3._kafli_C_Z - 3 MODETS OF SECURITIES PRICES IN FINANCIAL...

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Unformatted text preview: 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 ) : d o * d r S ' ( t ) + . . . + d N S r ' ' ( l ) Since x(0) : d0B(0) * dr,Sr(0) + .. + dNSN(0) , the previous equation can be written as x i r ; : x ( 0 ) + d 1 l s l ( 0 ) - s r ( 0 ) ] + . . . + d r l s r ( 1 ) - s N ( 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 ) : d 0 ( t ) + d r ( r ) S r ( r ) + . . . + d N ( r ) S N ( 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 ) + d 0 ( 1 ) - d o ( 0 ) + d 1 ( f ) ^ s l ( 0 ) - d , ( 0 ) s 1 ( 0 ) + . . . + d N ( t ) s N ( f ) - d N ( O ) s N ( 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 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 d o ( 2 ) - d 0 ( 1 ) + d l ( 2 ) s 1 ( 1 ) - d , ( 0 ) s 1 ( 0 )...
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3._kafli_C_Z - 3 MODETS OF SECURITIES PRICES IN FINANCIAL...

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