TAP CHi KHOA HOC DHSP TPHCM Nguyin Chi Long
CAC NGUYEN LI TOAN HOC QUAN TRONG CUA THI TRlTOfNG TAX CHINH
NGUYEN CHi LONG*
TOM TAT
Din cudi thdng 6 ndm 2013, ngdnh cdng nghiip phdi sinh tdi chinh thi gidi, cd gid tri danh nghia khoan 700.000 tl Dollar Mi vd ngdnh cdng nghiep qudn tri danh miic ddu tu, cd le cd gid tri cdn lan han. Do do, todn hgc tdi chinh la ngdnh quart trgng cua todn ung dung. Muc dich cua bdi bdo ndy Id tdm tdt cdc nguyin li todn hgc quan trgng nhdt trong thi trudng tdi chinh.
Tukhda: loan tai chinh, li thuyet dinh gia tai san, thi trudng diy du.
ABSTRACT
The important mathematical principles of financial markets
The derivatives industry worth totals in notional amount more than 700 trillion USD at end-June 2013 and the portfolio management industry is probably even bigger.
Therefore, the financial mathematics is an important branch of applied mathematics. The aim of this article is to summarize the most important mathematical principles in financial markets.
Keywords: Mathematical Finance, Theory of asset pricing, Complete market.
1. Gidi thieu
Hau het cac md hinh toan trong nganh tai chinh deu bit ngudn tir luan an Tiln sT nam 1900 ciia Louis Bachelier (1870-1946) cd ten "Li thuylt diu co tai chmh (Theory de speculation)" tai Dai hpc Sorborme (Paris), dudi su hudng dan cua nha toan hpc Iimg danh Henri Poincare'.
Luan an nay dupc nhieu nha khoa hpc thira nhan la cdng trinh khai sinh ciia nganh toan tai chinh. Tuy nhien cho den hon niia the ky sau, cac nha toan hpc nghien Cliu ling dung trong tai chinh mdi biet den cdng trinh nay. Nam 1953, Harry Markovitz va James Tobin da dua ra li thuyet "Lya chpn danh muc dau tu" tai chinh qua viec phan tich trung binh phuong sai trong li thuy§t xac suat. Nam 1965, cac nha kinh te hpc Paul Samuelson va Henry McKean da chiing td ring gia co phieu chiing khoan tang giam CO tinh ngau nhiSn va md hinh tdt nhat dien ta su thay ddi cua gia co phiSu la mo hinh chuyen dpng Brown hinh hpc. Nhimg cot mde quan trpng, danh dau thdi ki phat trien manh mg ciia toan tai chinh la su ra ddi ciia md hinh Black-Scholes nam 1973 ve tinh hpp li gia ciia cac quyln chpn (Pricing of Options and Corporate Liabilities).
Fisher Black va Myron S. Scholes, cung vdi nha kinh tl hpc lam viec ddc lap Robert Merton dua ra cdng thiic tinh gia cac quyln chpn. Giai Nobel kinh tl 1997 dupc trao cho R. C. Merlon va M. S. Scholes (luc dd Black da mit). Phuong phap ciia hp dS md
• TS, Trudrng Bgi hpc Su phgm TPHCM; Email: nguyen.c [email protected]
Tif lieu tham khao sd 9(75) nam 2015
ducmg cho viec xac dinh gia tri kinh te trong nliiSu linh vuc, tao ra nhiSu loai cong cu tai chinh mdi va tao dieu kien cho viec quan tri nil ro trong xa hoi hieu qua hem. Giai Nobel kinh te nam 2003 danh cho Clive Grange v^ phuang phap phan tich kinh te qua chuoi thai gian va Robert F. Engle III ve mo hinh dao dpng nglu nhien. Nganh cong nghe phai sinh tai chinh the giai uac tinh khoang 700.000 ti do la trong nam 2013 va nganh quan tri danh muc dau tu tai chinh co le co gia tri con cao han, dieu nay cho thay tam quan trong ciia nganh toan hoc tai chinh hien dai.
