Lessons from the Pros

Options

# Delta Gamma Theta Vega

This is not the name of an extra-geeky college fraternity. It’s a list of the greeks, a set of variables that describe how option prices change when certain things happen.

Last time, I mentioned Delta in connection with the hedging of option positions. This week, we’ll cover Delta in more detail.

Since options represent the right to buy or sell something, their value should change when the value of that something changes. The Delta of the option measures just how much, in dollars and cents,  the option’s per-share price should change, for the next one-point change in the underlying asset’s price. (For stocks, one point means one dollar, but this is not true for every underlying asset. Indexes, for example, do not represent dollar values – 1400 on the S&P in no sense means \$1400 – it’s just points, a mathematical calculation).

Back to option deltas. If an option has a delta of .48, that means its price should rise by 48 cents (48% of a point) when the underlying rises by a full point; and it should fall by 48 cents when the underlying falls by a point. All else being equal, call options must become more valuable as the underlying rises, as the right to buy the underlying at a fixed price provides more and more of a discount compared to the current price. Since calls have to move in the same direction as the underlying, their deltas are positive numbers. By the same logic, the deltas of puts are negative numbers – their values go down when the underlying rises. A put with a delta of  -.48 would drop by 48 cents (or you could say that it would rise by minus 48 cents) if the underlying rose by a point.

Delta always has to be a fraction. It can never be greater than 1.0 for calls, or less than -1.0 for puts. If it could, that would mean that the option could change by more than a dollar for a dollar move in the underlying. And this can’t happen. To see why, imagine a call with a delta of 1.5. For the next dollar increase in the underlying, the option will rise \$1.50. This would provide a risk-free profit opportunity: you could sell the stock short and simultaneously buy the call option. If the price of the underlying then rises by \$1.00, you lose a dollar on your short stock, but make \$1.50 on the option, for a \$.50 profit. There is no such thing as a risk-free profit in a free market. If such a thing did appear, everyone who saw it would immediately short stock and buy calls as fast as they could, driving the price of the calls up and the price of the stock down, until the normal relationships were re-established.

This may make more sense if we consider a secondary interpretation of the Delta: that is, that the Delta is the rough probability that the option will expire in the money. A delta of .48 indicates that the call buyers believe that there is a 48% chance of its expiring in the money. Note that I didn’t say that there really is a 48% chance – just that the option-buying public believes that. They express this belief by the price they bid for the call. Since there can never be a probability of more than 100% that something will happen, there should never be a delta that is more than 100%, or 1.0.

The delta of an option to buy the stock at its current price (an at-the-money option) is around .50. In probability terms, there is a 50% chance that the stock price will be higher in the future than it is now, and a 50% chance that it will be lower (the option pricing model can’t read charts. It assumes that the next penny movement is exactly as likely to be up as it is to be down, as is the move after that, etc.)

The delta of an option that is extremely far out of the money (OTM) is very low. Today (a Wednesday), the stock of Apple is around \$560. It moves on average about \$30 per week. If I have a \$590-strike call that expires this Friday, its probability of ending up in the money is very small. Apple would have to move by a weeks’ average range in just a couple of days. Not likely – and the \$590 call’s delta today is only .04, implying just a 4% chance that Apple will move up \$30 in two days.

The other side of that coin is that the deltas of options that are far in the money are very high. The \$530-strike call (about \$30 in the money) has a delta of over .95, implying more than a 95% chance that Apple will stay above that price – that it will not move down by more than \$30 in two days.  So the option market is saying that there’s only a 4 or 5% chance that Apple will move a whole week’s worth in two days, in either direction. A reasonable assessment.

There is a third interpretation of Delta, which is really the most important one of all: it represents the hedge ratio. Say that I sell one of those 590-strike calls, with a delta of .04. I know that if Apple then rises by \$1.00, I’d lose \$.04 times 100 shares, or \$4.00 total, on that call.  To offset that, simultaneously with my sale of the call I’ll need to buy 4 shares of stock. Four shares X \$1.00 gain = \$4.00, which equals my \$4.00 loss on the short call.

Why do I say that the hedge ratio is the most important aspect of Delta? Because without the ability to hedge, there would be no option market makers and therefore no option market. The possibility of hedging was the whole reason for developing the option pricing model.

Next time: Beyond Delta – the other Greeks.

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

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