## Where do options prices come from?

Since options contracts can offer substantial leverage, they tend to experience price swings that may seem unpredictable at first glance. Like any financial instrument, the value of an option is ultimately determined by the market – which is simply the constantly changing supply and demand (bids and offers) for that particular contract. However, options traders throughout the world use a common framework for evaluating options prices based on the original Black-Scholes pricing formula developed in 1973.

It’s not necessary to become an expert in the calculus that drives the Nobel Prize-winning Black-Sholes formula or the dozen or so variations that have been created over the last 40 years. The model creates a small set of variables, (aka ‘the Greeks’) that are extremely valuable to options traders and are readily available on trading platforms like City Index. But a good working knowledge of the Greeks is a helpful tool in becoming a profitable options trader and getting the most out of the product.

Before we dive into the Greeks, let’s look at some other terms you’ll need to know.

## ‘The money’: option trading lingo explained

### What’s an in-the-money option?

This is how we describe an option in which the underlying market has already moved beyond the strike price so that if it settled today, it would have some value. This amount is called the intrinsic value of the option. For example, if the S&P 500 is trading 3750 and you are long the June 3500 Calls, you are in-the-money. The option already has an intrinsic value of 250 (the difference between the strike price and the underlying market).

### What’s an out-of-the-money option?

This is what we call an option whose underlying market hasn’t moved beyond the strike price, so it has no intrinsic value. If the option expired today, it would be worth zero. For example, if the S&P 500 is trading 3750 and you are long the June 4000 Calls, you are currently out-of-the-money. However, if the option expires sometime in the future it will still be worth something today even though its intrinsic value is zero. This is because an option also has time value – known as extrinsic value – based on the possibility that it could eventually land in-the-money before it expires.

### What’s an at-the-money option?

These are the options whose strike price sits right at or near the underlying market. For example, if the S&P 500 is trading 3750 and you are long the June 3750 Calls, you are at-the-money. At-the-money options have no intrinsic value, however they are so close to being in-the-money that they have earned their own identity. These are often some of the most actively traded options in any market.

## Delta – ‘the fractional Greek’

Delta is probably the most commonly referenced option variable and is an excellent way to get an initial baseline for the expected changes in the value of an option contract that you might buy or sell. We call delta ‘the fractional Greek’ because it views the option as being a fraction of the underlying market. This variable assigns a value (between 0% and 100%) to every option contract that gives you an idea of the options relationship to price movement in the underlying market. So, an option with a delta of 25% would be expected to increase in value by $2.50 if the underlying market increases by $10.

Looking at the delta can tell us a lot about the characteristics of a potential trade. Low-delta options (less than 10%) are typically very far out-of-the-money and would need a very large price move to increase in value. An option with a 95% delta, on the other hand, is very deep-in-the-money and therefore behaves almost the same as a position in the underlying market.

At-the-money options usually have a delta of 50%. This means that if the value of the underlying market goes up by $10, the 50-delta call option would increase in value by around $5.

Delta polarity (+ / -): Long call options have a positive delta because they increase in value as the underlying market moves higher. Long put options have a negative delta because they increase in value as the underlying market moves lower. Of course, by selling a call, you end up with a negative delta position and selling a put gives you a positive delta position.

Sometimes it can be helpful to think of a non-option position in an underlying market as a 100 delta position. Here are some general rules of thumb for delta polarity:

- Positive delta (bullish positions): you’d be long on the underlying market using long calls or short puts
- Negative delta (bearish positions): you’d be short on the underlying market using short calls or long puts

## Vega – ‘the volatile Greek’

The Black-Scholes formula needed a way to account for the fact that options in highly volatile markets should be worth more than options in very stable markets. AUD/USD (Aussie Dollar) is a rather stable FX pair, whereas a stock like Tesla (TSLA) often moves quite a bit during any given time frame. So, an out-of-the-money option in TSLA is going to be worth a lot more than a comparable out-of-the-money option in AUD/USD.

‘Implied volatility’ is one of the variables that go into the Black-Scholes pricing model. This is simply the annual expected range of the underlying market. Underlying markets that have a very volatile trading range would have a higher implied volatility, which translates into a higher option price. It is important to note that implied volatility is a theoretical forward-looking value, after all, nobody really knows what the range of TSLA is going to be for the next year. Implied volatility is calculated by taking the market price of an option and then reverse-engineering the ‘implied’ volatility number in the Black-Scholes model.

