No-Arbitrage Pricing and Why Prices Are Not Expectations

# Understanding No Arbitrage Assumption


# Understanding No Arbitrage Assumption












I knew that If two portfolios have the same profit at maturity time T, then for all prior times $t<T$ the price of the portfolio's must be equal and this comes from no arbitrage assumption.

I am trying to understand the "price" we are referring here, is it an expected price?

For example, when I saw the proof for no early exercise for American call options by replicating a portfolio with a call option ($C_t$) with strike $K$ and zero-coupon bonds pays $K$ at maturity, by no arbitrage assumption, we later claim that

$$C_t + e^{-r(T-t)}K \geq S_t$$

Is this $S_t$ an expectation of the stock price?

The option prce $C_t$ is an expectation discounted and hence implying that the price of stock $S_t$ should also be a discounted expectation. Otherwise, it does not make sense to compare an expectation to a random variables right?

Another example I am thinking is simply having a portfolio that have the same price as stock at $T$ in Black-scholes world. It seems that the "price" of the portfolio and the stock is an expected price because the stock price $S_T$ is itself a random variable and it makes sense to referred to its expectation rather than any single realisation.

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/81415

The statement is that if two investments have the same payoff , in all states of the world , at some future time T , then they have the same price today. So there is no ‘expected price’ concept here.

## Answer by Rylan (score 0)

https://quant.stackexchange.com/a/81419

For a stock, a price is a value we can see and know today. A price does not depend on a distribution.

To use your second example, and I'm going to rephrase a bit, there is a stock $S$, at times $0 \leq t \leq T$ we can observe its price which we call $S_t$. There is also some portfolio $X_t$ which has the property $X_T = S_T$. In a sense we don't even need to know the distribution of $S_T$ to know that $X_t = S_t$. The only thing we need to know about the distribution is that $P(S_T = X_T) = 1$ (hopefully you can convince yourself that there are many different distributions for $S_T$ and $X_T$,indeed with different expectations, that have this property)

The no arbitrage argument from there basically boils down to saying that if $X_t \neq S_t$, then you can buy the cheap one, sell the expensive one, and for every realization of $S_T$ then you're perfectly hedged by your opposing position in $X_T$, so you can pocket the (strictly positive) difference and take on no risk.

When you say the option price is an expectation, well, that's technically true, but it's a bit circular. I personally find it beneficial to think that the option price is the value of the portfolio that hedges the option, and that taking the "expectation" (which is under the risk-neutral measure -- not a measure designed to model "real world probabilities") is simply a trick to find that value.