金融中的数学方法第五章

? ? (continuous-time fi nance)? ?Merton?1990? ?1992? ? ? ? ? ? ? ? ? ? ?Brown? ? ?BlackScholes? ? ? ? 5.1?Brown? ?(n)n=0,1,2,.,? ? 0= 0,n+1= n+ n,n = 1,2,., ?n, n = 1,2,.,?1/2?1? ? ? ? ? 1. En = 0, Varn = n, n = 0,1,2,.,N? 2.?m n j k 0,?m n?j k? 3.?m n = j k,?m n?j k? ? ? ?Brown?(Bt)t0? ? ?t?B0= 0,? 1. (?)? ?s 0? ? VarB2s=VarB2s Bs+ Bs B0 = VarB2s Bs + VarBs B0 =2VarBs B0 = 2VarBs. ?k,?VarBks = kVarBs,? ? ? 0,?VarB = VarB1.? ? ?t 0, VarBt = tVarB1.? ? ? ? ?Brown? ? ? ? ? ? ? (t,) = Bt() = EeiBt. ? (t + s,) (t,)=EeiBt+s EeiBt =E eiB t ei(B t+sBt) 1 =EeiBtEei(Bt+sBt) 1(?) =EeiBtEei(BsB0) 1(?) =(t,)EeiBs 1. ? EeiBs 1 = 1 2 2VarBs + o(s) = 1 2 2 + o(s). ? ?s 0,?s 0,? t (t,) = 1 2 2(t,). ?t? ? R?(0,) = E1 = 1. ? (t,) = Bt() = e 1 2 2t. 244 ? ? ? ? e 1 2 2t = Z eix e x2 2t 2tdx, ?Bt?0,?t?N(0,t).? ?Brown ? ? Brown?(Bt)?(V s t )t0= (Bt+sBt)t0?EV s t = 0, EV s t+V s t = 0, s, R(), s, ?R()? ?R(0) = VarBs = s.? ft?Eft = ?Eft+ft = r()?t? ?(? ?)?Brown? ? ? ? ? ? ? ?gt= asint + bcost? ? ? (?)?a,b? ?a?b ? ? ? ?(?)? ? ? ? ? ? ? ? ? ? ? ?(?)? ? ? (?)? ? ?(? ?)? Brown? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(?) ?Brown? ? ? ? ?Brown? ? ? ? ?Brown? ?-? ? ? ? ? ? ? ?Brown?(?)? ?s 0,? ?Ft?-? ? 0 t1 t2= F0 Ft1 Ft2 F. ?-? ?- ? ? ? ?t 0,T),? ? ? ? Ft= t0? ?t, Xt?Ft-? ?-? ? ? ?-? ? ?(Xs,s 6 t) = Ft(?Xs(s 6 t)? ?-?)? ?(Xt)?-? ? ?Ft? ? ?Ft? ? ?-? ?-? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(?)? ? ? ? ? ?t? ?Ft? ? ?(?)? ? ? ? ? ? ? ? ? ?Brown? 250 ? ?(,F,P)?(Bt)t0?Brown? ? 1. (?)? ?s t,?Bt Bs?Fs= (Bu, u 6 s)? ? 2. (?)? ?s 0?Brown? ?B0= 0, EBt = 0,?VarBt = tVarB1.?VarB1 = 1,? ?Brown?Brown? ? ?BtB0? ?t,?2t? ? ? ? ?0 6 t1 0? ?(? ?t, E|Mt| 0?Ft-?Brown? ? 1. (Bt)?Ft-? 2. (B2 t t)?Ft-? 3. exp(Bt (2/2)t)?Ft-? ?1.?s t?BtBs?Fs? ?EBtBs|Fs = EBt Bs = 0.?(Bt)?Ft-?. 251 2.?s 0,P)?Ft-? ?Brown? ?(Ht)? R t 0 HsdBs.?Lebesgue? ? ? ?(Ht)?(Ht)06t6T? ? Ht= i,t (ti1,ti,0 = t0 t10?Ft- Brown? ?(Xt)06t6T?It o? t 0,T,Xt= X0+ Z t 0 Ksds + Z t 0 HsdBsa.s., 254 ?X0?F0-?(Kt)?(Ht)?Ft? ? Z T 0 |Ks|ds +a.s., Z T 0 H2 sds + a.s. ? ? ?5.4?(Mt)06t6T? ? Mt= Z t 0 Ksds, Z T 0 |Ks|ds +a.s., ? t 6 T,Mt= 0a.s. ?It o? ? ? Xt= X0+ Z t 0 Ksds + Z t 0 HsdBs= X0 0+ Z t 0 K0 sds + Z t 0 H0 sdBs, ?X0, X0 0 ? ? Ks= K0 s,? X0= X0 0 a.s.? ?Hs= H0 s. It o?It o? ? ? Z T 0 |b(s,Xs)|ds +a.s., Z T 0 |(s,Xs)|2ds +a.s., x 7 b(t,x)?x 7 (t,x)?Lipschitz?2?b? ?EZ2 +,? ?