There are many options pricing models with complex mathematical foundations and variables that go into determining what an option is worth.

But in terms of the big-picture intuitive understanding of an option’s value is, it really boils down to two main factors:

- the probability that an option will be in the money (ITM) by expiration (i.e., have value), and
- by how much

Everything is essentially wrapped up under these two variables.

## Options and expected value

Options are especially valuable for managing risk, making bets in a risk-limited way, and capturing the part of the distribution that might be after.

Especially valuable in options trading is the concept of expected value.

Expected value, in a nutshell, is essentially:

What is your probability of being right multiplied by the reward for being right minus the probability of being wrong multiplied by the penalty for being wrong.

For example, if you’re playing a game and you have a **10 percent chance of being right** and a **reward of $1,000 for being right** and a **90 percent chance of being wrong** and a **$75 penalty for being wrong**, is that a risk that’s worth taking?

From an expected value standpoint, playing this would amount to:

**Expected value** = $1,000 * 0.10 – $75 * 0.90 = +$25

You have a positive expected value. So as long as you can cover the $75 loss in the chance you’re wrong, your chances of being positively rewarded are high if you play these odds an adequate number of times.

The game just described is akin to buying an out-of-the-money (OTM) option.

Your odds of being right are usually under 50 percent, sometimes significantly less, but your potential reward is so high it can sometimes justify making the trade.

To use a baseball analogy, it’s not your batting average that matters (how many times you put the ball in play to get a hit), but your slugging percentage (the value of the hit when you do put the barrel on the ball).

It’s okay to be wrong a lot so long as you’re adequately compensated for all your misses when you are right.

### Increases in the option premium

If any given variable works to increase the option premium, it’s because it increases one or both of the abovementioned factors.

So, a longer time until expiration or higher implied volatility will *increase* premiums because it increases the chance that the option will be in-the-money (ITM) by expiration *and *increases the odds of it being ITM by a larger amount.

Similarly, premiums will be lower the more an option is OTM and the lower the implied volatility because the odds of the underlying security, asset, or product reaching the strike price and going above it by expiration is low.

Prices and time are straightforward to measure. However, what can’t be known with a high level of precision is the volatility of the underlying asset.

Therefore, volatility is a key consideration and a big driver of options prices given its outsized impact on the probability of whether an option will be in the money or by how much.

Historical volatility is not necessarily a very good indicator of where volatility will be in the future. It can provide a reasonable approximation of future volatility where the future is likely to be similar to the past but big deviations are possible.

Various options pricing models have been created to more accurately determine what options should be worth, or to price them more effectively when they’re first created.

Let’s take a look at some of the main ones.

## Binomial option pricing model

The binomial option pricing model calculates what a call premium should be if the underlying asset can only be of one or two different prices by expiration.

A variable that can only take on two possible values is known as a binomial random variable.

By dividing time into smaller intervals with two possible prices that are closer together, a more accurate option premium can be derived.

As the number of time periods increases, the distribution of possible prices of the underlying asset, security, or product approaches something like a normal distribution – the familiar bell curve.

(Financial returns are more fat-tailed than what’s predicted by the normal distribution. Oftentimes, financial people will refer to big moves in markets as an ‘X-sigma’ move, where X is a number like 5-10, signifying a very rare move. But they’re only seemingly rare because of the flaws associated with using the normal distribution to describe financial returns.)

## Black-Scholes model

In 1973, US economists Myron Scholes and Fisher Black developed a mathematical formula for calculating options prices.

Their model was based on variables such as the current price of the underlying security, time until expiration, and volatility (by how much the prices varied over time).

Robert Merton later expanded on this options pricing model, and the Black-Scholes is sometimes also referred to as the Black-Scholes-Merton (BSM) model.

To simplify matters, they based their pricing model on certain assumptions:

- A “no arbitrage rule”: This means that prices reflect all information that’s known about the underlying asset today and into the future. Accordingly, an arbitrageur would not earn guaranteed profits by hedging against future risk.
- A call or put option contract can be used to offset any risk of any portfolio of securities or assets.
- Asset price fluctuations are random, but follow a normal distribution, which means that prices are assumed to not change much over the short run.