The actual, or historic, volatility does not factor into the price of an option. For example, if you look up the current implied volatility of TSLA options and you get a value of 50%, you might then look at a chart and notice that TSLA shares have moved over a 300% range over the past year. This makes no difference – if the implied volatility is 50% then you know that the options market is expecting a range of 50% over the next year.

Vega is simply a way to estimate the impact of changes in the option’s implied volatility to changes in the options price. So, an option that is currently worth $50 might have a vega value of $12. This means that each 1% increase in implied volatility will increase the option’s value by $12, regardless of whether or not the underlying market moves at all.

Studying the vega, or implied volatility, of an option allows you to make smarter decisions on whether or not an option is ‘cheap’ or ‘expensive’ relative to what is happening in the market.

For example, you might have an idea to start buying S&P500 Put options right before earnings season kicks off because you think there is a good chance that many companies will miss their earnings and the market will go down fast. You see that those put options are trading at an implied volatility of 40%, even though the usual implied volatility for the S&P500 is below 20%. This tells you that the options are considerably more expensive right now.

Buying those put options still might be a great trade, but now you can see that you would be paying a very big premium and would therefore need a very large move in the underlying market to get the payoff you’d like.

## Theta – ‘the sleepy Greek’

Options have an expiration date, so each minute that goes by reduces the probability that the option will have a high intrinsic value at settlement. Theta is how we measure the daily ‘time decay’, or the amount by which the option loses a little bit of value each day.

Options that expire within a week have the most theta. As an example, if we buy an out-of-the-money option today for $3 and it has three days remaining until expiration, how much will it be worth tomorrow? If the underlying market doesn’t move at all (no delta impact) and the implied volatility doesn’t change (no vega impact), then that option is still likely to lose around a third of its value because of the time decay.

As a contrast, an option that has three months until expiration would only lose 1/90^{th} of its value over the same period. Think of it this way: an out-of-the-money option has an intrinsic value of zero. If the underlying market doesn’t move, it will expire worthless. So, as time keeps ticking, the value of the option slowly grinds lower – even when you’re sleeping!

Theta can work for or against you, depending on whether you have bought or sold the option. If you are an option seller, theta is likely your favourite Greek because it continuously works in your favour, slowly but surely. Those who favour buying options often view theta as a cost of doing business – it’s the amount you steadily lose while you stay in the game trying to profit on delta and vega movement.

## Gamma – ‘the dynamic Greek”

Gamma is probably the least understood of the Greeks. By definition, it is a measure of the rate at which the delta of the option is changing as the underlying market moves. Think of it as the delta (rate of change) of the delta.

To visualise how gamma works, consider a 5% delta call option, with a strike price of 140 and an underlying market currently at 100. We know that if the underlying market moves up to 101, the value of the 140 Call option will increase by $0.05 (since it has a 5% delta on a $1 underlying move).

Now imagine that the market has moved all the way up to 140, so that the option is now at-the-money and therefore has a delta of 50%, at this level, a $1 underlying move will result in a $0.50 change in the same option’s price – representing a far larger exposure to the market. So, while the delta can help us predict the change in value across a small underlying price move, we really need to understand that the gamma is working to continuously change the delta during a protracted move.

Gamma helps us understand how options come alive in a fast-moving market. It makes it difficult to continuously hedge a short option position because it forces you to keep buying at higher prices (or selling at lower prices) to keep up with the changing delta. Gamma shows us the effects of compounding leverage in options, which can be extremely beneficial when you are long options, and very costly when you are short.

## Rho – ‘the boring Greek’

Rho is just how we measure interest rate sensitivity in options. We call it ‘the boring Greek’ because it is downright useless these days. Thanks to decades of central bank easing, interest rates are stuck at levels in which there is no meaningful impact on options pricing.

However, that could change in the future so it’s something to be aware of. There is an opportunity cost to own an expensive option for a long period of time if it could have been earning interest in a bank account. Also, it’s worth noting that just because interest rates are near zero in Europe, Japan, and most English-speaking countries, there are many nations with well-established financial trading markets where rates are still quite high. All else being equal, higher interest rates reduce the value of most options.

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