If a person owning faction stock dies, those stocks go to the owner of the faction. At least in the case of the person not owning more than 51% of the stock. I am unsure if it happens with 51+% of stock (but I imagine it does) or what happens if the dead owner is also the leader (in which case I'd guess it goes to the new leader, but again I am unsure).
"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."
Think the real question would have to be "Does a faction dissolve if the owner dies and it doesn't have a leader/2IC?" because, if a faction dissolves, stock owners get a percentage of the current faction value at time of dissolution based on percentage value of their stock-holding (sorry, but Hal's answer seemed to be regarding if a simple stock owner dies, not the faction owner)