# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Futures contracts

Forward Contracts

It turns out that all forward contracts we have discussed have some common features – in particular:

1. The farmer and the baker have to meet: Had the farmer (baker) not known the baker (farmer) s/he’ll still be faced with the risk of having to sell (buy) the wheat a lower (higher) price. So both counter-parties in the trade must find each other to even have a contract in the first place.
2. The contract is agreed upon before the delivery of the underlying asset (called the “deliverable instrument”): When the farmer and the baker meet, the farmer agrees to make the delivery of the wheat at some future date. It’s not an ‘on-the-spot’ transaction.
3. The time at which the delivery is to be made (called the “expiration date”) is agreed upon in advance: The farmer and the baker mutually agree upon the time (and often the location) at which the farmer will delver the wheat.
4. Forward contracts can be highly customized: The farmer and the baker may, for all they like, agree to the quantity of wheat till the last ‘gram’. They may settle that the baker will pick up the wheat from the farmer’s house at a particular time (say, before/after lunch) on the expiration date. Or the farmer may agree to deliver the wheat at a particular location at the baker’s house or warehouse. It’s a lot like you going to your tailor to get a custom suit stitched – right to the shape of buttons, it’s your choice.
5. The price at which the delivery is to be made (called the “delivery price”): Quite importantly, the farmer and the baker agree on the price at which the farmer will deliver the wheat to the baker on the expiration date (this is kinda really the whole point, isn’t it?).
6. No money changes hands at the time the contract is entered into.
7. Both parties cannot get out of the contract before the expiration date: By their very nature forward contracts are very illiquid.
8. At least legally, both the farmer and the baker are obligated to honour the contract: While this may not be an issue for a farmer and a baker in a village who know each other, this is of crucial importance in the real world.
9. In a primitive village economy it’s a farmer and baker coming together, but in the modern world, typically both the counter-parties are a financial institution, like a bank. (And they do so because most big banks deal with each other very frequently, and ‘know’ each other.)

Now this last couple of features above makes it little problematic for forward contracts. And why, pray this be so?

The problem is that if the expected price of wheat changes during the life of the contract (that is, before the delivery is made but after the contract is entered into), then both the farmer and the baker have an incentive to renege on their obligation. How?

Let’s say the farmer and the baker meet and agree on a price of 100, i.e. farmer agrees that she’ll sell the wheat to the baker at the end of, say, 1 month at 100 a bushel. And then, say, in a fortnight’s time both farmer and the baker find that there are lot of birthdays coming up in the village around the end of the month, and there will be a disproportionate demand for breads, i.e. the demand for wheat flour will be high.

Now, if our smart baker has already entered into the forward contract with the baker, he’ll be all smiles knowing that he has already locked-in the price at which he is going to procure the wheat. But the poor guy perhaps doesn’t know that our lady friend farmer has other plans. When the time comes, she may just, say, well too bad for you that we entered into a contract, but I am afraid I can’t sell you a bushel for just 100 , when the market price is 120. You might say that the farmer is spoiling her ‘trust quotient’, and she can be sure that she’ll have one less friend in the village, but, hey, it’s money we are talking about.

Now I am not saying she will do it, but there will be a mighty big temptation for her to do so. The fact that being a part of a community acts as natural enforcer doesn’t take away the problem. Again, consider the case, if for some reason at the end of a fortnight now both come to know that there’ll be lesser demand for wheat. Now it’s the baker who is faced with a dilemma.

So herein lies the problem. The fact that a forward contract requires both parties to honour their contract makes it difficult to operationalize it in the ‘real world’. While in a small village both parties may, and in all probability, will, know each other, in a place like Mumbai, even if you go and try hunting for a counter-party you may never find him/her in time because you’ll probably be stuck in a jam near Andheri (or these days even Bandra), or if you are going through an intermediary, your broker may just form a cartel with your counter-party and fleece you by offering a forward price that is clearly to your disadvantage (but which may still be the best you could do).

The problem, therefore, is not just about ‘trust’ or ‘honour’ . The fact that an unforeseen circumstance (say, a flood) can make one of the parties renege on their contract makes it practically a big problem. Not to mention the fact that even if the farmer wanted to get out of the contract, because say, she was robbed of all her wheat, and she needs the wheat for her consumption, she can’t do so even if it’s for a ‘right’ reason. The baker still would feel cheated as he is stranded if the farmer fails to deliver the wheat for whatever reason.

For people or entities who knew each other well they could still (and do) enter into forward contracts (for example, big banks who transact with each other on a daily basis they do have a fairly high turnover of outstanding forward contracts – but they ‘know’ each other and their businesses quite well), for a farmer and the baker (as for an exporter and an importer), there are these problems.

So, just like most good things in life, forward contracts also come with riders attached.

Let’s revisit our earlier example and make it concrete. Let’s say when the baker and the farmer meet after a fortnight, say, over a social occasion they agree that the fair expected price at the delivery time would be 120. That is, if they had settled on a forward contract on this day, both would agree that the fair price at the delivery time would be 120.

Our lady friend farmer, of course, isn’t too happy at the turn of the events and would prefer to sell the wheat in the market at the higher expected price in the market rather than deliver the wheat to the baker when the time comes at the agreed price of 100. She is definitely getting tempted to side with the devil on this one and deceive the baker.

But let’s say our lady friend is not bad at heart. She has good intentions, but is still concerned about her financial loss, and given a choice she would rather not lose baker’s friendship. If she expects the price of the wheat to still go higher than the today’s expected future price of 120, she may think, well alright the price in the future could still end up higher than 120, and maybe it’s a good idea to just gracefully ‘walk out of the relationship’ with the baker and just pay-up for his loss of 120 – 100 = 20.

It’s baker’s loss because it is the farmer who ‘wants out of the relationship’, and it is the baker who ends up ‘without a partner’ (ok, wheat) and so must be compensated. Or rather, more boringly, now the baker is stuck with the possibility of having to buy the wheat at the higher expected delivery price of 120 when he was promised wheat at 100.

So, if the farmer could pay up 20 to the baker, baker would say, ok, my friend I understand, but now that you’ve been honest and compensated me for the fact that I’ll have to enter into a new contract with another farmer, we are good. So long, and thanks for all the fish.

In fact, it’s also possible that the farmer and the baker may not have to ‘break-up’ at all. If the farmer is not sure that the wheat price is going to keep rising, and her best estimate is that it’ll end up near 120, then both could still ‘settle their disagreement’ if farmer pays up 20 to the baker. This is workable because if the expected price of the wheat, say, were to go down in a week’s time (but still before delivery), if the baker ‘honors the relationship’, and now pays to the farmer, it’ll be all good in the village. Everybody’s life can go on in the atmosphere of trust and good-will.

Imagine now that there are many such bakers and farmers in the village, and not enough social gatherings. This leaves us with a village with lot of worried bakers and farmers. And if they could not meet like our original couple, and if there were only a single rotten egg, it would lead to a vicious cycle of distrust and antipathy.

Human beings are good at resolving such situations by deciding on a neutral third party. In the village this would mean someone everybody knows and trusts, say someone from the gram panchayat. In that case, they may decide that they would monitor the expected future price every week after the forward contract is signed, and settle their profits and losses with the panchayat. So, if the price of wheat goes down, the bakers of the village would come and pay up their losses, and if the price of wheat goes up, the farmers would do the same. Existence of a neutral, trusted third-party again results in a nice (as against a lousy) equilibrium (I guess that is what, in some sense, a judicial system also ensures).

Of course, this arrangement in the village would work only if our panchayat member is a trusted fellow, and if for some reason either a baker or a farmer is unable to keep his/her end of the bargain, that is pay up the ‘margin’ of loss that is due, he should be able to compensate the other party still.

However, the first and an even bigger problem is that of existence. The question of a forward contract ‘coming into the world’ doesn’t even arise if the farmer (exporter) and the baker (importer) can’t meet. For them to even enter into a contract means they must find each other first. So there is a search (transaction) cost involved.

Fast forward this primitive village economy to the modern world, where as it is there is a deficit of trust for reasons of scarcity of time, resources, opportunities and what not, an entity like panchayat doesn’t suffice. But as it is often the case, left to their own devices, one can trust the financial market participants to come up with a workaround.

