How to Price Options (Options Basics Part 3)

Guest post by The Duomo Initiative

There are two popular types of options you can trade, American and European. A key difference between the two is that American options can be exercised at any time until the expiration date, whereas European options can only be exercised at the expiry, assuming they are in the money.

This means the calculator will be different depending on which one you’re trading. For now, we’re going to focus on pricing European options. However, even if you’re trading American options the same principles will apply.

To find the price for a European option we can use the Black-Scholes formula. This is used to determine the fair price, also known as the theoretical value of an option.

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Like we mentioned earlier, this formula is for European options. However, the explanations are also relevant for American options. So, let’s go through the key aspects and make sure we understand what each of these are: 

Call or Put Option Price

Firstly we have C & P, this is the price of the call or put option. 

If you have a call position, the strike price is the price you can buy the underlying shares at. If you have a put position, the strike price is what you can sell the underlying shares at. 

Unlike the underlying asset price though, the strike price is usually standardised to the fixed dollar or pound. For example, £10, £11, £12 and so on. 

So, let’s look at an example:

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Say the price of a stock is currently at £25, as shown in the middle of the diagram. When you look at the available options contracts there will be a range strike prices for you to choose from. In this example we have £10, £15, £20, £25, £30, £35, and £40.

If you buy the call option at the strike price of £25, this will also be the same price that you could have bought the stock at. This will also be the price that your options will be exercised at.

Before we go any further, there are three important terms to understand:

1. In the money
2. Out of the money
3. On the money

The stock price is currently trading at £25, so any calls that have a strike price above this will be out of the money. Essentially, if an option expires out of the money, it’s worthless.

If you buy at £30, £35 or £40, then you are buying the option above the underlying price of the asset. In order to make a profit, the price needs to be above the strike price (in the money) when the option expires. If the option is out of the money at expiration, it is worthless. 

So, the higher your strike price, the further the price has to move to make a profit, which means there’s a lower probability of that happening. 

In order to be 'in the money', you would need to buy a call below the market price, so £10, £15 and £20. 

The opposite is true for puts. Let’s say you buy a put because you expect the price of the asset to fall. If you buy the put above the current asset price, you are in the money, but if you buy a put below the asset price, you are out of the money.  

Again, this is because if you buy a put, you need it to close below the strike price to make a profit. 

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Finally, being on the money means that the strike price is the same as the current share price. 

So why can't we just buy options that are already in the money? The issue is with the cost of the premium. The more 'in the money' the option is the higher the premium will be, whereas the further 'out of the money' the option is, the cheaper it should be.

Interest Rate

Next up is a lower case r, which is the rate of interest., In theory this should be a short-term safe interest rate such as a one-year government bond. However, in reality this is likely to be the same rate that your options broker is going to be borrowing at.  

Time to Expiration

Then we have the time to expiration, also known as extrinsic value. This is one of the most important factors when considering which option to trade. 

So, imagine you are buying insurance for your car, it’s not typically a one-time payment, you may have to pay each month to keep your car insured. Options are similar in that the longer you hold an option, the more it costs in fees (the premium).

For example, for an option that lasts one week you may pay a premium of $100, whereas for an option that lasts a month you may pay a premium of $400. 

Since options suffer from time decay, as time passes, the value and premium of the options contract will decline. As you get closer to the expiration date, the time decay accelerates and, as we can see in the diagram, this will be exponential. 

That means if you buy an option further from expiry, the premium will be higher. However, as the price gets closer to expiry, the premium will fall as there’s less time until expiration. 

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Volatility

The volatility represents the degree of the asset's price swings. If you are trading an underlying asset that has high volatility, it will have much bigger price swings than an asset which has low volatility. 

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In the diagram, the pink line represents Asset A which has lower price swings than the green line which is Asset B. This means Asset B has more volatility. 

In the Black-Sholes formula 'N' represents the standard normal distribution, which can be between 0 and 1.

To calculate N, we would need to use the d1 and d2 formulas. In the d1 & d2 formulas, Ln represents the natural log of standard deviation and measures how wide the data spread is. It’s showing how close the data points are to the average. 

