To show that if the market contains a risk-free asset with price $B_t$ for $t \in [0, T]$, then this price must satisfy $B_t = B_0 e^{rt}$ or else arbitrage exists, we can analyze the behavior of a risk-free asset under continuous compounding and argue why deviating from this form leads to arbitrage opportunities.

Step-by-Step Solution

1. Define the Continuously Compounded Interest Rate: Let the continuously compounded interest rate be denoted by $r$. In a market where this rate applies, an initial investment $B_0$ in a risk-free asset (such as a bond) should grow according to the continuously compounded formula.

2. Determine the Price Function for the Risk-Free Asset: If the asset's price $B_t$ grows continuously at the risk-free rate $r$, its value at any time $t$ should be given by: $B_t = B_0 e^{rt}$ This formula reflects that the asset earns interest continuously at rate $r$, which is the standard behavior in such markets.

3. Assume $B_t$ Does Not Follow This Formula and Show Arbitrage: Suppose that $B_t$ does not follow the formula $B_t = B_0 e^{rt}$. In this case, there exists some alternative growth pattern or structure for $B_t$.

4. Construct an Arbitrage Opportunity: If $B_t \neq B_0 e^{rt}$, we can compare the actual growth of $B_t$ to what we would get by investing in an asset that follows $B_t = B_0 e^{rt}$. An arbitrage opportunity arises if one can make a risk-free profit by exploiting the difference between the actual price $B_t$ and the expected compounded value $B_0 e^{rt}$.

   • Case 1: If $B_t > B_0 e^{rt}$, an investor could:

     • Borrow money at the continuously compounded rate $r$.
     • Invest in the asset with price $B_t$, which grows faster than $e^{rt}$, leading to a guaranteed profit.

   • Case 2: If $B_t < B_0 e^{rt}$, an investor could:

     • Sell the risk-free asset short (i.e., sell it at the higher expected compounded price $B_0 e^{rt}$ and buy it back at the lower price $B_t$).
     • This setup would also guarantee a risk-free profit, resulting in arbitrage.

5. Conclusion: For there to be no arbitrage, the price of the risk-free asset must match the continuously compounded growth formula $B_t = B_0 e^{rt}$ for all $t \in [0, T]$. Any deviation from this formula allows an arbitrage opportunity, as it would imply a mismatch between the actual and expected growth of a risk-free investment.

Final Answer:

Thus, we have shown that in a market with continuous compounding, the price of a risk-free asset must satisfy: $B_t = B_0 e^{rt}$ for all $t \in [0, T]$ to prevent arbitrage.

Do you need more details, or have any questions?

---

1. What is arbitrage, and why is it significant in financial markets?
2. How does continuously compounded interest differ from discrete compounding?
3. Can you explain how to construct an arbitrage opportunity in more detail?
4. Why is the formula $B_t = B_0 e^{rt}$ the only solution that avoids arbitrage?
5. How would this argument change if interest were compounded discretely rather than continuously?

**