ch.3 solutions

# ch.3 solutions - 3 MODETS OF SECURITIES PRICES IN FINANCIAL...

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3 MODETS OF SECURITIES PRICES IN FINANCIAL MARKETS t 2. In the single-period model show that equation (3.3) holds. Solution: After dividing X(1) : doB(1) * dr,Sr(1) +'. . + dr.",Sri (1) with 1 * r, we get X(t) : do * drS'(t) + . . . + dNSr''(l) Since x(0) : d0B(0) * dr,Sr(0) + . . + dNSN(0) , the previous equation can be written as xir; : x(0) +d1lsl(0) - sr(0)] +... +drlsr(1) - sN(O)] : x(0) +G t 4. Show that equation (3.7) holds in the multiperiod model, or at least for the case of two assets and two periods. Solution: After dividing x(t) : d0(r)B(r) + d1(r)s1(r) +... + dN(t).9N(r) with (1 * ,)' , we get X(t) : d0(t) + dr(r)Sr(r) + . .. + dN(r)SN(r) Since x(0) : d0(0)B(0) +dl(0)s1(0) + . . +dri(0)sr,'(0) , the previous equation can be written as (since B(0) : 1) x(t):x(0)+d0(1)-do(0)+d1(f)^sl(0)-d,(0)s1(0)+. ..+dN(t)sN(f)-dN(O)sN(0) . (3.1) we have to show that the right-hand side is equal to X(0) + G(t). We have c(t) :Ia'(s)AS1(') + . + td1,(s)AS1'(s) s : l s : I Consider, for example, the terms in the sums corresponding to s : 1: dl(1)[s1(1) - s'(o)] + . . . + dr,'(1)[sr"(1) - sr(o)] .l

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Using the self-financing condition divided bv (1 * t)', that is, d o ( 1 ) * d r ( r ) S r ( / ) + . . . * d r v ( r ) S N ( r ) : d o ( f + 1 ) + d r ( t + 1 ) ^ 9 r ( t ) + . . . + d N ( f + 1 ) S , . . ' ( r ) we see that those terms can be written as do(2) -d0(1) +dl(2)s1(1) -d,(0)s1(0) +... +dN(2)sr,'(1) - d,.r(0)s,^r(0) (3 2) Consider now the next terms in G(f
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ch.3 solutions - 3 MODETS OF SECURITIES PRICES IN FINANCIAL...

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