CC BY
6URAL MATHEMATICAL JOURNAL, Vol. 2, No. 2, 2016
GROUP CLASSIFICATION FOR A GENERAL NONLINEAR MODEL OF OPTION PRICING1
Vladimir E. Fedorov
Laboratory of Quantum Topology, Mathematical Analysis Department, Chelyabinsk State
University, Chelyabinsk, Russia, kar@csu.ru
Mikhail M. Dyshaev
Mathematical Analysis Department, Chelyabinsk State University, Chelyabinsk, Russia,
MikhailDyshaev@gmail.com
Abstract: We consider a family of equations with two free functional parameters containing the classical Black—Scholes model, Schonbucher—Wilmott model, Sircar—Papanicolaou equation for option pricing as partial cases. A five-dimensional group of equivalence transformations is calculated for that family. That group is applied to a search for specifications' parameters specifications corresponding to additional symmetries of the equation. Seven pairs of specifications are found.
Key words: Nonlinear partial differential equation, Group analysis, Group of equivalency transformations, Group classification, Nonlinear Black—Scholes equation, Pricing options, Dynamic hedging, Feedback effects of hedging.
Introduction
In the paper a nonlinear model
ut w{t,x)uxx + r(xUx _ u) = 0_ (01)
2 (1 _ xv(Ux)Uxx)
from the theory of financial markets is considered. In the case of v = 0 it is generalized Black— Scholes equation [1], if, besides, w(t,x) = a2x2 (0.1) is the classical Black—Scholes model [2]. For arbitrary v and w(t,x) = a2x2 (0.1) is the Sircar—Papanicolaou nonlinear feedback pricing equation [1]. If v is arbitrary, w(t,x) = a2x2 and r = 0, it is the equilibrium pricing model or Schonbucher—Wilmott nonlinear feedback pricing model [3-6]. The last two models take into account a feedback effect of the presence of two types of traders. The programm traders are the portfolio insurers and the reference traders are the Black—Scholes uploaders.
The aim of the paper is to obtain a group classification [7] of equation (0.1) with free parameters v and w. The group of equivalence transformations [7,8] of equation (0.1) will be found. By means of this group symmetries for the equation with all specifications will be calculated. Further these results will be applied to the theory of financial markets, particularly, they will allow to calculate various exact solutions of equation (0.1).
The groups of classical Black—Scholes model and their accordance to the groups of the heat equation were found in [9]. Research of symmetries of Schonbucher—Wilmott model and of some other nonlinear pricing models was made in [10-13].
xThe work is partially supported by Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020).
1. Group of the equivalence transformations
Let us find the continuous group of equivalence transformations of equation (0.1) for the applying to the search of specifications of the functions v = v(ux), w = w(t,x) in the equation, that corresponds to additional symmetries for the symmetries of the kernel of principal Lie group for the equation. We rewrite equation (0.1) in the form
wuxx
ut +-----o + r(xux — u) = 0, (1.1)
2(1 — xvuxx)
where v, w are the additional variables, depending on t, x, u, ut and ux. Generators of a continuous group of equivalence transformations will be searched in the form Y = rdt + £dx + ndu + ¡dv + vdw, where the functions T,£,n depend on t,x,u, and i v depend on t,x,u,ut,ux,v,w. For brevity hereafter ^ = dt and similar notations are used. We add to (1.1) the equations
vt = 0, vx = 0, vu = 0, vUt = 0, (1.2)
wu = 0, wut = 0, wUx = 0, (1.3)
meaning that in the statement of the problem the function v depends only on ux and the function w depends on t, x.
We consider the system of equations (1.1)-(1.3) as a manifold N in an expanded space of corresponding variables. Let us act on the left-hand side of system (1.1)-(1.3) by the extended operator
Y = Y + ptdut + Vxxduxx + rfdvt + Ixdvx + Iudvu + Iut dvUt + vudWu + vut dWut + vUx , we restrict a result of the action on N and we obtain the equations
t , vwuxx£ , w(1 + xvuxx)^xx t uxxV , wx^x!
