Valuation Methods of Executive Stock Options

3.3 The optimal trading strategy

This section treats about the Executive's optimal strategy where in this case she is endowed by 1 unit of ESO. By using a similar way that in the EIP section we are going to solve this by Hamilton-JacobiBellman principle. The problem is stated as:

max è ?Èt,T

LG(u,x,s)=0 (39)

Where L is the inifinitesimal generator of (X,S) under P which is defined in the equation (20). By Hamilton-Jacobi-Bellman argument we have to solve:

(u - r) DG

DX + èó2D2G

DX2 + (ñçóS) D2G

DSDX = 0

And then we have a general form of the optimal trading strategy è^**attimeu:

è** = -( u - r) aG

ax + ( ñçóS) a2G

aSax (40)

ó 2 a²G

ax²

Now we can express all the partial derivative functions:

1. aG

ax = - ãer(T -u)G

2. a2G

ax2 = ( ãe r(T -u))2G

3. a2G

axaS = aG aS = ( ãer(T -u)) 2G ap

ap

axaS

by separation of variables argument.

Then we obtain the optimal strategy 9^** as a function of the differential of the Private Price:

9** =

-'y(u - r)e^r(T-^u)G + (ñióS)('yer(T -u))2G ap aS ó^2('ye^r(^T-^u))^2G

((u-r) ~

9** = _e-r(T -u)

ãó²

- ñçS ap

ó aS

9** = 9^* +ö(S,p,ñ,i,ó,'y)

9** = 9^* If ñ < 0 and 9^** <9^* otherwise.

Where ö(S, p, ñ, i, ó, 'y) = - ñçS aS <0. If ñ> 0 and the reverse otherwise

ap

ó

Since^ap

aS = 0 (we will see this assertion later)