CHAPTER
13

Arbitrage

In This Chapter

• The definition of arbitrage
• Theoretical arbitrage and practical arbitrage
• How to use arbitrage strategies
• Waiting for the high-frequency folks

By definition, arbitrage is a riskless profit. In theory, arbitrage is not possible. It’s a useful way to think about how markets work in a classroom, but the real world is very different. When some people talk about arbitrage, they are using the theoretical definition.

In the real world, the word arbitrage is used to mean a trade that’s similar to theoretical arbitrage, but that has a little bit of risk. This is not the strictest definition of the word. You’ll hear the word used in both its strict sense and its more real-world version when you talk to people about trading. It’s safe to assume that if an academic tells you arbitrage is impossible, she is talking about arbitrage in the theoretical sense. If a trader tells you it happens all the time, she is talking about the real-word definition of the word.

An understanding of arbitrage will help you understand the options markets and how they work.

All that happens in arbitrage is someone buys cheap in one market and immediately resells the item in another market offering a higher price.

Arbitrage is the key to the functioning of the financial markets. It is the process that forces markets to behave rationally, and so it is the foundation of many options trading strategies for both hedgers and speculators. After all, options draw their value from other securities. That, alone, creates opportunities for arbitrage. On occasion, price discrepancies in other markets can be played with options. For example, if stock prices have not fully responded to the announcement of a merger, an options trade might offer a higher payoff with less risk than working with the stocks.

Here’s what that would look like: Company A announces great earnings, and the stock price increases from $23 to $27 in one day, a $4 increase. A call option with a strike price of $23 has a premium of $2.04, despite news of the price increase. You could buy the call option, immediately exercise it to buy stock at $23, then sell those shares in the market at $27. Your profit is the $4 difference in the strike price and the market price, less the $2.04 premium, or $1.96. Nice, huh?

It’s so nice you might never see it.

True riskless arbitrage is rare. To the extent that riskless arbitrage exists at all, it is the province of electronic trading systems. The arbitrage strategies used by options traders have very little risk, but the risk is not 0. So why have I included this chapter?

First, arbitrage can happen. Second, many of these arbitrage strategies are available to professional traders, who have access to high-speed trading software and deep pockets. That means these strategies will come up in discussions of the options market.

If you understand what’s happening in the market, you’ll be in a better position to manage your own, plain-vanilla trades.

The Law of One Price

The law of one price says identical assets should have the same price in every market. For example:

• A share of IBM stock should cost the same in New York as it does in Tokyo.
• The euro/pound exchange rate should be the same in London as it is in Paris.
• An ounce of gold should be valued equally everywhere.

If you noticed any of these items were cheaper in one market than another, you would buy in the cheap market and immediately sell in the expensive one. The increased demand in the cheap market and the increased supply in the expensive market would push the prices together, exactly as you learn in any intro to economics class. Until the prices converged, you would collect a profit on the difference with no risk at all.

It doesn’t happen often, but when it does, there is money to be made.

Arbitrage and near-arbitrage transactions occur in the options market because of the number of different cash flows that can be replicated.

This section covers some of the most common strategies. Keep in mind most of these involve at least a little risk, so they aren’t arbitrage under the strictest definition of the term (which, you may recall, is a riskless profit). The risk relative to the return is less than it would be on a standard trade.

And a better return for a level of risk is a good thing.

Simple Arbitrage

The simplest way to take advantage of arbitrage is to look for situations in which the same asset has a different price in two different markets. Simple, yes?

The concept is simple, but it’s very hard to do in real life.

Still, it’s not impossible in the options market, in part because there are several exchanges competing for your business. Nearly identical options might trade at slightly different prices on one exchange than on another. If so, buy where it’s cheaper and sell where it’s more expensive.

These opportunities are rare, especially if you are a retail trader paying relatively high commissions and without a lot of computing power behind you. Still, you might see one someday. You are more likely to see arbitrage opportunities in times of great market stress, because they are not normal.

