The Risk-Free Growth Assumption in Black–Scholes Replication
Summary
The document asks why a deterministic portfolio in the Black–Scholes derivation is assumed to earn the risk-free rate, and whether advance knowledge of a market event could produce a higher return. The answers point to the model’s assumptions about market pricing and a common interest rate, and raise the possibility that event information could be nonpublic. They also stress that mathematical pricing models simplify real markets and may not describe every circumstance.
The exchange offers only a partial explanation: it does not show the replication argument that makes a locally riskless hedged portfolio earn the risk-free rate under the model’s no-arbitrage assumptions. Its remarks about market efficiency, expected returns, and bond maturities are broad and do not establish why the rate follows mathematically. The treatment is therefore useful as a prompt to distinguish model assumptions from practical information advantages, but not as a complete derivation or legal analysis.
Key ideas
- In the Black–Scholes framework, a hedged portfolio is treated as locally riskless under the model’s assumptions.
- No-arbitrage reasoning links the return on a riskless portfolio to the risk-free rate.
- The document raises event information as a challenge but does not work through the replication derivation.
- Its comments on market efficiency and interest rates are informal and do not fully justify the pricing assumption.
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Full text
# Why can a deterministic portfolio only grow at risk free rate # 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? ## Answer by Magic is in the chain (score 1) https://quant.stackexchange.com/a/41875 This is how Black Scholes justified it (copied from Black Scholes original paper) ## Answer by Bob (score 0) https://quant.stackexchange.com/a/41873 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
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