Options and Hedges

Russ Allen

In a couple of previous articles, I mentioned the role of Market Makers in the options market. These people make the market possible by standing ready to buy or sell options at any time. Of course, they’re not doing this as a public service. They make money at it (although not as much as they used to).

The option market makers’ profit comes primarily from the bid-ask spread on the options they deal in. They buy at the bid, sell at the ask, and make the difference.  Many times, this means selling options that they don’t own – being short the options.

Looking at just Call options for the moment, we know that people who buy them have potentially unlimited profit. The higher the price of the underlying asset (“the stock”) goes, the more the call buyers can make, with no limit. Since every option contract is a zero-sum proposition, the sellers of calls have potentially unlimited losses.

Then aren’t market makers taking on a huge amount of risk, by being short a lot of options?

No, they’re not. They take on no risk at all related to movement of the stock price. They avoid that risk by using positions in the underlying stock itself as a hedge against their option positions.  Here’s how it works:  An option market maker sells 10 call contracts, representing 1000 shares of stock. As of that moment, every penny that the stock rises is going to take money out of his pocket and put it into that of the option buyer. To negate this, the market maker immediately buys stock himself.  In that way, every penny that he loses on the option, will be made up for by a gain on the stock. Conversely, if the price of the stock goes down, the market maker makes money on the options, and now loses the same amount on the stock. The market maker is neutral as to the stock price – he doesn’t gain or lose if it goes up or down. His money is made on the option bid-ask spread.

Looking at our previous examples, can it be worthwhile for a market maker to buy stock worth $600, just to make a profit of 40 or 50 cents on the option? Isn’t that few pennies of option bid-ask spread a pretty small tail wagging a pretty big dog? Yes and yes. The market makers wouldn’t do the business if they didn’t make money at it.

Now that we know that an options market maker is always hedged, remember that he is continually buying and selling both put options and call options.  When he sells calls or buys puts, then he has made his total position more bearish, so he must offset this by buying stock. When he sells puts or buys calls, then he has made his position more bullish, so he must offset this by selling stock.  He is always doing all four things at once (buying calls, selling calls, buying puts, and selling puts).  So he is constantly adjusting his stock position, to keep his net options exposure hedged perfectly.

Does the market maker have to buy or sell exactly 100 shares of stock every time he buys or sells an option?  No, he doesn’t.  The amount of stock that he must buy or sell per option is almost always some number less than 100 shares.

How many?

The answer to that question is what made it possible for the listed options business to exist.

Before 1973, there was no mathematically determined formula to calculate the right size of a hedge against an options position. There were no options exchanges. The few dealers who sold puts and calls over the counter (or negotiated them individually with big customers) had their own proprietary ways of managing their risk, but there was no widely-accepted systematic method. Some early work was done on this by mathematician Dr. Edwin O. Thorpe. In 1967 he published a book titled Beat the Market.  This book was revolutionary (although not nearly as popular as his 1962 book Beat the Dealer, which described the mathematically correct strategy for playing blackjack).

Beat the Market described the math for determining whether listed warrants were overpriced. Warrants were somewhat like options. Often attached to convertible bonds as an equity “kicker,” they represented the right to buy the company’s stock at a certain price.  Some warrants could be detached from the bonds and traded independently.  In the 1960’s when Thorpe was writing, some of these warrants were listed on the NYSE, where they could be bought or sold short.  Thorpe’s work related the theoretical price of the warrants to certain factors: the stock price, the warrants’ exercise price, interest rates, and the volatility of the underlying stock. The basic premise was that at any point in time, a warrant should sell for a price that equates its probability of profit to that of someone who buys the stock at that time. Sound familiar? It should, because it’s the same basic idea that makes possible the calculation of the theoretical price of any option. And the same idea that market makers use as they build and tear down their hedges every minute of every day.

Work on the mathematics of options and other derivatives continued, and a 1973 academic paper by three mathematicians named Black, Scholes and Merton, called “The Pricing of Options and Corporate Liabilities” introduced the Black–Scholes equation. This formula describes the price behavior of an option over time. It allows anyone with enough computing power to calculate what an option should sell for at any given time. Wasting no time, the Chicago Board of Trade created the Chicago Board Options Exchange, which became the first exchange to list standardized, exchange-traded stock options on April 26, 1973.

The key idea behind the Black-Scholes model, echoing Thorpe, was to hedge an option position perfectly by buying or selling the underlying asset in just the right way, and consequently to eliminate the price risk of the option position.  Hedging in this way is called delta hedging, or delta-neutral trading.

That name is used because one of the byproducts of an option pricing model (including the original Black-Scholes and a handful of others now widely used) is the delta of an option. In mathematics, Delta is the Greek letter used to represent change in a variable’s value. In this usage, the delta of an option is how much its price should change, for a given change in the underlying asset, to keep the relationship between the option’s probability of profit and that of the stock constant. If the underlying price goes down, then its likelihood of reaching any given higher price is less than it was before the drop (It’s harder to get to $640 from $600 than it is to get to that same $640 from $630).  So after a price drop, all call options for the stock should drop in price too. And all put options should increase in price. The Delta tells us how much, and it’s different for each individual option on the chain.

That leads into a fuller discussion of Delta and the other Greeks, which are the variables that describe option price movement. Understanding these factors is what allows us to build profitable options positions.  More about Delta and the other Greeks next time.

For questions or comments on this article, contact me at rallen@tradingacademy.com

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