American Put–Call Bounds

Why American Options Do Not Have European Put–Call Equality

For European options with the same strike and expiration, put–call parity is an equality. For American options, the holder can exercise before expiration. That right matters: a put may be worth more as an American option than as a European option, while a call on a non-dividend-paying stock has no benefit from early exercise under the assumptions below. As a result, the American call–put price difference lies within bounds rather than satisfying European parity exactly.

The bounds provide a useful check on hypothetical quotes. They do not, by themselves, determine either option price or guarantee that a pair of prices is otherwise reasonable.

The nonnegative-rate assumption is important. It supports the usual result that early exercise of a call on a non-dividend-paying stock is not advantageous, and it is used in the put-premium bound derived below. The bounds are

The lower bound can be negative. It is not a claim that the call is always worth more than the put.

From European Parity to American Bounds

For European options on a non-dividend-paying stock, put–call parity is

One justification is that a European call plus a risk-free bond paying \(K\) at time \(T\) has the same expiration payoff as one share plus a European put. In either case the payoff is \(\max(S_T,K)\), where \(S_T\) is the stock price at expiration. The bond costs \(Ke^{-\delta T}\) today, so no-arbitrage gives the equality.

An American put is at least as valuable as the corresponding European put because it allows every European exercise strategy and also permits earlier exercise. Thus \(P_A\geq P_E\). In addition, for a non-dividend-paying stock and \(\delta\geq0\), the American call is worth the same as its European counterpart: \(C_A=C_E\). These facts immediately give the upper bound:

To obtain the lower bound, we need to limit how much extra value early exercise can give the put. Define the American put premium over its European counterpart as \(P_A-P_E\). Under the stated assumptions, it satisfies

Here is a justification. At any time \(\tau\) between 0 and \(T\), let \(S_\tau\) be the stock price and let \(u=T-\tau\) be the remaining time. The European put value with price \(S_\tau\) and remaining time \(u\), denoted \(p_E(S_\tau,u)\), is at least its lower bound from European put–call parity and the nonnegativity of the European call:

Compare the payoff from exercising the put immediately, \((K-S_\tau)^+\), with the value of continuing to hold the European put. If \(S_\tau\leq Ke^{-\delta u}\), the difference between the immediate payoff and the parity lower bound is at most \(K(1-e^{-\delta u})\). If \(Ke^{-\delta u}<S_\tau<K\), then \((K-S_\tau)^+\leq K(1-e^{-\delta u})\), while the continuation value is nonnegative. If \(S_\tau\geq K\), the immediate exercise payoff is zero. Therefore, in all cases,

This is a non-strict inequality. In particular, when \(\delta=0\) or \(\tau=T\), its right-hand side is zero; the argument still holds. Discounting the bound back to time 0 gives

The final inequality uses \(\delta\geq0\) and \(\tau\geq0\). Under risk-neutral pricing, the discounted value of the European put is a martingale. Consequently, for any exercise time chosen by the American put holder, the expected discounted gain from exercising instead of continuing is no more than \(K(1-e^{-\delta T})\). Taking the best permitted exercise strategy gives the stated bound on \(P_A-P_E\). This argument explains why early exercise can change put value, but cannot increase it without limit.

Now use European parity and \(C_A=C_E\):

Since \(0\leq P_A-P_E\leq K(1-e^{-\delta T})\), the largest possible difference is \(S-Ke^{-\delta T}\), and the smallest permitted difference is

This proves both bounds. The expression also shows precisely why European equality need not hold for American options: the difference from European parity is the American put’s early-exercise premium. If that premium is positive, then \(C_A-P_A\) is strictly below \(S-Ke^{-\delta T}\). The bounds do not say the premium must be positive in every situation.

Why the American Call Equals the European Call Here

The call result is part of the reasoning behind the upper bound, so it is useful to see its basis. Suppose the holder considers exercising at time \(\tau<T\), when the stock price is \(S_\tau\), and \(u=T-\tau\) years remain. If \(S_\tau\leq K\), immediate exercise pays zero, while continuing has nonnegative value. If \(S_\tau>K\), the European call value is at least \(S_\tau-Ke^{-\delta u}\), which is at least \(S_\tau-K\) because \(\delta\geq0\). Thus the European call continuation value is at least the immediate exercise payoff. Exercising early cannot improve the call’s value; the right to exercise early adds no value, so \(C_A=C_E\). At \(\delta=0\), equality of the two payoffs can occur, but early exercise still does not increase value.

This reasoning depends on the absence of dividends. A shareholder receives dividends, whereas a call holder who has not exercised does not. With dividends, early exercise may have a different comparison, and the bounds here should not be applied without adjustment.

Applying the Bounds

For a particular set of inputs, first calculate the present value of the strike, \(Ke^{-\delta T}\). Then compute the lower and upper limits for the price difference. Compare the quoted difference \(C_A-P_A\) with that interval. If it falls outside, at least one assumption or quote is inconsistent with the no-arbitrage relationships used here.

Common Mistakes and Interpretation

• Using equality instead of bounds. European parity is \(C_E-P_E=S-Ke^{-\delta T}\). For American options in this setting, the call equals the European call, but the put may be more valuable than its European counterpart. The resulting relationship is an interval, not necessarily an equality.
• Forgetting which option can have an early-exercise premium. Under the assumptions here, \(C_A=C_E\) and \(P_A\geq P_E\). A positive put premium moves \(C_A-P_A\) downward from the European-parity value.
• Using inconsistent time units. The rate \(\delta\) is annual and continuously compounded, so \(T\) must be measured in years. For example, six months is \(T=0.5\), not \(T=6\).
• Assuming a passing check proves everything. A price difference inside the interval satisfies these bounds, but other restrictions and market assumptions may still matter.
• Turning a valid weak inequality into a strict one. The early-exercise comparison uses \(\leq\), not necessarily \(<\). At \(\delta=0\) or at expiration, the bound can be exactly zero.