How Options Are Priced

Options trading can often be complex, but understanding how these financial instruments are priced is crucial for anyone diving into this market. Options derive their value from an underlying asset, typically a stock, and their price, known as the premium, is influenced by factors ranging from the present share price to the time left until expiration of the option.

Two primary models have emerged as the go-to tools for pricing options: the Black-Scholes and binomial models. While these mathematical formulas can look intimidating, their purpose is simple—to help traders determine fair market value for options contracts. We'll go through some of the math below, and everyone, even the most math-phobic, can follow along to gain key insights about options trading. By doing so, you'll learn the fundamentals of options pricing, how to manage risk more effectively, and how to uncover more profitable prospects in the options market.

What Is an Options Contract?

The value of options is derived from another asset, typically a stock or index, which is called the underlying asset. Buying or selling an option comes with a price called the premium. Buyers of an option pay it while sellers receive it.

There are two main types of options:

Call options:

• Buyer: Has the right to buy an asset at a preset price, called the strike price
• Seller: Has the obligation to sell an asset at a strike price

Put options:

• Buyer: The right to sell an asset at a strike price
• Seller: The obligation to buy an asset at a strike price

Investors buy options to profit from stock price moves without owning the stock outright, often to hedge existing positions. Sellers of options receive the premium as income but take on the risk of potential obligations.

For instance, if you buy a call option on XYZ stock with a strike price of $50, you have the right to buy XYZ shares at $50 each, even if the market price rises above that level before expiration.

Option Pricing Models

Options pricing models distill complex market dynamics into actionable information. However, it's crucial to remember that these models are merely prediction algorithms. They provide estimates based on present information, but the market often has its own surprises in store.

The factors determining the value of an option include the present stock price, the intrinsic value, the time to expiration or time value, volatility, interest rates, and cash dividends paid (if applicable).

While these factors may seem straightforward, determining the fair market value of an option has not been. To simplify this process, financial experts developed several pricing models. The most widely known and used are the following:

1. Black-Scholes: Developed in the 1970s, this model brought options pricing into the modern era. It's particularly useful for European-style options, which can only be exercised at expiration.
2. Binomial model: This uses a "tree" approach to map out possible price paths of the underlying asset. It's more flexible than Black-Scholes and can handle American-style options, which can be exercised anytime.

While the mathematics behind these models can be complex, investors don't need to do the calculations manually. Many trading platforms and online tools can quickly compute option prices based on them.

The Black-Scholes Formula

While the math behind Black-Scholes can seem daunting—ultimately, it isn't—its impact on options trading can't be overstated. It gives traders a standardized way to think about option value, even if they're using more advanced models in practice. Developed by Fischer Black, Myron Scholes, and Robert Merton in the 1970s, Black-Scholes is the most well-known options pricing method and earned Scholes and Merton the Nobel Prize in economics in 1997.

The formula attempts to calculate the fair price of an option based on several variables. For those comfortable with mathematical notation, the Black-Scholes formula for a call option is as follows:

$\begin{aligned} &C = S_t N(d _1) - K e ^{-rt} N(d _2)\\ &\textbf{where:}\\ &d_1 = \frac{ln\frac{S_t}{K} + (r+ \frac{\sigma ^{2} _v}{2}) \ t}{\sigma_s \ \sqrt{t}}\\ &\text{and}\\ &d_2 = d _1 - \sigma_s \ \sqrt{t}\\ &\textbf{where:}\\ &C = \text{Call option price}\\ &S = \text{Current stock (or other underlying) price}\\ &K = \text{Strike price}\\ &r = \text{Risk-free interest rate}\\ &t = \text{Time to maturity}\\ &N = \text{A normal distribution}\\ \end{aligned}$​C=St​N(d1​)−Ke−rtN(d2​)where:d1​=σs​ t​lnKSt​​+(r+2σv2​​) t​andd2​=d1​−σs​ t​where:C=Call option priceS=Current stock (or other underlying) priceK=Strike pricer=Risk-free interest ratet=Time to maturityN=A normal distribution​

You don't need to worry about the specifics of the equation. What's crucial is that the model uses these inputs to estimate the probability that an option will be profitable at expiration.

