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Cost of Carry, Convenience Yield, and No-Arbitrage Forward Pricing

Article Quant Q&A · Author: Aksel
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Summary

The document asks why a forward or futures price is expressed as spot multiplied by an exponential of net carry over time, and what convenience yield represents. The response points to standard pricing references and gives a brief no-arbitrage intuition: holding the underlying can provide benefits or income, such as lending income for a stock, which must be reflected in the forward price. Interest rates and dividends also affect the relationship between spot and forward prices.

This is only a concise conceptual answer, not a derivation of the formula. It does not show the replicating trade, specify assumptions about financing or storage, or explain how convenience yield is estimated for physical commodities. The stock-lending example illustrates the role of asset income but is narrower than the general cost-of-carry framework. Readers seeking a proof or practical estimation method would need a fuller treatment.

Key ideas

  • No-arbitrage links spot prices to forward and futures prices through carrying costs and benefits.
  • Income earned from holding an asset can reduce its forward price relative to spot plus financing.
  • Interest rates and dividends influence forward pricing.
  • The response offers intuition but does not derive the formula or explain how to estimate convenience yield.

Tags

FuturesDerivatives pricingArbitrageCarry
Full text
# how to derive the cost of carry formula


# how to derive the cost of carry formula












Can anyone explain why the cost of carry formula looks like this:

$$F_0 = S_0 \cdot e^{(c-y)T}$$ ,where $S_0$ equals the spot price when $T=0$, i.e. today. $c$ denotes the cost of carry and $y$ the convenience yield(?).

So I want to know the mathematical proof of why the Futures function looks like it does. Also I don't really understand how you find the convenience yield and what it is, except that it is the premium you get from having the asset close to the production(?) so that you save time?

## Answer by Ezy (score 1)

https://quant.stackexchange.com/a/43431

This is explained in Hull.

Alternatively you can check this link

https://web.ma.utexas.edu/users/mcudina/m339d-lecture-ten-forwards-pricing.pdf

Essentially the seller of the forward contract earns the income associated to the stock lending activity so it needs to be discounted from the forward price to ensure absence of arbitrage opportunity.

Absence of arbitrage is also what justifies the impact of rate and dividend also to the futures/forward formulae

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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