Forward Prices, Expected Values, and Risk-Neutral Valuation

# Is there an arbitrage opportunity if the forward price is different from the true expected value of the asset?


# Is there an arbitrage opportunity if the forward price is different from the true expected value of the asset?












Assume an arbitrage-free market. Let's say that the current price of an asset is $100$, its forward price in 1 month is $110$

Is it possible that the true expected value of the asset is not $110$? Sheldon Natenburg in Option Volatility and Pricing says that

> If we assume that the underlying market is arbitrage-free, the expected value for the underlying contract must be equal to the forward price.

Why would this be so?

## Answer by river_rat (score 2)

https://quant.stackexchange.com/a/58549

Arbitrage no, profitable yes. Remember arbitrage implies riskless, and given only the underlying and a bond you can't create a riskless profit. However, in this case just buying the forward and waiting for expiry would give you an expected positive return.

The forward price almost never matches the markets expected value of any given asset. It is one of the reasons people speculate with forwards. For example, in FX currencies with large interest rate differentials (like EM vs USD) tend to have very high forward prices. So selling forward as a carry trade is a very popular strategy as the spot price at maturity is generally lower than the forward price sold.

## Answer by KT8 (score 1)

https://quant.stackexchange.com/a/68717

What I think the quote is meant to say is the following:

> If we assume that the underlying market is arbitrage-free, the expected value for the underlying contract under the risk-neutral probability measure must be equal to the forward price.

The idea behind the risk-neutral (or risk-free) probability measure is that you can hedge it directly in the market, whereas for trying to profit from arbitrage via the real probability is somehow similar to betting.

## Answer by JoshK (score 0)

https://quant.stackexchange.com/a/68722

Just to add a little bit to the explanations. There's a difference between the price of a "forward" and the future price of an item.

Let's say that a nice guitar that you like costs \$500 today if we do the deal now and settle cash now. So what will the guitar be worth in the future? Who knows?

But, say that you want to buy the guitar from me but settle in a month from now. So we agree on a price now - but we only exchange cash flows in a month. That's what people call a "forward". It just means the settlement is later.

In this case maybe having a guitar for another month is really valuable as I'll get a lot of enjoyment out of it. So the market price for the forward will be lower than \$500.

But, what if instead people really want money now. Maybe interest rates are high and they can invest that cash. So if you want to get my guitar in a month and pay me later then the market price for the forward would be higher than \$500.

Does that make sense? The forward price has nothing to do with the future price of the underlying asset. A forward is just saying I'll pay you for it (and take possession of it) later.