Put-Call Parity With Negative Interest Rates

Why Negative Rates Change the Strike’s Present Value

Put-call parity links the prices of a European call and put with the same underlying asset, strike, and expiration. A negative risk-free rate does not change the relationship. It does change the direction of the comparison between the strike’s present value and its future amount: under a negative rate, the present value of a fixed payment due at expiration is greater than the payment itself.

That result can feel counterintuitive if you are used to positive rates, where discounting a future amount makes it smaller. The key is to keep the accumulation factor and discount factor distinct. With a negative effective annual rate \(i\), where \(i>-1\), the accumulation factor over \(T\) years is \((1+i)^T<1\), so the discount factor is \(v^T=(1+i)^{-T}>1\).

The assumptions matter. The options must be European and have matching terms; the underlying must have no dividends or other distributions during the option term; and the same risk-free rate must apply to the financing used in the replication. If there are known cash dividends or a continuous dividend yield, the parity relationship must be adjusted. The formula here does not cover those cases.

The rate \(i\) is an effective annual rate, and \(T\) is measured in years. If the rate is negative, it must still satisfy \(i>-1\), so the accumulation factor remains positive. A fractional \(T\) is allowed under the effective annual convention: for example, a term of six months uses \(T=0.5\).

Why the Parity Equation Still Holds

The justification is a payoff identity at expiration, not an assumption that rates are positive. At time \(T\), a European call pays \(\max(S_T-K,0)\), while a European put pays \(\max(K-S_T,0)\). For every possible value of \(S_T\),

The left side is the expiration payoff of a call plus a risk-free investment that pays \(K\) at time \(T\). The right side is the payoff of a put plus one share of the underlying asset. Since the payoffs agree in every state, no-arbitrage requires the initial costs of the two portfolios to agree.

At a constant effective annual rate \(i\), the amount required today to provide \(K\) at time \(T\) is \(K(1+i)^{-T}\). This remains true when \(i<0\). Substituting that cost and the current share price into the equality of initial costs gives \(C_0+K(1+i)^{-T}=P_0+S_0\). The proof therefore does not depend on the present value of the strike being less than \(K\).

Using Parity When the Rate Is Negative

The algebra is the same as at a positive rate. First calculate the present value of the strike using the stated rate convention. Then substitute into the form of parity that isolates the unknown quantity. The negative rate affects the value of \(K(1+i)^{-T}\), but it does not change the signs in the parity equation.

Worked Applications

Common Mistake: Multiplying Instead of Discounting

A negative rate can tempt a solver to multiply the strike by \((1+i)^T\), perhaps because the exponent looks like the standard accumulation factor. But the strike \(K\) is the amount due in the future. To express that future payment in today’s dollars, divide by the accumulation factor. Equivalently, multiply by its reciprocal.

For the two-year example above, the incorrect calculation would use \(120(0.95)^2=120(0.9025)=108.30\) as the present value of the strike. It would then produce the incorrect put price

The correction is to use \(120/(0.95)^2=132.9639889\), giving \(P_0=\$43.96\). The incorrect method treats the strike as a current amount to be accumulated forward; parity needs the current cost of a future payment. This distinction applies whether the rate is positive, zero, or negative. Only the comparison between present value and future amount changes with the sign of the rate.

What to Check Before Applying the Result

Negative rates make careful interpretation especially important, but they do not require a new parity formula. Before using the equation, make sure the option prices are time-zero prices, the expiration is measured in years consistent with the stated effective annual rate, and the strike and expiration match across the call and put. Then calculate the strike’s present value explicitly rather than relying on a mental rule about whether it should be above or below \(K\).

For a negative effective annual rate and positive term, \(0<1+i<1\), so \((1+i)^T<1\) and \(K(1+i)^{-T}>K\). For a zero rate, the present value is exactly \(K\). For a positive rate, the present value is less than \(K\). These observations are checks on the calculation, not replacements for it. Round only after substituting into parity, and use the unrounded present value when checking equality.

The payoff identity explains why the parity equation survives a change in the sign of the rate: the options and stock produce matching state-by-state payoffs when paired with a fixed payment of \(K\) at expiration. The rate determines the current cost of financing that payment. In particular, the negative-rate case is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it tests whether a candidate can keep the payoff relationship unchanged while correctly interpreting the strike’s present value.