# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Forward contracts

Forward on Stocks

Just like we could talk about interest rates and FX in the future, we are tempted to believe that perhaps we can do something of that sort with common stocks too. Well, indeed we can.

Locking-in a stock price in the future is not too different from locking-in the price of foreign exchange in the future. If anything, this is even simpler. By now, you should now how we would proceed in this case.

Since we want a stock in the future, what we would do is borrow money and buy the stock today. And if we don’t sell the stock we would have the stock for as long as we want. So by buying the stock today (say, at price $S_0$) and ‘carrying’ it forward, assuming no dividends, we have ensured that we’ll indeed have the stock at the end of the period as we wanted.

So what is our payable at the end of 1 period? Well, since we borrowed money to buy the stock, our payable is just the money that we owe to the bank at the end of the period. If the one-period rate is $r$, the our ‘forward price’ for holding the stock at the end of the period is just the amount that we return to the bank, that is:

$\boxed{F_S = \displaystyle S_0(1 + r)}$

And again, because people can do so, financial institutions offer forward contracts on stocks and indices that allows buyers/sellers to lock-in their buy/sell price in the future.

If the stock paid dividends too, then there is an extra income from the stock, and the forward price will have to be a little lesser than what it would have been otherwise, as ‘carrying’ a stock now gives dividends, and in that case we’ll have:

$F_S = \displaystyle (S_0 - D)(1 + r)$

where $D$ represents the present value of all dividends through the period.

Or if we use continuous compounding we would write it as:

$F_S = \displaystyle S_0e^{r - d}$

In general, for a contract expiring at time $t$, using continuous compounding, the forward price of stock will be:

$\boxed{F_S = \displaystyle S_0e^{(r - d)t}}$

(By the way, getting into a forward contract to buy a stock refers to ‘going long’, and getting into one to sell a stock refers to ‘going short’.)

Forward on Commodities

While financial institutions do provide a very useful service by allowing people to lock-in the prices of traded securities by offering forward contracts, as we have seen, assuming away transaction costs people could mighty well do that on their own. So, loosely speaking, at least for big players, a forward contract is redundant security as they can ‘replicate’ the payoff from a forward contract by buying/selling the stock today and borrowing/lending (just like we found the ‘fair’ price of FX and stock above).

The social utility of having forward contracts in the world really becomes apparent in the case of commodities.

Let’s take the example of a farmer and a baker in a small village. Our farmer sells wheat and the baker uses that wheat to bake and sell bread. Farmer worries that when the time comes to sell wheat the price would be low, and the baker worries the price will be high. If wheat were like a common stock, and if the baker wanted to buy wheat in the future, the baker would, just like in the example of forward contract on stock, buy stock today and ‘carry’ it forward and he would have locked-in his price.

But wheat is not a like a share, is it? First the wheat has to be available today for him to buy in the spot market and ‘carry’ it forward. If the farmer hasn’t harvested the wheat yet he can’t even do that. Even if the wheat has been harvested he may not have enough physical space or the wherewithal to store the wheat to ‘carry’ it forward.

Similarly, even if the farmer could ‘short’ the wheat today, if there is no wheat there is no way she can do that. The only way the baker and the farmer could achieve their objective of locking-in the price is by entering into a forward contract and agreeing on a price today.

And they can (and will) do so because when the baker and the farmer enter into the contract, their meeting together is a zero-sum game. Say, both agree that 100 is about the ‘fair’ price for wheat at the end of one period. Farmer worries the price of wheat at the end of 1 period is going to fall below 100, and the baker worries that at the end of the period the price is going to be more that 100.

While by entering into a forward contract they have reduce the uncertainty for them, if the actual price of wheat were to be more than 100 (say, 120), the farmer will probably curse herself for agreeing to sell the wheat at 100 when the actual price turned out to be 120. But the loss of farmer (by 20) is the gain of the baker. Baker will be thanking his stars that he entered into this contract, because now he is able to buy the wheat at 100 when the market price is 120. The farmer’s loss is exactly the baker’s gain in this case.

