# Why can a deterministic portfolio only grow at risk free rate

In black scholes derivation we assume that portfolio grows at risk free rate because the process is deterministic, my question is why is it riskfree rate? If i have information about some event in the market(deterministic), i can earn interest over riskfree rate. So why risk free rate?

• Because the process is known to be deterministic by a broad set of market participants. Is the simultaneous existence of two different risk free rates consistent with an economic equilibrium? If investors could invest at a risk free rate of $r_A$ or a risk free rate of $r_B$ where $r_B > r_A$, why would anyone invest at $r_A$? – Matthew Gunn Sep 24 '18 at 15:41

This is how Black Scholes justified it (copied from Black Scholes original paper)

The Black Scholes model assumes that the market is efficient. That is, the pricing of stocks is efficient and therefore you cannot out perform the stock market in the long run. Many but not all economists take this view. The Black Scholes model also assumes a single interest rate. Part of the justification for this assumption is that if long term bonds are offering a greater interest rate than short term bonds the reason for that must be that interest rates are rising and in a rising interest rate environment you do not want to hold long term bonds.

You wrote: If i have information about some event in the market(deterministic), i can earn interest over risk free rate.

One question is, how did you get this information? Did you have inside information? If it is the second then you cannot legally act on it.

The Black Scholes model makes certain assumptions. For example, it assumes that if a certain stock went down yesterday that does not tell us anything about what the stock will do tomorrow. Is that assumption right? I am not sure but it is close to right. When you model something with a set of equations it will not be perfect. You need to make simplifying assumptions. In the case of Black Scholes, one of the assumptions is that the excepted return in a money market and the stock market is the same. I do not really believe that but I would argue that the two are closely related. Therefore, the Black Scholes assumption about interest rates is okay.

I hope that helps.

Bob