Consider a market that has one bond and one stock, with the following price dyna

Question

Consider a market that has one bond and one stock, with the following price dynamics: dB(t) = B(t)r(t)dt; dS(t) = S(t)|1(t)dt + (t)dW(t)]. Let h(t) = (x(t), y(t)) be the portfolio process in which x(t) and y(t) are the amount of money invested in the bond and stock, respectively. Clearly, the total wealth of the investor at time t, denoted by V(t), should be defined by V(t)(t)y(t) a) Suppose that there is no consumption involved. Would you define a "self-financing" portfolio to be one such that dh(t) 0? Why?

Explanation / Answer

A self-financing portfolio is characterised by all exchanges are financed by offering or buying resources in the portfolio. No cash is pulled back or embedded after the intial shaping of the portfolio.

Suppose two asset prices are given follows i= 1,2

dSi(t) = Si(t)(u(t)dt+ a(t)dW(t))

Consider self financing h = (h0,h1,h2) with initial Vh= 0

where hi(t) = 1/Si(t)ai(t) for i = 1,2

dVh(t) = (b1(t)/a1(t) - b2(t)/a2(t) + h0(t)S0(t)r(t))dt

on the other hand,

= V(t) = x(t) + y(t)

= Vh(t)= 1/a1(t) - 1/a2(t)+h0S0(t) = 0

which implies V1(t)= V2(t)

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