Equivalent Definitions of Self-Financing Portfolio

Question

Consider a multi-period model with $t=0,...,T$. Suppose there is a bond with $B_0=1$ and $B_t=(1+R)^t$ and a stock with $S_0=s_0$ and

$$ S_{t+1}=S_t\,\xi_{t+1}, $$

with $\xi_t$ iid random variables. I indicate with $(\alpha_t,\beta_t)$ the predicable components of the portfolio. I have found two different definitions of self-financing portfolio that I would like to re-conciliate. In the book by Tomas Bjork ("Arbitrage Theory in Continuous Time") it is said that the value of the portfolio at time $t$ is

$$ V_t^{(\alpha,\beta)} = \alpha_t\,S_t+\beta_t\,(1+R) $$

and the self-financing condition is expressed as

$$ \alpha_t\,S_t+\beta_t\,(1+R) = \alpha_{t+1}\,S_t+\beta_{t+1}. $$ This is quite intuitive form me since, in Bjork, $\alpha_t$ (resp. $\beta_t$) is, by definition, the amount of money we invest in the stock (resp. in the bond) at time $t-1$ and keep up to time $t$. So if I buy $\beta_t$ unit of the bond at time $t-1$ then I gain $\beta_t\,(1+R)$ at time $t$.

Nevertheless, in the book by Andrea Pascucci ("PDE and Martingale Methods in Option Pricing") it is said that the value of the portfolio is

$$ V_t^{(\alpha,\beta)} = \alpha_t\,S_t+\beta_t\,B_t = \alpha_t\,S_t+\beta_t\,(1+R)^t $$

and the self-financing condition is expressed as

$$ V_t^{(\alpha,\beta)} =\alpha_t\,S_t+\beta_t\,(1+R)^t = \alpha_{t+1}\,S_t+\beta_{t+1}\,(1+R)^t. $$ Pascucci define $\alpha_t$ (resp. $\beta_t$) as the amount of the asset $S$ (resp. of the bond $B$) held in the portfolio during the period $[t-1,t]$. Are the two definitions equivalent? I am pretty sure that the the solution is in the fact that in Bjork it is defined as the amount invested while in Pascucci as the amount held. Nevertheless I miss which kind of relationship is in between the two.

Answer 1

You have already answered your question. In Bjork, the $\beta$ terms represent the value in the money market, or deposit, account, while the $\beta$ terms in Andrea Pascucci represent the units in the money market account. Then, the two definitions are basically the same. More specifically, \begin{align*} \beta^{Bjork}_{t+1} = \beta^{Andrea\, Pascucci}_{t+1} (1+R)^t. \end{align*} We assume that the initial face money market account value is 1. Then the initial money market account value, in Bjork, and the initial units of the money market account, in Andrea Pascucci, are the same.