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ELEMENTARY PRINCIPLES OF DERIVATIVE PRICING

In this post, we take a preliminary glance at how derivative contracts are priced. First, we introduce the concept of arbitrage. Arbitrage occurs when equivalent assets or combinations of assets sell for two different prices. This situation creates an opportunity to profit at no risk with no commitment of money. Let us start with the simplest (and least likely) opportunity for arbitrage: the case of a stock selling for more than one price at a given time. Assume that a stock is trading in two markets simultaneously. Suppose the stock is trading at $100 in one market and $98 in the other market. We simply buy a share for $98 in one market and immediately sell it for $100 in the other. We have no net position in the stock, so it does not matter what price the stock moves to. We make an easy $2 at no risk and we did not have to put up any funds of our own. The sale of the stock at $100 was more ! than adequate to finance the purchase of the stock at $98. Naturally, many market participants would do this, which would create downward pressure on the price of the stock in the market where it trades for $100 and upward pressure on the price of the stock in the market where it trades for $98. Eventually the two prices must come together so that there is but a single price for the stock. Accordingly, the principle that no arbitrage opportunities should be available is often referred to as the law of one price. Recall that we mentioned in earlier posts that an asset can potentially trade in different geographic markets and, therefore, have several spot prices. This potential would appear to violate the law of one price, but in reality, the law is still upheld. A given asset selling in two different locations is not necessarily the same asset. If a buyer in one location dis- covered that it is possible to buy the asset more cheaply in another location, the buyer would still have to incur the cost of moving the asset to the buyer’s location. Transportation costs could offset any such price differences.
We have just had a taste of not only the powerful forces of arbitrage but also a pricing model for one derivative, the forward contract. In this simple example, according to the pricing model, the forward price should be the spot price increased by the interest rate. Although there is a lot more to derivative pricing than shown here, the basic principle remains the same regardless of the type of instrument or the complexity of the setting: Prices are set to eliminate the opportunity to profit at no risk with no commitment of one’s own funds.
There are no opportunities for arbitrage profits.
Lest we be too naive, however, we must acknowledge that there is a large industry of arbitrageurs. So how can such an industry exist if there are no opportunities for riskless profit? One explanation is that most of the arbitrage transactions are more complex than this simple example and involve estimating information, which can result in differing opinions. Arbitrage involving options, for example, usually requires estimates of a stock’s volatility. Different participants have different opinions about this volatility. It is quite possible that two counterparties trading with each other can believe that each is arbitraging against the other.
But more importantly, the absence of arbitrage opportunities is upheld, ironically, only if participants believe that arbitrage opportunities do exist. If market traders believe that no opportunities exist to earn arbitrage profits, then they will not follow market prices and compare these prices with what they ought to be, as in the forward contract example given above. Without participants watching closely, prices would surely get out of line and offer arbitrage opportunities. Thus, eliminating arbitrage opportunities requires that participants be vigilant to arbitrage opportunities. In other words, strange as it may sound, disbelief and skepticism concerning the absence of arbitrage opportunities are required in order that it hold as a legitimate principle.
Markets in which arbitrage opportunities are either nonexistent or are quickly eliminated are relatively efficient markets. Recall from your study of portfolio theory and investment analysis that efficient markets are those in which it is not possible, except by chance, to earn returns in excess of those that would be fair compensation for the risk assumed. Although abnormal returns can be earned in a variety of ways, arbitrage profits are definitely examples of abnormal returns, relatively obvious to identify and easy to capture. Thus, they are the most egregious violations of the principle of market efficiency. A market in which arbitrage profits do not exist is one in which the most obvious violations of market efficiency have been eliminated.
Throughout this blog, we shall study derivatives by using the principle of arbitrage as a guide. We will assume that arbitrage opportunities cannot exist for any significant length of time. Thus, prices must conform to models that assume no arbitrage. On the other hand, we do not want to take the absence of arbitrage opportunities so seriously that we give up and believe that arbitrage opportunities never exist. Otherwise, they will arise, and someone else will take them from us.