoted as uu and we can define its derivative as distribution on, since

3 Applied Mathematics 4, 4(): equation (.) is not an element of LL ( ). Hence, we introduce the Sobolev space. Definition 3.: We call Sobolev space of order on the space HH ( ) = {uu LL ( ), LL ( )}. The space HH is endowed with the norm associated with the inner product:, ( ) u, v = uv + u v dx and we note the corresponding norm: u = u, v = u dx u dx,, + Definition 3.: Let mm N. A function uu LL ( ) belong to the Sobolev space of order m, denoted HH mm ( ), if all the derivatives of uu up to order m, in the distributional sense, belong to LL ( ). By convention, we note HH ( ), = LL ( ). Theorem 3.: The space HH mm ( ), mm endowed with the following inner product are Hilbert spaces:, m, with the associated norm α α ( ) ( ) (3.) uv = u x v x vv mm, = α m αα vv LL ( ) αα mm (3.) Proof: The expression (3.) defines an inner product. It is sufficient to show that the space is complete for associate norm. Let consider a Cauchy sequence (vv nn ) nn mm for this norm. We have lim jj,kk αα vv jj αα αα mm vv kk =, LL (3.3) ( ) which means that, for each αα such that αα mm, the sequence αα vv jj is a Cauchy sequence in LL ( ). Since LL ( ) is complete, there exist uu αα such that αα vv jj uu αα in LL ( ). Since the convergence in LL implies the convergence in the distributional sense, we have vv jj uu. Hence, we have αα vv jj αα vv, in the distributional sense and thus αα uu = uu αα LL ( ) and consequently vv HH mm (). Moreover, we have uu vv jj = uu mm, αα αα mm vv jj, LL (3.4) ( ) and uu j vv for the HH mm ( ) Definition 3.3: More generally, defining for every pp and for every mm N, mm, the Sobolev space is given as WW mm,pp = uu LLpp ( ), αα uu LL pp ( ), αα N pp (3.5), αα mm endowed with the norm: p p α (3.6) α m u, p = u dx W We shall consider herein the spaces WW mm, () = HH mm (). Note: The inclusion HH mm () LL ( ) is continuous. Moreover, since DD() is dense in LL ( ) and DD() HH mm (), we have that HH mm () is dense in LL ( ). Theorem 3.: Suppose is an open set of class CC and its boundary is bounded (or = R dd + ). Then, for every mm N, there exists a linear extension operator PP: HH mm (OOOOOOOOOO) HH mm (R dd ) such that for every uu HH mm (), we have. PPPP Ι = uu,. PPPP HH mm R dd CC uu HH mm () wwheeeeee CC dddddddddddddd uuuuuuuu oooooooo. Proof: The proof is established in the particular case where NN =, mm = aaaaaa iiii aa ssssssssssss oooo R +. Let consider = ], [ aaaaaa HH mm ().We define and we pose (PPPP)(xx) = uu uu(xx) xx (xx) = uu( ) xx <, vv(xx) = uu (xx) xx > uu ( xx) xx <. It is easy to see that vv LL. To conclude, we have to show that (PPPP) = vv iiii iiii DD (R). Indeed, under this assumption, we deduce that PPPP HH (R) and that PPPP HH (R) uu HH (R). To prove that the hypothesis (PPPP) = vv iiii DD (R) hoooooo, we introduce the sequence (ηη kk ) of functions of CC (R) defined for every kk N by ηη kk (tt) = ηη(kkkk), tt R, kk N, where ηη CC (R) is a given function such that nn(tt) = tt <. tt > Let us consider vv DD(R), we have uu (xx)vv (xx) dddd, which implies χχ(xx) = vv(xx) vv( xx). Since ηη kk (xx) DD(, ), we have uu(ηη kk χχ) dddd = uu (ηη kk χχ)dddd. (3.7) However, we have the inequalities (ηη kk χχ) (xx) = ηη kk (xx)χχ (xx) + kkkk (kkkk)χχ(xx) and kkkkkk kk kkkk(xx)ηη (kkkk)χχ(xx)dddd xx uu(xx) dddd MMMM kk uu(xx) dddd, with the constants C= Sup tt [,] ηη(tt) aaaaaa χχ(xx) MM xx, hence

