Introduction
A put option is a contract arranged between two parties, the option seller or writer and the option buyer or holder. The agreement usually involves the buyer paying the seller an agreed upon price for the option, resulting in the writer granting the buyer the right, but not the obligation, to purchase a specified asset (e.g. a share of some company's stock) from the seller for a specified price K, called the strike price at some future time. For a European-style option, the buyer of the option exercises this right at a specified future date, called the expiration date of the option. There is also a more flexible American-style option which allows the buyer to exercise the option any time before the expiration date.
Example
A concrete example will help to explain the basic idea. On March 4th 2008, at 2:10PM, a share of Microsoft stock traded at $27.04 per share. At the same time, an (American) put option with a strike price of K = $25 and an expiration date of March 21st 2008 traded at a price P of fifteen cents. At any time during the 17 day period between March 4th and March 21st, if the price of Microsoft falls below $25, the buyer of the option can purchase a share at the market price and sell it to the option seller for $25, netting the difference. If the option buyer has no other interest in Microsoft, he or she hopes the stock will fall significantly below the strike price, at least enough to justify the initial investment of fifteen cents. If the price fails to fall below the strike price, there is no point in the buyer exercising the option, so the buyer loses out on their initial fifteen cent outlay. Thus, such an option buyer is betting that Microsoft shares will drop below price during the period, and the seller is taking the opposite position.
Put options have multiple uses, but one clear benefit of obtaining a put option for the owner is to insure against a precipitous drop in the stock price. In the example above, for fifteen cents the owner of a share of Microsoft stock is guaranteed to be able to receive at least $25 for their share of stock, no matter what happens to the market price.
At any given time t, the strike price K of the put option can exceed the current price S(t) of the underlying asset (e.g. the share of stock) making the option at least worth the difference K - S(t), in which case the option is said to be "in the money." Otherwise, the option is said to be "out of the money." In the Microsoft example, with the current stock price at S(t) = $27.04 a put option with a strike price of K = $17.50 is considerably out of the money. For someone to profit from owning such an option, the price will have to fall to roughly 65% of its current value in a period of 17 days. The chances of this happening seem rather remote and in fact the price of the option of about 1 cent reflected this reality. Perhaps landing in the money is such a remote possibility that the one cent that the seller receives can be viewed essentially as risk-free. Does it make sense for the seller to take such a risk? What can we say about the probabilities of such extreme events? What do the markets reveal about investors' attitudes about these extreme events?
The purpose of this article is to present a simple inequality for what the market is saying about the probability p that a put option lands in the money. The result is:
p ≥ P / K
So, for example, in the Microsoft example, the probability of the stock falling below the strike price of $25 is at least .15/25, that is, 1 in 167, and the probability of the stock falling below the strike price of $17.50 is at least .01/17.50, that is, 1 in 1,750.
The Market as a Bookmaker
We introduce a simpler and perhaps more familiar situation so that the reader can appreciate the basic idea. A bookmaker creates opportunities for gamblers. We can think of the bookmaker as representing the intereste of a casino, henceforth referred to as the house. This house agrees to enter wagers of an amount of money P to any potential takers on some event F. The house will be paid an unknown reward R if F occurs, but the reward is known to be at most RU. If F fails to occur the house receives nothing.
The fact that the house is willing to play such a game reveals information about its views about the likelihood of F occurring. The house would want to never play an unfair game, so it adjusts the price in order to ensure that its expected payoff is nonnegative. In a single play of the game, whether F occurs or not, the house pays P, but in addition, if F occurs, the house receives R. If p(F) represents the probability the house assigns to the event F, then the house would take steps to ensure that its expected return is nonnegative, i.e.:
p(F)(-P + R) + (1 - p(F))(-P) ≥ 0
Since R ≤ RU this implies:
p(F)(-P + RU) + (1 - p(F))(-P) ≥ 0
Observe that it must be the case that RU ≥ P otherwise this expression will be negative. This last inequality leads to the conclusion that:
p(F) / (1 - p(F)) ≥ P / (RU - P) = (P / RU) / (1 - (P / RU))
which is equivalent to:
(1)
Put Options
In the analysis of put options, we can think of the market as playing the role of the bookmaker, available to buy a put option for anyone interested in selling one. Just as the house reveals information about its views of probabilities of events when it declares certain bets to be in its interests, the markets reveal information about investor's attitudes about payoff probabilities when equilibrium is reached and prices are established. From our perspective, the complex process under which these prices are arrived at can be ignored, and we can think of the market as a casino in which bets with various payoffs are made available for the public. The fact that we can sell an option at a given price to a market of potential buyers reveals information about these buyer's attitudes about probabilities of the option landing in the money.
