May someone please help me understand the difference between the phi_t and psi_t in Chapter 16 vs Chapter 17? In other words, the difference between the interpretation of holdings phi_t and psi_t in the discrete time sense (Chapter 16) versus in the continuous time sense (Chapter 17). In chapter 16, I understood that we hold phi_t units of the share from time t-1 to time t and psi_t units of cash from time t-1 to time t. 1. Is this correct? That's why we need phi_t and psi_t to be previsible so that we set these up at t-1 in order to replicate the derivative at time t. The notes on page 8 of Chapter 17 say that we hold phi_t and psi_t in the period [t,t+dt], I'm failing to wrap my head around this. 2. Shouldn't we hold phi_t and psi_t in the period [t-dt,t]? I thought the aim of requiring that phi_t and psi_t be previsible was so that we could rebalance the portfolio before time t so that it can replicate the derivative at time t. Then at time t, we rebalance with holdings phi_t+dt and psi_t+dt before time t+dt to replicate the derivative at time t+dt. As we did in the discrete case.
Hi, Firstly sorry for a late reply. 1. This is partially correct. In the discrete case we're aiming to replicate the payoff at some future time U 0<t<U. At time t we need to hold a portfolio of shares and cash that correspond to the value of the derivative at that time. So phi_t and psi_t are the holdings at time t to hold for the next discrete time step before we rebalance our portfolio. We must determine these values in advance i.e at time t-1 or else we're doomed to fail when it comes to replicating the derivative. 2. Then in the same manner in the continuous case we hold the portfolio from time t to t+dt, i.e we're continuously rebalancing our portfolio, but we would be determining the phi_t and psi_t to hold a moment before we need to actually hold them i.e as you have suggested at time t-dt. So I guess the simple answer is that we are holding these assets at time t, for the next discrete or continuous time period, but we determine the values at either time t-1 in discrete case or t-dt in continuous case.