For example, let’s suppose a stock falls on news, but the prices implied in the option market have not fallen as fast. In that case, the trader buys the underlying asset and a put, then sells a call with the same strike price and expiration date. The strike price is higher than the current stock price. The synthetic short means the trader has bought cheap in the stock market and sold for a profit in the options market, a neat trick that generates a riskless profit.

Scalping

Despite the law of one price, option prices change a little each order placed. Sure, the difference from the right price might be only a few cents, but that’s enough of a difference for the scalper.

Scalpers look to profit from changes in an options price, especially in the bid-ask spread. (You might remember from Chapter 2 this is the difference between the price the broker buys the option from a customer and the price at which she sells it to another customer.) There was a time when scalping was a common trade strategy for individual investors, but increased electronic trading has made it very difficult for all but the largest traders. Still, you may hear about it, so you should know what it is.

In normal market conditions, the bid-ask spread for a given option will be steady. As long as supply and demand follow usual patterns, the spread will be the same number of pennies apart. If the spread is a little wider or a little narrower than usual, the scalper swoops in to make a profit.

Here’s how it works:

If the spread is wider than usual, that means the ask is higher and/or the bid is lower than it should be. This happens because slightly more people want to buy than want to sell. The scalper views this as a sign to sell the option.

If the spread is narrower than usual, meaning the ask is lower and/or the bid is higher than normal, then slightly more people want to sell than to buy. The scalper views this as a sign to buy the option.

Scalpers must work fast. They make a lot of transactions with very small profits on each of them. There’s little risk under normal market conditions. The problem is if supply and demand are starting to change because of some new information coming to the market, the scalper will end up on the wrong side of the market and be crushed. As a result, scalping is sometimes referred to as picking up nickels in front of a steamroller.

In the equity markets, scalpers have been decapitated by high-frequency and electronic trading systems. These identify and close out small price discrepancies so quickly it is almost impossible for an independent trader to keep up. (We’re talking fractions of a second here.)

And guess what? The electronic trading systems moved on to the options markets. You might see scalping taking place, but you probably can’t participate.

Synthetic Securities

Synthetic securities are a form of arbitrage. The combinations of payoffs are designed to mimic the payoff of options. The reason for doing a synthetic is it is often a cheaper way to achieve the same outcome. A synthetic increases your return for the same level of risk, which means some of your return is in the form of a riskless profit. And that’s what you want!

There are enough different patterns that I devote a whole chapter to the topic—Chapter 14. For this discussion, the important points are that options and underlying assets can be thought of as payoffs, and there is more than one way to get the same payoff.

Conversion Arbitrage

Conversion arbitrage is a way to lock in a riskless profit if the price of an at-the-money call is higher than the price of an at-the-money put. You go long a put and short a call (with the same strike price and expiration) on an underlying asset you already own.

If the price of the underlying asset goes above the call before expiration, then the short call will be exercised. That offsets the position in the underlying asset, and the long put expires. The trader’s profit is the cost of the put less the premium received for the short call.

If the underlying price falls below the put strike price, the long put can be exercised, and the short call expires unexercised. The profit is the exercise price of the option less the price of the underlying asset, adjusted by the amount of the initial premium position.

Did you see what happened here? As long as the premium received on the put is higher than the premium paid on the call, then the position will have no risk—and a profit is locked in.

The wrinkle is the price paid to acquire the stock. If you end up selling it for a lower price than you paid to buy it, you’ll have a loss on the position. Conversion arbitrage can be powerful, but it can’t work miracles.

The following tables show the payoff of a conversion on 100 shares of a $45 stock and at-the-money options. The Calculation column shows how the cost was found, and the Cash Inflow (Outflow) column shows the net.

Opening Transaction

Closing Transaction If the Stock Falls to $40 at Expiration

Closing Transaction If the Stock Is at $45 at Expiration

Closing Transaction If the Stock Is at $50 at Expiration

Yes, the profit is small. But it also comes with very little risk. In academic theory, all returns are a function of risk, so a return with no risk is, essentially, free money.