Importantly, you don't need to calculate these values manually. Most trading platforms and many financial websites offer options calculators that do the heavy lifting for you. These tools allow you to input the variables and quickly see how any changes in these factors affect an option's theoretical price.

While powerful, the Black-Scholes formula has limitations. For instance, it assumes that stock prices follow a normal distribution and that volatility remains constant over the option's life—major assumptions that don't always hold in reality. Despite these limitations, the Black-Scholes model remains a fundamental tool in options pricing and a starting point for more complex models.

Intrinsic Value

Intrinsic value is like the foundation of a house. It's the solid, measurable part of an option's worth. Everything else built on top—that's the time value covered in the next section.

Intrinsic value is the price a given option would have if it were exercised today. Intrinsic value is calculated differently for calls and puts.

The equations to calculate the intrinsic value of a call or put option are as follows:

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

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

The intrinsic value of an option reflects the financial advantage of exercising it—it's the option's minimum value. Thus, the result of the formulas above give you the option's moneyness.

Example

Let's look at a simplified hypothetical: Suppose XYZ stock is trading at $55. A call option with a strike price of $50 would have an intrinsic value of $5 ($55 - $50). However, a call option with a strike price of $60 would have no intrinsic value since exercising it would mean buying the stock for more than its present market price.

Suppose we look at a real-life scenario instead, taking shares of GE Aerospace (GE), selling at $187.40 per share, as our example. On a site like TradingView, you see the following options chain (a list of present market values at different strike prices):

The GE 180 call option would have an intrinsic value of $7.32 per share:

$187.40 (present price) - $180 (options strike price) - 0.08 (the option price per share) = $7.32

Thus, the option holder can exercise the option to buy GE shares at $180 and then turn around and automatically sell them in the market for $187.40 for a profit of $7.32.

The GE 187.50 call option would have an intrinsic value of zero (187.40 - 187.50 - 0.03 = -$0.13) because the inherent value can't be negative. This works the same way for a put option.

Time Value

Time value is critical to an option's price, representing the premium investors are willing to pay for the option's potential to increase in value before expiration.

The formula for time value is straightforward:

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

Two primary factors influence time value:

1. Time until expiration: The more time left, the higher the time value.
2. Implied volatility: Higher volatility increases time value.

For example, if a call option with a strike price of $50 is trading at $5 when the underlying stock is at $52, its time value would be $3: $5 (option price) - $2 (intrinsic value).

Time value is often called an option's extrinsic value because it's how much of the price of an option exceeds the intrinsic value.

In addition, it's essentially the risk premium the option seller requires to provide the option buyer with the right to buy or sell the stock until the option expires. It's like an insurance premium for the option. The higher the risk, the higher the cost to buy.

Understanding time value helps traders assess whether an option is overpriced or underpriced. An option will generally lose one-third of its value during the first half of its life and two-thirds during the second half. This is important for investors because the closer the option gets to expiration, the more the underlying security needs to move to affect the value of the option.

Time decay is particularly important for strategies involving selling options, where profiting from it is often the key objective.

Example of Time Value

Suppose we have the following information in front of us as we look to buy options:

• Stock price (S): $50
• Strike price (K): $55
• Option type: Call option
• Market price of the option (premium): $3
• Time to expiration: Two months
• No dividend expected

Since the stock price ($50) is below the strike price ($55), the intrinsic value of the call option is $0.

The time value is the amount by which the option's market price (premium) exceeds its intrinsic value. It represents the extra value attributed to the potential for the option to become profitable before it expires:

Time Value = $3 - 0 = $3.

This $3 reflects the market's expectation of the potential for the stock to rise above the strike price within the next two months.

Volatility

Volatility is a measure of how much a stock's price fluctuates and is another crucial factor in options pricing. There are two types of volatility involved:

1. Historical volatility (HV): A measure of past price shifts.
2. Implied volatility (IV): The market's forecast of future volatility.

IV has a significant impact on options prices, particularly on their time value. When it's higher, this leads to higher options prices, all else being equal.

Understanding and predicting changes in volatility is key to many options strategies. For instance, some traders profit by buying options when volatility is low and selling them when it increases.

Historical Volatility

HV tells you where you've been—but not where you're going. Nevertheless, HV is still crucial for contextualizing the present market conditions and option prices. Also known as statistical volatility, HV measures the speed and size of recent price changes in the underlying asset. It's calculated using the standard deviation of price changes over a specific period, typically in the past 10, 20, or 30 trading days.