Tai Viet Nam, toan tai chinh chi dugc quan tam va nghien cuu khoang han 10 nam gan day, nhtmg so nguofi nghien cuu, quy mo, tai lieu con qua nho, chua dap ung dugc yeu cau hoi nhap cua Viet Nam vao nen kinh te the giai. Dac biet la cong tac dao tao chua dap ung dugc nhu cau ve nhan su cua cac cong ty tai chinh va chung khoan thanh lap a Viet Nam. Do do cac thuat ngir, khai niem, cac nguyen li can ban cua toan tai chinh can dugc Iam sang to va trinh bay chac che, co tinh su pham de giup cac sinh vien, hpc vien cao hpc, cac nghien ciiu sinh de tiep can, tir do quan tam nghien ciiu Iinh vuc mai va dat biet quan trpng nay.
2. IVIpt so khai niem cc ban
Chiing ta xet thi truemg tai chinh mot chu ki tdng quat, ma nha dSu tu (NDT) dugc phep 6&u tu trong tai khoan ngan hang (tai khoan tiSt kiem) va mpt tap hgp hiru ban cac co phiSu chiing khoan S',..'.,S''. Gia cua c6 philu thii i, S' tai thai dilm t = 0 la Si, va tai thai t = I la s;. Gia sii rang, tai thcri diem t = I, thS gidi tai chinh co thS a mpt trong k trang thai ffl|,(Bj,...,ffl^ vai xac suit ducmg V(o),)>0, i - 1, ...,k. Dodo, thS giai tai chinh cd khong gian trang thai la:n := [a^, ..., m^}. Gia co philu S'l: n -> K dugc xem nhu mpt bien nglu nhien xac dinh tren khong gian xac su4t (£i, F, P), trong dd F:= {A : A c n } . Vay Sl(ffl) la gia cua c6 philu thti i, tai thdi dilm t = I khi the gidi tai chinh a trang thai ra e i2. DT nhien md hinh tai chinh mpt chu ki la khdng thuc te, nhung nd nhu tl bao trong mdt cku tnic kinh tl, cho phep chiing ta hilu va giai thich nhilu nguyen li quan u-png trong toan tai chinh.
Gia tri thoi gian cua tiln te; 1 USD trong tay hdm nay thi cd gia trj han su ki vpng nhan dugc I USD d mpt ngay nao dd trong tuang lai, do dd viec vay tiln khong the tu do. Ngudi vay phai tra chi phi, dugc ggi la lai suit, cho ngudi cho vay. Gpi r > 0 la lai bdi rdi rac khdng rui ro, ma mdt don vi tiln te dupc giii trong tai khoan ngan hang se tang thanh (1 + r) don vi trong mdt chu ki thdi gian T. Khau hao gia tri tiln theo thoi gian vdi thira so khiu hao c := cho phep ta so sanh gia tri tiln te d nhOng thdi diem khac nhau. Vay mpt sd tiln X tai thdi dilm T cd thi xem nhu la s6 tiln cX ngay hdm nay.
Mot chien liryc kinh doanh (hay mgt phuang an diu tu) la mdt cap (x, H), trong dd H = (Hj,H,,...,H^)eM''*' (doi khi dl dem gian ta vilt mpt chiln luge kinh doanh la
TAP CHi KHOA HOC DHSP TPHCM Nguyen Chi Long
H) la mpt vec ta (N + I)chieu, x la tong sd vdn ban dau (tai thai diem t = 0) va H,,i = l,2,..., N la so lugng cd phieu cua chiing khoan thii i. Gia sii ring S^ = 1 va S" I + r. Cho trudc mdt chien luge kinh doanh (x;H) nhu tren, ta ludn gia sii ring sd tiln cdn lai X-(H,S]|+HJS;+...+H„ Sj) = H„ dugc diu tu khdng riii ro trong tai khoan ngan hang. Vay gia tri V„ (x,H) ciia (x, H) tai thdi dilm t = 0 dugc cho bdi
V.(x,H):=2;H,Si=x
1=0
Gia tri V, (x, H)cua chien lupc kinh doanh (x, H) tai thdi diem t ^ 1 la mpt bien ngau nhien
V , ( X , H ) : = | ; H , S ; (I)
Qua trinh lai (hoac 16) G(x, H) dugc dinh nghia bdi
G(x,H) = H„r + ^H,AS' (2) trong dd AS' := S| -SQ la su thay ddi cua gia cd phieu chimg khoan thir i.
