Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 3 / c

A previsible strategy selects $({\varphi }_n,{\psi }_n)$ using only ${\mathcal{F}}_{n-1}$, before the price changes from time $n-1$ to time $n$ occur. Thus these holdings cannot exploit future information.

The self-financing portfolio condition values both the old and newly rebalanced holdings at the same time-$n$ prices:Their equality says that rebalancing uses only the proceeds of selling existing assets: there is no external deposit or withdrawal. In particular, the changes in holdings have zero net cost at the current prices. The holdings for the following period may use the newly available information in ${\mathcal{F}}_n$.

At time $n-1$, the self-financing portfolio can be valued using the holdings about to be carried over the next period:Subtracting this from $V_n={\varphi }_n{S}_n+{\psi }_n{B}_n$ gives the gains identityConversely, suppose the gains identity and the valuation $V_n={\varphi }_n{S}_n+{\psi }_n{B}_n$ hold. Rearranging the identity gives $V_{n-1}={\varphi }_n{S}_{n-1}+{\psi }_n{B}_{n-1}$. Comparing with the valuation using $({\varphi }_{n-1},{\psi }_{n-1})$ shows that rebalancing at $n-1$ costs nothing. Applying this at each rebalancing time recovers the self-financing condition.

Assume the bank account stays positive and write ${\tilde{S}}_n=S_n/B_n$ and ${\tilde{V}}_n=V_n/B_n$. Dividing the time-$n$ rebalancing equality by $B_n$ givesThe discounted bank account has constant price one. This is exactly the self-financing equality in the discounted market. Equivalently, subtract the corresponding valuation at $n-1$ to obtain the discounted wealth equation in discrete timeThis algebra uses only positivity of $B_n$, and does not require it to be deterministic.