(Xt)06t6T, ?E sup 06s6T|Xs|2 0, 0,? ? k = a2 log x K ,x = K exp ? k a2 . 261 ? ? y(s,v) = K 2s Z v exp ?(v w) a2 1 exp ? w 2 4s dw. ?w = q2s? ? y(s,v) = K 2 Z v/2s exp v + q2s a2 1 # exp ? q 2 2 dq. ?s(t)?v(t,x)? ? v 2s= a2 log x K + ? r 2 2 (T t) 1 a2T t = log x K + ? r 2 2 (T t) T t = d2. ? ? a2 v + q2s = a2 ( a2 ! log x K + ? r 2 2 (T t) + qa2T t ) =log x K + ? r 2 2 (T t) + qt T. ?y(s,v)? ? y(s(t),v(t,x) = K 2 Z + d2 exp log x K + ? r 2 2 (T t) + qT t q2 2 dq K 2 Z + d2 exp ? q 2 2 dq. ? u(t,x)=f(t)y(s(t),v(t,x) = xexp h 2 2 (T t) i 2 Z + d2 exp(qT t)exp ? q 2 2 dq Ke r(Tt) 2 Z + d2 exp ? q 2 2 dq. 262 ?p = q? ? u(t,x) = x 2 Z d2 exp ?2 2 (T t)pT t p2 2 dpKer(Tt)N(d2). ?z = p + T t? ? x 2 Z d2 exp ?2 2 (T t)pT t p2 2 dp = x 2 Z d2 exp ? 1 2 p + T t 2 dp = x 2 Z d2+Tt exp ? z 2 2 dz = xN(d1). ? u(t,x) = xN(d1) Ker(Tt)N(d2). ? d1= log x K + ? r + 2 2 (T t) T t , d2= log x K + ? r 1 2 2 (T t) T t , N(d) = 1 2 Z d exp ? 1 2z 2 dz. ? ?BlackScholes? 5.1.2? ? ?(?)? ? ? ? ? ?(? ?)? ? ? ? ? ? ?4? ? ?(term structure of interest rates)? ? 4? ? ? ? 263 ? ?(? ?)? ?(yield curve) y = y(s)? ? ? ? ? 3?6?1?2?30? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(coupon bond)? ? ? ?(zero-coupon bond)? ?y = y(s)?s? y? ? ? ? ?s = 0,1,.,T,?s = 0? ?s = 1? ?(1+y(1)? ?s = 2? ?(1+y(2)2? ? ? ?s? ?(1 + y(s)s? ?s = 1? ?s = 0?1/(1+y(1)?s = 2? ?s = 0? 1/(1 + y(2)2? ?s? ?s = 0? ?1/(1+y(s)s? ?(discount bond,? ?)? ?y(s)? ? ? ?s = 1,2,.,T? ? ? ?1/(1+y(1), 1/(1+y(2)2, . 1/(1+y(s)s, ., 1/(1+y(T)T ? ? ? ? ? ? ? x?s = T? ?K? ? s = 1,2.,T 1? ?(?) R? ?x? ? p( x) = T1 X s=1 R (1 + y(s)s + K (1 + y(T)T . ?T?T?x = (x1,x2,.,xT)? ? ?y = y(s)? ? p(x) = T X s=1 xs (1 + y(s)s . ? ? ? ? 264 ? ?y(s)? ? ? (?)?r(t)?t? ? ?V = V (t)? dV (t) dt = r(t)V (t),V (0) = V0. ? V (t) = V0e Rt 0 r(s)ds. ? ?V (T) = VT,? V (t) = VTe RT t r(s)ds. ? ?t( 0;i = 1,2,.,K. ?B1 t,B 2 t,.,B K t ?Brown? ?(B1 t,B 2 t,.,B D t )?D ?Brown? ?-?(Ft)?D?Brown ? ?Brown? ? ? ? ? ? ?r(t), ?(?) 1(t),2(t),.,K(t)? ?(ij(t)i=1,.,K;j=1,.,D?t? ? ?(Ft)? ? ? ?It o? ? Z T 0 |r(t)| + K X i=1 |i(t)| + K X i=1 D X j=1 |ij(t)|2 ! dt 0, a.s. 271 ? ?4.8?4.9,? ? ? ?4.8?4.9? ? ? ?4.64.8? ? ? ? ? ? ? ? ? ?4.8?4.9? ? ? ?(CM,L1,L)? ? ? ? ?(CM,L1,L)?(no free lunch)? clA0 T L1 +(,F,P) = 0. ? ? ? ? ? ? ? ? ? ?5.2?(CM;L1,L)? ?P,?(Sk t)? ? ? 0 6 t 0;?M(t)?(?x) ? ?(tame)? ? ? ?,?M(t) 0,? ? ? 0,? ? ? ? ?(?)? ?I. Karatzas?S. E. Shreve 1998? ?6? ?5.3?CM? ? ?D?:0,T RD,? (t) r(t)eK= (t)(t),t 0,T, a.s. ? ?,? Z T 0 k(t)k2dt ,a.s. ? E exp Z T 0 (t)TdB(t) 1 2 Z T 0 k(t)k2dt ? = 1, ?CM? 6Karatzas, I. and S. E. Shreve, Methods of Mathematical Finance, Springer Verlag, New York, 1998. 274 ? ? ? ? ?5.2? ?1.20?3? ? ? ?(t),? ?(t)T(t) = 0,? ?(t)T(t) r(t)eK) = 0,? ? ? ? ?D?K? ? ? ?K? ? ?K? ? ? ? ? ?D?K? ? ? (t)?D? ? ? ? ? ? ? ? ? ? ? ? ? ?5.3? ? ? ? Z(t) = exp Z t 0 (s)TdB(s) 1 2 Z t 0 k(s)k2ds ,0 6 t 6 T. ? ? ? ? ? (,F) (FT= F)? ?P:?A F, P(A) = EZ(T)1A, ?1A?A? ? 1A() = 1, A, 0, / A. ? E exp 1 2 Z T 0 k(t)k2dt ? , ? ? ? ?Girsanov? ?I. V. Girsanov?1962? ? ?5.4?(t)06t6T? R T 0 k(s)k2ds , a.s.?D? ? ?(Z(t)06t6T? Z(t) = exp ? Z t 0 (s)TdB(s) 1 2 Z t 0 k(s)k2ds , 275 ?(B(t)06t6T?D?Brown? ?Z(T)? ?P? ?W(t) = B(t)+ R t 0 (s)ds?D? ?(W(t)06t6T?D?Brown? ? ? ? ? ? ? ? ? ? ?0? ?1? ? f(x) = 1 2exp(x2/2),? exp ? x 1 2 2 1 2exp(x2/2) = 1 2exp ? (x + ) 2 2 ? ? ? ?Girsanov?5.3?Brown? ? ?Girsanov?5.4? ?5.3?5.4? ?Brown?W(t) = B(t) + R t 0 (s)ds,? (t)X(t)=x + Z t 0 (s)(s)T(s) r(s)eK)ds + (s)dB(s) =x + Z t 0 (s)(s)T(s)(s)ds + (s)dB(s) =x + Z t 0 (s)(s)T(s)dW(s) ? ?X(t) = Si(t)? ? d(t)Si(t) = (t)Si(t) D X j=1 ij(t)dWj(t),i = 1,2,.,K. ?(t) = 1/S0(t),? dSi(t) = Si(t) r(t)dt + D X j=1 ij(t)dWj(t) # ,i = 1,2,.,K. ? ? E exp 1 2 Z T 0 k(t)k2dt ? ? ?(t)Si(t)06t6T? ?P? ?P? 276 ? ?5.3? ? ?CM? ? ? ? ? ? ? ? ? CM? ?(?)? ? ? ? ? ? ? ? ? ? ?r(), (), ()?t? ? ?3.5? ?BlackScholes? ? ? ?BlackScholes? ? ?BlackScholes? ? ? ? ?t?S0 t ? ?(?)? dS0 t = rS0 tdt, S0 0 = 1, ?S0 t = ert.? ? ? dSt= St(dt + dBt), ?(Bt)?Brown? ? ? ? = (t)06t6T= (H0 t,Ht), ?(?Brown? ?(Bt)?-?Ft)? ?H0 t ?Ht? ?t? ?t? Vt() = H0 tS 0 t + HtSt. ? dVt() = H0 tdS 0 t + HtdSt. ? ?(Vt()?It o? ? ? Z T 0 |H0 t|dt + a.s., Z T 0 H2 tdt + a.s. ? ? Z T 0 H0 tdS 0 t = Z T 0 H0 tre rtdt ? Z T 0 HtdSt= Z T 0 (HtSt)dt + Z T 0 HtStdBt 277 ? ? ? = (H0 t,Ht)? ? 1. R T 0 |H0 t|dt + R T 0 |Ht|2dt +a.s., 2.?t 0,T,? H0 tS 0 t + HtSt= H0 0S 0 0 + H0S0+ Z t 0 H0 udS 0 u+ Z t 0 HudSua.s. ? ? St= ertSt? ? ? ?5.5? = (H0 t,Ht)06t6T ? ? R T 0 |H0 t|dt+ R T 0 |Ht|2dt +a.s. Vt() = H0 tS 0 t +HtSt, Vt() = ertVt().? ? t 0,T, Vt() = V0() + Z t 0 HudSua.s. ? 6= 0? ? ?BlackSholes?5.3, ? = ( r)/? ? ? (St) = (ertSt) = ? S0exp ? ( r)t 2 2 t + Bt ? 6= r? ?Girsanov?5.4,? ? ?(St)? ? ? ?(Z(t)?(?)? ? ? Wt= Bt+ ( r)t/,?P?Brown ? ?