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T = (0.014, 0.23) = tT r = trT + (1 - t)rf t A 1 An c T = n c ri = 0.09 + 700ciT ciT A 1 An i T u P{ = 50 + 2u + 2(4 - u)} = C4 (3/5)u(2/5)4-u R 1 Rn 48.13 A 1 An tw(t) = w(t)t + tw ri Ai C [1.90, 210.13] Ri Ai C0 = 19. i Ri S(6) N(10, 25) Ai (i, ri) 6% (, r) A1... An c 1... n i -c Ai P (t) T = 1Aa +... + nAn P (T )er(T -t) - max{K - S(T ), 0} 0.56 R 0. = e-rt(up + dq) = = -R = 1R1 +... + nRn, = 2. 1 +... + n = 1 r = M R = D R n n e(r1 -r2)t-d 2 1 e-rT = b(, p; n)e-r T + b(, p; n)e-r T p = (, r) (, r) 2 2 u-d 1-r2)t p = pue-(r A 1 An r n r = iri.

i= FS(T )(x) S(T ) n 2 S-2r/ = ijijij, S r i,j= r B ij 0. Ri Rj S() A B - + C 0.08 0.05 r 0.0. A Q = 10 AB 0 0.1 0.2 0.3 0.4 A B AB = AB AB -1 = r V V2 AB = -ACB 0 AB P{V1 -V2 > 0} P{V1 -V2 > AB ACB 0.25Q} P{V1 - V2 > 0.5Q} P{V1 - V2 > Q} P{V1 - V2 < -0.25Q} ArB + BrA P{V1 - V2 < -0.5Q} P{V1 - V2 < -Q} r =.

A + B = 0. -1 < AB < 1 (, r) (r) r = 20% = (, r) A = 1.75 B = -0.05 C = -0. 1 = (, r ) 1 1 = 0.85 2 = 0.24 0 = -0. r > r, <. 1 3 = A + B 11 = 0.25A + 0.75B E = 0.5A + 0.5B (r) rf = (1 - 0.9)/0.9 = 0.09 r1 = 0.18 r2 = 0.32 1 = 0. (r) 2 = 0.14 Cov(A1, A2) = -0. (a) (b) r r E _ a b c = e-r(T -t) Ster(T -t)(d1) - K(d2) = St(d1) - K(d2)e-r(T -t), l2 * l T t f = S -e-r(T -t) S = S(t) t f E a2r - b2(r2 + 2) U(, r) (, r) U(, r) $40 15% r 30% S(T ) r() U(, r) = C U S(T ) = t + X. U(, r) < 0 Ur(, r) > S(T ) U r (S(T ) - S0)/S0 = S(T )/S0 - 1 F(x) S(T ) F(x) = P - 1 < x = r S U = P S(T ) < (x + 1)S0 = FS(T ) (x + 1)S0, U = U(, r) (a) (b) r r Er ln S(T ) - m K - m T K (x)dx = K P S(T ) > K = K P >, T - t T - t T K * l m = ln S0 + (r - 2/2)T (ln S(T ) - m)/ T - t l F 0 F d ln S(T ) - m K (x)dx = K P > -d2 = K 1 - (-d2), T - t K Er F T F (x) T F T Er - (-d2) = (d2) [F T ) K (x)dx = K(d2) K m = ln S0 + (r - 2/2)(T - t) s = T - t (ln x-m)x(x)dx = e- 2sdx.

2s K K = (ln x - m)/s - s F = (0, r F ) ln K - ln S0 - (r - 2/2)(T - t) - T - t = -d1.

F T T T - t F T Er T (+s)2 2 s+m x(x)dx = e- 2 es+s +md = e e- 2 d.

r K -d1 -d r 0 -d1 1 - (-d1) (d1) r r - r0 2 2 -1 (T -t)+ln S0+ r- (T -t) 2 = V ( - r0 I ), r x(x)dx = e (d1) = S0er(T -t)(d1).

- - -( - r0 I )T V ( - r0 I ) K r r - = (1,..., n)T V = n Cov(Ri, Rj) Ri M max S(T ) - K, 0 = S0er(T -t)(d1) - K(d2).

i,j= - r = (r1,..., rn)T r0 I = (1,..., 1)T ln(S(t)/K) + (r + 2/2)(T - t) d1 =, T - t ln(S(t)/K) + (r - 2/2)(T - t) d2 = = d1 - T - t.

T - t f(S(t), t) = c(t) F P () T K F T T P () = Ke-r(T -)(-d2) - S()(-d1). Ai rT - rF ri = rF + Cov(Ri, RT ).

T c(t) = e-r(T -t) M max S(T ) - K, 0, Cov(Ri, RT ) Ri Ai RT T y = rF + (rT - rF )/T x ln S(T ) N ln S(0) + (r - 2/2)(T - t), 2(T - t).