If the deviation is close to 0 then it has low volatility. The higher it is, the higher the volatility is. In our example, the pink asset would be closer to 0 than the green asset.

In the d1 and d2 formulas we also have a lower case 's', which is the volatility of the underlying asset. 

If you’re new to options, the main thing to take away from all this is that if the underlying stock has higher volatility, it will have a higher premium. That’s because a more volatile asset is more likely to be in the money before it expires. 

As I mentioned earlier, one of the limitations for the formula is that it can only be used on European options. It also has some other limitations such as implying that volatility, dividends and interest rates will remain constant. However, that’s not always the case.  

Greeks

We previously covered the Black-Scholes formula to determine the fair value of an option. Now we start to incorporate more advanced pricing.

The Greeks are a set of five factors that might affect the price of an options contract. These are:

• Delta
• Gamma
• Theta 
• Vega
• Rho

Let’s go through each of these in more detail. 

Delta

Delta is the measurement for how much an option price changes compared to the underlying asset. This means that although options and the underlying asset move together, it’s not at the exact same rate. 

For example, if an option has a Delta of 0.5, the options price should theoretically move 50p for every £1 move in the stock price. Delta can be used by traders to determine the likelihood of expiring in the money.

For a call option, the Delta will be between 0 and 1. The more in the money the price is, the higher the Delta should be. However, the more out of the money, the lower the Delta. 

Put options will have a negative Delta 0 and -1. The more in the money the price is, the lower  the Delta should be. However, the more out of the money, the higher the Delta.

Gamma

Gamma is kind of similar to Delta, except it measures the rate of change in Delta over time.

Based on the previous example, if the option has a Delta of 0.5, and the option moves 50p alongside the asset price moving £1, the Delta would no longer be 0.5. If the option moves closer to being in the money, the Delta may have increased to 0.6, and this would mean the Gamma is 0.1, that’s the rate of change. 

Since Gamma has a limit of 1.00, the closer it gets to 1 the slower the Gamma will change. Gamma is also typically lower for options deep in or out of the money, as they are less sensitive to price changes compared to options closer to being in the money. 

Theta

Theta measures how much the price of an option should decrease as the option approaches its expiration. You may remember this when we discussed time decay in the Black-Scholes formula. 

Options which are in the money will see the price increase as the expiration date gets closer. However, options which are out of the money will usually decrease as there’s a higher probability they will expire worthless. 

As we previously talked about, time decay isn’t linear. The rate of time decay increases as the option approaches expiration, which means that options lose value more rapidly as they get closer to expiration.

Vega

Vega measures an option's sensitivity in the volatility of the underlying asset. It’s measuring the rate of change in the option’s price for a one-unit increase in the implied volatility of the asset. 

This is the market’s expectation for future volatility over the lifetime of the option, until expiration. If the asset has a higher Vega, it shows the options price is more sensitive to volatility, whereas a lower Vega shows the options price is less sensitive to volatility.

Volatility is one of the most important factors in options pricing. If the option is more sensitive, there’s a higher chance it will expire in the money.  

Rho

Rho measures an option’s sensitivity to changes in the interest rate, the rate we covered in the Black-Scholes formula. Specifically it’s the change in the options price per a one-percentage-point change in the interest rate.

Typically, as interest rates increase, the value of a call option will increase, whereas the value of a put option will decrease. 

Generally Rho isn’t a big factor in options pricing, particularly for shorter-term options, but can be more important when interest rates in the economy are expected to change, such as a rate hike or cut from the central bank. 

Implied Volatility

Although technically not a Greek, implied volatility is an important concept in options pricing, volatility also ties in with the Greeks. 

Implied volatility is a measure of the market's expectation of the future volatility of an underlying asset, as reflected in the price of an option.

When traders buy or sell options, they are making a bet on the future movement of the underlying asset. Implied volatility reflects the market's expectation of how much the underlying asset will move over a given period of time. If the market expects the underlying asset to be highly volatile, the implied volatility will be high, and vice versa.

The implied volatility of an option is a key input in options pricing models, such as the Black-Scholes model, which is used to estimate the fair value of an option. A higher implied volatility will result in a higher option premium, and a lower implied volatility will result in a lower option premium.

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