# Understanding How Options Are Priced

You might have had success beating the market by trading stocks using a disciplined process anticipating a nice move either up or down. Many traders have also gained the confidence to make money in the stock market by identifying one or two good stocks posed to make a big move soon. But if you don't know how to take advantage of that movement, you might be left in the dust. If this sounds like you, maybe it's time to consider using options.

### Key Takeaways

• Options contracts can be priced using mathematical models such as the Black-Scholes or Binomial pricing models.
• An option's price is primarily made up of two distinct parts: its intrinsic value and time value.
• Intrinsic value is a measure of an option's profitability based on the strike price versus the stock's price in the market.
• Time value is based on the underlying asset's expected volatility and time until the option's expiration.

This article will explore the factors to consider if you plan to trade options to take advantage of stock movements. Options are derivatives contracts that give the holder the right, but not the obligation, to buy (in the case of a call) or sell (in the case of a put) an underlying asset or security at a pre-determined price (called the strike price) before the contract expires. Hence, the term "derivative" simply means that an option's value is derived primarily from the underlying asset it is associated with.

It is important to note, however, that there are two parties to an options contract: a buyer and a seller. As mentioned, a buyer of an options contract has rights, but the seller of an options contract, on the other hand, has an obligation. It can become confusing, so to summarize:

• Buyer of a call: the right to buy an asset at a pre-determined (strike) price
• Seller of a call: the obligation to sell an asset at a pre-determined (strike) price
• Buyer of a put: the right to sell an asset at a pre-determined (strike) price
• Seller of a put: the obligation to buy an asset at a pre-determined (strike) price

Buying or selling an option comes with a price, called the option's premium. Understanding how to value that premium is crucial for trading options, and essentially rests on the probability that the right or obligation to buy or sell a stock will end up being profitable at expiration. So buyers of an option pay the premium, and sellers of an option receive the premium

## Option Pricing Models

Before venturing into the world of trading options, investors should have a good understanding of the factors determining the value of an option. These include the current stock price, the intrinsic value, time to expiration or the time value, volatility, interest rates, and cash dividends paid.

There are several options pricing models that use these parameters to determine the fair market value of an option. Of these, the Black-Scholes model is the most widely known.﻿﻿ In many ways, options are just like any other investment—you need to understand what determines their price to use them effectively. Other models are also commonly used, such as the binomial model and trinomial model.

Let's start with the primary drivers of the price of an option: current stock price, intrinsic value, time to expiration or time value, and volatility. The current stock price is fairly straightforward. The movement of the price of the stock up or down has a direct, though not equal, effect on the price of the option. As the price of a stock rises, the more likely it is that the price of a call option will rise and the price of a put option will fall. If the stock price goes down, the reverse will most likely happen to the price of the calls and puts

## The Black-Scholes Formula

The Black-Scholes model is perhaps the best-known options pricing method. The model's formula is derived by multiplying the stock price by the cumulative standard normal probability distribution function. Thereafter, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation.

In mathematical notation:

\begin{aligned} &C = S_t N(d _1) - K e ^{-rt} N(d _2)\\ &\textbf{where:}\\ &d_1 = \frac{ln\frac{S_t}{K} + (r+ \frac{\sigma ^{2} _v}{2}) \ t}{\sigma_s \ \sqrt{t}}\\ &\text{and}\\ &d_2 = d _1 - \sigma_s \ \sqrt{t}\\ &\textbf{where:}\\ &C = \text{Call option price}\\ &S = \text{Current stock (or other underlying) price}\\ &K = \text{Strike price}\\ &r = \text{Risk-free interest rate}\\ &t = \text{Time to maturity}\\ &N = \text{A normal distribution}\\ \end{aligned}

The math involved in a differential equation that makes up the Black-Scholes formula can be complicated and intimidating. Fortunately, you don't need to know or even understand the math to use Black-Scholes modeling in your own strategies. Options traders and investors have access to a variety of online options calculators, and many of today's trading platforms boast robust options analysis tools, including indicators and spreadsheets that perform the calculations and output the options pricing values.

Below, we'll dig a little deeper into options prices to understand what makes up its intrinsic vs. extrinsic (time) value, which is a bit more straightforward.

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## Intrinsic Value

Intrinsic value is the value any given option would have if it were exercised today. Basically, the intrinsic value is the amount by which the strike price of an option is profitable or in-the-money as compared to the stock's price in the market. If the strike price of the option is not profitable as compared to the price of the stock, the option is said to be out-of-the-money. If the strike price is equal to the stock's price in the market, the option is said to be "at-the-money."

Although intrinsic value includes the relationship between the strike price and the stock's price in the market, it doesn't account for how much (or how little) time is remaining until the option's expiration—called the expiry. The amount of time remaining on an option impacts the premium or value of an option, which we'll explore in the next section. In other words, intrinsic value is the portion of an option's price not lost or impacted due to the passage of time.

### The Formula and Calculation of Intrinsic Value

Below are the equations to calculate the intrinsic value of a call or put option:

\begin{aligned} &\text{Call Option Intrinsic Value} = USC - CS\\ &\textbf{where:}\\ &USC = \text{Underlying Stock's Current Price}\\ &CS = \text{Call Strike Price}\\ \end{aligned}

The intrinsic value of an option reflects the effective financial advantage resulting from the immediate exercise of that option. Basically, it is an option's minimum value. Options trading at the money or out of the money, have no intrinsic value.

\begin{aligned} &\text{Put Option Intrinsic Value} = PS - USC\\ &\textbf{where:}\\ &PS = \text{Put Strike Price}\\ \end{aligned}

### Example of Intrinsic Value

For example, let's say General Electric (GE) stock is selling at $34.80. The GE 30 call option would have an intrinsic value of$4.80 ($34.80 -$30 = $4.80) because the option holder can exercise the option to buy GE shares at$30, then turn around and automatically sell them in the market for $34.80 for a profit of$4.80.

