Probability is generally defined as the likelihood of an event happening, within a certain time frame, expressed as a percentage. With options probability, the event may be the likelihood of an option being in the money (ITM) or out of the money (OTM), and the time frame might be the expiration of the option.
Take a look at the Option Chain in figure 1. The underlying stock is trading around $132, so the 135-strike call is OTM, and its 0.22 delta implies it has about a 22% chance of finishing ITM at expiration. Another way of expressing this is to say the option has about a 78% chance of expiring worthless.
These numbers assume the position is held until expiration. Depending on your objectives, you could try to close or adjust this trade prior to expiration. But when structuring your trade and considering adjustments prior to expiration, understanding these probability calculations can help you more objectively manage your risk.
Options are not suitable for all investors as the special risks inherent to options trading may expose investors to potentially rapid and substantial losses. Options trading subject to TD Ameritrade review and approval. Please read Characteristics and Risks of Standardized Options before investing in options.
Delta is a measure of directional exposure, mathematically the first derivative of option premium with respect to underlying price. Simply said, an option's delta represents the dollar amount by which the option's value changes when underlying price rises by one dollar.
Delta of a call option can reach values from 0 to +1. It is never negative, as call option prices increase when underlying price rises. It is never greater than 1, because an option's value can't change by more than the underlying price change (see why for ITM and OTM options).
Delta of a put option ranges from -1 to 0, as put options tend to appreciate when underlying stock goes down. Again, the rate at which the option price moves is never greater than the underlying price change, therefore put delta is never lower than -1.
Therefore, if the absolute value of an option's delta is lower than 0.50, the option is out of the money. If it is higher than 0.50, it is in the money. This simple rule doesn't work 100% of time, especially for deltas very close to 0.50 and options with longer time to expiration. In reality, an option can be exactly at the money and have a delta of 0.54. But when you see an option's delta of 0.80, you can be virtually sure that this option is in the money.
If you have a call and a put option, both for the same underlying, with the same strike price, and the same time to expiration, the sum of absolute values of their deltas is 1.00. For example, you can have an out of the money call with a delta of 0.36 and an in the money put with a delta of -0.64.
Sometimes delta is used as a proxy for the probability that an option will expire in the money. According to this technique, an out of the money call with a delta of 0.36 has a probability of expiring in the money of 36%. An in the money put with a delta of 0.64 has a 64% chance of expiring in the money (for puts you take the absolute value of delta).
This is in line with the above mentioned relationship between call and put delta (their absolute values summing up to 1 for the same strike). For any pair of call and put options on the same underlying and with the same strike price, one of them will always expire in the money and the other will expire out of the money. Therefore, the sum of the probabilities should be 100% (and the sum of the absolute values of deltas should be one).
Note: We do not assume the possibility of underlying price ending up exactly at the strike, whose probability is infinitely small when we work with continuous (as opposed to discrete) prices, which most option pricing theory does.
If you use a risk-neutral pricing model and consider the probability there, then you get the probability with respect to a risk neutral measure, in addition that probability depends on the chosen numeraire. For example, in Black-Scholes model taking the risk-neutral measure with respect to the bank account $B$ gives
If you like to have a real world probability you have to consider the market price of risk and a real estimate for the volatility (not the implied one). Both are not listed in your parameters. If you like to get this probability use the first formula, but replace the interest rate $r$ with the drift of the stock (which contains the market price of risk) and the implied volatility with an appropriate estimate (you might consider historic volatility or assume that implied vol is an appropriate estimate or have a different view).
You can certainly calculate the probability of changes in variation but I have not come across a model that only looks at an isolated iVol and its associated term and then deriving a directional probability.
However, what you can do, and what options traders do all the time is to look at changes in skew which involves a range of implied data points. In Fx traders look at risk reversals. Also, in the short term, where trades in the option relative to the book take place has a bearing on directional probabilities. I am not gonna provide a formula, because I use some of that as part of my own business, just trying to push you into the right direction.
Deltas for owning call options always range from 0 to +1, because there is a positive relationship between changes in the underlying stock price and the value of the call option. A jump in the price of the underlying security should result in an increase in the value of the call option (assuming implied volatility and time-to-expiry remain essentially flat).
As a simple example, if a call option has a Delta of 0.25 and the underlying stock increases by $1, the value of the call option should increase by about $0.25. (note that we're speaking of dollars and cents here. In performance terms the call option movement would deliver a higher % change than the % move of the underlying).
Conversely, Deltas for owning put options always range from -1 to 0 because when the underlying security increases in price, the value of put options decreases (again, assuming essentially unchanged implied volatility and time-to-expiry)
Option delta behavior depends on the relative position of the strike price in comparison to the current price of the underlying asset. An in-the-money option, one that could be currently exercised for value, will have a higher Delta score than an out-of-the-money contract. An out-of-the-money option has no intrinsic value, meaning that it would make no financial sense to exercise an out-of-the-money option.
For out-of-the-money options, there is no intrinsic value regardless of how near or far the underlying security is trading from the strike price. In such case, 100% of the option price is related to time value (extrinsic value), which is the value assigned to the possibility that the option moves into the money.
The same is true for put options, but with negative numbers for delta scores. The delta of an in-the-money put option gets closer to -1 the deeper it becomes in-the-money. The delta of out-of-the-money put options approaches 0 as it becomes deeper out-of-the-money.
Professional option sellers determine how to price their options based on sophisticated models that often resemble the Black-Scholes model: a mathematical equation that estimates the theoretical value of options by taking into account the impact of time and other risk factors. The formula for delta can be derived by dividing the change in the value of the option by the change in the value of its underlying stock.
For example, suppose stock XYZ was trading at $520 per share and a call option with a strike price of $500 was trading for $45. This call option is in-the-money because the stock price is above the strike price. If the price of XYZ stock rises to $523, and the value of the call option rises to $46.80, the delta of this option was:
Important: Delta is used to measure the theoretical change in the value of option price given a change in the price of the underlying security. In reality, an option price, as reflected by its more recent trade price, may not change exactly as predicted by the delta, and could possibly not move at all if there is a wide bid-ask spread or limited trading volume.
In addition to predicting option price movement, delta values can also be used as a probability measure. Delta measures the expected probability that an option will end in-the-money at expiration. Remember, the deeper a call option is in-the-money, the closer the delta value will be to +1. Delta values are important to consider when assessing the risk and price volatility of option contracts.
Both call options are trading out-of-the-money here. Since the first call option, with the $550 strike price, is closer to the market price of XYZ stock, it will carry a higher price & higher Delta than the second call option. The $550 strike call option might have a delta value of 0.25, while the $600 strike call option might have a delta value of 0.15. This corresponds to the fact that the $550 strike call option has a greater probability of expiring in-the-money.
Since options carry delta scores that approach +1 as they move further into-the-money, and approach zero as they move further out-of-the-money, it's evident that the delta for any specific option is never fixed. In fact, delta values constantly change on the basis of:
The option delta for an initial $1 move in the underlying stock will likely be different than an additional $1 move in the same direction, although perhaps only marginally so, depending on various other factors, like the absolute level of the share price. For instance, if an in-the-money call option rises in value by $1.80 on the basis of the underlying security price increasing from $500 to $503, its delta score was +0.6.
The reason for this is due to the fact that a delta score increases as an option becomes further in-the-money. So as the underlying security rises higher and higher, the delta itself increases. In the above scenario of a $200 increase in the underlying security, the effective delta for this move might be +0.80. Furthermore, after the underlying security price reached $700, the delta of the call option at this point might be very close to +1.0, since it is heavily in the money at that stage. 59ce067264