4 44 Bright O. Osu et al.: A Solution to a Non Linear Black Schole s Equation with Transaction Cost and Volatile Portfolio Risk in Sobolev Space lim kk kkkk(xx)ηη(kkkk)dddd =. Thus, we can deduce from the previous relations: + + uχ dx = - uχ dx = u' vdx From the relation (3.7) and previous relations, we deduce that (PPPP) = vv iiii DD (R). We can show that this result still holds if the boundary is only piecewise CC continuous. (Dauge, []). Corollary. (Stein [6]) Consider a Lipschitz domain. There exists a linear extension operator PP bounded of WW mm,pp () iiii WW mm,pp R dd such that theorem 3. holds. Using the extension operator introduced above, we deduce the following result. Lamma 3.: If CC with bounded (or if = R dd + ), then for every uu HH () there exists a sequence of functions (uu nn ) nn N DD(R dd ) such that uu nnι uu, nn. HH () This lemma states that only the restriction of DDR dd are dense in HH () and not the functionsdd().actually, in general DD() is not dense in HH (). 3.. Characterization of Sobolev Space Using Fourier Transform When = R dd, we can characterize the Sobolev space HH kk R dd using the Fourier transform [Adam and Fourier]). Givena uu HH kk R dd, by the obvious identity DD αα uu = (iiii) αα uu and the plancherel s theorem, we deduce that Thus, by definition DD αα uu,r dd = ξξ αα uu,r dd. α u ˆ ( ) d = ξ u ξ dξ k, d α k Using the elementary inequalities ( + ξξ ) kk ( ξξ ) αα αα kk ( + ξξ ) kk, (3.8) we conclude that uu kk,r nn ( +. ) kk uu kk,r dd. This relation shows that the Sobolev space HH kk (R dd ) may be equivalently defined by HH kk (R nn ) = uu LL R dd, ( + ξξ ) kk uu LL R dd. This alternative definition can be used to characterize Sobolev space with real index. Namely, for any given ss [, ), tthee Sobolev space HH ss R dd can be defined as follows: HH ss R dd = uu LL R dd, ( + ξξ ) ss uu LL R dd with a norm defined by ffffff aaaaaa ss [, ) (3.9) uu ss,r dd ( +. ) ss uu ss,r dd (3.) 4. Weak Formulation If there exists a function uu CC () CC oo () satisfying equation (.-.), we call uu a classic solution. We need to seek a weak solution in broader spaces Sobolev spaces.it means the smoothness is imposed by weak derivatives. Recall the basic idea of Sobolev space is to treat function as functional. Let us try to understand the equation (.-.) in the distribution sense. We seek solution uu DD αα () such that for CC,. uu. uu, = ff,, which suggest a weak formulation: Find uu HH () such that ( ) u. dx = f dx, C (4.) But we do not discretize 4. directly since it is impossible to construct a finite dimensional subspace of CC (), we first extend the action of uu on CC () to a broader space. Let us define a bilinear form on HH () CC (): ( ) a u, = u. dx By Cauchy-Schwarz inequality, aa(uu, ) uu. Thus aa(uu,. ) is continuous in the HH topology. Thanks to the fact CC () is dense in HH (), the bilinear form aa(.,. ) can be continuously extend to UU VV = HH () HH (). Here the space U is the one we seek a solution and thus called trial space and V is still called test space. In (4.) the right-hand side ff LL () LL (). After CC () is extend to HH (), we can take ff HH () = HH () = VV, We are in the position to present the variational (or so called weak) formulation of equation (.). Given an ff VV, find a solution uu UUsuch that aa(uu, vv) = ff, vv ffffff aaaaaa vv VV (4.) The weak solution u can be proved to be a solution of equation in a more classic sense if u is smooth enough such that we can integrate by parts back. First we assume right side ff LL which will imply uu LL ().