Reversal

A reversal, also known as a reverse conversion, is another form of option arbitrage. In a reversal, the trader buys the call, sells the put, and sells the stock short. It offers a risk-free profit if the prices of the options are out of alignment relative to the underlying position. This is a rare situation, and it requires plenty of margin in your account. But, if this applies to you, read on!

If the underlying rises above the strike price of the call before expiration, then the call will be exercised. That offsets the position in the underlying asset, and the short put expires. The trader’s profit is the put premium less the cost of the long call.

If the underlying price falls below the short-call strike, the call can be exercised, and the long put expires unexercised. The trader’s profit is the put premium less the cost of the short call.

As long as the premium received on the put is higher than the premium paid on the call, then the position will have no risk—and a profit is locked in.

The following tables show the reversal on a 100 shares of a $45 stock and at-the-money options.

Opening Transaction

Closing Transaction If the Stock Falls to $40 at Expiration

Closing Transaction If the Stock Is at $45 at Expiration

Closing Transaction If the Stock Is at $50 at Expiration

As with the conversion, the reversal is likely to generate a very small profit, but a profit coming with very little risk.

Box Spreads

A box spread is also known as a delta-neutral hedge. Delta, of course, is the rate at which an option’s price changes as the price of the underlying asset changes. It is a combination of a bull-call spread and a bear-put spread.

A bull-call spread is the purchase of a call with one strike price, then writing a call with a higher strike price. The premium received will be less than the premium paid because the call with the higher strike price will be further out-of-the-money.

The maximum profit on a bull-call spread is the difference in the strike prices, less the net debit position of the premiums paid. The maximum loss is the premium paid.

A bear-put spread involves buying a put with one strike price and then writing a put with a lower strike price. The premium paid on the purchased put will be higher than the premium received on the written put because the purchased put will be closer to the money in value.

The maximum profit on a bear-put spread is the difference in the strike prices minus the net debit position of the premiums involved. The maximum loss is that net debit position—much less than with a naked put.

The bull-call spread posts its maximum profit when the underlying price is greater than the strike price on the short call. The bear-put spread has its largest profit when the underlying price is less than the strike price on the short put.

With a box spread, the profit will be even lower, but the loss is contained. Remember, one of the spreads will lose money, and it’s possible both will.

Interest Rate Arbitrage

An interest rate, also known as a yield, is nothing more than the price of money. However, the price of money is a component of the value of many different types of underlying assets. Hence, many traders look to the derivatives markets to speculate or hedge on interest rates. This is known as interest rate arbitrage.

The interest rate itself is driven by three factors:

• The level of opportunity cost in the economy as a whole
• Inflation
• The level of risk for the underlying asset

Differences in interest rates for different types of securities are affected by differences in these specific factors.

For example, the difference in interest rates (known as the spread) between U.S. government bonds and Japanese government bonds is due to differences in inflation and opportunity cost. This is true because these countries are as close to certain to repay their loans as is possible; there is not risk that they will not repay their loans. The difference in interest rates between U.S. government debt and emerging markets debt includes inflation, opportunity cost, and the risk that the loans will not be repaid.

If one of these three factors changes, then the spread must change, too. If the change in the spread is out of alignment with what it should be, then there’s an opportunity to buy futures or calls on the bond with the lower-than-expected yield. There’s also an opportunity to sell futures or calls on the bond with the higher-than-expected yield.

Sure, a trader could buy and sell the actual bonds, but using futures and calls will be cheaper and easier.

Index Arbitrage

A market index is a collection of stocks used to measure the performance of the stocks in a particular country, industry, or investment strategy. They are used as a performance benchmark and as an investable asset. Traders can use options, futures, and exchange-traded funds to hedge, speculate, and set up index arbitrage strategies.

As a simple matter, the fact that there are different ways to play the same underlying asset means there might be occasional price discrepancies. Price discrepancies are especially likely to occur on days with a high level of market volatility, in which the price of an exchange-traded fund (ETF) on an index might not reflect the price of the stocks in the fund because of the speed and magnitude of the price changes. You could go long an underpriced ETF or short an overpriced ETF and wait for the price to move into alignment before closing out the position.