In a different example, the GE 35 call option would have an intrinsic value of zero ($34.80 -$35 = -$0.20) because the intrinsic value cannot be negative. Intrinsic value also works the same way for a put option. For example, a GE 30 put option would have an intrinsic value of zero ($30 - $34.80 = -$4.80) because the intrinsic value cannot be negative. On the other hand, a GE 35 put option would have an intrinsic value of $0.20 ($35 - $34.80 =$0.20).

## Time Value

Since options contracts have a finite amount of time before they expire, the amount of time remaining has a monetary value associated with it—called time value. It is directly related to how much time an option has until it expires, as well as the volatility, or fluctuations, in the stock's price.

The more time an option has until it expires, the greater the chance it will end up in the money. The time component of an option decays exponentially. The actual derivation of the time value of an option is a fairly complex equation.

As a general rule, an option will lose one-third of its value during the first half of its life and two-thirds during the second half of its life. This is an important concept for securities investors because the closer the option gets to expiration, the more of a move in the underlying security is needed to impact the price of the option.

### The Formula and Calculation of Time Value

The formula below shows that time value is derived by subtracting an option's intrinsic value from the option premium.

﻿ $Time\ Value = Option\ Price-Intrinsic\ Value$﻿

In other words, the time value is what's left of the premium after calculating the profitability between the strike price and stock's price in the market. As a result, time value is often referred to as an option's extrinsic value since time value is the amount by which the price of an option exceeds the intrinsic value.

Time value is essentially the risk premium the option seller requires to provide the option buyer the right to buy or sell the stock up to the date the option expires. It is like an insurance premium for the option; the higher the risk, the higher the cost to buy the option.

### Example of Time Value

Looking again at the example from above, if GE is trading at $34.80 and the one-month-to-expiration GE 30 call option is trading at$5, the time value of the option is $0.20 ($5.00 - $4.80 =$0.20).

Meanwhile, with GE trading at $34.80, a GE 30 call option trading at$6.85 with nine months to expiration has a time value of $2.05. ($6.85 - $4.80 =$2.05). Notice the intrinsic value is the same; the difference in the price of the same strike price option is the time value.

## Volatility

An option's time value is also highly dependent on the volatility the market expects the stock to display up to expiration. Typically, stocks with high volatility have a higher probability for the option to be profitable or in-the-money by expiry. As a result, the time value—as a component of the option's premium—is typically higher to compensate for the increased chance that the stock's price could move beyond the strike price and expire in-the-money. For stocks that are not expected to move much, the option's time value will be relatively low.

One of the metrics used to measure volatile stocks is called beta. Beta measures the volatility of a stock when compared to the overall market. Volatile stocks tend to have high betas primarily due to the uncertainty of the price of the stock before the option expires. However, high beta stocks also carry more risk than low-beta stocks. In other words, volatility is a double-edged sword, meaning it allows investors the potential for significant returns, but volatility can also lead to significant losses.

The effect of volatility is mostly subjective and difficult to quantify. Fortunately, there are several calculators to help estimate volatility. To make this even more interesting, several types of volatility exist, with implied and historical being the most noted. When investors look at volatility in the past, it is called either historical volatility or statistical volatility.

### Historical Volatility

Historical volatility (HV) helps you determine the possible magnitude of future moves of the underlying stock. Statistically, two-thirds of all occurrences of a stock price will happen within plus or minus one standard deviation of the stock's move over a set time period.

Historical volatility looks back in time to show how volatile the market has been. This helps options investors to determine which exercise price is most appropriate to choose for a particular strategy.

### Implied Volatility

Implied volatility is what is implied by the current market prices and is used with theoretical models. It helps set the current price of an existing option and helps options players assess the potential of a trade. Implied volatility measures what options traders expect future volatility will be.

As such, implied volatility is an indicator of the current sentiment of the market. This sentiment will be reflected in the price of the options, helping traders assess the future volatility of the option and the stock based on current option prices.

Collectively, the factors that help measure the impact on an option's premium are referred to as Option Greeks.

## Examples of How Options Are Priced

Below, you can see the GE example already discussed. It shows the trading price of GE, several strike prices, and the intrinsic and time values for the call and put options. At the time of this writing, General Electric was considered a stock with low volatility and had a beta of 0.49 for this example.

The table below contains the pricing for both calls and puts that are expiring in one month (top section of the table). The bottom section contains the prices for the GE options that expire in nine months.

In the figure below, the pricing for both calls and puts expiring in one month and nine months are listed for stock of Amazon.com Inc. (AMZN). Amazon is a much more volatile stock with a beta of 3.47.

Let's compare the GE 35 call option with nine months to expiration with the AMZN 40 call option with nine months to expiration.

• GE has only $0.20 to move up before the nine-month option is at the money, ($35 strike - $34.80 stock price). • On the other hand, AMZN has$1.30 to move up before its nine-month option is at the money ($40 strike -$38.70 stock price).
• The time value of these options is $3.70 for GE and$7.50 for AMZN.

The significant premium on the AMZN option is due to the volatile nature of the AMZN stock, which could result in a higher likelihood the option will expire in-the-money.

An option seller of GE will not expect to get a substantial premium because the buyers do not expect the price of the stock to move significantly.

On the one hand, the seller of an AMZN option can expect to receive a higher premium due to the volatile nature of the AMZN stock. Basically, when the market believes a stock will be very volatile, the time value of the option rises.

On the other hand, when the market believes a stock will be less volatile, the time value of the option falls. The expectation by the market of a stock's future volatility is key to the price of options.

Article Sources
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1. The Options Industry Council. "Black-Scholes Formula."

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