Academically, the Black-Scholes formula is the most commonly used formula to calculate options prices.

Part of its popularity is the ease of using it. It still depends on assessing the volatility of the underlying security, which is denoted by the standard deviation, σ, of the underlying asset price.

### Black-Scholes Formula

C_{0} = S_{0}N(d_{1}) – Xe^{-rt}N(d_{2})

- C
_{0}= current call premium - S
_{0 }= current stock price - N(d1) = the probability that a value in a normal distribution will be less than d
- N(d
_{2}) = the probability that the option will be in the money by expiration - X = strike price of the option
- T = time until expiration (expressed in years)
- r = risk-free interest rate
- e = 2.71828, the base of the natural logarithm
- ln = natural logarithm function
- σ = standard deviation of the stock’s annualized rate of return (compounded continuously)

d_{1 }= ln(S_{0}/X) + (r + σ^{2}/2)Tσ√T

d_{2 }= d_{1} – σ√T

Note that:

- Xe
^{-rt}= X/e^{rt}= the present value of the strike price using a continuously compounded interest rate

Requirements for the Black-Scholes:

- The stock or instrument does not pay a dividend before expiration.
- No changes in interest rates and variance before expiration.
- No discontinuous jumps in stock price (which might otherwise be hedged through gamma hedging).
- The assumption of instant, cost-less trading.

The Black-Scholes formula calculates the premium for a call, but the put premium can be calculated by using the put-call parity formula.

From the Black-Scholes formula, the standard deviation, **σ**, which measures volatility, can be calculated if the other variables are known.

This is what’s known as **implied volatility** because it is *implied* (or essentially backed out) by the other variables.

Some traders will compare the implied volatility with the observed volatility to help determine whether an option is fairly priced.

On many trading platforms, when trading options (especially equity options), the broker will provide implied volatility (IV) readings to the trader.

## Why Volatility Increases Time Value and Option Premiums

Let’s consider a couple hypotheticals.

### Hypothetical #1

Volatility is the change in the price of the underlying security over time. It’s a big part of what gives options value.

For example, consider a hypothetical security that never changes in price.

An option based on a security of this nature would never have any time value because the underlying is always the same price.

According, if the option was out of the money (put or call) no one would want the option because it would always expire worthless.

At the same time, no one would write (sell) an option based on this security if it was in the money because it would be exercised.

### Hypothetical #2

Now let’s say there was a scenario where the price of the security changed according to some formula that everyone knew about.

In other words, anyone could calculate the stock price with certainty at any time.

In this case, again, no option on this stock would have any time value because the price would be known by anyone ahead of time.

For example, if the stock were $100 and it was known that before a certain expiration date that it would be $120, nobody would want to write a 100 call unless they were being compensated $20 per share. (The exception would be if they could charge a little less in order to get the money sooner, in line with current risk-free interest rates.)

The only way the option would have value is if it was discounted adequately to at least equal current interest rates.

Accordingly, the unknown movement in the underlying price of a security is what gives options their value.

### Hypothetical #3

Consider two hypothetical securities.

Both trade at $100.

Stock A has ranged between $80 and $120 over the past year. In other words, it’s reasonably stable and this price behavior is expected going forward.

Stock B, on the other hand, is more volatile. It has ranged from $60 to $140 over the past year. Like Stock A, this price behavior is also expected going forward.

If one wanted to buy a call option at a 100 strike, which would have a higher value?

Stock B would have a higher call premium because it has a greater potential payoff.

Even though both have a roughly equal chance of expiring worthless because they’re both at the same current price, from an expected value standpoint, Stock B can move further, giving the option on it a higher intrinsic value.

### Criticisms of Black-Scholes

The normality assumption of the Black–Scholes model does not capture extreme movements such as crashes in markets.

Warren Buffett, in a 2008 Berkshire Hathaway shareholder letter wrote:

“I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long-term variety are being valued… The Black–Scholes formula has approached the status of holy writ in finance… If the formula is applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula.”

## Black Model

The **Black model** – sometimes known as the **Black-76 model **(in reference to the year it was introduced to separate it from other Fischer Black models) – is a variation of the Black–Scholes option pricing model.

