How arbitrage is a crucial cog in derivative trading

Written by Sunil K Parameswaran | Updated: May 17 2013, 05:58am hrs
Arbitrage is an integral part of the pricing of derivative securities like futures and options. As the saying in economics goes, there is no free lunch. The absence of arbitrage is a sin qua non for deriving a pricing relationship between the spot price of an asset and its futures price. If the futures contract is overpriced, then traders will resort to cash and carry arbitrage.

This requires that they borrow and acquire the asset in the spot market and, simultaneously, go short in a futures contract to lock in an assured selling price at a future date. If the contract is underpriced, they will resort to reverse cash and carry arbitrage. These strategies require them to short sell the asset and invest the proceeds at the risk-less rate. Simultaneously, they need to go long in a futures contract to cover the short position at a future date at a price that is known since inception.

While cash and carry arbitrage poses relatively less obstacles in practice, reverse cash and carry arbitrage need not always be an impediment-free task. Every underlying asset may not be easy to sell short. In derivatives parlance, we make a distinction between pure or investment assets and convenience or consumption assets. A pure asset is one that is held largely for investment purposes. Financial assets and some physical assets fall in this category.

Let us assume that you have a stock and your desired holding period is one month. If you were to be assured that a potential borrower will return the asset at the end of the holding period, and will also compensate you for any income that you are likely to lose in the interim by lending the security, such as cash dividends, you will not be reluctant to lend the asset to facilitate a short sale.

However, consider an asset like wheat. A wheat mill owner may build up his stocks prior to a harvest. From an economic standpoint, this may appear senseless. For, if the harvest was to be good, then, subsequently, the spot price of the commodity will decline. Consequently, the mill owner not only has to incur storage costs, he also faces the risk of a potential capital loss. However, the mill owner may choose to hold the asset to avoid a potential stock-out due to a monsoon failure or a natural calamity such as a cyclone.

Thus, holders of agricultural commodities may not easily lend such assets to facilitate short sales. While agricultural commodities are typically convenience assets, metals may fall in either of the two categories. Precious metals like gold, silver, and platinum are usually investment assets.

Industrial metals like copper are often consumption assets. Futures contracts can be replicated by a combination of a position in the spot market and a concomitant position in a risk-less asset. We say that Spot T-bills is equivalent to a synthetic futures position. In other words, a long spot position, coupled with a risk-less borrowing, is equivalent to a long position in a futures contract. Consider the import of this relationship. While a spot position in isolation is risky, as is a futures position, their long-short combination is risk-less. The variables can be repositioned to derive equivalent relationships. Thus, T-bills plus futures is equivalent to a synthetic long spot position. That is, if two of the positions are natural, the third can be artificially generated.

The ability to make a more attractive investment in a synthetic security as compared to a natural position in it is termed as quasi-arbitrage. For instance, take the case of a trader who wants to make a risk-less investment. One possibility is to invest in treasury bills. This will yield an assured return. However, there could be a situation where a long position in an asset such as Reliance shares, coupled with a short position in futures contracts on the same asset, yields a higher assured return. Thus, while pure arbitrageurs engage in arbitrage to earn a free lunch, quasi-arbitrageurs perceive such strategies as an alternative to natural market positions.

A pure arbitrageur, who engages in cash and carry arbitrage by going long in spot and short in futures, will compare his borrowing rate with the return from the synthetic T-bill that he is generating. A quasi-arbitrageur will, however, compare the return from the synthetic investment with the return from a long position in natural T-bills. What often happens in practice is that while the return from synthetic T-bills is not high enough to justify the borrowing of funds to implement such a strategy, the return from such artificial positions may be higher than what the trader could earn, were he to invest in natural treasury bills. Thus, quasi-arbitrage may be profitable in a situation where pure arbitrage is not.

Pure arbitrage opportunities usually do not last for a long time. One of the assumptions behind cash and carry and reverse cash and carry strategies is that traders can borrow and lend unlimited funds at the risk-less rate of interest.

In other words, the inherent assumption is that there is no budget constraint. A quasi-arbitrageur, however, faces a funds constraint. The ability of a trader to invest in a synthetic T-bill is constrained by the amount of funds at his disposal. Taking further positions would require him to borrow, and would, therefore, be tantamount to pure arbitrage.

And, as we have seen arbitrage may not be profitable in a situation where quasi-arbitrage is. Thus, the potential for profit through a quasi-arbitrage strategy is likely to linger for a longer period compared to a profitable arbitrage opportunity. In other words, while arbitrage opportunities are ephemeral, quasi-arbitrage opportunities typically persist for longer periods.

Pricing of derivatives

* If the futures contract is overpriced, traders will resort to cash and carry arbitrage

* This requires that they borrow and acquire the asset in the spot market and, simultaneously, go short in a futures contract to lock in an assured selling price at a future date

* If the contract is underpriced, they will resort to reverse cash and carry arbitrage. These strategies require them to short sell the asset and invest the proceeds at the risk-less rate.

* Simultaneously, they need to go long in a futures contract to cover the short position at a future date at a price that is known since inception

* The ability to make a more attractive investment in a synthetic security compared to a natural position in it is termed as quasi-arbitrage

The writer is the author of Fundamentals of Financial Instruments, published by Wiley, India