T r Ai Cov(Ri, RT ) = ln m ri s2 Ai (ln x-m) 1 (x) = e- 2s2sx Cov(Ri, RT ) < 0, x > 0 x 0 Ai c(t) max S(T ) - K, 0 Cov(Ri,RT ) Ai iT = T M max S(T ) - K, 0 = (x - K)(x)dx, K Ri = i + iT RT + i, (x) S(T ) i iT M max S(T ) - K, 0 = x((x)dx - K (x)dx. K K i Ai Cov(Ri, Rj) = iT jT T.

A B A B 0.1 0. 0.2 A B A B = 0.25A+ [,T ] 0.75B R RB A T S(T ) - [,T ] A B F () RA 0.1 0.2 RB 0 0. P 0.5 0.5 P 0.5 0. R RB A F () = e-r(T - ) M(S(T ) - [,T ]) = = e-r(T - ) M S(T ) - [,T ]e-r(T - ).

rA = M RA = 0.1 0.5 + 0.2 0.5 = 0.15, rB = M RB = 0 0.5 + 0.3 0.5 = 0.15.

S(T ) S()er(T - ) A B F () = S() - [,T ]e-r(T - ).

RA RB\ 0.1 0. 0 0 0. 0.3 0.5 [,T ] = S()er(T - ) = 0.25A + 0.75B r = 0.25rA + 0.75rB = 0.15, 2 = 0.252A + 0.752B + 2 0.25 0.75 Cov(RA, RB).

c(t) = S(t)(d1) - Ke-r(T -t)(d2), -1 (RA-M RA)2 (RB-M RB)2 f/S 0.052 0.152 1 f/S 2 A = M(RA - M RA)2 = 0.0025, B = 0.0225.

f = -f + S.

RARB 0 0.03 0.5 S M(RARB) = 0.015 Cov(RA, RB) = M(RARB) t M RA M RB = 0.0075 0. f = -f + S.

S A(0.1, 0.1) B(0.2, 0.3) F (0, 0.09) f 1 2f r = 0.15 AB = 0. = - - 2S2 t.

t 2 S r 2 = jjijij - min i,j= = rt.

0 + iri = r.

i=f 1 2f f - - 2S2 t = -r f - S t. t 2 S2 S Cov(A, B) = 1 0. ABAB = 0.01 V = 0.01 1 -0. -1 1 V = det V = 0.f f 1 2f det V -0.01 + rS + 2S2 = rf.

t S 2 S T r - r0 I = (0.01, 0, 21) 0. -1 - V ( - r0 I ) = r 0. 0. f t = T S - 0. - -1 ( - r0 I )T V ( - r0 I ) = (0.01, 0.21) = 0.r r 0. 0. = (0.0127, 0.2861) f(S(T ), T ) = max{S(T ) - K, 0}, U(, r) = 0.6r - r2 - U(, r) = C = f(S(T ), T ) = max{K - S(T ), 0}, -(r - 0.3)2 - 2 = C - 0. r 7% p q p + q = A(0.1, 0.1) B(0.2, 0.3) S(t) AB = 0. t 40 = 20% U(, r) = 0.6r - r2 - t = = tA+(1 t)B K = 41 r 9% r = trA + (1 - t)rB = 0.3 - 0.2t r% 2 = t2A + (1 - t)2B + 2t(1 - t)0.5AB = $1 = 30 $1 = = 0.03t2 - 0.06t + 0.04.

r r U(, r) r2 r n = tA + (1 - t)B dr dr d k = = : =.

d dt dt 0.3 - 0.3t U(, r) = 0.6r - r2 - 2 -(r - 0.3)2 - 2 = C - 0. dr/d S(t) t f(S(t), t) dr w(t) 2(r - 0.3) + 2 = 0.

(t) d f f 1 2f f dr/d = /(0.3 - r) f = S w + S + 2S2 + t.

k S S 2 s2 t w =.

0.3 - r 0.3 - 0.3t k S(T ) T r = 0.3 - 0.2t r t = 3/ $ (0.141, 0.214) $3 $ A1 A2 T = 0.2A1+0.8A Cov(R1, RT ) = 200 Cov(R2, RT ) = T T = Cov(RT, RT ) = Cov(0.2R1 + 0.8R2, RT ) T = 0.2 Cov(R1RT ) + 0.8 Cov(R2, RT ), 38 42 T 216 14. 41 r A B C 7% rA = 20% rB = 10% rC = S(t) t 30% $40 = 20% 1/2 A B 0.2 c(t) 1/2 0 0. $42 r A B 9% 40 A B = 0.5(A+ 39 B) 20% r A B 7% 32 30 A 0.12 0. B 0.20 0. = tA + (1 - t)B t 0 0.1 0. 32 30 0.3 0.4 0.5 0.6 0, 7 0.8 0.9 AB -1 -0.5 0 0.5 40 20% A B A B $ B r 7% u = e t d = 1/u = 0. B t = 0.4 u = 1.1348 d = 0. n S(t + t) t + t S(t)u = A 45.3936 S(t)d = 35. 0.5 max S(t + t) - K, cu = max S(t)u - K, 0 = 4.3936 cd = max S(t)d - K, 0 = (, r) ert - d u - ert c = e-rt(cup + cdq), p =, q =.

