We will review the definition of the self-financing, and give some related examples. One example shows that if all asset prices being only dependent to the same 1-d Brownian motion must have the same Sharpe ratios.
Let the -dimensional price process be given to denote the asset prices in the market. In particular, the th component of the vector , denoted by , is the th asset price at time . Let be an -dimensional process, where its th component stands for the number of shares of the th asset in the investor’s portfolio. This implies that, if the investor’s portfolio consists of shares of th asset at time , then the value at is the sum
By Ito’s formula, the local change of the portfolio value is
In other words, the change of portfolio value consists of two parts: the change due to the stock price change
and the change due to exogenous capital injection
In this below, we will give mathematical definition of strategy and self-financing strategy. Loosely speaking, self-financing strategy is a strategy with zero exogenous injection for all .
Definition 1 Let the -dimensional price process be given stock price process.
- A portfolio strategy (most often simply called a portfolio) is any -adapted -dimensional process , and the value process corresponding to the portfolio is given by (1).
- A portfolio strategy is called self-financing if the value process also satisfies the condition
Usually we assume there is one risk-free asset with short rate , i.e.
and risky assets with drift and volatility , i.e.
where is a Brownian motion under a given filtered probability space . In this context, Sharpe ratio is defined as
where is an adapted process. Then it must hold that for all , otherwise there exists an arbitrage opportunity. In particular, if of the above form satisfies , then for all .
Example 2 Suppose two stocks with positive volatility both follow Ito process dependent on the same BM, then their Sharpe ratio are equal.
Proof: Suppose two asset prices follow, for
Consider self-financing with initial , where for . Then, we have
By Example~1, we conclude for all . So, can be determined from
On the other hand, we have
The above two equations implies .