De dang kilm chirng ring V,(x,H) = V„(x,H) + G(x,H)
Qua trinh gia co philu khau hao dugc dinh nghia Sj := Sj and % := cS;
Khi i = 1,..., N. Va qua trinh gia khiu hao tucmg ling ciia (x, H) la V„(x,H):=x and V,(x,H):=H„+2]H,S;
Qua trinh lai khau hao G(x, H) la biln nglu nhien G(x,H):=2]H,AS'
Vdi AS' := s; - Sj. Bing phep tinh dan gian ta cd V„(x,H)-V,(x,H) and V,(x,H) = cV,(x,H) V,(x,H) = V„(x,H) + G(x,H)
Binh nghia 1.
Moi chien lupc kinh doanh (x, H) val H = (H„,H ,H^) duac goi la co ca hoi chenh lech thi gid (hay gol tdl la chenh lech thi gid) neu
/. X = V,(x, H) = 0.
Ty lieu tham khao Sd 9(75) nam 2015
2. V, (x, H) > 0 .
N
3. E[v,(x,H)]~2]P(<i),)V|(x,H)(«)>0. Dieu Iden ndy thi tuang ducmg v&i:
[=1
Tin tgi meO.sao cho V,(x,H)(o))>0.
Ghi chu / Vi c := > 0, nen dl dang suy ra kit qua (x, H) la mpt chiln luge 1 + r
kinh doanh chenh lech thi gia nlu va chi neu 1. V''(x, H) = 0.
2. V'(x, H)>0.
3. E[V'(X, H ) ] > 0 .
Bing sir tinh toan dem gian chiing ta cung cd ket qua sau: mot chien luge kinh (x, H) cd CO hdi chenh lech thi gia nlu va chi neu
G(x,H)>0 (3) E[G(x,H)]>0 (4) Mot each true giac, chien luge diu tu cd eg hdi chenh lech thi gia la chien lupc diu tu khdng gap bit cir nil ro nao, va xac suat kiem dugc lgi nhuan la duang. Sir hien hihj cua mdt ca hdi chenh lech thi gia nhu vay cd the xem la thj trudng tai chinh khdng hieu qua, theo nghTa la chac ehan tai san khdng dugc dinh gia mdt each hpp li. Trong cac thi trudng thuc te, ca hgi chenh lech thi gia rat hiem khi tim thay. Do dd, su vang mat ciia ca hpi chenh lech thi gia se la gia thiet then chdt.
Sir vang mat ciia ca hdi chenh lech thi gia dan den S', triet tieu P-iikn khi Sj, = 0.
Do dd khdng mat tinh tdng quat neu chiing ta gia sir rang
S;>0, i = l,2,...,N (5) Bo de sau dau chimg td rang, khi khdng xuat hien co hpi chenh lech thi gia, thi thi trudng cd tinh chat sau: Mpi dau tu vao tai san nil ro ma nd cd ket qua tot ban dau tu d tai san khdng rui ro, thda man vdi xac suat ducmg, thi chien luge diu tu nay phai chap nhan nhupe diem la cd thi gap rui ro.
Bddi L
Cdc phdt bleu sau day la tuong duong nhau (a) Thi truemg tdi chinh co ca hoi ciienh lech thi gid.