5 Applied Mathematics 4, 4(): Theorem 4.: Let u be the solution of equation (4.). If ff LL (), i.e. ( ) u.v dx = f. vdx v L (4.3) For the choice of UU = HH (), VV = HH (), u satisfies the boundary condition uu = oooo and for the boundary condition UU = HH /R, VV = HH, uu. nn = oooo in the trace sense. (Dauge, []). Proof: When ff LL (), ff, vv = (ff. vv). CChoooooooooooo vv CC () VV in (4.) implies that uu, vv = (ff, vv), vv CC (). That is uu as a distribution is equal to ff LL (). Since CC () is dense in LL () and both side are continuous in LL topology. We conclude equation (4.3). When uu LL (), we are allowed to use a generalised Gauss theorem, that is for any vv HH : u u. u dx = u. v dx + v dx n (4.4) For the choice of UU = HH (), VV = HH (), the boundary condition is built into the space. Now we prove the choice of UU = HH /R, VV = HH will lead to the boundary condition uu. nn = oooo. We have already proved 4.3 and thus from 4.5 and 4.3, we conclude u v ds = n Using the fact that Υ: HH () is onto, we conclude uu. nn = in HH () Next we will show uu HH () iiii uu LL () provided the domain is smooth. We first give a formal derivation for the model problem uu = ff iiii R nn and assume u is smooth and vanishes sufficiently rapidly as xx to justify the following integration by parts: ( ) n f dx = u dx = ii jjudx n n i, j= n n ii jjudx D u dx i, j= n n = = We can also see it from Fourier transform for the Laplacian Operator on the whole space.given a distribution vv defined on R nn such that vv LL, by the properties of Fourier transform, we have DD αα vv(ξξ) = (iiii) αα vv(ξξ) = (iiii) αα ξξ vv (ξξ) The function (iiii) αα ξξ is bounded by unity if αα =, hence DD αα vv,r nn vv,r nn. By Planchel identity, we have DD αα vv,r nn vv,r nn ffffff aaaaaa αα =. The above inequalities illustrates an important fact that v is a function such that vv LL, then all its second order derivatives are also in LL. This is rather significant fact since vv is a very combination of second order derivative of v. 5. Conclusions We have sought a solution of u in the function space HH () which is a completion of CC () for the norm.,. HH () is a Hilbert space for the extension of., and (.,. ), and is the Sobolev space of order. The Sobolev spaces appear naturally in the solution of problem (.) in the sense that CC () can be extended by continuity to HH () and (4.) is thus the problem. Finding uu HH (). (4.) requires less regularity on the solution u and on the data f than the classical problem (.). (4.) is thus the weak formulation of (.). Any solution of (.) is a solution of (4.) but the converse is not true in general. Solutions uu CC () of (4.) if they exist are referred to as classical solution while solution uu HH () of (.) that are not solutions of (.) are called weak (or generalized vibrational) solutions. Solutions of (4.) exist in bounded domains (see []), but in our ongoing research, we are constructing a solution of (4.) in the whole domain, then we will employ the services of the Riesz representation theorem (powerful results of functional analysis) to try toobtain the converse of the solution of (4.) which is a solution of (.). REFERENCES [] F. Black and M. Scholes. The principle of option and cooperate liabilities. Journal of political economic (98) [] Black, F., Scholes, M. The valuation of options contracts and test of market efficiency. Journal of Finance (97), [3] H.M. Soner, S.E. Shreve, and J. Cvitanic. There is nonontrivial hedging portfolio for option pricing with transaction costs. [4] H. E. Leland. Option pricing and replication with transactions costs. The journal of finance, vol. 4, No.5.(985), pp [5] G. Barles and H. M. Soner. Option pricing with transaction costs and nonlinear Black-Scholes equation. Finance Stochast. (998), [6] P. Amster, C. G. Averbuj, M.C. Mariani and D. Rial. ABlack-Scholes option model with transaction costs. Journal of Mathematical Analysis and Applications. 33() (5), [7] M. Avellaneda, A. Levy and A. Paras. Pricing and Hedging

6 46 Bright O. Osu et al.: A Solution to a Non Linear Black Schole s Equation with Transaction Cost and Volatile Portfolio Risk in Sobolev Space derivative securities in markets and uncertain Volatilities. Applied Mathematical Finance, (995), [8] B. O. Osu and C. Olunkwa A solution by stochastic iteration method for nonlinear Black-Scholes equation with transaction cost and volatile portfolio risk in Hilbert space. International Journal of Mathematical Analysis and Application: (3) (4), [9] B. Mawah. Option pricing with transaction costs and a nonlinear Black-Scholes equation. Department of Mathematics U.U.D.M. Project Report 7:8.Uppsala University. [] R.A Adam and J.F Fourier. Sobolev spaces Academic Press, 978. [] M.C. Maraiam, E.K. Ncheuguim and I. Sengupta. Solution to a nonlinear Black-Scholes equation. Electronic Journal of Differential Equations. 58(),-. [] M. Dauge Elliptic boundary Value problems on corner domain. Lecture note in Mathematics, (34).988 [3] L. Chem Sobolev space and Elliptic Equation. Lecture note by Jinchaoxu in Pennsylvania State University 3.

Financial Market Models. Lecture 1. One-period model of financial markets & hedging problems. Imperial College Business School

Financial Market Models. Lecture 1. One-period model of financial markets & hedging problems. Imperial College Business School Financial Market Models Lecture One-period model of financial markets & hedging problems One-period model of financial markets a 4 2a 3 3a 3 a 3 -a 4 2 Aims of section Introduce one-period model with finite