Option Pricing and Valuation: Option Premium, Theoretical Value

OPTION PRICING AND VALUATION

Understanding what options are and how they work is certainly a prerequisite to trading them. But the real key to trading options or utilizing them to hedge is in understanding how they are valued. The price relationship between options and their underlying securities, between options and other options, or between options and time, volatility, or other factors, is not always intuitive, nor is it necessarily linear. Given the leverage involved in options, even minor pricing discrepancies or misunderstandings can magnify themselves into substantial amounts of money.

This chapter describes the factors that affect option prices, how much they affect them, and in what direction. Understanding each of these factors individually is important, though in reality, your option holdings will vary in value from the combination of all these factors simultaneously.

As a result of the wide range in strike prices, durations, and underlying instruments, option pricing can be quite complex and is certainly dynamic. We begin by explaining the various components of an option’s value and what they depend on. From there, we discuss how various factors affect an option’s price.

Option Premium                                                

An option’s price, also called its premium, is determined by market participants, just as that of a stock, and is quoted on a per-share basis. Since each contract represents 100 shares (unless adjusted due to a split or other corporate action), the amount of money you actually pay for or receive from an option is 100 times the quoted price. Thus, if a call is quoted at 2.25, the contract will have a total premium of $225.

The price or premium of an option has two components: intrinsic, or cash, value and extrinsic, or time, value. Intrinsic value is defined as the amount, if any, by which an option is in the money (ITM)—in other words, how much money it is worth if it is immediately exercised and the stock is sold (for calls) or purchased (for puts) at this moment in the market. Thus, for a call option, this value is determined by how far above the option’s strike price the stock is currently trading, and for put options, it is how far below the strike price the stock is currently trading. The intrinsic value changes with every tick of the underlying stock, and out-of-the-money (OTM) options have zero intrinsic value. Any premium value over and above the intrinsic value is the option’s time value. The simple relationship between intrinsic and time value can be expressed as follows:

Intrinsic value + time value = total value (premium)

As an illustration, consider the following hypothetical scenario:

QRS stock is selling at $42.50/share.

QRS Jun 40 calls are 2.80.

QRS Jun 45 calls are 0.55.

The QRS Jun 40 call is in the money and has a current intrinsic (cash) value of $2.50. This stems from the fact that one could hypothetically pocket $2.50 per share by exercising the call, thereby purchasing stock at $40, and then selling the purchased shares for the market price of $42.50 each. The option’s time value is therefore $0.30, the amount by which the premium exceeds the intrinsic value. The QRS Jun 45 calls have a strike higher than the current share price, so they are out of the money (OTM). Their premium, $0.55, is thus entirely time value. An option could have a negative time value if selling below its intrinsic value, although this is rare. Options do, however, frequently show time values of zero or almost zero as they get close to expiration. (Note: Option exercises always occur at night, and subsequently take one more day to settle. So, in reality, one could not exercise a call option in the middle of the day and have the stock to sell at that very moment. Nonetheless, the concept of intrinsic value assumes that one could.)

The key valuation terms are as follows:

• Premium: The price of an option or, more accurately, the money you pay or receive when you buy or sell it (per-share price times 100 shares).
• Intrinsic, or cash, value: The amount, if any, by which an option is in the money. Example: An ABC call option with a strike of 40 has $3 of intrinsic value if ABC is trading at $43. The same option has zero intrinsic value if ABC is trading anywhere below $40.
• Time value: The amount, if any, by which an option’s premium exceeds its intrinsic value. Example: If ABC is trading at $43, an ABC call option with a strike of 40 that is trading at 5 has $2 of time value. The same option has zero time value if it is trading at 3.
• Parity: An in-the-money option that trades exactly at its intrinsic value is said to trade at parity.

Theoretical Value

Although options’ actual prices, like those of stocks, are determined by the fluctuating dynamics of supply and demand in the marketplace, participants benefit by having a sense the fair, or theoretical, value of an option. There is a widely accepted formula for calculating the theoretical value for any listed option, called the Black-Scholes formula (after the two University of Chicago professors, Fisher Black and Myron Scholes, who first proposed it in 1973). This formula, for which Myron Scholes later earned the Nobel Prize for Economics, has become the accepted standard for valuing options throughout the industry. Ask 10 knowledgeable research analysts for the fair value of a stock they follow and you are likely to get 10 very different answers. That’s because a myriad of factors can be used to determine the fair value of a stock, and there is no universal agreement on which factors to use or exactly how to evaluate them. But, ask 10 different option traders what the fair value of any option is and you will get 10 very similar, if not precisely the same, answers. That’s because the factors determining an option’s price are well known, well defined, and calculated the same way by most sources.

