Delta: A Little Goes a Long Way

Black-Scholes and delta—they’re kind of a “which came first, the chicken or the egg” situation. Delta is a derivative of the Black-Scholes Option Pricing Model, so you’d think Black-Scholes came first. But delta is also a key part of the Black-Scholes equation, so maybe delta did come first. Even the chicken could be scratching her head on this one.

Either way, delta is the most critical “greek” when it comes to options trading. The stock price is the main determinant of whether an options trade is profitable or not, and delta is the metric that tells you how much the price of an option may theoretically respond to changes in the stock price (see Figure 1). Sure, the other greeks (like vega and theta) are important, and we covered those in the previous two issues. But it’s delta that tells you where most of your risk and potential opportunity lie.

Delta Works in Mysterious Ways

Say FAHN is trading for $140 per share. The September 145 call has a theoretical value of $3.80 and a delta of 0.39. If the stock price rises from $140 to $141, theoretically the 145 call will go up from $3.80 to $4.19, all other things (like time and volatility) being equal. If FAHN drops from $140 to $139, theoretically the 145 call will drop from $3.80 to$3.41. So, delta is the theoretical measure of how much the price of an option will change when the underlying stock, index, or futures changes $1.

Delta is an instant metric, meaning it’s accurate only for current data. Let’s say FAHN increases $10 to $150. If you multiply the 0.39 delta by $10, you might think that 145 call would grow from $3.80 to $7.70 ($3.80 + $3.90). But that 145 call might actually be worth $9.50. That’s because as the stock price rises, the delta of that 145 call increases as well, from 0.39 to some higher amount—say 0.80—thanks to changes in gamma. (We looked at this in Unpacking Gamma—Delta’s Evil Enabler). So the impact on the 145 call’s price when FAHN rose from $146 to $147 is greater than when FAHN rose from $140 to $141.

Also, delta changes as time passes. If that 145 call has 90 days to expiration when its delta is 0.39, and the stock price doesn’t change, it might have 0.20 delta with 30 days to expiration. Because delta represents the options position’s sensitivity to a change in the stock price, that 145 call is becoming less and less “bullish” on FAHN. Remember, positive deltas are bullish, and long stock, long calls, and short puts have positive deltas. Negative deltas are bearish, and short stock, short calls, and long puts have negative deltas.

As far as bullish or bearish speculations go, delta can be interpreted as how many shares of stock the options position’s risk represents. When that 145 call had a delta of 0.39, it theoretically had the same directional risk as 39 shares of FAHN. When the 145 call had a delta of 0.20, it theoretically had the same directional risk as 20 shares of FAHN. This is the genius of Black-Scholes in recognizing that an options position’s price is determined by the number of shares required to hedge it, plus a loan to borrow that number of shares.

Technically, 39 shares of FAHN is a riskier, more bullish position than 20 shares, simply because when FAHN moves up or down $1, the first position makes or loses $39, while the second position makes or loses $20. Therefore, the underlying has to move nearly twice as much for the 0.20 delta call to increase by the same amount as the 0.39 delta call. That’s why we can say that a 0.20 delta call is less bullish than a 0.39 delta call, and that a -0.20 delta put is less bearish than a -0.39 delta put.

Putting Delta to Work

There are two important points that need to be considered here.

Delta is always changing. As stock price,time, and volatility change, delta sometimes changes a lot or little. You need to monitor the deltas of your trades regularly to make sure they’re giving the risk exposure you want—not too high, not too low. For more on how delta changes, see thinkTank.

Delta is probability. Understanding delta in this way offers insight into the probability that an option will be in the money (ITM) or out of the money (OTM) at expiration. To get into the theoretical weeds a bit, delta depends on how far (expressed as a percentage) the strike price is away from the current stock price.

For example, with FAHN at $140, the 145 call is 3.57% OTM. If you plopped the current stock price and strike price onto a normal distribution (bell curve), the stock price would sit at the peak of the mean and the 3.57% OTM call strike price would sit to its right. At the mean, there’s a 50% probability of being above or below (which is why the at-the-money [ATM] option has a delta close to 0.50).

Because that strike price is to the right of the mean, the probability of being to the left of the strike price is higher than the probability of being to the right. So the probability of the stock being above that OTM call strike price is lower than the probability of the stock being below.

