The Mysterious Smile of Volatility

We have seen so far very direct connections of option prices with the underlying price: delta, which provides a linear connection, and gamma, which makes the connection non-linear (our provider of convexity). We also discovered that we needed to pay a price in order to enjoy said convexity, and it is reflected in a parameter called theta. However, all of those three Greek letters are not enough to describe the full price dynamics of options. Even though we can design some basic strategies now that we know how they behave, we are still missing a big part of the puzzle. 

It is hard to imagine what else is missing. Options are a derivative, so it is natural that their price is tied to the price of the underlying, but the current price of the underlying alone fails to capture an important component of options pricing, and that is: the future.

A window into the future

One of the most salient features of options is that they expire at some point in the future: days, weeks, and even years ahead. In fact, the price of the underlying at that time in the future is what will dictate the final payout of the option (if we hold them until expiration, that is). So here is an interesting problem: the future behavior of the underlying price necessarily has to influence the current price of the option. After all, if we know that one option will be ITM by expiration day, then it should be more valuable than another one that surely will be OTM the same day. But then, how can we quantify that future behavior? I mean, can we even predict what the price of the underlying will be in the future at all?

The answer is of course not, we can't predict the price of the underlying at any future date. If we could, we could be immensely rich already. However, we need to be able to price our option right now in the present somehow. We need to introduce expectations of the future for the underlying into the option price, and we need to do it in such a way that doesn't require us to assign probabilities of any kind to a particular underlying price at expiration date. The problem seems very tough, and in fact it remained opened for quite a long time. The solution to this issue involved a very ingenious insight that managed to get rid of the predictions about individual price levels and instead exchanged them for a single prediction for a single quantity. The nature of that quantity is all of the story that follows.

The present value of the Future

As all good stories this one needs a little detour first. Because we are talking about a very complex derivative that has a future payout, it could be better if we start with something simpler first, and then start building up from that point. And what could be simpler than a derivative that only has a future payout? So let's start with that, and talk a bit more about futures.

I imagine a lot of you my readers trade futures in a regular basis, or are at least familiar with popular futures products like ES mini, or GC (gold futures). Those products are very simple to trade, they are in fact linear derivatives and they are connected to the spot price of the underlying (SPX or gold in this case). The spot price being the price or level of the underlying for immediate delivery. If you pay spot price you get the underlying right away. In a futures contract of course we don't want immediate delivery of the underlying, we are entering into a contract to sell or buy an asset at a certain price at a certain point in the future. This can be really helpful for many market participants. For holders of the underlying it is a great way to lock in a certain price today for delivery in the future which removes uncertainty about future cash flows. For someone that uses the underlying it could also be advantageous to secure a good price today even if the need for it will only appear later on. And finally for speculators is a good way to make money with a leveraged instrument that moves one to one with the underlying. The interesting appeal of futures contracts is to lock in the price of the underlying right away, so it is not a surprise that the price of a futures contract changes when spot price changes, however why the two prices are not the same? In fact why does the price of a futures contract varies with the settlement date ? 
The answer to those questions is closely related to the concept of efficient markets. In an efficient market ideally there should not be any builtin advantages for any of the two sides of a transaction. In other words, buyers and sellers should not have an advantage over their counterpart, the transaction should be fair to both. A more formal way to express this concept is that there should not exist any opportunities for arbitrage in a transaction. The A word finally starts to appear in this book, arbitrage is defined as risk-less profit, the holy grail, making money with 100% guarantee of success. If one of the parts has the change to perform arbitrage on the transaction (profit without risk) the other part needs to be crazy or clueless to enter into such a contract, therefore in an efficient market, the transaction should never occur as both parts would be aware of the potential arbitrage opportunities available to the counterpart and would not even dare to enter into a transaction if they are still there.

So here comes a very cool concept, for a transaction to occur, buyer and seller must agree on a fair price, that is, the transaction has to occur at a price that removes any possible chance of arbitrage by any of the two. Of course lots of readers must be having some cynical thoughts about this concept, pure efficient markets don't actually exist and in fact there are plenty of clueless uninformed traders out there that would jump head first into a transaction no matter the price. Yes, that is correct, however, the US equities and option markets are indeed very efficient, as well as the futures market for a few commodities. What this means is that those clueless retail traders will be "arbed" away rather quickly and price will settle around fair price very rapidly. Also the efficient market hypothesis is self-fulfilling very quickly, there are lots of huge hedge funds and big banks looking for arbitrage opportunities consistently (with powerful computers and ultra fast software) so anytime one opportunity appears it is exploited right away by one of those "big boys" and with the pass of time the market starts to look like an efficient one.

Fair Price

Returning the the concept of fair price, let's focus on how we can compute it for a futures contract derivative. Imagine for a moment that you are in possession of 1 Oz of gold for instance. Let's also imagine that you are a very uncomplicated person and you keep it in your home in some place, like a drawer in your desk because you believe nothing will really happen to it. Now, if you wanted to sell that gold right now you would be given about $1,224.60 (Feb 26 2016 price) for it, however you don't really want to part from it right away for some reason, but instead you want to lock-in that price until December 2016 for instance, perhaps you think prices will fall in the future and it is better to get something decent today. If that is the case then a futures contract is a good option. So, let's imagine that a potential buyer for that future contract comes along a offers to buy that gold from you in December 2016 for exactly 1,224.60 (the spot price) would you enter into that contract? It seems fair at first, after all you locked-in the price that you wanted, so why am I even asking this question? The answer is that a futures contract like that would be unfair to you the seller because you are allowing the buyer to realize a risk-less profit. I'll let this sink in a moment and see if any of you my readers can come up with the explanation for this.