For a financial market participant the solution is very clear – create markets. Create a platform to bring the farmers and bakers of the world together. Having a common marketplace reduces transaction costs by bringing the participants together. And when finance people talk about markets, more often than not they have in mind a clearing exchange, and in the case of forward contracts they are called futures exchanges.

Futures Contracts

The way the exchange manages the ‘trust’ of the counter-parties is exactly the same way as the panchayat in the village. They settle the profits and losses ‘every so often’. But because in a city like Mumbai, one can’t afford to run around and find the person who refuses to settle his end of the bargain, and there are more crooks in Mumbai, the mechanism of “margining” in exchanges are run by strict rules.

But the fact that having a market, or an exchange, allows buyers and sellers to come together doesn’t solve the problem of default. If anything, perhaps, it exacerbates it. If even a few people started defaulting on the exchange, for lack of trust, the whole system will very quickly break down. So what is the solution?

At this point one of my colleagues draws a nice analogy of this problem with that of a pawnbroker’s in a big city. A pawnbroker is someone who gives loan to people who can’t get it otherwise, and his (a pawnbroker typically is a “he”) obvious worry is that his money won’t come back. And for a right reason – the kind of people who go to a pawnshop to borrow money are exactly the ones who have a bad (or no) credit history. And what is his solution? His solution is that he asks for a collateral of value much more than the value of the loan that he is giving out. (Contrast this with the problem of a moneylender’s in a village – would he ask for the same collateral as a pawnbroker in Mumbai?)

The solutions that markets came up for this problem is that they agreed to make the exchange as the counter-party for all forward contracts (either entered into by a farmer or a baker). And, like a pawnbroker, exchanges made a system of asking for collateral from the counter-parties in the forward contract, and this new kind of forward contract in which exchange was the intermediary counter-party for the farmer and the baker, they called the futures contracts (not “future”, but futures, with an “s”) – very similar to a forward, but different enough to warrant a new name.

The exchange-traded futures contracts offered almost everything what forward contracts did, but in a stroke of genius, exchanges came up with a way that prevented default exactly where it was most likely to occur – among traders who did not know each other. And what is truly a marvel of modern finance is that while banks have had a history of defaulting (and very much so in the recent past), very rarely have exchanges collapsed the way banks have.

As by now you should be able to guess, for futures contracts to be traded on an exchange they must be standardized. So, unlike forward contracts which could be customized right down to a T, futures contracts, by their nature of being traded on an exchange, must be standardized. In fact, it’s now a good time to list down the features of futures contracts:

1. Exchange Traded: Futures are exchange traded. By construction all futures contracts are traded on exchange, and as a consequence, just like stocks, everybody can see what is being traded at what price and so on. Unlike, forward contracts, then, they are not opaque and highly transparent. An implication is that futures contracts are highly liquid. At any point the buyers and sellers of the futures contract can buy from or sell to the exchange. This means that if a farmer ‘sold’ wheat on the futures exchange, if she now desires, she could unwind her contract by entering into an offsetting contract with the exchange at the going price. We’ll elaborate on this point later.
2. Standardization: Futures contracts are highly standardized. For a commodity/investment asset to be traded on the exchange, it has to standardized. So if wheat/rice is harvested in particular months, the delivery dates of the futures contracts will be closer to those dates – that is, for a specific asset futures contracts have a fixed/limited number of delivery dates. Not only delivery dates, often the location and the ‘grade’ of a good (different grades of wheat are grown and priced differently) is also pre-specified exactly by the exchange.
3. Anonymity: The farmer and the baker need never know about each other. For both the farmer and the baker, the counter-party is the exchange. That is, both the farmer and the baker could protect themselves against the risks they face by entering into forward contracts with the exchange that are mirror images of each other. So while exchange is the counter-party for both the farmer and the baker, on a net basis the exchange has exactly offset its exposure. That is, unlike in a forward contract the farmer and the baker are not dealing with each other at all. The consequence is that all the default risk is assumed by the exchange.

And how does an exchange protect itself from default by the farmer or the baker? This goes back to the ‘stroke of genius’ we referred to on part of the exchange. The key idea here is “margining”.

