From time to time I write about the basics of options for the benefit of newer traders. Today we’ll start a series on the Greeks. These are measurements of how an option’s price will change as a result of certain events.
– Current change in price of the underlying asset
– Changes in market expectations for future changes in price of the underlying asset
– The passage of time
– Changes in interest rates (minor to negligible effect)
About forty years ago an option pricing model, or formula, was devised, which takes these influences into account in a very precise way. Options had existed before then, but there had been no general agreement as to how they should be valued. Consequently there were no options exchanges and therefore no listed options. The publishing of the Black-Scholes model in 1973 coincided with an increase in available computing power. The model is a fairly complex set of partial differential equations. Without computers, computing a single option’s price would take so long that it would be obsolete long before it was done. But with the formula and the computers to calculate it quickly, an options market became possible. From the original twelve stocks with listed options in 1973, the number of stocks with associated options has grown to over four thousand, or roughly half of all stocks and ETFs listed in the United States.
The Black-Scholes model is not the only option pricing model. Others have followed it, with names like the Binomial Model and the Bjerksund-Stensland model. What they all have in common, though is the assumption that an option’s price is based on the factors listed above. All models calculate values for an option’s price, and for how it will change based on those factors.
All of the models take these items as inputs:
– Underlying Stock Price
– Option Strike Price
– Option Type (Put or Call)
– Historical Stock Volatility
– Current Risk-Free Interest Rate
– Expected dividend payments during the option’s life
– Option’s market price (actual trading price, determined by supply and demand for the option)
And produce these outputs:
– Theoretical option price
– Implied Volatility, which is the expected rate of stock volatility implied by the actual price
– Expected change in the option price for a 1% change in the stock’s implied volatility (Vega)
– Expected change in the option price for a $1.00 change in the price of the underlying (Delta)
– Expected change in the option price for the passage of one day’s time (Theta)
– Expected change in the option price for a 1% change in interest rates (Rho)
– Expected change in the option’s Delta for a $1.00 change in the price of the underlying (Gamma)
The last five items listed above are the variables called the Greeks. Like any variables dreamed up by proper math nerds, they are named with Greek letters, hence the term.
Let’s look first at the best-known of the Greeks, which is Delta.
Delta is given in pennies per share of option price change, for the next one-point increase in the price of the underlying asset. For stocks, one point equals one dollar. So an option that has a Delta of .50 should increase in value by fifty cents if the underlying stock’s price moves up by one dollar; or decrease by fifty cents if the underlying moves down by a dollar.
Notice that above I said that Delta is the number of pennies per share of option price change, for the next one-point increase in the price of the underlying asset. I used those words intentionally. There are two different types of options: Call options, which are the right to buy the underlying stock; and Put options, which are the right to sell the stock. Both types of options have Delta values, and they are of opposite signs.
Since a put option is the right to sell the stock at a fixed price (the strike price), the put becomes more valuable as the stock price goes down. That is because it then gives the right to sell the stock at a price which is above market. The farther above market something can be sold, the more valuable the right to sell it. So put options go up in value when their underlying stock goes down, and conversely down in value when the stock goes up. Therefore the Delta of a put is always a negative number; and the definition of Delta as, “the number of pennies per share of option price change, for the next one-point increase in the price of the underlying asset” applies equally to puts and calls. Since a put’s change in value is negative when the stock price increases, the put’s delta is a negative number. A put with a delta of -.50 would go down in price by $.50 when the stock goes up by $1.00. The put would go up in price by $.50 if the stock went down by $1.00.
OK. The Delta of calls is a positive number, and the Delta of puts is a negative number. But more specifically, the Delta of an option can only be within a specific range. For calls, the Delta cannot be a negative number, which is another way of saying that the Deltas of a call cannot be less than zero. Likewise the Delta of puts must be a negative number, so their value cannot be more than zero.
A Delta value of zero for an option means that the next $1.00 move in the stock will have no effect on the price of the option. How can this be? Well, say that you have a call option that gives you the right to buy IBM stock at $300 for a month, when IBM is at $180. There is so little chance of IBM going to $300 in a month, that the option to buy it for $300 would be worthless – there would be no bids for that option. If IBM moved that next dollar, from $180 to $181, the change in its chances of reaching $300 in a month will not move the needle. It’s still a probability of zero, so the $300 call will still be worth nothing. Since a one-dollar change will make no difference in the price of this option, moving it from zero all the way to zero, its delta would be zero.
At the other end of the probability spectrum, let’s look at the maximum value that delta could have. The short answer is that an option can never move more, in pennies per share, than the underlying stock does, no matter how big the underlying stock move is. If this were not true, then there would be an opportunity for a risk-free profit. And that condition cannot persist for any longer than the milliseconds it takes for computers to detect it and trade it out of existence.
So the maximum value that the Delta of an option can have is 100% of the speed at which the stock moves, and not a penny more. 100% is also expressed as 1.0, and that is the maximum value that the delta of a call can have. Puts’ deltas are negative, so -1.0 is the maximum value for the delta of a put. Options that are deeply in the money have deltas of plus or minus 1.0.
A call is deeply in the money when it gives the right to buy the stock at a strike price that is so much lower than the stock’s current price, that there is no chance of the stock dropping to that strike price before the option expires. In our IBM example, the right to buy IBM at $100 (when it was trading at $188) is such an option. That call’s delta would be 1.0, and it would change in value penny for penny with IBM stock.
A put is deeply in the money when it gives the right to sell the stock at a strike price that is so much higher than the stock’s current price, that there is no chance of the stock rising to that strike price before the option expires. In our IBM example, the put that gives the right to sell IBM at $300 (when it was trading at $188) would be deeply in the money. That put’s delta would be -1.0. It too would change in value penny for penny with IBM stock, but in the opposite direction to the stock.
Since Delta shows us how an option’s price will change when the underlying changes, it tells us how many options we would need to hedge a stock position of any size. Deltas are given in values per share, and option contracts are for 100-share lots. So a Delta of .5, for example, means that a 100-share option contract would have a total delta of 100 * .5 = 50. The price movement of such an option contract would be fifty dollars when IBM moved by one dollar per share ($50 per 100 shares)
Say we owned 1000 shares of IBM, and wanted to completely neutralize our position for the very next penny’s worth of movement. We had Puts available with Delta values of -.25, -.5, and -1.0. The Delta values of each put tells us how many of those puts we’d need. For the put with the -.25 Delta, each contract would give the same amount of price movement as -.25 * 100 = -25 shares of stock. Since we need to have -1000 shares worth of coverage, we would need -1000 / -25 = 40 option contracts. If we used the puts with the .5 delta, we’d need -1000/-50 = 20 option contracts. And if we used the puts with the -1.0 delta, we would need just -1000/-100 = 10 option contracts. This usage of the Delta is as a hedge ratio.
Finally, the Delta is said to give the approximate probability of the option finishing in the money. Notice that I wrote that the Delta is said to give that. In fact this is an extremely rough measure, which is only approximately true at best, and then only when the stock’s volatility is very low. We should not rely on this common interpretation of the Delta.
That’s all we have space for today. Next time we’ll continue our exploration of the Greeks.
For comments or questions on this article, contact me at firstname.lastname@example.org.