CHAPTER
5

The Basics of Options Valuation

In This Chapter

• Put-call parity: the foundation of valuation
• Basic forms of return
• Adding risk through leverage

People trade options for many different reasons. Regardless of why someone trades an option, the trader needs to know what a particular contract should be worth and why. Options are a type of derivative, and derivatives were named that because their value is derived from the value of the under-lying security.

The value of the underlying is important, but it is not the only factor involved. The same is true with the time to expiration. Options and other derivatives have features that the underlying asset does not. Those features add value.

Options add value to a portfolio by offering choices about whether and how to conduct transactions in the underlying asset. How much value do these things add? That’s what will be covered in this chapter.

Put-Call Parity

Put-call parity is the most basic relationship between the value of the underlying asset, the value of a put, and the value of call. It gives you a way to think about how much an option should be worth without getting into the weeds of calculus and statistics. It is simple, and it serves as a great check on how well the market is valuing options on any given day. It is the relationship between the price of a put and the price of a call.

Put-call parity holds means the relationship between the price of a call, the price of a put, and the price of the underlying will remain fixed for European options with the same strike price and same expiration date. In easy-to-remember equation form, this is:

Underlying asset price = strike price + call premium – put premium

For European options with the same strike price and expiration date, the strike price plus the call premium minus the option premium should equal the stock price.

Stock price = strike price + call premium – put premium

This means that you can create the same payoff of the stock by buying a call and a put, then exercising whichever is in the money.

It also means if one component of the equation changes, then the others should, too. For example, if a stock pays a dividend, then the stock price will go down. This will make the call premium decline, too. Also, it will make the put premium go up.

The put-call parity equation has a big effect on arbitrage, which is the process of making a riskless profit because an asset is at the wrong price. If a trader notices the put premium is higher than it should be, given put-call parity, he can sell puts and buy calls to make a profit with no risk. It does not happen often—in fact, it’s rare—but it does happen.

The put-call parity relationship is used to design trading strategies. Also it tells you what the price of an option should be without running through all the math of Black-Scholes or running a Monte Carlo simulation. Those complex models are covered next.

Basics of Valuation

Options markets didn’t take off until people understood how they could be valued. Options valuation is a complicated process because there are several variables involved, namely these:

• Price of the underlying asset
• Volatility of the underlying asset
• Strike price
• Time remaining until expiration
• Option type (American or European)
• Current interest rates

Obviously, this is not a back-of-the-envelope calculation. The math involved in options valuation is complicated, and many options trading firms hire actuaries and mathematicians to develop valuation models.

You probably won’t be doing that, but you will be using models that others have created, or you will be trading against people who use them. That’s why it’s important to know what goes into the valuation systems that are used most often.

At the most basic level, there are two factors that affect an option’s price:

• Intrinsic value, or the value based on the underlying asset
• Time value, also known as extrinsic value, which is the value based on the fact that the option’s moneyness will change until the time to expiration.

These aren’t the only two factors, of course. That’s why options valuation gets a lot more technical.

Intrinsic and extrinsic values are the most important factors in options valuation, and so they are the most important for a new trader to learn. Most of the changes in an option’s price are related to changes in the price of the underlying asset and to the passage of time. However, other factors do come into play, especially with complex trading strategies.

Understanding Pricing with the Binomial Option Model

The binomial model isn’t really a usable model for options pricing, but I’m putting it in here anyway. That’s because it’s a good way to think about how options prices should work. This will help stimulate the mathematical part of your brain without throwing too many equations or terms at you all at once. Also, the binomial model is the basis for Monte Carlo simulation models that are used by some traders.

Let’s assume that you buy a call option that will be worth either $10 or nothing at expiration. This is a really simplistic assumption, but that’s okay. We can build on it. When the option in this scenario expires, it will be worth either $10 or nothing. If those two events are equally likely, then the option is, on average, worth $5:

($10 × .50) + (0 × .50) = $5

Of course, these two states won’t be equally likely. So then what?

The value of the option is determined by the value of the underlying asset during the time that you hold the option. What are the possible prices that the stock could reach in that time, and what is the probability of each? Suppose the stock price could lead to an option value at expiration of $15, $10, and $0, all of which are equally likely. If that’s the case, then the option is worth $8.33:

($15 × .333) + ($10 × .333) + ($0 × .333)

The idea behind the binomial model is that you continue finding all the outcomes and probabilities until you finally reached the right price. It’s unreasonable to try to do this by hand, but many computer algorithms used to value options follow the binomial process to reach a value in seemingly no time at all.

Pricing in Detail with the Black-Scholes Model

The Black-Scholes model is the fundamental theory for the valuation of options. Like all models, it explains how everything works in a perfect world. The world is hardly perfect, but the model gives traders a good starting point for how an option’s price should behave when everything goes right. Then, traders can determine what will happen when reality interferes.

Options pricing is difficult to do by hand. One reason the exchange-traded options were not introduced until 1973 is the market could not function without enough computing power to handle the valuation. Experienced traders often internalize the math, meaning that they’ve seen enough options trade to know how they should be trading without running through the calculations.