The Black model is used most heavily for pricing:

The more popular Black-Scholes model is more commonly used for equity options.

The Black model is known as a log-normal forward model.

## Finite Difference Methods

**Finite difference methods for option pricing** involve using numerical methods in the valuation of options. Eduardo Schwartz first applied the method in 1977.

Finite difference methods involve pricing options by approximating the differential equation that best describes how the value of the option changes over time through a series of difference equations.

Discrete-time difference equations can then be solved to find a price for the option.

Finite differences came about because an option’s value can be modeled through a partial differential equation as a function of price (of the underlying security) and time (and potentially other variables).

Once in the form of a partial differential equation, a finite difference model can be derived, which can then yield the valuation of the option.

The finite differences approach can be used to solve option pricing exercises that have the same level of complexity as those solved, perhaps more commonly, by tree approaches.

## Garman-Kohlhagen

The value of a European option on a foreign exchange rate is commonly found by assuming that the rate follows a log-normal process. This is similar to the Black–Scholes model for equity options and the Black model for some types of interest rate options.

An FX options pricing model was first published by Biger and Hull in 1983. The model came before Garmam and Kolhagen’s model, which is more popularly referenced today. That same year, Garman and Kohlhagen extended the use of the Black–Scholes model to help solve FX options pricing when there are two interest rates involved in a currency pair – i.e., one associated with each currency.

Suppose that is the risk-free interest rate (to expiry) associated with the domestic currency and is the foreign currency’s risk-free interest rate.

The domestic currency is the currency where the value of the option is referenced.

The formula also requires that the currency – both strike and current spot price – be quoted in terms of units of domestic currency per unit of foreign currency.

The results are, in turn, expressed in the same units. To have meaning, they need to be converted into one of the currencies.

## Lattice model

A lattice model involves the valuation of options and other derivatives in cases where discrete time models are needed.

American options differ from European options in that the former can be exercised at any time while the latter can only be exercised at expiration.

In these cases, a discrete time lattice model would be appropriate for American options because the decision to exercise is always open (at least during market hours) before and including maturity.

A continuous model would be more appropriate for European options where the option’s maturity date is the only exercise period. Black-Scholes is a continuous model.

Exotic options are often valued using lattice models because of unique features of their payoff, such as the path dependency in lookback and Asian options and that impact on their valuations.

Interest rate derivatives and their pricing works better with lattice models as well due to the issues continuous models have with capturing some of their features (e.g., pull to par).

In general, lattice models may have to inform traders that terminating the option early is optimal rather than waiting until expiration.

Continuous models don’t account for this.

## Put-call parity

Put–call parity relates to the definition in the prices between a call option and put option (as it pertains to European options), holding expiry and strike price constant.

Under put-call parity, a long call option and a short put option should have the same price as a contract in the forward market at the same strike and expiry.

This should be true because if the price at expiry is above the strike price, the call option will be exercised. If it’s below, the put option will be exercised.

For put-call parity to hold, certain assumptions have to be met. But in reality, financing costs (interest on margin debt/leverage), transactions costs (e.g., spreads between bid and ask), the relationship will not be exact.

In highly liquid markets, such as the SPY options market, put-call parity will be close to holding true.

## Monte Carlo

Monte Carlo is typically used as a method of last resort as an options pricing model.

It is commonly used when conducting a valuation with:

- multiple variables that are unknown and can’t be known
- complicated features associated with the derivatives, where continuous models like Black-Scholes or other models that could normally account for exotic options (e.g., lattice models) won’t be effective

Monte Carlo enables customization to fit the appropriate features of the option.

The assumption of normality often underlies options models as a matter of convenience and rough approximation. Monte Carlo enables this to be modified to any distribution.

So when it comes to real options analysis (as it pertains to capital spending decisions), Asian options, and lookback options, Monte Carlo is more common.

Nonetheless, if there is a numeric technique (e.g., pricing tree) or analytical/computational technique for valuing the option that already exists, Monte Carlo is generally the last method to be used.

## Real options value or Real options analysis (ROV or ROA)

Real options value, also called real options analysis (ROV or ROA) has to do with options and derivatives valuation as it pertains to corporate budgeting and capital spending decisions.

