American vs European Options: Valuation and Exercise Differences and Their Implications for Investment Strategies
In the world of finance, options are a popular type of derivative instrument that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price and time. There are two main types of options: American options and European options. The main difference between these two types of options is the time at which the option can be exercised. American options can be exercised at any time before expiration, while European options can only be exercised at a single pre-defined point in time.
Assuming an arbitrage-free market, the Black-Scholes equation can be used to derive the price of derivative securities as a function of a few parameters. Under the widely adopted Black model, the Black-Scholes equation for European options has a closed-form solution known as the Black-Scholes formula. However, no corresponding formula exists for American options. Instead, a variety of approximation methods are available, including the Roll-Geske-Whaley, Barone-Adesi and Whaley, Bjerksund and Stensland, binomial options model by Cox-Ross-Rubinstein, and Black’s approximation. There is no consensus on which method is the best.
Investors holding American-style options seeking optimal value will only exercise the option before maturity under certain circumstances. Owners who wish to realize the full value of their option will mostly prefer to sell it as late as possible, rather than exercise it immediately, which sacrifices the time value.
When an American and European option are otherwise identical, the American option will be worth at least as much as the European option. If it is worth more, then the difference is a guide to the likelihood of early exercise. In practice, one can calculate the Black-Scholes price of a European option that is equivalent to the American option, except for the exercise dates. The difference between the two prices can then be used to calibrate the more complex American option model.
To account for the higher value of American options, there must be some situations in which it is optimal to exercise the option before the expiration date. This can arise in several ways. For example, an in-the-money call option on a stock is often exercised just before the stock pays a dividend that would lower its value by more than the option’s remaining time value. A put option will usually be exercised early if the underlying asset files for bankruptcy. A deep in-the-money currency option where the strike currency has a lower interest rate than the currency to be received will often be exercised early because the time value sacrificed is less valuable than the expected depreciation of the received currency against the strike.
An American bond option on the dirty price of a bond may be exercised immediately if it is in the money and a coupon is due. Similarly, a put option on gold will be exercised early when deep in the money, because gold tends to hold its value whereas the currency used as the strike is often expected to lose value through inflation if the holder waits until final maturity to exercise the option.
In conclusion, while American and European options share many similarities, their differences in exercise time and valuation make them unique. The Black-Scholes equation can be used to derive the price of European options, but no corresponding formula exists for American options, which require more complex approximation methods. Understanding the circumstances in which it is optimal to exercise American options early can help investors make better-informed investment decisions.