A B u - d u - d r rB 0.1 0.4 B 0.2 0.A A p = 0. c = 2. A A2 0.1 0.2 0.1 0. 0. $ $ 10% r A B rA = 0. 8% 3% rB = 0.3 A = 0.1 B = 0, 4 AB = 0.5 F 0.6600 rF = 0. $0. r = 0. A A2 0.1 0.1 $ 0.1 0. 8% 1% F 0. A1 A F 0.2 S(t) A B C (, r) T K P (t) A = (0.1, 0.1) B = (0.14, 0.20) C = (0.3, 0.3) P (t) Ke-r(T -t).

A C u d ln(S(t + t)) - ln(S(t)) 2t S(t + t) U(r, ) = r - - 2 r r [0, 0.4] S(t + t) uS(t) dS(t) P p q A B rA rB 0.1 0.4 B 0.2 0.A U(r, ) = r - r2 - 2 r D(S(t + t)) = M S(t + t)2 - (M S(t + t))2 = pu2S2+ [0, 0.4] A B + qd2S2 - (puS + qdS)2 = S(t)2(pu2 + qd2 - p2u2 - q2d2 - 2pqud).

r rB 0.2 0.3 B A A 0.1 0.2 AB A B 0. p + q = U(r, ) = r - r2 - 2 r D(S(t + t)) = S(t)2(u2pq + d2pq - 2udpq) = pqS2(u - d)2. [0, 0.4] A B p r rB 0.2 0.4 B A A 0.1 0.3 AB D(S(t + t)) = S(t)2 ert - e- t e t - ert.

A B A A F 1 0. ert - e- t = t + (r - 2)t + O (t)3/2, q = 0.6 q = 0.3 q = 0. F 1.00 1.00 1.e t - ert = t - (r - 2)t + O (t)3/2.

A1 1.05 1.20 1. A2 1.30 1.00 1. 0.9 A1 D(S(t + t)) = S(t)22t + O(t)2.

1.20 0. S(t+t) S(t+t)-S(t) ln = ln 1 + rf F r r2 S(t) S(t) A A2 Cov(R1, R2) 1 1 O (t)2 S(t + t)/S(t) - A1 AO(t)2 = 1R1 + 2 RD ln(S(t + t)) - ln S(t) = 2t.

2 2 2 2 r = 1r1 + 2r2, = 11 + 2 2 + 212 Cov(R1, R2).

S(t) t $40 = 20% s(1, 2) c(t) F = 1A1 + 2 A2 s = r -t) Ke-r(T rF / 1r1 + 2r2 - rF 1,c + Ke-r(T -t) p + S(t) - max, 2 2 2 2 11 + 2 2 + 212 Cov(R1, R2) 1 + 2 = 1.

S(T ), S(T ) > K max{S(T ) - K, 0} + K =.

K, S K S(T ), S(T ) > K max{K - S(T ), 0} + S(T ) =.

K, S(T ) K 9% T 0.2 0. t A1(1, r1 ) A2(2, r2) F = (0, rf ) K A1 A2 T F S(T ) T S(T ) K S(T ) > max S(T ) - 100, 0 - 1r1 + 2r2 - rf L = + (1 + 2 - 1), T K T - ST 2 2 2 T = 1 1 + 22 + 212 Cov(R1, R2) 90 95 100 105 8% T S(T ) T T -1 = T 11 + 2 Cov(R1, R2) 1 S(T ) K S(T ) > 100 T 1 r1T - 11 + 2 Cov(R1, R2) (1r1 + 2r2 - rf ) 5e0.082/12 5.L T = +.

max S(T ) - 100, 0 - 5.1 T 1 r1T - 11 + 2 Cov(R1, R2) (1r1 + 2r2 - rf ) T + = 0, () = e-q(T - )S(), (T ) = S(T ) + [,T ] - S(T ) = [,T ]. T r1T - 22 + 1 Cov(R1, R2) (1r1 + 2r2 - rf ) (t) T + = 0, T (T ) = ()er(T - ) T 1 + 2 = 1.

[,T ] = S()e-q(T - )er(T - ).

(t) 2 T 1 (t) -C 1r1 + 2r2 rT C 0 t [, T ] 2 2 r1T - 11 + 2 Cov(R1, R2) (rT - rf ) + T = 0, (t) = e-q(T -t)S(t) - F (t).

2 2 r2T - 22 + 1 Cov(R1, R2) (rT - rf ) + T = 0.

(t) = e-q(T -t)S(t) - S(t)e(r-q)(T -t) - [,T ] e-r(T -t) = = [,T ]e-r(T -t) > [,T ]e-r(T -). 2 2 2 r11T + r22T - T (rT - rf ) + T = 0.

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