(b) Co mot vecta H' = (H,, ..., H^jsIR" sao cho
H'S, ^2]H,S;>(l + r)H'S„:=(r + l)2]H,S; P-a.s. (6) Vd P[H'S, >(r + l)H'S„]>0
TAP CHl KHOA HOC DHSP TPHCM Nguyin Chi Long
Chung minh.
Dl chiing minh (a) suy ra (b), liy (x, H) voi H = (H„, H ' ) = ( H , , H , , . . . , H„)la phuang an dau tu cd ca hpi chenh lech thi gia, thi
0>V„(x,H)sH„+H'S, (7) Dodd,
H'S,-(l + r)H'S, >H'S.+(l + r)H„»V,(x,H) (8) Vi V, (X,H) khong am P-hiu khip noi va ducmg ngat vdi xac suit duong, do dd ta cung cd kit qua tuong tu cho H'S|-(l + r)H' S„.
Chiing minh (b) suyra (a): Liy H = ( H , , H ' ) vdi H' = ( H | , . . . , H ' ' ) nhutrong(b).
Ta khang dinh rang phucmg an dau tu (x, H) voi Hg = -H|S| la mpt phuang an chenh lech thi gia. That vay, V„(x,H) = H„+H'So= 0 theo dinh nghia. Mat khac, V|(x,H) = H„(l + r)+H'S, = - ( l + r)H'S, + H'S|ma nd khdng am hiu khip nai va
duan; ngat vdi xac suat duang. D Bjnh nghia 2.
Mot do do xdc sudt Q tren (D, F, P) duac gal Id do do rui ro trung tinh hay dg do martingale neu
L Q(o) > 0, Vai moi co envd
2. En fcS] | = SJ,, 1 = 1,2,...,N Dieu kien ndy thi tucmg duang vai vdi dieu ici^n E^[AS'] = 0,i = l,2,...,N.
Vidu 1. (Md hinh thi trudng tai chinh, hai trang thai, mpt chu ki).
Xet md hinh tai chinh rit don gian gdm hai trang thai va mpt chu ki nhu sau
• Mgt tap hpp thdi gian giao dich T := {0;1}. Thdi dilm hien tai la t = 0, thdi diem bit diu giao dich va thdi dilm T = 1 la thdi dilm dao ban, kit thiic giao dich. Tai thdi dilm T = 1 gia sir ring khdng gian tai chinh chi gdm hai trang thai (hay kjch ban): Q :=
{o),, Oj}, «), bilu diln thi trudng tdt, va Oj bilu diln thi trudng xiu.
Dp do xac suit P tren Q xac dinh bdi P(ffl,) = p , ( 0 < p < l ) vaP(a)2)= 1 - p = q.
Ta dinh nghia u : = - i i ^ ; d— '^ " va gia stt ring 0 < d < l < u . Dilu nay cd
So SQ
nghia 14 gia chirng khpan cd thi len khi o, xay ra va giam khi o>^ xay ra, nhung trong mpi trudng hgp u va d vin duang. Ta ndi ring thi trudng khdng chenh lech gia (arbitrage tree) neu khdng cd ca hgi chenh lech thj gia trong md hinh.
Tu- lieu tham khao Sd 9(75) ndm 2015
Menh de 1.
Md hlnh tdi chinh hai trgng thdi, mdt chu ki la khdng chinh lech gid neu vd chi neu d < l + r < u .
Chung minh. Xem [5]
Gpi P^ la tap hpp tit ca cac dp do xac suat rCii ro trung tinh ma nd tuong duong vdi dp do P. Nhic lai ring hai dp do Q va P dupc gpi la tucmg duong nhau (Q ~ P) nlu, Vdi A e F, Q(A) ^ 0 nlu va chi nlu P(A) ^ 0.
3. Nguyen li dinh gia tai san
Dinh li sau day la mpt trong nhiing nguyen li quan trpng nhat cua nganh loan hpc tai chinh
Dinh li 1. (Nguyen If dinh gia tai san)
Md hinh tdi chinh mgt chu ki tong qudt Id khdng chenh lech gid neu vd chi neu
Chdng minh. Xem [3].
