Why Forward and Spot Prices Share Volatility Under Constant Rates

Summary

The document explains why an asset forward price has the same percentage volatility as its spot price when the interest rate is constant and the cost of carry equals that rate. In this setting, the forward price is the spot price multiplied by a deterministic time-dependent factor. Applying Itô’s lemma to that relationship shows that the forward’s proportional change is driven by the same Brownian shock and volatility as the spot.

The explanation is a short derivation rather than empirical analysis. Its conclusion depends on the stated assumptions, including a fixed rate and the particular carry relationship. It addresses a forward price, despite the question referring to futures; futures and forwards need not behave identically when rates are stochastic or other market features matter.

Key ideas

• With constant rates and the stated carry assumption, the forward price equals spot multiplied by a deterministic factor.
• Applying Itô’s lemma shows the forward price has the same percentage volatility as spot.
• The result concerns forwards and relies on fixed interest rates and the specified cost of carry.
• Futures may differ from forwards when rates or other relevant market conditions vary.

Tags

# Volatility of Futures


# Volatility of Futures












Apparently:

Under a constant interest rate, the futures price is given by a deterministic time function times the asset price (I think I understand this). This means that the volatility of the futures price should be the same as that of the underlying asset price.

Not really sure how this is true. Is there any more intuitive explanation as to why this would hold?

## Answer by Bram (score 2, accepted)

https://quant.stackexchange.com/a/14435

To avoid confusion (the term futures price could be a type for "future price" or "future's price"), it seems to me that you are talking about the forward price of an asset for which the cost of carry is equal to the interest rate. In that case, indeed, with a fixed interest rate r and an spot S, the forward price F for a time T is given by $F=Se^{r(T-t)}$. If $S$ is governed by the SDE: $\frac{dS}{S} = rdt+\sigma dW$ and by noting that $\frac{\partial F}{\partial t} =-re^{r(T-t)}S$, $\frac{\partial F}{\partial S} = e^{r(T-t)}$ and $\frac{\partial^2 F}{\partial S^2} = 0$, we can apply Ito's lemma. Doing so results after some simple algebra, in the following process for $F$:

$$dF = e^{r(T-t)}S\sigma dW$$

which is equivalent to:

$$\frac{dF}{F} = \sigma dW$$

This is probably what you had in mind?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.