P +--3 +--3--1--2 +--3 +
(1 — xvuxx) 2(1 — xvuxx) 2(1 — xvuxx) (1 — xvuxx)
+r(ux£ + xpx — n) =0, (1.4)
n
= 0, ix|n = 0, iu|n = 0, iut |n = 0, (1.5)
v u|n = 0, vut |n = 0, iux |n = 0. (1.6)
From (1.2) and (1.3) it follows that
Yt = dt + wtdw + wttdwt + wtxdwx + ..., Yx = dx + wxdw + wtxdwt + wxxdwx + ..., Yu = du, Yut = dut, Yux = dux + v'(ux)dv + v"(ux)dv'(ux) + ..., !t = it + wtlw — v'(ux)^x = it + wtlw — v'(ux)(ntx + uxntu — utJtx — u^Ttu — ux£tx — ^{to),
1 — ¡x + wx^w v (ux)px - ¡x + wx^w v (ux)(nxx + uxVxu utTxx utuxTxu ux£xx ux£xu^
1 - ¡u v (ux)pu - ¡u v (ux )(nxu + uxnuu utTxu utuxTuu ux£xu ux£uu)i
!ut = ¡¡ut — v' (ux)pxut = Iut + v' (ux)Tx + uxv' (u^Ty, vu = Vu — wtTu — wx£u, vut = Vut, vux = Vux + v'(ux)Vv. Therefore, equations (1.5) and (1.6) have the form
It + wtlw — v' (ux )(ntx + uxntu — utTtx — utuxTtu — ux£tx — ^{tu^n = 0, (1.7)
ftx + wxftw v (ux)(nxx + uxVxu UtTxx UtUxTxu Ux{xx Ux{xu )|n = (1.8)
ftu _ v'(Ux)(n xu + Uxn'uu UtTxu UtUxTuu Ux{xu Ux^UU)lN - 0J (1.9)
Hut + v' (Ux)Tx + Uxv' (Ux )Tu = 0, (1.10)
Vu _wtTu_ wx^u = 0, Vut = 0, VUx + v'(Ux)Vv = 0- (1.11) By equality (1.1) equations (1.7)-(1.9) can be rewritten in the form
ftt + wtftw _ v'(Ux) (vtx + Uxntu _ UxCtx _ U2xCtu+
WUxx(Ttx + Ux Ttu) , f w A n (1.12)
+--:-"2--+ (rxUx _ rU)(Ttx + UxTtu) = 0,
2 (1 xvUxx)
ftx + wxftw _ v'(Ux) (n xx + Uxnxu UxCxx Ux Cxu + 'xx(txx + UxTxu
xx
. wUxx (Txx + UxTxu) . / w . \\ r\ (1.13)
+--:-To--+ (rxUx _ rU)(Txx + UxTxu)) = 0
2 (1 xvUxx)
ftu _ v'(Ux) (n xu + Uxnuu Ux{xu Ux {uu+
. wUxx(Txu + UxTuu) . / w . \ 1 „
(1.14)
+--:-To--+ (rxUx _ rU)(Txu + UxTuu ) = 0.
2(1 _ xvUxx) J
By means of the equality
^ - nxx + 2Uxnxu + Ux nuu + Uxxnu UtTxx 2UtUxTxu 2UtxTx UtUx Tuu
2UxUtxTu UtUxxTu Ux{xx 2ux {xu 2uxx{x Ux {uu 3uxuxx{u
equation (1.4) is rewritten as
nt + Utnu _ UtTt _ u2Tu _ UxXt _ UtUxXu +--:-T3 (2vwu2xx{ + 2xwuxxft +
2(1 _ xvuxx)
+UxxV _ xU2xxW + w(1 + xvUxx)(nxx + 2Uxnx u + Ux nuu + Uxxnu
2
UtTxx 2utUxTxu 2utxTx UtUx Tuu 2uxUtxTu UtUxxTu Ux{xx 2ux {xu 2uxx{x Ux {uu 3uxuxx{u)) +
+rUxC + rx(nx + Uxnu _ UtTx _ UtUxTu _ Ux{x _ ^{u) _ Hn =
= nt + v^in-nu> + {rXUx _ ru)(T, _ nu) _ f^" .
2(1 _ xvUxx) 4(1 _ xvUxx)
, ,2 wuxx(rxux _ ru)Tu .