How do we find the forward price in this case – the ‘fair’ price they should agree upon when the contract is entered into?

It turns out we can still use the insights we developed when we were talking about common stock. While it maybe difficult for the baker to store the wheat at his place, he may yet hire the services of a professional warehouse and pay the storage cost (assuming the wheat is available in the market), and the baker will proceed exactly the way we did in the case of the common stock.

The baker would borrow money today, take the delivery of wheat today, store it in the professional warehouse and take it out at the end of the period at which time he’ll return the money to the bank. That is the baker will literally ‘carry forward’ the wheat. And the forward price for wheat would be just what we found in the case of stock, but now we’ll have an extra cost of physically ”carrying’ the wheat into the future, that is:

$F_C = \displaystyle S_0(1 + r) + \mbox{Storage Cost}$

But is it all? Can not baker use the wheat in the mean-time, if he runs out of his inventory’s ‘safety stock’ (the minimum amount of wheat he needs at all times)?

Yes, by holding wheat physically the baker is getting the ‘convenience’ of using the wheat if he needs to in emergency. Obviously, assuming no dividends, there is no direct utility to holding a share certificate. For one, we can’t ‘consume’ our share certificate the way we can consume wheat. So to that extent the ‘carrying’ cost of wheat is lower than it would have been otherwise. And we revise the forward price for wheat as:

$\boxed{F_C = \displaystyle S_0(1 + r) + \mbox{Storage Cost - Convenience Yield}}$

If we assume that the storage cost and convenience yield can be measured as a percentage of the amount/value of wheat to be stored and that loss of wheat due to any wastage is counted as part of the storage cost, using continuous compounding we could write this more succintly as:

$\boxed{F_C = \displaystyle S_0 e^{r + x - y}}$

where $x$ is the per-unit storage cost and $y$ is the per-unit convenience yield – all continuously compounded. As you may guess, typically it’s not at all obvious how to quantify convenience yield.

In general, for a contract expiring at time $t$, using continuous compounding, the forward price of a commodity will be:

$\boxed{F_C = \displaystyle S_0 e^{(r + x - y)t}}$

Time to GeneralizeThe Cost of Carry Model

If you notice carefully, what has been common to valuation of forward contracts across asset classes is that we are ‘replicating’ the payoff of a forward contract by ‘financially carrying’ the FX or the stock to the future time period, or in the case of wheat, physically carrying the wheat to the future time period.

A useful comparison is obtained by writing the forward price in each case by taking logs. Taking log of the three forward prices (and here continuous compounding is really a boon), we get:

\begin{aligned} ln(F_{FX}) &= ln(S_0) + r_{IN} - r_{f} \\ ln(F_{S}) &= ln(S_0) + r - d \\ ln(F_{C}) &= ln(S_0) + r + x - y \end{aligned}

That is, in each case the forward price is a sum of the spot price and carrying cost net of any yield.

We could interpret the borrowing cost as the cost of ‘financially carrying’ the FX or the stock forward. In case of wheat it’s the most obvious. We are literally borrowing money (financial ‘carry cost’) and physically carrying the wheat to the next time period. So in all cases, one could write the forward price as the sum of the spot price and a ‘Cost of Carry’, net of any income/yield that may be available while carrying the underlying asset.

In case of FX, that yield is the rate of return on USD when’carrying’ USD into the future. In the case of stock, it is the dividends, and in case of commodities it is an unobservable convenience yield.

To generalize, then, we could write as:

$\boxed{\mbox{Forward Price = Spot Price + Cost of Carry}}$

where the ‘Cost of Carry’ is to be interpreted as the net of any yield from carrying the underlying asset.

This way of finding the forward price is referred to as the ‘Cost of Carry’ model. As should be clear, it works best for financial/investment goods, and not as well for consumption goods. For consumption goods convenience yield is a fuzzy thing, highly time varying (you need oil in the winters more than in the summers) and hard to pin-down.

Written by Vineet

October 25, 2016 at 12:35 am

Posted in Teaching: FM