MATLAB FINANCIAL DERIVATIVES TOOLBOX Manuel d'utilisateur télécharger pdf (Page 116)

115

Since this equals zero, we add it to the objective function that we want to

minimize without affecting the results. Finally, we minimize the quadratic

function without any constraints. Thus, we have to minimize a function, f (so

it can also be solved with methods that we have seen before). So, after

substitution we have:

))r(E)ww()r(Ew)r(EwR(?s)ww(w

s)ww(wswws)ww(swswf

CBABBAABCBAB

ACBAAABBA

BBAA

+−−−−++−+

+−+++−++=

112

1221

222222

We need to solve this problem to find the weights of the three assets in the

portfolio. We thus first find the solution to

w ,

w and ?. The way to solve

the above problem is now easy. We take the partial derivatives each time in

respect to one of the three unknowns and set them equal to zero, and we will

derive a set of three equations with three unknowns. The first is the partial

derivative of f w.r.t

w , the second w.r.t

w and the third w.r.t the

Langrangean multiplier ?.

The first equation gives:

224222222

−−−−++−++

)r(E?)r(E?

sws)ww(sws)ww(sw

BCBACBAABB

The second equation gives:

422222222

−−−+−+−++

)r(E?)r(E?

s)ww(swsws)ww(sw

BCBAACAABA

The third equation gives:

01 =−−−−− )r(E)ww()r(Ew)r(EwR

CBABBAA

To work with this case study, assume the following data:

1 2 ... 111 112 113 114 115 116 117 118 119

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