These are called real options. These provide the right, but not the obligation, to undertake certain business decisions.

This pertains to anything involving a capital spending project, such as contracting, deferring, stopping, expanding, purchasing, and so on.

For example, it could mean the decision to expand a firm’s productive output while at the same time the option to sell productive output (such as a piece of machinery, factory, and so on).

Real options are unique because they are not like traditional financial options that are underwritten on securities that are traded in the markets.

Also, management teams – the holders of these options – can also influence the value of the option on an underlying project. This is less true as it pertains to options on a financial security.

On top of that, management teams also have no distinct measure of volatility. They must rely on their own knowledge of what uncertainty comes along with the project.

And real options don’t naturally exist. It’s not as simple as buying or selling an option that already exists on an underlying security.

So they have to be created and knowing how to do this is a task in itself – finding a counterparty (e.g., bank, investor) and how to price it.

The value of a real option is highest when uncertainty governing a project is highest.

Management has the power to influence the course of a project through the way they make decisions and allocate resources. So they can move the project along in a favorable way and also be willing to exercise the options on it.

The analysis of real options is a subset of corporate finance and making decisions under uncertainty.

The theory behind financial options can also extend to real options.

Purchasing managers in a company can use real options analysis to help them decide how to make certain decisions given all the known and unknown/can’t be known factors governing how they make purchasing decisions and allocate resources.

### Other applications of real options analysis

Another application of real options analysis outside corporate finance and spending decisions might include whether a student joins the workforce immediately after high school or college, or gets a degree/graduate degree first (or even alongside working).

There are the costs of time, all-in educational expenses, and foregone income to consider in getting this degree and whether the extra income that comes of the extra skills procured is worth in excess of these costs. Then there are also discounting considerations – the value of earning money now versus earning money several years in the future.

Whatever the exact business circumstance, it forces those making the decisions to be clear about the assumptions they’re making that lead to their projections. Once all factors are considered, then a business strategy can be formulated on how to deal with this particular problem.

It might also involve building out support systems to deal with these decision-making needs.

## Trinomial tree

Developed by Phelim Boyle in 1986, the trinomial tree is a lattice-based computational model used to value options.

The trinomial tree works off of the binomial options valuation model and is essentially the same as the explicit finite differences model discussed earlier in this article.

The trinomial method is used to produce more accurate results than the binomial model when there are fewer discrete time steps to model. So if time or other resources may be an issue, the trinomial model may be preferred.

For standard vanilla options, the binomial model is commonly implementeded because results converge when the number of time steps increases, so the method is easier to use.

The trinomial model is often used when modeling exotic options or complex options that have path dependencies that make their valuation more difficult to accurately model with the binomial or time continuous models.

## Vanna-Volga pricing

Vanna-Volga pricing is used for pricing exotic options in currency derivative markets.

Vanna-Volga entails adjusting the Black-Scholes theoretical value by the cost associated with the main risks related to an option’s volatility:

- vega (sensitivity of the price of an option with respect to implied volatility)
- vanna (sensitivity of vega to a change in the spot price of the underlying)
- volga (sensitivity of vegas to a change in implied volatility)

## Conclusion

Options pricing models are used as methods to value options contracts.

There are several different types of options pricing models, including the Black-Scholes, the binomial options pricing model, the trinomial options pricing model, more niche applications like the Vanna-Volga pricing model, and broad applications to real-world business decisions (real options analysis).

Each of these models has its own strengths and weaknesses and is used for different sorts of markets.

The binomial options pricing model is simple and fast, but may not produce accurate results for exotic options.

The trinomial options pricing model is more accurate than the binomial options pricing model, but is also more complex and takes longer to compute.

The Vanna-Volga pricing model adjusts the Black-Scholes theoretical value by accounting for the cost of a portfolio that hedges three main risks associated with the volatility of the option.

Monte Carlo methods are used when a trader or analyst doesn’t want to be constrained by the distribution assumptions used in a standard mathematical model. However, it is typically used as a method of last resort, especially if existing models provide relatively robust estimations for options pricing.