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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?

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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

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  • $\begingroup$ Okay so it probably wasn't clear but the proof I am looking for is for consumption commodities and not stocks etc... Perhaps I am misunderstanding, but I do not feel like your article explain why they even use e... They just suddenly say that the FV of all the discrete Dividends D, can be written as D*e^(r(T-t)) $\endgroup$ – Aksel Jan 11 at 16:19
  • $\begingroup$ @Aksel It does not matter whether it’s commo or equities. By no arbitrage if the short receives any cash flow until expiry then this has to be discounted from the forward price. The detailed no-arbitrage argument is provided at the end of the doc i linked. $\endgroup$ – Ezy Jan 11 at 16:47
  • $\begingroup$ the use of e is as a method of description of interest rate compounding known as continuous compounding. technically you associate with that description a continuously compounded rate. You can equivalently formulate the equations with a discretely discounted rate i.e. in the form 1/(1+Tr), but then your derivatives just get messy to work with. $\endgroup$ – Attack68 Jan 11 at 17:13
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One could (1) buy the underlying at the spot price and store it at a cost expressed by the yield y, or (2) don't buy the underlying and invest the spot price in an interest bearing account and earn the yield c and then take that amount and buy it at the point in the future.

By arbitrage arguments these two portfolios should be equal.

c and y are expressed here as continuously compounded yields which can be added in the exponent of e. Therefore the forward price today would be the spot price * e^rt, where r=(c-y), the amount of carry you would earn in the interest bearing account, less any carry costs from storing the asset (in equities this would be in the form of foregone dividends.)

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