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I hope the previous two articles about options have been useful and somewhat intuitive. The overall document contains more information if you want to dig a little bit deeper and that’s cool with me.

To finish it off nicely now, let’s look at the “greeks” and again try to use a good pinch of common sense as to what matters most. I really hope this will help your understanding of options.

#### Delta

The most often-used term isn’t that difficult to grasp. It basically is the change of the price of the option for a price change in the underlying security.

Going through the table example is very useful.

First, let’s look at call options. You should be familiar with the overall payoff profile below. So as the option moves more in the money (higher than the strike) the delta increases.

Also, quite importantly is the “accrual” of deltas as time progresses. Here it’s important to assess whether the option is in-the-money or not.

The same story for puts really while the payoff profile flips.

**Key points on Delta:**

Delta is the change in the price of an option for a 1-unit move in the underlying. A delta of 0.5 means that a one-cent increase in the underlying price will cause a one-half of a cent increase in the option price. Hence, the option price moves only half as much as the underlying price.

Delta can be used as the linear hedge ratio. Since delta is a measure of how sensitive an option's price is to changes in the underlying, it is useful as a hedge ratio. A bond futures option with a delta of 0.5 means that the option price increases by roughly 1/64th for every 1/32nd increase in the bond futures price. For small changes in the futures price, therefore, the option behaves like one-half of a futures contract. Similarly, a long position in 10 at-the-money call options behaves like a long position in 5 futures. Remember that at-the-money call options have a delta of 0.5.

Constructing a delta hedge for a long position in 10 ATM calls, each with a delta of 0.5, would require you to sell 5 futures contracts. (The delta of a futures contract is always 1.)

Delta can be interpreted as the probability that an option will end up in-the-money. An at-the-money option, which has a delta of approximately 0.5, has roughly a 50/50 chance of ending up in-the-money. Out-of-the-money options have deltas less than 0.5; in-the-money options have deltas greater than 0.5.

Call option deltas range between 0 and 1. Put option deltas range between 0 and -1.

The delta of a long option position increases as the underlying price increases. (This is true for both calls and puts.) This reflects one of the main benefits of being long options compared to holding a position in the underlying. A long option position becomes longer the market (delta increases) when the market is rallying and shorter the market (delta decreases) when the market sells off.

**Time to expiration**Deltas of in-the-money call options increase towards 1 as time passes.

Deltas of out-of-the-money call options decrease towards zero as time passes.

Deltas become more changeable for at-the-money options as time passes.

**Volatility**An increase in volatility raises the delta of out-of-the-money call options and lowers the delta of in-the-money call options. At higher volatility levels, out-of-the-money options have a greater chance of ending up in-the-money; in-the-money options have a greater chance of ending up out-of-the-money.

A doubling of volatility has roughly the same effect on an option's delta (and its price) as a quadrupling of time. For example, the 100 strike call option delta is 0.92 with 1 month to expiration, futures = 102, and volatility = 5%. If volatility increases to 10%, the delta falls to 0.76, which is about the same delta that a 4-month option would have with volatility at 5%.

**Symmetry**There is an enormous amount of symmetry at work in option pricing models. For example, as shown in the above table, the 100 strike call has a delta of 0.76 when it is 2 points in- the-money (futures = 102), there is 1 month to expiration, and volatility = 10%. This tells us that there is roughly a 76% chance the option will end up in the money and a 24% chance the option will end up out of the money at expiration. With the option 2 points out-of-the-money (futures = 98) on the other hand, the delta is 0.24 suggesting the option has a 24% chance of ending up in-the-money and a 76% chance of ending up out-of-the-money.

if you have grasped Delta, we can now tackle its quirkier cousin Gamma.

#### Gamma

Again, let’s look at the table and go through it.

**Gamma** is the change in an option’s delta for a one-unit change in the price of the underlying. The numbers above show the change in delta for a one-point increase in the underlying futures price. For example, a gamma of 0.14 on a one-month at-the-money option means that the delta would increase from 0.50 to 0.64 for a one-point increase in the underlying.

Important to remember that ATM (at-the-money) options are most impacted by gamma. This should make intuitive sense as the sensitivity of the option will be mostly felt around the strike price, especially as it approaches expiry.

**Key points on Gamma:**

**Time to expiration**

The gamma of an ATM option increases as time passes. The gamma of in-the-money and out-of-the-money options will converge to zero at expiration.

**Volatility**

An increase in volatility will lower the gamma of ATM options and raise the gamma of deep ITM and OTM options.

A doubling of volatility has roughly the same effect on an option's gamma as a quadrupling of time. For example, the gamma of the 100 strike call option is 0.42 with 1 week to expiration, futures = 100, and volatility = 5%. If volatility increases to 10%, the gamma falls to 0.27, which is about the same gamma that a 1-month option would have with volatility at 5%.

#### Time Decay (Theta)

**Intrinsic value**

An option's intrinsic value or exercise value is the amount by which the option is in-the-money. ATM and OTM options have no intrinsic value. The intrinsic value for ITM options is simply the difference between the underlying price and the option's strike price.

**Time value**

Time value is simply the difference between the price of an option and the intrinsic value of the option. For example, with futures = 102, a 100-strike call with a price of 2.40 has an intrinsic value of 2 points, and time value equal to 0.4.

**Theta**

Theta is the rate of decay in the time value of an option. It is usually expressed as the change in the value of an option for one day's passage of time.

Again, let’s look at the table.

That is in cents / day. In the above example, with futures = 100, volatility = 10%, and 1 week remaining to expiration, theta equals -4.09 meaning the option price will fall from 0.55 to 0.51 if one day passes.

Study those details in detail. Theta is one of the largest & fastest detractors of your long option positions. It has tripped me up a few times.

**Key points on Theta:**

**Moneyness**

ATM options have the greatest time value and the greatest rate of time decay (theta). The further an option goes in-the-money or out-of-the-money, the smaller is theta.

**Time to expiration**

Theta increases as time passes for at-the-money options.

**Volatility**

The time value of an option increases as volatility increases and so too does theta.

#### Vega

Not that often used greek letter on options. Vega is the change in the price of an option for a one percentage point change in implied volatility. For example, the vega of a 100-strike 4-month call with futures = 101 and volatility = 10% equals 0.224. The price of this call is 2.80. If volatility increases to 11%, the option price will increase by 0.224 to 3.024.

**Key points on Vega:**

**Moneyness**

ATM options have the greatest vega. The further an option goes in-the-money or out-of-the-money, the smaller is vega.

**Time to expiration**

Time amplifies the effect of volatility changes. As a result, vega is greater for long-dated options than for short-dated options.

**Volatility**

An increase in volatility increases vega for in-the-money and out-of-the-money options.

A change in volatility has no effect on the vega of ATM options. This means that ATM option prices increase linearly with changes in volatility.

Aggregating greeks in a portfolio context is also quite a useful thing to keep in my mind. Work through the below examples.

The rest of the presentation is littered with more examples and certain trading strategies. For the sake of keeping it simple I advise those who are interested to work through those example in their own time.

Hope this was useful! Please leave comments or suggestions!

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