_ (rxUx _ rU) Tu +----"2--Ux{t+
(1 xvUxx)2
wUxUxx{u . , x t . 1 /„ 2 t , o 2 (1.15)
'TTT,-\2 + (rxUx _ ru)ux{u + —-T3 i2vwUxxi + 2xwUxxft +
2 (1 xvUxx) 2 (1 xvUxx)
,2
+UxxV _ xU2xxW + w(1 + xvUxx) (n xx + 2uxnxu + Ux nuu + Uxxnu+ wUxxTxx . r \ . wUxUxxTxu in/ \
+--:-"2 + (rxUx _ ru)Txx + 7-"2 + 2(rxUx _ ru)UxTxu_
2(1 _ xvuxx) (1 _ xvuxx)
2 2 wuxUxx Tuu . / \ 2 n i wuxxTu ,
_ 2UtxTx + —--o + (rxUx _ ru)u2xTuu _ 2UxUtxTu + —-"o +
2(1 _ xvuxx) 2 (1 _ xvuxx)
+ (rxUx ru)uxxTu Ux{xx 2ux {xu 2uxx{x Ux u 3uxUxx{u) +
+rUx{ + rx(nx + Uxnu _ Ux{x _ U2x(u) _ rn+
rxwUxx (Tx I Ux Tu) / \ / \
+--:--o--+ rx(rxUx _ ru)(Tx + UxTu) = 02 (1 xvUxx)
We differentiate the last equations with respect to utx and obtain w(1 + xvuxx)(Tx + uxTu) = 0, consequently, t = t(t), if w = 0. Therefore, equations (1.11)-(1.15) have the form
¡ut =0, vu — wx£u = 0, vut = 0, vux + v' (ux)vv = 0, (1.16)
It + wt^w — v'(ux) (ntx + uxntu — ux£tx — u2x£tu) = 0, (1.17)
¡x + wx^w — v'(ux) (vxx + uxVxu — ux£xx — u2x£xu) = 0, (1.18)
¡u — v'(ux) (n xu + uxnuu ux£xu ux£uu) - 0, (1.19)
wuxx (t' (t) — nu) . , w , t .
nt +--:-To + (rxux — ru)(T (t) — nu) — ux£t +
2 (1 — xvuxx)
wuxuxx£u , \ 1 ( 2 A 2
^-\~2 + (rxux — ru)uxiu + —--3 (2vwuxx£ + 2xwuxxJ
2 (1 — xvuxx) 2 (1 — xvuxx)
xx
+ uxxv xuxxvv + w(1 + xvuxx )n xx + 2uxnxu + ux nuu + uxxnu ux£xx 2ux £xu 2uxx£x ux £uu 3uxuxx£u^ + rux £+
+rx(nx + uxnu — ux£x — u2x£u) — rn = 0. We multiply by 2(1 — xvuxx)3 the last equation, then
2(1 — xvuxx)3(nt + (rxux — ru)(T (t) — nu + ux£u)) + (1 — xvuxx)wuxx(T (t) — nu) —
\3„, A , n \..............I o.......2 f , o™.....2
—2(1 — xvuxx) ux£t + (1 — xvuxx)wuxuxx£u + 2vwuxx£ + 2xwuxx¡+ xxv - xuxxvv + w(1 + xvuxx) inxx + 2uxnxu + ux nluu i "xx1
+uxxv — xu2xxW + w(1 + xvuxx) {nxx + 2uxnxu + ux n'uu + uxx nu ux£xx (1.20)
2ux £xu 2uxx£x ux £uu 3uxuxx£u) +
+2(1 — xvuxx)3(rux£ + rx(nx + uxnu — ux£x — u2x£u) — rn) = 0.
Equation (1.20) for the case v = 0 has at u3xx multiplier
nt + rxuxT (t) — ru(T (t) — nu + ux£u) — ux£t + rux£ + rx(nx — ux£x) — rn,
after its splitting with respect to ux, we obtain two equations
nt + rxnx + runu — rn — ruT (t) = 0, (1.21)
rxT (t) — £t — rx£x — ru£u + r£ = 0. (1.22)
After the splitting with respect to ux of the multiplier at uxx in zero degree it follows that
£ = A(t,x)u + B (t,x),
n = Ax(t, x)u2 + C(t, x)u + D(t, x)
and by (1.21), (1.22)
2nt — 2ruT + 2runu + 2rxnx — 2rn + wnxx = wnxx = 0, 2rxT — 2ru£u — 2£t + 2r£ — 2rx£x — 2w£xx + 4wnxu = —w£xx + 2wnxu = 0.
The last equality implies that
Axx = 0, A(t,x) = Ai(t)x + Ao(t), C (t,x) = 1 Bx(t,x) + E (t),
£ = Ai (t)xu + Ao(t)u + B (t,x), n = Ai(t)u2 + 1 Bx(t,x)u + E (t)u + D(t,x).
(1.23)
Then from (1.23) it follows that
Bxxx = 0, Dxx = 0, £ = Ai (t)xu + Ao(t)u + B2(t)x2 + Bi(t)x + Bo(t), n = Ai(t)u2 + B2 (t)xu + 2 Bi(t)u + E(t)u + Di(t)x + Do(t).