Futures Exchanges and “Margining“: The idea of Mark-to-Market

So the ‘stroke of genius’ by futures exchanges was to modify the system adopted by bakers and farmers in the village, and implement it in the context of the modern financial markets as this is what make futures contract really work in the market place. The key idea is Mark-to-Market.

1. Mark-to-Market: Exchanges made the system of settling profits and losses on part of the bakers and the farmers more systematic  ‘every so often’ and called it mark to market:

• Settlement every business day: First of all exchanges made ‘every so often’ specific and this is one of the crucial feature of futures contracts traded on the exchange. The profits and losses of the bakers and farmers are settled every business day. This makes sense, because it may be easy for a panchayat member to locate a particular baker or a farmer who fails to pay up at the end of a week or so, in modern cities this is not only prohibitive but also may not be possible.
• Anonymous trading: This above feature of marking to market, in fact, is what allows the exchange to become a counter-party to any entity. It could be anyone who needs the wheat, need not just be the baker. So, the counter-party may never know each other at all. And in today’s modern world, they could even be situated in different geographical regions. As long as they can transfer money to the exchange’s bank account when the margin payment is due that’s all what is needed. This is made possible because exchanges require settling the profits and losses in the contracts every day.
• Settlement price: Just like in the case of the baker and farmer who settled their profits and losses on the basis of the future expected price at the time of delivery (120), the futures contracts are also settled on the basis of the futures price prevailing on the settlement day. So once the futures contract has been entered into between the exchange and the baker, and the next day the futures price changes (because perhaps the spot price and the cost of carry have changed), the profits and losses are settled on the basis of the futures price prevailing on that day. Think of it this way. If the contract was signed yesterday, and futures price for the same delivery date today has gone up to 120, the farmer would have been better off had she got into agreement with the baker today rather than yesterday. So her loss based on changed futures price is 120 – 100 = 20. The amount/loss settled is referred to as the “margin money” or “variable margin” or “mark-to-market margin”.
• Settling profit and losses: Since futures exchange is the counter-party for both the baker and the farmer, when the price of the wheat goes up, the exchange gains vis-a-vis the farmer (collects 120 – 100 = 20 from the farmer), but loses viz-a-viz the baker (pays up  120 – 100 = 20 to the baker received from the farmer). But because both parties are obligated to settle their losses every day, exchange is in a position to transfer the proceeds it receives from the losing party.
• Obligation of the exchange: Profit and loss settlement by the exchange subsumes that the exchange is never going to default on its obligation. Even if the baker defaults, the exchange must still honor the farmer’s contract, and still ‘buy’ the wheat at the agreed futures price on the delivery date. (The fact that exchange never has to do so is another matter. See point 3 below.)

2. Initial Margin: Although daily mark to market helps reduce the default risk for the exchange, it does not completely eliminate it. So, just like a pawnbroker, exchanges also require both counter-parties to keep a collateral with the exchange. As you would expect, since the exposure of the exchange is only to the extent of the margin money every day, the collateral required by the exchange is just the maximum amount it expects the wheat price to fluctuate every day. This margin is called the initial margin, and both the baker (if the expected future price of wheat falls) and the farmer (if the expected future price of wheat rises) have to pay this to the exchange before they can enter into any futures contract with the exchange.