The factors incorporated into the Black-Scholes formula are:

• Price of the underlying stock
• Strike price of the option
• Amount of time left until expiration
• Volatility of the underlying stock
• Dividends on the underlying stock
• Interest rates

Plug the values of these variables into the formula (as you can easily do on one of the online calculators such as the one at CBOE.com) and you will get a theoretical value for that option. The first two factors—stock price and strike price of the option—affect the option’s price by determining whether or not there is any intrinsic value. By definition, if UVW stock were trading at, say, 36.60, the 35 strike calls in each month would all have 1.60 of intrinsic value, and the 40 strike calls would all have zero intrinsic value, as they would be out of the money. In fact, intrinsic value is independent of all other variables—expiration, volatility, interest rates, and dividends. Time value, or simply time premium, on the other hand, varies with all the other variables, and it is important to understand how.

Time Value

How Time Value Varies with Strike Price

Keeping all other variables constant, time value is greatest for the strike that is closest to the current market price of the underlying and diminishes as one gets further in or out of the money (see Table 2.1).

The fact that time value is greatest for the ATM option is true for any month and any volatility, and it is an important factor in deciding which strike price to use in different strategies.

TABLE 2.1 Time Value versus Strike Price

 

As an option buyer, you want to pay no more time value than is necessary as that is the part of an option’s value that diminishes each day you hold it. On the other hand, sellers look for as much premium as they can get for the same reason.

How Time Value Varies with Volatility

Time value has a somewhat complex relationship with volatility. For an ATM option, time value varies in direct proportion (i.e., linearly) with volatility. Therefore, as shown in Table 2.2, the time value of a 50 strike option when the underlying is 50 will double when volatility doubles. That means the ATM option of a stock with a volatility of 50 percent will be twice the time value of a 50-strike option on a stock with a volatility of 25 percent, all else being equal. But remember that volatility (or, more appropriately, implied volatility) can change, even on the same stock, over time. A big news item might cause the options on a particular stock to spike upward in implied volatility. Such a change in volatility on a given stock will therefore affect the ATM options the most.

How Time Value Varies with Time

The relationship between time value and time itself is not just important, but one of the defining characteristics of options. You would certainly expect that an option’s time value decreases as it gets closer to expiration, and conversely, that longer dated options have greater time value.

TABLE 2.2 Time Value versus Volatility

FIGURE 2.1 Time Value Decay in an Option 

Both, of course, are true, but if you expect that the time premium of a 60-day option would be twice that of a 30-day option on the same stock and strike price, you’d be incorrect. The relationship is not linear. In reality, you should expect to get somewhat less than twice the time premium in the 60-day call because time value varies with the square root of time remaining until expiration. As such, the all-important rate of decay in time value on an option occurs more slowly in the early months and more rapidly as they approach expiration. Figure 2.1 represents this relationship graphically.

A call with four months until expiration will therefore not have four times as much premium as one with a similar strike price and one month to go. Instead, it will have approximately twice as much premium, since  = 2. This means that the rate of decay in the time value of an option is (theoretically) fastest in the last month before expiration, slowing with each successive month farther from that date. Therefore, while you either pay or receive more total premium with the purchase or sale of a more distant option, you take in proportionately less for the extra time. This can easily be seen by taking the time value of any option and dividing it by the number of days remaining until expiration. (Most option chains will tell you the number of days to any expiration.) By doing this, one can see how much time value per day there is in an option, and can then compare that with the time value per day for closer or more distant options. Bear in mind that the square root relationship is theoretical.

TABLE 2.3 Comparison of Time Premium in Different Expiration Months

Thus, it should be used as a guide and not held as a given (see Table 2.3).

Time value decay is an important concept in the development of option strategies. If one plans to hold an option for, say, one month, one would simply buy a one- or two-month option under the rationale that paying for the extra time value of say a six-month option is unnecessary. But in the next 30 days, the six-month option might lose less time value than the one- or two-month option (price of the underlying being constant), and might therefore actually represent the better choice.