Now the equality (1.22) implies that Ai(t) = Fe-rt, Ao(t) is a constant,
B2 (t) = Ge-rt, Bi(t) = rT (t) + H, Bo (t) = Jert, £ = Fe-rtxu + Aou + Ge-rtx2 + rT (t)x + Hx + Jert, -rtu2
By (1.21) Di is a constant,
n = Fe-rtu2 + Ge-rtxu + 1(rT (t) + H )u + E (t)u + Di(t)x + Do(t).
Do (t) = Kert, E(t) = 2 rT (t) + P,
n = Fe-rt u2 + Ge-rtxu + rT (t)u + Pu + Dix + Kert.
From (1.16) it follows that v = wx(Fe-rtx + Ao)u + 5(t, x, ux, v, w).
The coefficient at uxx in equation (1.20) is equated to zero and we obtain the equation
—6xvnt — 6xv(rxuxT' (t) — ruT (t) + runu — ruux£u) + wt' (t) + +6xvux£t + wux£u + v — 2w£x — 3wux£u + 2xvwuxnxu + xvwu2xnuu— —xvwux£xx — 2xvwu2x£xu — 6xv(rux£ + rxnx — rxux£x — rn) = 0.
Let us substitute in it the expressions for £, n, v that were found before, and splitting with respect to the variable u leads to the equations
—2Fe-rtw + wx(Fe-rtx + Ao) = 0, (1.24)
S = 4Ge-rtxw — wt' + 2rwT + 2Hw + 2Fe-rtxuxw + 2Aouxw.
The last of them implies the equalities vv = Sv = 0, consequently, by (1.16) we obtain
vux = Sux = 2Fe-rtxw + 2Aoxw = 0, Ao = F = 0.
Thus,
£ = Ge-rtx2 + rT (t)x + Hx + Jert, n = Ge-rtxu + rT (t)u + Pu + Dix + Kert, v = 4Ge-rtxw — wt' + 2rwT + 2Hw.
Analogous calculations are made with the coefficient at uxx in equation (1.20), we obtain the equation
6x2v2(nt + rxuxT' — ruT + runu) — xvw(t' — nu) — 6x2v2ux£t + 2vw£ + 2xw¡1 — xvv+ +xvw(nu — 2£x) + 6x2v2(rux£ + rxnx — rxux£x — rn) = 0,
that implies the equality ¡1 = v (H — P — xert + 2Ge-rtx) . Therefore ¡u = ¡w = 0, and for the case v' = 0 obtain G = 0 from equation (1.19). Then equation (1.18) implies that ¡x = 0, hence J = 0. From equation (1.17) it follows that ¡t = 0, it corresponds to the resulting formula H = (H — P)v. Thus, t(t) is an arbitrary function,
£ = Hx + rT (t)x, n = Kert + Dix + Pu + rT (t)u, H = (H — P )v, v = 2Hw + (2rT (t) — t' (t))w.
Let us formulate the result in the form of theorem.
Theorem 1. The Lie algebra of infinitesimal generators of the equivalency transformations groups for equation (0.1), is generated by operators
Yi = xdu, Yo = er du, Y3 = xdx + udu + 2wdw, Y4 = xdx + vdv + 2wdw,
Y5 = t(t)dt + rT(t)xdx + rT(t)udu + (2rT(t) _ t'(t))wdw, when v', w are identically unequal to zero.
Remark 1. It is easy to check that the infinitely-dimensional part of the Lie algebra from Theorem 1 consists of operators of the form Y5 only.
The extensions of the operators Yk, k = 1,2,3,4, 5, are
Yi = xdu + dux, Y2 = ertdu, % = xdx + udu + 2wdw,
(1.25)
Y4 = xdx + vdv + 2wdw _ Uxdux, Y5 = Tdt + rTxdx + rTudu + (2rT _ t')wdw-
Therefore, the kernel of the principal Lie algebras for equation (0.1) is one-dimensional with the basis Y2, because the corresponding group only doesn't transform the additional variables v, w and their arguments t, x, ux.
Corollary 1. The kernel of the principal Lie algebras for equation (0.1) is spanned by the operator Xi = ertdu when v', w are identically unequal to zero.
2. Group classification
Consider Lie algebra of projections of operators (1.25) on the subspace of the variables t, x, ux, v, w, i. e. the algebra generated by
Zi = dux, Z2 = vdv _ Uxdux,
x (2.1) Z3 = xdx + 2wdw, Z4 = Tdt + rTxdx + (2rT _ t )wdw.
It is the direct sum of subalgebras (Zi, Z2) and (Z3, Z4) that corresponds to two different functions v and w and their different arguments. Therefore, the subalgebras can be considered separately.
