Reconciling Self-Financing Portfolio Conventions for a Bond and Stock
Summary
This document compares two ways of writing a self-financing portfolio in a discrete-time market with a stock and a money-market account. In one convention, the bond position is expressed as the amount of money invested; in the other, it is expressed as the number of bond units held. Since the bond account grows over time, these quantities differ by the account value at the relevant date.
The accepted explanation relates the conventions by multiplying the unit holding in the account by the accumulated bond value. Under the stated setup, the initial account value is one, so the initial money amount and initial unit count agree. The portfolio-value and rebalancing equations can therefore describe the same economic strategy once holdings are translated consistently. The result is specific to the indexing and timing conventions used in the question; readers must check whether positions refer to money invested or units held before comparing formulas.
Key ideas
- A self-financing condition requires rebalancing to be funded from within the portfolio.
- A bond position can be measured as money invested or as units of the money-market account.
- The account value converts a unit holding into its monetary value at a given time.
- The two formulations agree when their bond-position variables are translated using the accumulated account value.
- The initial money amount equals the initial unit count when the account starts at one.
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# Equivalent Definitions of Self-Financing Portfolio
# Equivalent Definitions of Self-Financing Portfolio
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 by Gordon (score 1, accepted)
https://quant.stackexchange.com/a/21909
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.