The calculations are complex, too. In 1973, economists Fischer Black and Myron Scholes published a paper called “The Pricing of Options and Corporate Liabilities” and economist Robert Merton published “Theory of Rational Option Pricing.” Scholes and Merton received the 1997 Nobel Prize for Economics for their work; Black had died and so was not eligible.

The Black-Scholes model explains how European call options on nondividend-paying stocks better than anything before or since. In a way, it’s a binomial option model with the factors that influence the underlying price thrown into the mix.

Here, in its gory glory, is the Black-Scholes model.

No, you don’t need to be able to calculate this in order to trade options! But you might find it helpful to understand the factors that go into the model to get a better understanding of how they affect both the price of an option and the rate at which the price changes (you know, the Greeks, as we discussed in Chapter 4).

The variables are as follows:

S₀ = the price of underlying stock today.

N(d) = the probability that a random number in a normal distribution will be less than d.

approximates the percentage amount by which the option is currently in or out of the money.

r = the risk-free rate of interest, usually assumed to be the U.S. Treasury bill rate. (Yes, it has some risk, but if the U.S. government fails, we have bigger problems than the price of a call option.)

T = the number of years until the expiration of the option. If it expires in 3 months, then T would be .25.

σ = the standard deviation of the annualized return on the stock. The combined term is a measure of how much the option is in or out of the money.

Finally, ln and e are related to natural logarithms. They are used to work the calculation and are not related to the options themselves.

With that in mind, the first equation needed to solve the Black-Scholes model is more or less the call price multiplied by a change in the price of the underlying. It is a measure of intrinsic value.

The second equation is used to determine the contribution of time value.

Intrinsic value and time value are the most important components of option value, but they are not the only components.

Combining all of this, the Black-Scholes equation says that the current price of a call option is determined by interest rates, time to expiration, and how much the current stock price varies from what it is expected to be given the historic rate of return. The Black-Scholes equation incorporates delta, which is the change in the option price based on a change in the price in the underlying security. It also incorporates theta, vega, and rho.

The Black-Scholes model was revolutionary at the time, but it has some flaws:

• It only applies to European options on stocks that do not pay a dividend. That excludes most of the equity options trading today.
• Then there’s the theoretical issue about what a risk-free rate really means. Most models assume that U.S. government bond rates are the next-best thing to risk free, but next-best thing leaves a lot of leeway in markets that trade almost instantaneously and in fractions of a cent.
• It assumes that the interest rate and the standard deviation of the stock returns remain consistent. That doesn’t happen.

Value of the Underlying

An option allows you to make a transaction on the underlying asset at a given price. The closer the underlying price is to making the option in the money, the more valuable the option.

The value of the underlying is the most important part of option valuation, but not the only factor.

Here’s a way to think about the value of the underlying asset using the Black-Scholes model. Suppose you buy a call on an underlying asset with an underlying price of $50. If the price of the underlying asset goes to $51, the call should increase in value, all else being equal.

Now, how much will that call price increase? That depends on the delta because delta is the rate of change. Delta isn’t a static number, though. How fast delta changes depends on the gamma—the rate of the rate of change.

What we do know is that the price of the call will go up in this example. Because a put has the opposite payoff, an increase in the underlying asset’s price will make the put less valuable, all else being equal.

In other words, Black-Scholes accommodates intrinsic value as well as other factors.

It accommodates intrinsic value another way. Think about what happens when the exercise price of a call option is high relative to the price of the underlying asset. If the price of the underlying is $20 and a call has a strike price of $50, it probably won’t be exercised. A call with a strike price of $25 is more likely to be exercised, so it will be more valuable. For puts, the situation is reversed. A put with a high exercise price relative to the price of the underlying asset is more likely to be exercised than a put with a really low strike price, so—all else being equal—it will have a lower price.

Time Value

The longer the time to expiration, the more likely it is that the option will end up being worth something. After all, the more time you have, the more that can happen. Every day is a new day with a new opportunity for a company to have a news announcement that will drive the option’s value up or down.

Time value is second to the price of the underlying in determining an option’s valuation. This is the option’s extrinsic value. The longer the time, the greater the extrinsic value and the more valuable the option.

Interest Rates

The role of interest rates in option valuation might not seem logical at first, but think of it this way: if you weren’t buying or writing options, you would be doing something else with the money. That something would probably involve investing or borrowing, so the return or the cost would be determined by market interest rates.

Generally, an increase in interest rates is good for call prices and bad for put prices. That’s because a call writer is receiving income and will expect a higher premium in times when interest rates are high, in order to make options writing more attractive than putting money in the bank.

Keep in mind that this is an average tendency, not an absolute. Interest rates affect time value, intrinsic value, and economic risk, so a change in rates might end up affecting options in different ways. On average, an increase in interest rates is good for calls and bad for puts. However, there are plenty of situations in history where the opposite has happened. Interest rates are affected by the overall level of economic activity, the amount of inflation in the economy, and the risk of a particular investment. Black-Scholes is a model that helps traders think about valuation, not a road map to profits.