Nonzero structure constants of (Zi,Z2) are ci2 = _1, = 1. Therefore, the inner automorphisms are Ei : e1 = e1 _e2ai, E2 : e1 = ele"2. Here e%, i = 1,2 are the coefficients at Zi respectively in the basis decomposition of Z. If e2 = 0, then e1 = 0 by the acting of Ei. Therefore the optimal system of one-dimensional subalgebras consists of subalgebras with bases Zi and Z2.
In the subalgebra (Z3,Z4) there are no nontrivial inner automorphisms, consequently, the optimal system of one-dimensional subalgebras has a form 6i = {(Z2), (bZ2 + Z4), b € M}. For operators Z from optimal systems we calculate the expressions
Z(V(Ux) _ v) lv=v = 0, Z(W(t, x) _ w) |w=w = 0.
Note, that if Z contains Zi with a nonzero coefficient and doesn't contain Z2, then v' = 0. Such case doesn't correspond to the conditions of Theorem 1. If an operator Z has nonzero coefficients at Zi and at Z3, then by Ei the coefficient at Zi can be equated to zero for equivalent operator to Z. Therefore, the operator Zi can be excluded from further considerations. We have
Zo(V(Ux) _ v)lw=w = _v _ UxV' = 0, V = P/Ux
for arbitrary 3 G R. Further,
Zs(W(t, x) - w)\w=w = xWx - 2W = 0, W = D(t)x2 for arbitrary function D(t). Finally,
(bZ3 + Z4(W(t, x) - w)\w=w = t(t)Wt + (rr(t) + b)xWx - (2rr(t) - r'(t) + 2b)W = 0,
e2rt+2b / Tv rt b f dt
W = e---p(xe-rt-bJ ^)
r (t)
for arbitrary functions p = 0, r = 0.
Optimal system of two-dimensional subalgebras consists of (Z2,Z3), (Z2,bZ3 + Z4), (Z3,Z4). In the first two cases we have the simultaneous specifications for v and w that are already known. In the last one specification we have the form W = yx2/t(t).
For the Lie algebra (Z2, Z3, Z4) the specifications are V = 3/ux, W = yx2It(t). For every basis operator from the optimal systems calculate the projection of the corresponding generator of the group of equivalency transformations on the space of the variables t, x, u. Then Z2 corresponds to pr^txu)(Y4 - Y3) = -udu, for the operator Z3 it will be pr(txu)Y3 = xdx + udu, and pr(t, xu)(bY3 + Y5) = r(t)dt + (rr(t) + b)xdx + (rr(t)u + b)du corresponds to bZ3 + Z4. It implies the next theorem.
Theorem 2. Let v', w be identically unequal to zero i. Then next assertions are true.
1. The principal Lie algebra of the equation
ut +-w(t,x)uxxx 2 + r(xux - u) = 0, 3 = 0,
2(1 - 3xuxx]
ux
is generated by the operators Xi = ertdu, X2 = udu.
2. The principal Lie algebra of the equation
T'(t)e2rt+2bT(tV(xe-rt-bTM)uxx , , , , , , ,
ut + W ^--,2 xx + r(xux - u) = 0, T'(t) = 0, p(z)=0,
2(1 - xv(ux)uxx)
is generated by the operators Xi = ertdu, X2 = t^)dt + ^^7^ + ^ xdx + ^+ ^ udu-
3. The principal Lie algebra of the equation
T' (t)e2rt+2bT (t)p( -rt-bT (t))u
ut + ^---)2-)uxx + r(xux - u) = 0, T'(t) = 0, p(z)=0, 3 = 0,
2(1 - 3xux)
ux
is generated by the operators
Xi = ertdu, X2 = udu, X3 = T7(t)dt + (TT-y + b^jxdx + (Trty+ b^ udu-
4. The principal Lie algebra of the equation
ut + D(t)xx uxx + r(xux - u) = 0, D(t) = 0, 2(1 - xv(ux)uxx)
is generated by the operators
1 r r
Xi = ertdu, X2 = xdx + udu, X3 = Df dt + ^^ xdx + ud^
5. The principal Lie algebra of the equation
Ut + —D(t)x Uxx 2 + r(xUx _ u) = 0, D(t) = 0, 2(1 _ px^)
Ux
is generated by the operators
1 r r
Xi = ert du, X2 = xdx, X3 = udu, X4 = Df dt + Df xdx + Df udn
Remark 2. Theorem 1 and Theorem 2 are valid for the case r = 0.
3. Conclusion
Further Theorem 2 will be applied to the search of exact solutions of the option pricing nonlinear models. Specification W(t,x) = a2x2 as partial case of D(t)x2 corresponds to the Sconbucher— Wilmott model, if r = 0, and to Circar—Papanicolaou model for r = 0.
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