4. Nguyen li thi trudng day du
Mpt quyen chon hay mpt quyen tai chinh (a contingent claim) (cdn dupc gpi la san pham phai sinh (derivatives)) la mpt bien ngau nhien X, xac dinh tren khdng gian tai chinh co sd (Q, F, P), bieu dien thu hoach cua nha diu tu tai thdi diem dao ban T=l. Chii y rang mot quyen chpn la mdt hpp ddng tai chinh giira ngudi mua va ngudi ban, ki tai thdi diem t ^ 0. Ngudi ban cam ket se tra cho ngudi mua mdt sd tien X(o)) t^i thdi diem T = 1 neu co G O la trang thai tai chinh liic nay. Do dd, khi xem xet tai thdi diem t = 0 thi thu hoach X la mdt bien ngau nhien, va vin dl dupc quan tam la:
xac dinh, lai thdi diem t = 0, gia tri cua thu hoach X nay.
Dinh nghia 3.
Cho X Id mdt quyen tdi chinh, mgt phuomg dn ddu tu (x, H) dugc ggi Id phucmg dn ddp ting (a replicating strategy) hay mdt bdo hg (a hedge) cho X niu W^{x,H)=X tgi thai diem t = 1.
Dfnh nghia 4.
Mot quyen tdi chinh X dugc ggi Id dgt dugc (attainable) hay mua bdn dugfc (marketable) neu cd mgt phuang dn ddu tu (x, H) bdo hg cho X (nghia Id V,(x,H)=vY). Thj truang tdi chinh dugc ggi Id day dii (complete) niu mdi quyin tdi chinh X, deu cd the tim dugc phuang dn ddu tu (x, H) bdo hd cho X.
Ta CO mpt nguyen li quan trpng tinh loan gia tri ciia quyln tai chinh X tai thdi diem t = 0 (dupc gpi la nguyen li gia tri nii ro trung tinh (Risk neutral valuation principle)), qua dinh li sau
194
TAP CHi KHOA HOC DHSP TPHCM Nguyin Chi Long
Dinh li2. (Risk neutral valuation principle)
Niu thi trudrng tdi chinh mgt chu ki tong qudt khong chenh lech gid, thi gid tri cda mgt quyin tdi chinh mua hdn dugc X tgi thai diem ki hgp ddng, t = 0, co the dirge tinh qua cdng thuc
Vo=E^[cX] (9) trong do Q do xdc sudt rui ro trung tinh bdt ki.
C/?img wm/i. Lay (x, H) la mdt chien lupc diu tu bao hp X, i.e.V,(x,H) = X va QePfj,tac6
Vo-V„=E^[VJ = E.[V,-G]
= E , [ V , ] - E J X H , A S ' ]
= E j y ] - j ; H , E [ A S ' ] - E^tV,]-0 = EJV,] = E^[cX]
Ghi chu 2. Trong thi trudng tai chinh lanh manh (nghia la khdng co chenh lech gia), neu X la mdt quyen tai chinh va (x, H) la phuong an dau tu dap iing cho X, thi x la gia ciia quyen tai chinh X tai thdi diem hien tai t ^ 0.
