Discrete self financing strategy

Question

Let $H$ be an investment strategy in a discrete price model. Proof $H$ is self financing if and only if the following holds for the portfolio process $P_t$: $$P_t = P_0 + \sum_{s=1}^tH_{s-1}(X_s-X_{s-1}) \quad \forall t=1, \dots,T$$

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$\textbf{Definition:}$ $H_t$ self financing strategy $\iff (\Delta H_t)^TX_{t-1}=0\ \forall t=1, \dots,T$.

We did not define what a portfolio process is so I guess the portfolio value process is meant here: $V=V(H)=H^TX$ with prices $X$.

I tried $$(\Delta H_t)^TX_{t-1}=0 \iff H_t^TX_{t-1}=H_{t-1}^TX_{t-1} \iff \Delta(H_t^T)_t=\Delta(H\circ X)_t \\ \iff H_t^TX_t=H_0^TX_0+(H\circ X)_t$$

$\forall t=1,...,T$. With $P_t:=H_t^TX_t$ I get $$P_t = P_0 + \sum_{s=1}^tH_s(X_s-X_{s-1})\ \forall t=1,...,T$$ but I need $H_{s-1}$ instead of $H_s$. I also tried integration by parts and got the same result... How do I proof the claim?

Thank you in advance!

Answer 1

It is enough to consider one time step. The portfolio value changes by \begin{align} P_t-P_{t-1}&=H_t^\top X_t-H^\top _{t-1}X_{t-1}\\ &=\underbrace{H_t^\top(X_t-X_{t-1})}_{\textstyle(A)}+\underbrace{(H_t^\top-H^\top_{t-1})X_{t-1}}_{\textstyle (B)}\,. \end{align} The term $(B)$ reflects changes in the portfolio value that are due to reallocations and/or withdrawals resp. additions of assets to the portfolio with newly added funds. The first term is the value change due to the assets having new values at the end of the period.

When the strategy $H$ is self-financing only reallocations are allowed, that is, $B$ must be zero. Nothing is allowed that adds or withdraws value from the porfolio. An example with two assets having prices $$ X^1_{t-1}=100\,,\quad X^2_{t-1}=80 $$ and current allocations $$ H^1_{t-1}=8\,,\quad H^2_{t-1}=10\,. $$ Then you are allowed to sell five shares of the second asset to buy four of the first: $$ H^1_t-H^1_{t-1}=4\,,\quad H^2_t-H^2_{t-1}=-5\,. $$ Conclusion: $H$ is self-financing if and only if one of the two equivalent conditions hold

• $P_t-P_{t-1}=H_t^\top(X_t-X_{t-1})\,,$

• $(H_t^\top-H^\top_{t-1})X_{t-1}=0\,.$