# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Options as unbundled forwards

Let’s go back to our baker-farmer story and consider what happens at the termination/expiration of the forward contract. At expiration, if the ‘spot’ price of the wheat has gone up it is the farmer’s loss and the baker’s gain (because she is delivering wheat at the contracted delivery price instead of the now-higher spot price, and the baker is the lucky bugger who is getting to buy wheat at the lower contracted price when the market price at the end of the period is higher). That is, a forward contract is a zero-sum game.

At the time of the settlement, one side will realize a gain at the expense of the other side. But if the point for both parties was to lock a price in advance, it is not a problem and both would not have too much of a problem resigning to their fate. After all, there will be times when farmer will gain at the expense of the baker.

Let’s quantify that gain/loss now. If the price of the wheat is $S_T$ at the termination time $T$, and the forward contract delivery price was set at $K$, the baker’s payoff is $S_T - K$ (the baker is happy when $S_T > K$) and the farmer’s payoff is $K - S_T$ (the farmer is happy when $K > S_T$).

Options

While the zero-sum game and all that is fine, farmer nonetheless would like to have a choice to sell at the market price when the price of wheat has increased, and not necessarily feel obliged to sell at a lower price as she has to in a forward contract. Similarly, baker would also like to have a choice to buy the wheat directly from the market if the price of wheat has gone down.

Forward contracts being legally binding do not obviously serve this desire. Again, well-functioning markets provide such felt needs to be addressed.

This is how Options come into the picture. Options serve exactly this felt need of both the farmer and the baker. Options are products that provide farmer the choice to sell the wheat at the contracted price when it suits her interest- that is, exactly when the price has gone down (a “Put Option”), and the baker the choice to buy the wheat at the contracted price when it suits his interest – exactly when the price has gone up (a “Call Option”).

If you think about it an Option basically provides an ‘escape clause’ from the forward contract – it allows the farmer to escape when the price has gone up, and the baker to escape when the price has gone down. The natural question then is how do we make it work. If the farmer has to have a choice, then there has got to be some baker out there who is ready to buy from her. Similarly, if the baker has to have a choice and that choice has to be respected, then there has got to be some farmer out there to sell it to him. Otherwise the whole system kinda breaks down.

Put Option

Let’s consider a Put Option first. It gives the farmer the right to sell at the contracted price $K$ if the price of wheat at expiration has gone down, i.e. when $S_T < K$. That is, her payoff from the Put Option is $K - S_T > 0$ – i.e. when the price of the wheat is lower it is in her interest to sell at the contracted price.

But when the price of the wheat has gone up, she is well within her rights not to use the ‘option’ and sell wheat at the prevailing higher market price ($S_T > K$), so there is no profit no loss – just the piece of paper with Put Option written on it which expires worthless. So her payoff then is simply the larger of $K - S_T$ and $0$, or simply $\mbox{max}(K - S_T, 0)$.

Call Option

A Call Option provides baker the choice to buy at the contracted price $K$ when the price of wheat has gone up. Again, the story works similarly. That is, his payoff from the Call Option is $S_T - K > 0$ – i.e. when the price of the wheat is higher it is in his interest to buy at the contracted price.

But, again, when the price of the wheat has gone down, he is well within his rights not to use the ‘option’ and buy the wheat at the prevailing lower market price  ($S_T < K$), so there is no profit no loss – just the piece of paper with Call Option written on it which expires worthless. So his payoff then is simply the larger of $S_T - K$ and $0$, or simply $\mbox{max}(S_T - K, 0)$.

Jolly good things, aren’t they then, these Options? Keeping only give the upside while keeping the downside to zero. Well, yes. And like all good things in life, they also come with a price attached – that is they are not free. Both must pay a price to buy such Options. And who, pray, must they pay this price to?

For a farmer’s Put to work, and the baker’s Call to work, there has to be a baker to buy from the farmer when she uses her Put and there has to be a farmer when baker uses his Call. That is, for a farmer’s option to sell to be respected there must be a baker who is forced to buy. Similarly, for a baker’s choice to buy to be respected, there must be a farmer who is forced to sell. It should be clear now who these prices must be paid to. When a farmer buys a Put, she pays a price to a baker who is then forced to buy from her when she wants to sell at the contracted price $K$. Similarly, a baker would buy a Call from a farmer who is then forced to sell it to him when he wants to buy at $K$.

So now we have four sides. Farmer’s choice to sell is matched by the baker’s obligation to buy from her, and the baker’s choice to buy is matched by the farmer’s obligation to sell. Nice little trick isn’t it?

So what financial market participants have done is that they unbundled/broken-down a forward contract into two parts. A baker now either has a choice to buy (when he buys a Call) or is being forced to buy (when he has sold a Put). Similarly, a farmer either has a right to sell (when she buys a Put) or is being forced to sell (when she has sold a Call).

It is market parlance to speak of a baker’s forward contract to buy as Long Forward, a seller’s forward contract to sell as Short Forward, a baker’s choice to buy as Long Call, a farmer’s choice to sell as Long Put and so on.

That is for a baker we have,

$\mbox{Always Buy (Long Forward) = Choice to Buy (Long Call) + Forced to Buy (Short Put)}$

which mathematically is

$S_T - K = \mbox{max}(S_T - K, 0) - \mbox{max}(K - S_T, 0)$

and for a farmer we have:

$\mbox{Always Sell (Short Forward) = Choice to Sell (Long Put) + Forced to Sell (Short Call)}$

or mathematically,

$K - S_T = \mbox{max}(K_T - S, 0) - \mbox{max}(S_T - K, 0)$

It is also useful to ‘draw’ the diagram of their payoffs as the spot price at termination $S_T$ varies. These are very useful and handy, but this we leave it for the next post.