3. Early Close-out

• Default on the margin payment: If one of the counter-party fails to keep its side of the bargain, and does not pay up the margin money at the end of the business day, the exchange confiscates its initial margin, and closes out that side of the contract by entering into an offsetting contract with another counter-party. Since the futures are on standardized contracts and exchanges offer liquidity, it’s never a problem for the exchange to find another counter-party at the going futures price (remember that if there are not enough takers at a particular, the supply-demand dynamics ensures that the futures price would adjust). If the baker fails to pay up when the price of wheat has fallen down to say, 80, the exchange will first of all say good-bye to the baker’s initial margin and use some of that money to transfer the profit due to the farmer. Now because the exchange is stuck with the obligation of having to buy the wheat at 100, when the prevailing futures price is 80, the exchange will enter into a futures contract with another counter-party to sell the wheat at the prevailing futures price, i.e. it will find another baker. By doing so it has taken care of its obligation to buy the wheat from the farmer. From next business day on-wards, the original situation would return, as both the original farmer and the (new) baker would resume mark to market. Note that in the entire process, the farmer need never know about the default on the margin by the baker.
• Early exit: An advantage of this possibility of early closing out is that any counter-party can get out of the futures contract by paying up the mark to market margin. Let’s continue with the same example. So, if the next day after the contract has been entered into, the price of wheat has gone down to 80, the baker has reasons to worry that he may be in a soup if the price of wheat keeps falling (as he’ll have to pay up margin money every day). Having an exchange as the counter-party, and the fact that the futures contracts are liquid, gives the baker a choice to walk out of his contract without any impact to anyone as long as he keeps paying his margin money. So, in this case, he may just pay up his loss of 20 loss and enter into an offsetting contract with the exchange to sell wheat at 80 with the same time to delivery. So exchange, in its account books now, would just have an offsetting entry and baker can go live his life in peace without having to bother about the margin payments to the exchange. And if the price is going up, the farmer is free to do the same thing. If the price goes up to 120, she may just pay up her loss of 20 and walk out of the whole thing by just entering into an offsetting contract (to buy) with the exchange.
• Choice not to take delivery: Extend the idea of early exit to the delivery date, and its clear why both parties may choose not to take/give delivery of the wheat at all. On the day of the delivery, both parties may just choose to settle the last margin payment and at the same time enter into an offsetting futures contract with the exchange for delivery on the same date. So, ignoring time value of money, the net cash for each would be exactly the same as 100, and by doing so both parties have exactly hedged their exposure to the rise (for baker) or fall (for farmer) of wheat price

4. Hedging and Basis Risk: A natural question to ask at this point is why would all the bakers and farmers wish to come to exchange when they may need to hedge exposure to very specific kind of wheat when the exchange provides liquidity and contracts for only  very specific kind of wheat. The answer to this lies in correlation. If we are talking about the same commodity (wheat), it’s very likely that the price movements in different qualities and grades of wheat will be highly correlated. So even if prices of different grades of wheat may not move in perfect lock-step with each other, they would still be safely assumed to change very closely together.

So, in that case, say, if the price of the wheat traded on the exchange has gone down by $20\%$, it’s likely that the price of the wheat that the baker wants to hedge himself against would be trading say, at $23\%$ or $18\%$. In that case, if the baker has a futures contract with the exchange, while he is gaining as a counter-party with the exchange, in the markets he is losing. And wasn’t that his really worry in the first place. What if the price of wheat goes up in the future? By entering into the futures contract he is better of by $20$, but in the real market place he is worse off by either $23\%$ or $18\%$ depending on the ‘state of the world’. So on the ‘net basis’ he is either gaining by $20 - 18 = 2\%$ or losing by $23 - 20 = 3\%$.

If the two grades of wheat were moving in a perfect lock-step with each other his net gain/loss would have been zero. But because the futures exchange does not provide exactly the same kind of wheat that he wants to hedge, there is a residual exposure still left, but more-or-less he is still protected, because exposure in futures contract and the spot are highly negatively correlated, the diversification effect implies that the variance of his losses would be a lot lower compared to if he hadn’t hedged. The residual exposure because the asset to be hedged is not exactly the same as the instrument used for hedging is known as the basis risk.

We close this post with a simple example of marking to market.

Example

 Time t = 0 (F = 100) t = 1 (F = 80) t = 2 (F = 90) t = 3 (F = 110) t = 3 (Delivery at F = 110) Baker’s Inflow/Outflow – 80 – 100 = -20 90 – 80 = 10 110 – 90 = 20 -110 (Net = 0-20+10+20-110 = -100) Farmer’s Inflow/Outflow – 100 – 80 = 20 80 – 90 = -10 90 – 110 = -20 110 (Net=0+20-10-20+110 = 100) Exchange’s Net – -20 + 20 = 0 10 – 10 = 0 20 – 20 = 0 – 100 + 110 = 0

Needless to say, futures contract requires that the users have enough liquidity to meet daily margin payments. Liquidity constraints applicable to most retail investors means that it’s mostly companies or institutions that typically hedge using futures contracts.