Answers for Stochastic Calculus for Finance I

Answers for Stochastic Calculus for Finance I; Steven Shreve VJul 15 2009 Marco Cabral mapcabral@ufrj.br plane section of Mathematics Federal University of Rio de Janeiro July 25, 2009 Chapter 1 ?? 1.1: Since S1 (H) = uS0 , S1 (T ) = dS0 , X1 (H) = ?0 S0 (u ? (1 + r)) and X1 (T ) = ?0 S0 (d ? (1 + r)). accordingly, X1 (H) verifying implies X1 (T ) contradict and vice-versa. ?? 1.2: Check that X1 (H) = 3?0 ? 3/2?0 = ?X1 (T ). Therefore if X1 (H) is positive, X1 (T ) is negative and vice-versa. ?? 1.3: By (1.1.6), V0 = S0 . ?? 1.4: mutatis-mutandis the proof in Theorem 1.2.2 refilling u by d and H by T . ?? 1.5: m either computations. ?? 1.6: We have 1.5 ?V1 = ?0 S1 +(??0 S0 )(1+r). We posit ?0 = ?1/2. So we should sell short 1/2 stocks. ?? 1.7: see last exercise. ?? 1.8: (i) vn (s, y) = 2/5[vn+1 (2s, y + 2s) + vn+1 (s/2, y + s/2)] (ii) v0 (4, 4) = 2.375. One can resist that v2 (16, 28) = 8, v2 (4, 16) = 1.25, v2 (4, 10) = 0.25 and v2 (1, 7) = 0. (iii) ?(s, y) = [vn+1 (2s, y + 2s) ? vn+1 (s/2, y + s/2)]/[2s ? s/2]. ?? 1.9: (i) Vn (w) = 1/(1+rn (w))[pn (w)Vn+1 (wH)+qn (w)Vn+1 (wT )] where pn (w) = (1 + rn (w) ? dn (w))/(un (w) ? dn (w)) and qn (w) = 1 ? pn (w). (ii) ?n (w) = [Vn+1 (wH) ? Vn+1 (wT )]/[Sn+1 (wH) ? Sn+1 (wT )] (iii) p = q = 1/2. V0 = 9.375. One can check that V2 (HH) = 21.25, V2 (HT ) = V2 (T H) = 7.

5 and V2 (T T ) = 1.25. 1 Chapter 2 ?? 2.1: (i) A and A are disjoin and their union is the whole space. Therefore P (A ? A [= whole space]) = 1 = P (A) + P (A ) [they are disjoint]. (ii) it is bounteous to show it for two events A and B. By induction it follows for any ?nite set of events. Since A ? B = A ? (B/A), and this is a disjoint uni on, P (A ? B) = P (A) + P (B/A). Since B/A i! s a subset of B, P (B/A) ? P (B) by de?nition (2.1.5). ?? 2.2: (i) P (S3 = 32) = P (S3 = 0.5) = (1/2)3 = 1/8; P (S3 = 2) = P (S3 = 8) = 3(1/2)3 = 3/8. (ii) ES1 = (1 + r)S0 = 4(1 + r), ES2 = (1 + r)2 S0 = 4(1 + r)2 , ES3 = (1 + 3 r) S0 = 4(1 + r)3 , median(a) rate of growth is 1 +...If you extremity to get a full essay, order it on our website: OrderEssay.net

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