1.4.2 Delta
Delta is the first derivative of the option price with respect to spot price, and roughly corresponds to the probability that an option will end up in the money. Delta ranges from 0 to +1 for calls and 0 to -1 for puts, corresponding to an assignment risk between -100 to +100 shares of the underlying. Buying and selling options and the underlying stock such that total portfolio delta is zero eliminates assignment risk and is commonly practiced by market makers.
Example
The SPY is currently trading at $595. A market participant sells 12 SPY calls at the $600 strike expiring in 45 days for $7.00 per contract with a delta of 0.7. The total expected assignment risk is for 70 x 12 = 840 shares of the underlying at expiration, corresponding to a delta dollar value of $499,800. To hedge this risk, the call seller can buy 840 shares of SPY at $595 per share.
Summary of Terms
= Contract Value
= Contract Delta
= Spot Price
= Strike Price
= Implied Volatility
= Years to Expiration
= Risk Free Rate
= dividend yield
Calculation
Delta vs. Strike Price
The figure below shows delta curves for calls and puts for an underlying equity with a spot value of $100. Each curve is an S shaped curve going from an absolute value of magnitude 1 when deep in the money, to zero when deep out of the money. The put delta curve is exactly identical in shape to the call curve, just shifted down by 1. When the spot price is equal to the strike price ($100 in this case), then the delta value is usually about 0.5, which means that the proposition that the contract will go in the money is basically a coin flip. In general, this “coin flip” point can be different than the strike price of the contract. For example, long dated contracts on indices tend to be above spot, as it is expected that spot will rise over time. The “coin flip” point coincides with the minimum value of IV on the IV smile, and can be regarded as where the market expects the future value to drift towards over time. At expiration, the delta on a contract can only be either 0 (out of the money) or 1 (in the money).
Because the delta on a contract determines the majority of the hedging done by market makers who open them, it’s important to understand how delta changes with changes in spot price, implied volatility, and time to expiration, as that will dictate how market makers will buy and sell the underlying during changing market conditions. The sensitivities of delta to changes in spot, IV, and time to expiration are referred to as gamma, vanna, and charm.
Delta vs. Spot Price
Delta vs. spot price is the most commonly tracked change in delta, and is measured using the slope of delta with respect to spot price, called gamma (see 1.4.5 gamma). Look at the figure above, it’s clear to see that market maker hedging is the most pronounced at a delta of 0.5. For example, a move in spot from $100 to $105 drops the call delta from 0.5 to about 0.2, a change of roughly 30 shares. If the market maker is short that contract, that means that they must buy 30 shares as the price rises! However, if the price of the contract goes from, say, $110 to $115, the market maker only needs to buy about 3 shares. Thus there is a simple rule, most of the market maker delta hedging occurs near the strikes of the options they hold on their books.
There is another interesting dynamic that can occur. If there are enough long calls and long puts in the market, where an increase in price causes the market maker to buy and a decrease causes them to sell, the MM buying pressure can become large enough to make the hedging unstable. A rise in spot causes buying, which raises spot, which causes more buying, and this process continues until there are not enough open contracts remaining at spot. This runaway event is called a gamma squeeze.
Delta vs. Implied Volatility
The change in delta in response to a change in IV is measured as the greek quantity vanna (see 1.4.6 vanna).
The spread of the “coin flip” region around the contract strike price is determined by both implied volatility and time to expiration, where when either are higher, the spread of the S curve is wider. The figure below shows what happens to delta as IV changes for a set time to expiration. There are a couple of interesting features here. Perhaps the most obvious response is that options that are out of the money undergo a rise in delta, as it becomes more probable they will go into the money later. Second, when spot is below the strike price of the contract, increasing volatility drops the delta! This is because higher volatility increases the risk that a contract that is in the money will go out of the money later. Thus, increases in volatility drops the delta of in the money contracts with strikes near spot, and raises delta of out of the money contracts near spot.
This behavior can result in strange hedging situations. If the market delta is made up of mostly in the money long calls and out of the money long puts, a rise in IV will dramatically drop the delta of the calls and raise the delta of the puts, creating massive downward selling pressure from delta hedging. However, if most of the contracts sensitive to IV are out of the money long calls and in the money long puts, a rise in IV will increase the positive delta of the calls and decrease the negative delta of the puts, causing massive upward buying pressure. When this selling or buying pressure becomes large enough to create further increases in IV, the hedging goes unstable and results in a vanna squeeze.
Delta vs. Time
The change in delta in response to a change in IV is measured as the greek quantity charm (see 1.4.7 charm).
The figure below shows how delta evolves with a change in days to expiration. The spread in the “coin flip” region again expands as time increases, similarly to increases in IV, with the key difference being that it expands based on the square root of time vs. roughly linearly with IV. In practice this means that the expansion has limited returns. Extending your expiration from 1 day to 5 days makes a big difference, while extending your time from 1 year to 1 year and 5 days makes almost no difference. This has useful implications for various trading strategies. For example, if you are long a contract but very uncertain of when your expected move will occur, buying additional time means that the delta, and therefore the value, of your contract will decay more slowly. So your position bleeds less value over time while waiting for the expected move. Conversely, if you are selling options, choosing shorter dated expirations means that the value of the contract decays more rapidly towards zero, so the position realizes more profit more quickly. Of course the catch for these shorter dated contracts is that they also can go up in price more quickly due to an increase in volatility or big move in spot, which can cause the trader to lose much more than 100% of their potential payout.
A rough estimate of the percent that a contract will decay in a given day is simply 1 divided by the number of days remaining until expiration. A contract with 16 days to expiration will lose roughly 6 percent in value that day just from theta decay. A contract with 2 days to expiration will lose roughly half of it’s value that day.
The market maker hedging of this decay often leads to either steady buying or selling over time, and this very steady behavior is applying small, consistent pressure on price over time, even in the absence of a lot of trading. This effect increases as you approach large expirations like quarterlies and leaps, resulting in strong, consistent trends that require extra volume on the market to alter. Note again that it depends on if the contracts are in the money or out of the money. If calls or puts are largely in the money, then they decay to a delta of magnitude 1. If out of the money, they decay to a delta of 0, or worthless. For example, if the market is mostly in the money long calls, delta decay will cause the market maker to consistently buy the underlying as the delta of those calls decay to 1.