Here are two straightforward ways of interpreting HV:

• A stock with 20% HV can be expected to trade within 20% of its present price about 68% of the time over the next year.
• Higher HV percentages indicate a more volatile stock.

In addition, traders often compare HV to IV to identify potential mispricing. If IV is significantly higher than HV, options might be overpriced, presenting potential selling prospects. Meanwhile, if IV is much lower than HV, options might be underpriced, suggesting buying opportunities.

Implied Volatility

IV is a forward-looking measure representing the market's expectation of future price changes in the underlying asset. Unlike HV, which looks at past price movements, IV is derived from present options prices.

Here are the most important things to remember about IV:

1. It's expressed as a percentage and annualized.
2. A higher IV means the market expects larger price swings, increasing option premiums.
3. IV can change rapidly based on market sentiment and events.
   

IV is particularly useful for the following:

• Comparing options on stocks or with different strike prices and expiration dates.
• Identifying overvalued or undervalued options.
• Gauging market sentiment and potential upcoming volatility.

The Role of Beta

Beta is used to measure the volatility of a stock compared with the overall market. Volatile stocks tend to have high betas because of the uncertainty of the price of the stock before the option expires. However, high-beta stocks also carry more risk than low-beta stocks.

Volatility is a double-edged sword. It allows investors the potential for significant returns but it can also lead to significant losses.

The Option Greeks

Beta isn't the only options Greek. Collectively, the options Greeks are risk measures named after (mostly) Greek letters. They help traders understand how different factors affect the price of an option. Besides beta, the main ones are the following:

1. Delta (Δ): Measures the rate of change in the option's price related to the change in the underlying asset's price.
2. Gamma (Γ): Measures the rate of change in delta with respect to the change in the underlying asset's price. It's higher for at-the-money options and as expiration approaches.
3. Theta (Θ): Measures the rate of change in the option's price related to time (time decay). • Typically negative for bought options and positive for sold options.
4. Vega (V): Measures the rate of change in the option's price related to changes in IV. It's higher for options that have more time until expiration.
5. Rho (Ρ): Measures the rate of change in the option's price related to any changes in interest rates. This generally has the least impact of the Greeks.

The table below provides more information on interpreting the major Greeks.

Understanding the Greeks allows traders to do the following:

• Predict how their options might behave under different market conditions.
• Construct positions with specific risk profiles.
• Manage and adjust their positions more effectively.
  

Examples of How Options Are Priced

To illustrate how various factors affect option prices, let's look at two hypothetical stocks: Low Volatility Inc. (LVI) and HighVol Corp. (HVC). LVI has low volatility, with a beta of 0.5, while HVC is more volatile, with a beta of 1.5.

Both stocks are trading at $50, and we'll examine call options with a $55 strike price for two different expiration dates, one month and six months out.

LVI Options:
One month to expiration: $0.75
Six months to expiration: $2.25

HVC Options:
One month to expiration: $1.50
Six months to expiration: $4.50

The time value of the six-month options is more expensive because of the greater time until expiry. Meanwhile, HVC options are pricier because of higher IV. Lastly, since the options are out of the money, their entire value is time value.

Now, suppose a major event causes a spike in market volatility. Here's what to expect:

• LVI option prices would increase moderately.
• HVC option prices could see a more significant jump, especially for the longer-dated options.

The Bottom Line

Understanding how options are priced is crucial for anyone venturing into trading them—otherwise, it's like hiking in treacherous terrain without a map. While the Black-Scholes model and other pricing formulas provide a mathematical foundation, options pricing in the real world continually shifts because of the interplay of multiple factors. Intrinsic value, time value, volatility, and the Greeks are all vital in determining an option's premium.

For novice traders, focusing on the basics—like the relationship between stock price moves and option values, the impact of time decay, and the role of volatility—is the best and most worthwhile way to begin. More experienced traders can amble up to using the Greeks while developing more sophisticated strategies.

Remember, options pricing is not just about formulas—it's also about market sentiment, supply and demand, and reacting to the unpredictability of the market during times of high volatility. As with any financial product, it's essential to understand the risks of options trading thoroughly and seek advice from financial professionals before making significant investment decisions.