Forward Prices, Risk-Neutral Expectations, and Real-World Returns
Summary
The note examines a claim that an arbitrage-free forward price equals the expected future price of its underlying security. It distinguishes the expectation used for pricing from a real-world forecast: under the risk-neutral pricing measure, the expected future price is consistent with the forward price, while under the real-world measure, investors may expect a different value because the asset carries risk.
The explanation points to the Capital Asset Pricing Model as an example of why real-world expected returns can include compensation for risk. It does not derive the forward-pricing relationship or discuss assumptions such as dividends, funding, or convenience yields. Its central lesson is that a forward price is not generally a forecast of the underlying’s realized or real-world expected value; the equality applies in the risk-neutral framework.
Key ideas
- A forward price corresponds to a risk-neutral expectation under the pricing measure.
- The real-world expected future price can differ because investors require compensation for risk.
- The Capital Asset Pricing Model illustrates how risk can affect expected returns.
- Arbitrage-free pricing does not imply that a forward price is a real-world forecast.
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Full text
# Why must the forward price be equal to the expected value for an underlying security # Why must the forward price be equal to the expected value for an underlying security In page 59 of his book Option Volatility and Pricing, Natenberg argues that the forward price of an underlying security is essentially the market's consensus expected value for that security, otherwise an arbitrage opportunity exists: > In a sense, the marketplace must think that the forward price is the most likely future value for the underlying contract. If we assume that the underlying market is arbitrage-free, the expected value for the underlying contract must be equal to the forward price. So if the two month forward price for a stock with a spot value of `$100` with interest rates at `12%` is `$102`, that's the expected value of that security in two months. This assumption makes no sense to me. The forward price of a stock takes into account only interest rates and time and is only a measure of what I'd pay now to receive the stock in two months. It seems to me that the expected value of that stock in two months should take into account volatility and other factors. Why does this assumption hold? ## Answer by dm63 (score 1, accepted) https://quant.stackexchange.com/a/76383 Strictly speaking the book is not accurate. The forward price is equal to the expected future price only in the risk- neutral world (usually denoted by the pricing measure $Q$). In the real world (often denoted by $P$) the expected value of the stock at some future date is generally higher, to account for the riskiness of the stock (see the Capital Asset Pricing Model for example).
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.