In this example if the December futures for gold are priced at the spot price then a risk-less profit could be realized when you buy them. There are many angles to explain this but for simplicity I'm going to use the point of view of the buyer of the futures contract. This buyer actually already holds 1 Oz of gold, however despite how shiny and beautiful the bar looks, it doesn't produce any dividends, or pays any interest to the owner. So being as smart as he is, and seeing how the futures price is exactly at spot he decides to do an EFP operation, which stands for exchange for physical. In this operation the smart trader sells his 1 Oz of gold at spot (pocketing $1,224.60) and simultaneously enters into a long futures contract for December with our original seller. Then he uses the money from the sale and invests it at the risk free rate. Later on, in December, he pays $1,224.60 for the 1 Oz of gold that his future contract entitles him to, so now he is holding the same 1 Oz of gold which he had before but also he gets to keep the earned interests to himself. Our very smart trader discovered a way to earn interest from a gold bar (the holy grail of gold bugs all over the world). That interest that he is pocketing was earned risk free, he started with 1 bar of gold, ended with 1 bar of gold an extra money. That my readers is a classic example of arbitrage.

By now you are starting to develop a mental model of fair pricing of futures. With this example we now imagine that the price for a December future should be higher than the spot price but by how much? Well, from the example we can see that the futures price should be one that removes any advantages for any of the parts so in this case it should include any risk free interest earned on the equivalent sale at spot price. If the risk free rate is r then the futures should be priced at:

Fair price = exp(rT)*Spot
r is the risk free interest rate
T is the time period until the contracts settles

Forward thinking

I'm using the exp() function because when you have continuously accruing interest that is the way to compute the interest earned. You can see from that equation that the longer the time period T is the more expensive the futures contract must be. This was a very simple example, in real life there are other things to consider too, for instance there are storage and insurance costs that also need to be added to the price of the futures to make the trade fair to the seller. You can think of a more general equation for the price along the following lines:

Fair price = exp((r+s)T)*Spot
r is the risk free interest rate
s is the storage cost (which might include insurance).
T time.

The term that multiplies the spot price is called  the carry factor. So now we have introduced the concept of carry which are all the costs associated with holding the asset for a period of time, and that includes storage, insurance and also the cost of money (the potential interest earned if we sold the asset today). For commodities the carry factor is usually a number bigger than one and it increases with time, that is why you see the well known contango shape in the futures prices, where short term futures are cheaper than longer term ones. In general carry always adds to the fair price of the futures however there are situations when carry can actually lower the price of the futures contract and that happens when there are shortages of the commodity (which requires the introduction of a convenience yield) and also for assets that pay dividends. So here is finally a more convenient definition:

Forward = exp((r-q)T)*Spot
r is the cost rate, which includes the risk free rate, storage costs, insurance.
q is a yield of the asset, that could be dividends paid, or a convenience yield for a commodity.
T is time.

So finally, the F-word makes an appearance. When the spot price is adjusted to reflect the cost of carry, the result is called the forward price, which also happens to be the fair price of a futures contract. From now on we will call this carry adjusted price, the forward price. When carry is positive we get contango in the futures, when carry is negative we get backwardation in the futures, that is when the near term futures are more expensive that the longer term ones. 

Hopefully the concept of fair pricing and forward price is now clear to everyone, but how this concept ties with options at all? Well, we'll move into this in great detail in the following sub section. This could be a good time to take a break, drink some coffee and come back to the book with renewed energy and attention. 

Options and Forwards

Now that we have a very straightforward relationship between future prices and present prices then we can move into the interesting problem of pricing options. Conceptually it is very simple, if we know the price of an option at some point in the future, then we can use the forward price formula to find the corresponding spot price now and that is the one we use to buy and sell it. However, the main problem with this simple approach is that the price of an option in the future can be one of these two prices:
  • 0 if the option is out-of-the-money
  • Or the amount that the option is in the money based on underlying price and strike.
As you can see, the price of the option in the future depends on the price of the underlying at expiration, and as we mentioned at the very beginning of this section, it is impossible for us to predict it. I guess I understand that it could be very tempting to price options based on the probabilities of them being ITM or OTM, and also by how much ITM they can be at expiration. However, such an approach is too arbitrary, and it also becomes impossible to check against arbitrage possibilities. It is clear that we need to find something better - something that takes advantage of the no arbitrage rule and also of the relationship between forward and spot prices.

Drunkards and Walks

The big insight that can help us resolve this problem is to admit that we can't predict the future when it comes to the underlying price. More specifically, we need to recognize that the future price of an underlying doesn't depend on the current price, nor on any other price from the past. In other words, the price of an underlying basically follows a stochastic process. Once we are willing to go this way, we can design a pricing strategy that is not concerned with the particular path that the underlying follows, we can price the option without really caring about whether price goes up and down. To see more clearly this point, let's look at the following example: