Consider a self-financing trading strategy of a portfolio with a dividend paying asset and a money market

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Consider a self-financing trading strategy of a portfolio with a dividend paying asset and a money market account over the time horizon [0,T ]. Under the risk neutral measure Q, let the dynamics of the asset price St be governed by

d St St (r-q)dt + od Zt,

where q is the dividend yield, q = r. We adopt the trading strategy of holding nt units of the asset at time t, where 

nt -q(T-1)  e-r(T-1)]. F[e-907 - 1 (r = q) T

Let Xt denote the portfolio value at time t, whose dynamics is then given by

dXt = nt dSt +r(Xt - nt St) dt + qn, St dt.

The initial portfolio value X0 is chosen to be

Show that Xo = no So - e-T X. T T = =  [

where Zt = Zt − σt is a Brownian process under Q-measure with eqtSt as the numeraire. Note that the price function of the fixed strike Asian call option with strike X is given by

C fix (So, 0; X) = e-T Eg[max(X7, 0)] = Soer Eq*[max(Y, 0)], with Show that Yo =  - Xo So (r-q)T e-qTe-rT = t

with u(y,T ) = max(y, 0).

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