The problem with using the ETF alone is the market could continue to move—your ETF might be underpriced, but if the market as a whole continues to fall, it will reach the correct price lower than where you bought it.

This is where the multitude of securities comes in. You could buy the underpriced ETF and sell a call on the index. That means you have the premium from the call and protection if the price of the ETF falls. This is very much the province of high-frequency traders. You might see opportunities like this on volatile market days, but as an individual, you will have a hard time executing them.

Merger Arbitrage

News flow creates volatility in the underlying asset. That leads to occasional periods where an asset is not priced properly. And in the options world, volatility adds value.

When two companies announce a merger, there are a lot of details to attend to and a lot of things that can change between the day the merger is announced and the day it closes. For example, changes might affect the price, the currency used to make the acquisition (i.e., cash, debt, or stock), or the date the transaction closes. That assumes that the deal goes through and no other company pops in to make a bid.

In this situation, the options of the buyer and seller company will probably increase in value because volatility will increase until the day the deal closes. That means there might be opportunities to buy in the equity market and to sell in the options market, or vice versa, for a profitable and low-cost trade. There are hedge funds and traders who specialize in these trades, which makes it difficult for individual traders to play in these markets.

Commissions, Taxes, and Other Considerations

There’s one very big problem with seeking out arbitrage transactions—the costs of putting them into play, including the taxes.

The shorthand is this: if the call price less the put price is greater than the interest expense and commission, less any dividend received, then you should go ahead with the transaction.

Commissions

There’s a term in trading called the alligator spread. It’s a position that looks great based on market prices. However, the profits will be eaten up by the commissions.

You must know what your broker charges, and keep that in mind when you are planning a trade. The more actively you trade, the more important the commission will be to the profitability of your strategy.

Most stocks paying dividends do so once a quarter. The value of the stock falls on the day the dividend is paid. All else being equal, the difference in value between a long call and a short put should be the difference in the dividend before the dividend is paid.

If you have a conversion on a dividend-paying stock when the dividend is paid, then you will receive the dividend. This will reduce the cost of carrying the position.

Interest

If you borrow money to establish a position, then you pay interest. If you use money you already have, then you give up the interest you could have earned if you had kept it on account instead. On the other hand, if you earn a premium from your option, then you have cash you can put in your account to earn interest.

Consider the earlier conversion example. The underlying asset is purchased on margin, so there’s no upfront cost. However, there will be interest charged on that position. (If you don’t have enough money in your account to meet the margin requirements, the broker isn’t even going to let you borrow the additional funding.) On the other hand, there will be interest earned on the premium received for the short put.

A lot of transactions make more sense when interest rates are high then when they are low. When rates are high, the money earned on cash in the account can be enough to help offset other costs.

Taxes

Benjamin Franklin said the only two sure things in life are death and taxes. You’ll probably owe taxes on your riskless profit, so take that into consideration.

Taxes on options can be tricky. Chapter 18 has tons of information on the tax treatment of options, and that can help you when you plan your trade. More than one trader has been tripped up by a mismatch of long-term losses and short-term gains. Don’t be one of them.

Problems and Downsides of Arbitrage

The biggest problem with arbitrage is that profits are very small. The risks are very small, too, but they are rarely exactly 0. That means it doesn’t take much to wipe out a lot of hard work finding and placing a trade.

Pin Risk

Many arbitrage strategies call for the purchase of an at-the-money position, under the assumption the price of the underlying asset will be different at the time of expiration. But what happens if the underlying price is exactly equal to the strike price at the time of expiration?

That position loses money. It doesn’t happen often, but it does happen.

In trader terms, the underlying price is pinned to the strike price, and a strategy that would be harmed by this is said to have pin risk.

High-Frequency Trading

Professional trading firms know all about arbitrage. They play it by running powerful computer programs with ultrafast connections to find mispriced investments, then buy the cheap and sell the expensive in no time.

How fast? Faster than you can blink your eye.

The problem for traders who are not machines is they can’t compete.

High-frequency trading is a significant factor in pretty much all financial markets these days. It has eliminated a lot of the opportunities that day traders had to make money.