Interest Rates and European Put Values

Why Interest Rates Affect European Puts

A European put gives its holder the right to sell an asset for the exercise price \(K\) at expiration \(T\). When the put finishes in the money, the holder receives \(K\) in exchange for an asset worth less than \(K\). If the risk-free rate rises, the present value of that future exercise-price payment falls. This is one reason a higher rate generally reduces the value of a European put on a non-dividend-paying stock.

The conclusion depends on the model and on what is held fixed. In the standard Black–Scholes model, with the stock price, volatility, and expiration fixed, the put value decreases as the constant risk-free force of interest increases. The same direction follows in a one-period binomial model when the up and down factors are fixed and the risk-free gross return remains between them. These are model results, not a claim that every observed put price must fall whenever a market rate changes: other inputs, such as volatility or the stock price, may change at the same time.

Present Value and Risk-Neutral Valuation in a Binomial Model

Consider a one-period binomial model. The stock price today is \(S_0\); at expiration it is either \(S_0u\) or \(S_0d\), where \(u\) is the up factor and \(d\) is the down factor. Let \(R=1+i\) be the one-period risk-free gross return. Assume \(d<R<u\), so the risk-neutral up probability lies between zero and one. In this model, the risk-neutral probability is a pricing weight, not necessarily the probability of the stock actually rising.

If the put’s payoffs in the up and down states are \(P_u=\max(K-S_0u,0)\) and \(P_d=\max(K-S_0d,0)\), its value is the risk-neutral expected payoff discounted one period:

Why does a higher rate reduce the put value in this setup? First, increasing \(R\) increases \(q\) and decreases \(1-q\). Because \(u>d\), the put payoff in the up state cannot exceed its payoff in the down state: \(P_u\leq P_d\). Increasing the weight on the up state therefore reduces or leaves unchanged the risk-neutral expected payoff. Second, dividing by a larger \(R\) further reduces the present value of a nonnegative expected payoff. Both effects point in the same direction, provided the model remains valid.

Black–Scholes Formula and the Rate Effect

For a more direct rate comparison, assume the Black–Scholes model: the option is European, the stock pays no dividends, the stock price follows a geometric Brownian motion with constant volatility \(\sigma>0\), and the risk-free force of interest \(\delta\) is constant. Assume a frictionless market and the usual no-arbitrage pricing conditions. With current stock price \(S\), exercise price \(K\), and time to expiration \(T>0\), the put value is

Here \(N(z)\) is the cumulative distribution function of a standard normal random variable. The term \(K e^{-\delta T}\) is the present value of the exercise price under continuous compounding. The rate also changes \(d_1\) and \(d_2\), so the rate effect is not captured by looking at the discounted exercise price alone. Differentiating the complete formula shows that, with \(S,K,T,\sigma\) fixed, the put decreases as \(\delta\) increases.

Why the Black–Scholes Sensitivity Is Negative

A short derivation makes clear why changing both the discount factor and the normal probabilities does not overturn the result. The standard normal density is denoted by \(\phi\). From the definitions of \(d_1\) and \(d_2\), both have rate derivative \(\sqrt{T}/\sigma\). Differentiating the put formula gives

The last two terms cancel because \(S\phi(d_1)=K e^{-\delta T}\phi(d_2)\). To verify this identity, use \(d_1=d_2+\sigma\sqrt{T}\) and the normal density \(\phi(z)=(2\pi)^{-1/2}e^{-z^2/2}\). Substitution into the ratio gives

where the final equality follows by substituting \(\sigma\sqrt{T}\,d_2=\ln(S/K)+(\delta-\sigma^2/2)T\). Thus the density terms cancel, leaving the negative sensitivity stated above. This derivation applies to the specified Black–Scholes model; it should not be applied without checking assumptions such as the absence of dividends.

Reading the Present-Value Effect Carefully

The exercise price is paid only if the put finishes in the money. Its present value is therefore an important intuition, but the put is not simply a risk-free payment of \(K\): whether that payment occurs depends on the stock price at expiration. The binomial risk-neutral weights and the Black–Scholes terms account for that uncertainty.

The rate result also has boundaries. In the one-period binomial model, if an increased \(R\) reaches or exceeds \(u\), the risk-neutral probability no longer lies strictly between zero and one; the stated model setup is no longer valid. For a stock with dividends, the standard no-dividend Black–Scholes formula above is not applicable as written. Rates and dividends can have interacting effects, so do not extend the conclusion to a different model without establishing its assumptions.

• Hold inputs fixed. A meaningful rate comparison keeps stock price, exercise price, volatility, maturity, and the model structure unchanged.
• Use consistent rate conventions. The binomial formula uses a one-period gross return \(R\); the Black–Scholes formula uses a continuously compounded force \(\delta\).
• Separate pricing weights from forecasts. The risk-neutral probability is used for pricing in the model; it is not automatically a forecast of the stock’s real-world chance of rising.