Think of the market as a bookmaker letting you sell it a put option with a strike price of Kat a price P. Take F to be the in-the-money event, i.e, the event that the option lands in the money. If F occurs, that is, the strike price exceeds the stock price S, the house can buy a share of stock at S and sell it to you for K, leading to a net gain of K - S. Thus, the reward R to the house has an upper bound of RU = K. Using the reasoning leading to inequality (1), we conclude that the market/house is behaving as if it believes that:
(2)
Someone purchasing a put option is in essence betting that the underlying asset will diminish in value, and we can interpret the right hand side of the inequality as a measure of market pessimism about the future value of the asset. For example, if the strike price is 10% of the current asset price, and probability of an option with this strike price landing in the money is 1 in 100, we can interpret the inequality as saying that, in the market's view, the asset has a 1 in 100 chance of losing 90% of its value before the expiration date of the option.
Discounting
Betting on options is not perfectly analogous to casino gambling in the traditional sense for two reasons. First, the buyer of the option can experience a significant time delay between the purchase of the option and the ultimate payoff. As a consequence, in the argument above we need to account for this when we bound the reward to the seller. Second, while the payoff for a European option is determined based on the price of the underlying asset at the expiration date, the holder of an American option can choose to exercise the option any time before the expiration date.
In either case, the argument presented above requires slight modification. To properly account for the time value of money, the payoff needs to be expressed in present-value terms. If a European option lands in the money at an expiration time T units from now, the present-value of the payoff to the buyer is not K - S but instead it is:
e(-rT)(K - S) ≤ e(-rT)K
So, while K remains an upper bound for the reward to the option holder, and the inequality above holds, a sharper inequality takes the form:
(3)
for a European option. Since an American option can be exercised any time before the expiration date, the best upper bound we have for the payoff to the holder is K so the inequality (2) is the sharpest inequality available.
Assumption-Free Probabilities
One of the attractive features of the put option inequality described in these pages is the lack of assumptions. Of course, a lower bound on the probability of an option landing in the money is considerably less informative than knowing the probability distribution of an option price at some future date, which in turn requires knowledge of the distribution of the underlying asset price. The situation is strikingly different for one who wishes to price an option, which requires information about the volatility of the underlying asset's price. The more volatile the asset price is, the more liklely it is that the option will land in the money, making the asset more valuable. Conversely, an asset whose price is quite stable and currently out of the money is less likely to land in the money, so the price of the option should be lower. The methodology used for pricing options relies on making use of historical information about volatilities, and requires, at the very least, an assumption concerning the relevance of past behavior for predicting the future. Beyond that, there are issues related to the form of the distribution of the future asset price. In practise, the distribution is taken to have a lognormal distribution, but such an assumption is often questionable.
Weakness of the Inequality
While the put option inequality provides a lower bound on the in-the-money probability, this bound can in some instances be relatively useless. It might not be sharp, that is, the bound need not be close to the true probability, and it is not terribly difficult to see why this is the case. The key observation leading to the inequality is that if and when the option is exercised, the option holder's gain, which is the difference K - S between the strike price and the stock price, is at most K. In other words, to get a lower bound we assume that when the put option lands in the money, the best possible result for the holder is obtained, namely, that the stock becomes worthless. As a consequence, the inequality will be sharpest when the strike price is closest to zero.