Option replication

[WinterIsComing]

While constructing a replicating portfolio \(X(t)\) for some option on \([0,T]\), I have that \(X(t_1)=cS(t_1)\) where \(0< t_1 < T\) and \(c\in\mathbb{R}\).
My question is, what should be \(X(0)\)? Presumably \(cS(0)\), but there doesn't seem to be a cash position in the money market. How can such a portfolio be self-financing then? Thanks!

 

[Yike Lu]

You're selling the option for \(X(0)\), that's your cash position to buy \(c S(0)\).

 

[WinterIsComing]

Hi Yike Lu. Thanks for your response. So we know, in the notation of Shreve, that \(dX(t)=\Delta(t)dS(t) + r(X(t)-\Delta(t)S(t)) dt\). In our case, \(\Delta(t) = c, \qquad \forall t\in [0,T]\). We also know (by assumption) that \(X(t_1)=cS(t_1)\). Notice the position at \(t=t_1\) is only in the stock... I still fail to see where an MMA position comes into play.

 

[Yike Lu]

Then the initial price should be \(c S(0)\) discounted at \(r\).

I don't understand your ambiguity with this problem. You sell the option for \(c S(0) e^{-r t_1}\), which is exactly the amount of cash you need to put on your hedge. There doesn't have to be an additional cash position, self financing merely means you never inject cash from outside the system.