Vi du 2. (tiep theo vi du 1) Ta xet mpt quyen tai chinh la quyen chpn mua kieu chau Au (viet tac QCMKCA). QCMKCA la mpt hpp dong ki ket giiia ben viet hpp dong (de ban) va ben mua hpp ddng (giii no) tren co sd tai san co ban (nhu chiing khoan, trai phieu, cac loai tien te...), quy dinh ngudi giii hpp dong co quyen, nhung khong bat budc mua tai san trong thdi diem dao han trong tuong lai T vdi mpt gia thyc thi quy dinh trudc la K. Tai san, thdi diem dao han T va gia thyc thi K la cac yeu td quan trpng cua hpp dong nay. Ngudi gitr hpp ddng se lam nhu sau d thdi diem dao han:
NIU gia chiing khoan S, tai thdi diem T ^ 1 cao hon K thi ngudi giii hpp d6ng se mua cua ngudi vilt hpp d6ng va dem ban ngay lai cho thi trudng tai chinh vdi gia S, va thu dupc mon lpi la S, - K. Neu gia chung khoan khoan S, tai thdi diem T ^ 1 thap hon K thi ngudi giii hpp dong se khong thuc thi, vi don gian la gia ben ngoai thi trudng re hon. Trong trudng hpp nay, ngudi giQ hpp ddng khdng thu dupc mon lpi nao. Vi li do tren nen ta cd thi xem QMKCA la mot tai san ma lpi nhuan ciia no tai thdi diem dao h a n T ^ 1 la max(S|-K,0).Cau hdi tu nhien la: Gia cua mpt QMKCA tai thdi diem t ^ 0 la bao nhieu? Ldi dap ciia cau hdi nay la mpt ap dung cua nguyen li dap ling de bao hp (Replication Principle) ma ta xem xet sau day.
Gia sii ta cd mot quyen tai chinh tdng quat hon mot QCMKCA vda xet, nic la san phim cd dang h( S,), trong dd h: M ^ M la mpt ham s6 sao cho h( S,) cung la mot biln ngau nhien, QMKCA co thi chpn mot ham rieng cho h nhu h{x):=max(x-K,0).Cd rit nhieu kha nang khac nhau khi chpn ham h dl co nhilu quyln tai chinh khac nhau.
Tu' lieu tham khao Sd 9(75) nam 2015
Theo Dinh nghia 3, mdt phuang an dau tu bao hp cho h(S,) la mpt chien luge kinh dpanh (x. H) (vdi H:=(H„,H,)) thda man dilu kien V,(x, H) = h(S,).dilu kien nay thi tucmg duang vdi
H,(l + r)+H,S,(ffl,) = h(S,(«,)) (10) H„(l + r) + H,S,(ffl,) = h(S,(»,)) (11) Hay
H„+cH,S,(ffl,) = ch(S,(«,)) (12) H„+cH,S,(o,) = ch(S,(ffl,)) (13) trong dd c := la thira sd khau hao.
* r + 1 Menh de 2.
Trong mo hinh Idi chinh mot chu kl, hai trang thdl vd khong chenh lech gid; gid strh(S,) la mot quyen tai chinh vd (x, H) Id phucmg dn ddu tu bdo ho cho lj(S,), thi x la gid ciia quyen tdi chinh h(5,) tal th&i diem 1 = 0.
Chung minh. Tir (12) va (13) ta cd H h(S,(<a,))-h(S,(«3))
S,(«,)-S,(o,) ' Ta dinh nghia
p : = ^ (15) u - d
Vi d < 1 + r < u suy ra ring 0 < p < 1 va ta cd
l-p:=Hziz£ (,,)
c(pS,(ffl,) + (l-p)S,(a',)) = S„ (17) Ta nhan phucmg trinh (12) vdi p, phucmg trinh (13) vdi 1 - p va cpng vl ddi ve, ta dupc
x + H,[c(pS,(ffl,) + ( l - p ) S , ( « , ) ) - S , J = c[ph(S,(<y,)) + (l-p)h(S,(«,))] (18) Tir (17) suyra
X = c[ph(S,(a),)) + (l-p)h(S,(ffl,))) (19) Vi u - d * 0, nen ta ludn luon cd the tim dugc phuang an dau tu bao hp cho mdt quyen tai chinh trong md hinh mpt chu ki, hai trang thai. Md hinh tai chinh cd tinh chit nay dugc ggi la md hinh diy du (complete) va ngugc lai, ta gpi la md hinh tai chinh
TAP CHl KHOA HOC DHSP TPHCM Nguyen Chi Long
khdng diy dii. Cdng thirc (14) thudng dugc gpi la cdng thirc bao hd Delta (Delta Hedging Formula). Trong md hinh tai chinh khdng day du, ta khdng thi dimg kl thuat dinh gia phai sinh theo nguyen li dap ling de bao hp.