The effect of interest rates is small, but it is not zero.

Volatility

Volatility is a measure of how much an underlying asset moves around in price. If you had statistics, it’s simply another word for standard deviation.

If you haven’t had statistics, this little illustration will help. Stock A has a return of 3 percent in year 1, 3 percent in year 2, and 3 percent in year 3. The average return for those 3 years is 3 percent, and the return never varies from that 3 percent number.

Stock B has a return of 6 percent in year 1, –2 percent in year 2, and 5 percent in year 3. The average for return for those years is also 3 percent, but the return bounces around a lot and, in fact, is never exactly 3 percent.

Because the price of Stock B is less predictable than the price of Stock A, Stock B is more volatile. Even though the expected return is the same, Stock B’s return is going to vary a lot more in any given year than will Stock A’s return.

The more volatile a stock price is, the more a hedger will want insurance on it and the more opportunities a speculator will have to make money. The greater the volatility, the more likely an option is to be exercised. That means that the greater the volatility, the more valuable the option. And, to get back to the earlier discussion of the binomial model, the greater the volatility, the more price points to consider when evaluating the probability of the option being in the money.

The option doesn’t have to be in the money for long to be profitable to exercise either. Plenty of traders have made big money on a position that was in the money for 10 minutes. In general, both puts and calls are more valuable if the underlying asset is volatile than if the asset is a steady performer.

The big criticism of standard deviation, which is the most common measure of volatility used in valuing common stock, is that it is calculated with historic information. That’s all well and good, but expiration is a point in the future. The future may be very different from the past. Options traders care about volatility between the purchase day and the expiration, and all sorts of things could happen—or not—that are very different from the historic level of volatility.

Now, if you calculate the option’s theoretical price using price of the underlying, the strike price, the time to expiration, and interest rates, you’ll get a number lower than the price of the option. The difference in value is based on the implied volatility, which is the volatility that the market is using to justify the current price of the option. Your job is to compare this to the information you have to see if the option is cheap or expensive.

The Black-Scholes model was developed by looking at European options because the analysis is simpler. This model can be used as a basis to analyze American options, with the difference being that the American option will be worth more. After all, the more things you can do with an option, the more valuable it will be.

The Effect of Dividends

A dividend is a payment that a company makes to shareholders out of its profits. Not all companies pay dividends, but most of the largest do. When a dividend is paid, the value of the stock declines by the same amount as the dividend. That decline, of even a few cents, makes a call option less likely to be in the money and a put option more likely to be in the money. The result is that a dividend makes a call option less valuable, all else being equal, and a put option more valuable.

The effect of the dividend payment is not included in the Black-Scholes model, but it has a very real effect on the price of an option.

Putting Together the Black-Scholes Model

To help you keep all this straight, the following table gives a simple overview of the factors in the Black-Scholes model and their effects on the direction of call and put option prices. It also includes stock dividends, which are not in Black-Scholes but have an effect on options prices, too.

The Black-Scholes Model, Summarized

Market Factor	Call Prices	Put Prices
Underlying price	Increases	Decreases
Exercise price	Decreases	Increases
Volatility	Increases	Increases
Time to expiration	Increases	Increases
Interest rate	Increases	Decreases
American option	Increases	Increases
Dividend	Decreases	Increases

Keep in mind that these relationships assume that all else is equal, so there will often be situations where you notice deviations from Black-Scholes. Black-Scholes is a model that shows what would happen in a perfect world, and real-life trading isn’t perfect.

Also, these factors don’t play out one by one. They often operate in conjunction with each other. A major news event might cause a stock price to fall and increase the expected volatility. The decrease in the price of the underlying will cause the price of the call option to fall, but the increase in volatility will cause the price of the option to rise. Does one of these factors become more important than another? Will they cancel each other out? I don’t know!

That’s the problem. The Black-Scholes model was a major revolution in option valuation because it broke out all of the different factors, but it can’t provide an exact price useful for a trader. Use it as a framework to think about prices, not to find out where to buy and where to sell.

Heston Volatility Model

In 1993, Steven Heston, a professor at the Yale School of Organization and Management, published a paper titled “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” It was a revision of Black-Scholes with one change—it assumes that the rate of volatility is itself changeable. The Black-Scholes model, meanwhile, assumes that the rate of volatility is fixed for every asset.

This refinement, known as the Heston Volatility model, has been incorporated into many options trading models. Like Black-Scholes, it can’t be used by itself for options trading, but it has influenced the work of other trading systems.

The Heston Volatility model is one of several models that have been developed to address problems in the Black-Scholes model, by the way. If you’re inclined to learn more, make a trip to your library and check out databases of academic journals. It’s interesting, although not necessarily practical unless you are programming your own trading models.

The Least You Need to Know

• Options valuation requires advanced math, heavy computing power, or both.
• The Black-Scholes model is the basis for most option valuation programs, but it has some flaws that make it impractical for use on its own.
• Other mathematical models have been developed to address problems with Black-Scholes, but the math is even more complex.
• The simplest option pricing relationship is put-call parity, and the only math it requires is arithmetic.