Lower-delta options have a lower probability of being ITM at expiration. Delta can range from zero to 1, so an option with a zero delta has a low theoretical probability of being ITM at expiration, while an option with a delta of 1 has a high theoretical probability of being ITM at expiration. Option swith a delta of 1 don’t always wind up ITM because there is never a zero probability a stock could crash to zero or soar infinitely high. Delta is a theoretical metric based on current market conditions, not a guarantee.

When you create an options trade, delta reflects how much risk the position has in terms of shares of stock and its probability of profit at expiration. Here are three examples to consider.

Long Stock or Short Puts

Are you bullish on a stock and deliberating between buying shares or shorting (selling) puts? A position that’s long 100 shares of stock has +100 deltas. If an ATM put has a delta of -0.50, shorting two of them would create a position with +100 deltas, giving you the same risk exposure as the long 100 shares. So what’s the difference?

Well, if the stock price doesn’t change by the options position’s expiration, the short puts can be profitable, while the long stock would have no profit or loss (unless it pays dividends). In this case, the short puts have positive theta. In exchange for that positive theta, as the stock price moves up or down a point or two, the profit and loss of the short puts will be similar to the long stock. But if the stock drops sharply, the put deltas will move from -0.50 to closer to -1, and the short two puts will have a delta closer to +200. So the options position has more risk and greater potential loss than the long 100 shares. Conversely, if the stock price rises sharply, the put deltas will move from -0.50 to closer to zero, and the short two puts will also have a delta closer to zero. In this case, the long 100 shares can be more profitable than the short two puts.

Married Puts

This leads us to the next strategy, the married put. That’s when you buy 100 shares of stock because you have a long-term bullish outlook and want to collect any dividends, but you also want to fine-tune your delta risk by buying puts.

Say a stock is $150 and you buy 100 shares to give you +100 deltas. If you bought an ATM put, that would generate approximately -50 deltas, giving the married put strategy a net +50 of delta exposure. But say you want between +25 and +30 of delta exposure. In that case, you could buy two puts that each have 0.35 delta. They generate -70 deltas and combined with the +100 deltas of the long stock would give the married put strategy +30 deltas.

By selecting a certain OTM strike, and depending on how many puts you buy, you can get the delta you want without having to buy an odd number of shares. As the stock moves up and down, you can adjust the puts to higher or lower strikes, or roll to further expirations to maintain the same delta exposure and continue the trade if you choose. Keep in mind that rolling strategies can incur significant transaction costs.

Synthetically, a married put is similar to a long call, which has the same risk profile. The reason you might consider the married put is that the long stock never expires, so you can treat the long stock and long put independently for a longer-term trade. If the stock price drops, the long puts could be profitable and help to offset the loss on the long stock. You may want to consider selling the puts, taking the profit, and holding the long 100 shares, hoping they’ll bounce back. You could even take things a step further by selling an OTM call against the stock. This strategy gives you a lot of flexibility but will limit the stock’s upside potential.

Beta Weighting

Finally, delta finds one of its most powerful uses with beta weighting on the thinkorswim platform from TD Ameritrade. Beta weighting is a way to convert the deltas of individual stock positions to be comparable to the deltas of an index, say, the S&P 500 Index (SPX). This way you can see how many deltas your FAHN position has in SPX terms and also compare it to the deltas for another position in SPX terms. When you compare your beta-weighted deltas, you can see which of your positions has more delta risk than the others.

In short, beta is a measure of a stock’s risk relative to an index like the SPX. Beta weighting uses that information to normalize individual deltas into the delta of a single symbol for comparison purposes.

For example, Figure 2 shows a portfolio composed of 100 shares each of a bunch of different stocks, as seen in the Position Statement section of the Monitor tab.

Before beta weighting, each position has 100 deltas. It’s difficult or impossible to judge which positions have more risk than others. But if you select the Beta Weighting box at the top and type “SPX” in the field, the deltas will be converted into SPX deltas.

The first symbol on the list has the equivalent of 7.46 SPX deltas. Another position has the equivalent of 41.12 deltas—about 5.5 times bigger. Clearly, the 41.12 delta position has more delta risk, so if you’re looking to reduce the exposure of your portfolio, you might consider reducing the deltas of the higher-delta position first. That could be done with short covered calls or long married puts, for example.

It’s easy to toggle between beta weighted and non-weighted deltas as you drill down into the risk of your portfolio. And a deep understanding of delta can help you make informed and confident choices as you manage your trades and portfolio.