Dieu dang luu y la gia x cua quyen tai chinh theo cdng thirc tren thi khdng phu thudc vao xac suit p hay q:= 1 - p cua su xuit hien trang thai tai chinh m, hoac » , . Dac biet, lay P la mgt dp do xac suat khac tren f2 : - {(W,, o^} vdi
P(ffl,)-p, P(ffl,)-l-p
Thi X chinh la gia tri ki vpng lgi nhuan da khau hao, liy theo dp dp xac suit mdi P, nghia la
x = Ep[ch(S|)] (20) Dp do xac suat P la dp do xac suat nii ro trung tinh, vi dudi dp do nay, gia quyen tai chinh chi phu thudc vao ki vpng ciia lgi nhuan ma khdng chi phdi bdi riti ro nao.
Bd ii 3.
Gid sic mo hlnh tdi chinh mot chu ki long qudt la idnh mgnh (bay khong co chenh lech gid); thi thi tru&ng ndy la ddy dii khi vd chi khi so trgng thai cua thi tru&ng trong Q bdng v&i so vec ta doc lap tuyen tinh trong i^S^ S|, ... ,S^}, nghia lama Iran k hdng, (N + 1) cot A cho b&i
'5° s;((i),) ... S,*'(ffl,)"
Sj Si(a),) ... S"(«,) Sj S;(;BJ ... s;'(ft)J_
phdi CO hgng Id k.
Chung minh.
Theo kit qua tir dai sd tuyln tinh, ma tran A cd hang la k khi va chi khi, vdi mdi X e IR\ phuang trinh AH = X cd mdt nghiem duy nhit H e M"*'. Mat khac ta cd
Sf Si(ffl,) S» S!(ffl,)
sr(«,) sr(«,)
V,(x,H)(ffl,) V,(x,H)(ffl,) S° SKfflj) ... S['(fflOjLHNj [V,(x,H)(a)J_
Dilu nay chimg td ring tim mdt phuang an diu tu biio hp cho quyen tai chinh X
la tuang duang vdi viec giai he phucmg trinh AH = X. D BS d& 4. (Farkas Lemma)
Cho ma Iran A, m hdng, n cot vd mot vec ta cot m chiiu b, thi hogc Id
AX = b, x>0, xeK" (21)
Ty lieu tham khao Sd 9(75) nam 2015
CO mot nghiem, hode
b^•<0, A V > 0 . ye?."' (22) CO mot nghiem, nhung kliong the cd hai cimg luc xdy ra.
Chung minh. (ed the xem [4]).
Ghl chii 3.
Tir Bd de Farkas. ta cd the kiem chirng de dang rang neu he (21) vd nghiem, thi tdn tai y e R"" sao cho
b ^ > 0 , and A^y = 0 (23) Bd ai S.
