# How can we argue that the "economic" risk-neutral argument doesn't introduce arbitrage?

I am wondering why when use the "economic" risk neutral argument, we don't introduce arbitrage. By "economic" I mean an argument that doesn't use stochastic calculus or equivalent martingales measures etc. I will explain in more detail what I mean. I also have a question about the "economic" risk-neutral argument. I will mark both questions in bold.

To make things simple I assume that the stock prices follow a Geometric Brownian Motion, $$S_t=S_0e^{(\mu-\sigma^2/2)t+\sigma B_t}$$, we also asume there is a risk-free interest rate $$r$$. The "economic" risk-neutral argument is this(please feel free to correct the argument if something is missing or it is wrong):

1. We assume that all investors are risk-neutral: they only care about the expected value when they consider an investment.
2. Because of point 1., the prices of the stocks will be pushed in a way such that the expected return is the risk free interest rate. Because for example if the expected return on a stock was higher, a lot of people would want to buy it, pushing the current price up, so we can then not expected that the stock gives us a much bigger return in the future.(Is the last sentence correct, is this how the argument works?, I am very unsure about this because why can we say that expected future return will go down after the current price has gone up?)
3. Assume you have an European option with payoff $$f(S_T)$$. Let $$\tilde{S_T}=S_0e^{(r-\sigma^2/2)T+\sigma B_T}$$. By point 1. and 2. we must have that the start price of the European option is $$e^{-rT}E[f(\tilde{S}_T)]$$.

My last question is: when we move from the risk-neutral world to the real world, how do we know that by selling an European option with start price $$e^{-rT}E[f(\tilde{S}_T)]$$ and payoff $$f(S_t)$$ we have not introduced arbitrage? I know that we can do it if we restrict us to self-financing, admissible trading strategies and use martingale methods. But since the argument above didn't use stochastic calculus, is it possible to answer the second question without invoking stochastic calculus? Does there exist an "economic" answer?

• I think the economic argument must go along the lines of the possibility of hedging the future claim (the option's payoff). Only in that case can we replace the expected return with the risk free rate. The asset itself is still assumed to accrue at $\mu$, on average, in order to reimburse investors for the assumed risk. IIRC, the economic argument requires a) no-arbitrage for a risk-neutral measure to exist, and uniqueness requires the market to be complete, i.e. every claim to be attainable by replication. Aug 2 at 8:56