Gid su rdng md hinh tdl ehinh mot chu ki long qudt la lanh mgnh, thi quyen tdi chinh Xld mua bdn duac neu vd chi neu E^ [cX] lay ciing mot gid tri v&i moi Q G P^,
Chung minh. Gia sir quyen tai chinh X la hudn ban dugc, thi theo Dinh li 2 va Ghi chu 2
E [cX] = V„ (constant)
ddi vdi mgi do do xac suat nil ro trung tinh Q. Xgugc lai, gia sir rang X Ichdng budn ban dugc. ta se chiing minh rang cd hai do do xac suat nil ro trung tinh Q, va Q-,tren Q sao cho
E;[cX]*E ,[cX]
Neu X khdng budn ban dupc thi he (21) khdng cd nghiem, theo Bd dl 4, Ghi chii 3.cdmptvecta n = ( n , n t ) thda n A = 0 va n x > 0 . Cho trudc mdt dp do xac suat nii ro trung tinh Q, tren n. Bay gid ta xem Q, dugc dinh nghia bdi
C:(f'',):=2,('»,) + /in,s;'
vdi /.> 0 sao cho Q,(»,)>0vdi mpi o^eCi. Vi tinh chit FI A = 0, din din
Z ;;(«.) :=Z:,(«)+^Zn,s:=i
Do dd Q, cung la dp do xac suat nii rp trung tinh tren Q. Mat khac,
E;icx] = |;:;,(«,)[cx(ffl)i
= Zc:.(ffl,)X(o,)+;.2;n,x(ffl,)
= E;[cX] + AnX Tir XnX > 0 su> ra ring
E-[cX]#E^ [cX]
198
TAP CHi KHOA HOC DHSP TPHCM Nguyin Chi Long
Binh lis.
Gid sir mo hinh thi tru&ng tdi chinh mot chu ki long qudt khong chenh lech gid, thi thi tru&ng la ddy dd neu vd chi neu co dung mot do do xdc sudt rui ro trung tinh, nghia la \V„\ = L
Chdng minh. (=>): Gia sir thi trudng Ichdng chenh lech gia va day du, theo nguyen li dinh gia tai san, tdn tai mgt do do xac suat nii ro trung tinh. Gia sir rang P^g chiia hai dp do xac suat riii ro trung tinh Q, va Q^ • Ta se chimg minh rang Q, = 0 , .
Vdi mdi i = 1,2,..., k, lay quyen tai chinh X dinh nghia bdi fsj, co = o>,
X'(«) =
[0, CO=^Q),
Thi X' la quyln tai chinh vdi moi 1=1,2 k. Hon nira Q,('»,) = E „ [ ^ X ' ] = E,J^X]=Q,(ffl,).
Dodd, Q, = Q , .
(«=): Gia sit ring thi trudng la khdng chenh lech gia va cd dung mdt do do xac suit riii ro trung tinh. Ta se chiing td ring thi trudng la day dii, nghia la liy X la mot quyln tai chinh bit ki, ta chimg minh X la mua ban dupc. Dieu nay dung. That vay, vdi gia thilt thi truemg cd dung mdt dp do xac suat rui ro trung tinh Q, thi
Eg [cX] cd mgt gia tri duy nhit. Tir Bd dl 5, suy ra ring X la mua ban dugc. Djnh li
dugc chirng minh. ° TAI LIEU THAM KHAO
1. Nguyen Van Hiru va Vuong Quan Hoang (2007), Cdc phuang phdp todn hoc trong tdl chinh, Nxb Dai hgc Qudc gia Ha Npi.
2. Nguyen Chi Long (2008), "Xac suat thdng ke va qua trinh ngau nhien", Nxb Dai hpc Qudc gia TP Hd Chi Minh, Tai ban lin L
3. Nguyen Chi Long (2010), "Nguyen li can ban dinh gia tai san trong thi trudng tai chinh", Tgp chi Khoa hoc Truang Dgi hoc Suphgm TPHCM, 21(55), tr. 38-51.
4. Nguyin Chi Long (2011), "Bd de Farkas va ap dung trong thi truemg tai chinh", Tgp chi Khoa hoc Trudng Dgi hoc Suphgm TPHCM, 27(61), tr. 41-53.
5. Nguyen Chi Long (2011), "Dinh gia tai san trong mo hinh nhi thirc", Sd chuyen dl ciia Trudng Dai hpe Sai Gdn: Hoi thdo Khoa hoc Qudc ti Gidl tich vd Todn ifng rfM«g,tr. 513-525.
6. Nguyen Chi Long (2011), "Mo hinh dinh gia tai san tu ban". Tap chi Khoa hoc Trudng Bgi hoc Suphgm TPHCM, 30(64) tr. 25-41.
