Interest Rates and European Call Values

Why Interest Rates Matter to a Call

A European call gives its holder the right, but not the obligation, to buy an asset for a fixed exercise price \(K\) at a specified future date \(T\). If the stock price at that date is \(S_T\), the call’s payoff is \(\max(S_T-K,0)\). For a call on a non-dividend-paying stock, a higher risk-free rate generally raises the call’s value, all else held constant.

One useful intuition is that the call holder pays \(K\) only at expiration if the call is exercised. A higher rate reduces the present value of that future payment. But this intuition is not, by itself, a complete option-pricing calculation: changing the rate also changes the probabilities used to price future payoffs in a risk-neutral model. Both effects must be handled consistently.

The result in this tutorial concerns a European call on a non-dividend-paying stock. The stock price, exercise price, time to expiration, volatility, and other model inputs are held fixed while the risk-free rate changes. It is a comparative-statics result: it describes how the model value responds to a change in one input, not a guarantee that an observed market price will rise whenever a quoted interest rate rises.

Present Value of the Exercise Price

Begin with the certain cash flow, rather than with the full option. If \(K\) dollars are due in \(T\) years and the effective annual rate is \(i\), then the amount that would grow to \(K\) over that period is its present value:

For fixed positive \(K\) and \(T\), increasing \(i\) increases the accumulation factor \((1+i)^T\), so it decreases \(PV(K)\). In the continuous-compounding convention, \(PV(K)=Ke^{-\delta T}\), which likewise decreases as \(\delta\) increases. This makes the delayed payment of the strike less costly in present-value terms.

A Paired Price Comparison in a Binomial Model

A one-period binomial model makes the full comparison concrete. Assume a stock currently costs \(S_0\), pays no dividends, and will have one of two possible prices at the end of the period: \(S_u\) or \(S_d\). The risk-free investment grows by a gross factor \(1+i\) over the period. Assume the model permits no arbitrage, so the risk-free gross return lies between the stock’s down and up gross returns. In this model, the risk-neutral probability of the up state is

where \(u=S_u/S_0\) and \(d=S_d/S_0\). The call’s price is the discounted risk-neutral expected payoff:

Here \(q\) is a pricing probability implied by the model, not a claim about the stock’s actual chance of rising. When the interest rate changes, \(q\) changes as well as the discount factor. The correct paired comparison recalculates both.

This rate also satisfies the no-arbitrage condition because \(0.80<1.10<1.20\). The price is

To the nearest cent, the call is worth $13.64 at 10%, compared with $10.00 at 0%. The higher rate lowers the discount factor, but it also raises the risk-neutral weight on the up state in this model. The correctly repriced call is higher. The common mistake this example avoids is changing the discount rate while incorrectly keeping the risk-neutral probability fixed.

Black–Scholes Rate Sensitivity

A standard continuous-time model gives a precise result. Assume a European call on a non-dividend-paying stock, no arbitrage, continuous trading without transaction costs or trading restrictions, a constant risk-free force of interest \(\delta\), and a constant volatility \(\sigma>0\). The model assumes the stock follows geometric Brownian motion, so its continuously compounded returns have constant volatility. For \(T>0\), the Black–Scholes call value is

where \(N(x)\) is the cumulative distribution function of a standard normal random variable. This model formula does not directly apply without adjustment when the stock pays dividends or when its assumptions are not appropriate.

The sensitivity of the call value to \(\delta\), often called call rho, is positive. The short derivation below shows why; it also makes clear why simply discounting the strike is not the entire argument. Write \(\phi(x)\) for the standard normal density. Differentiating the pricing formula with respect to \(\delta\), while holding \(S_0,K,T,\sigma\) fixed, gives

The two density terms cancel. To verify the cancellation, use \(d_1=d_2+\sigma\sqrt{T}\). Then

Since \(\phi(x)=e^{-x^2/2}/\sqrt{2\pi}\), this identity implies

Substituting this equality into the derivative removes the density terms and leaves

For positive \(K\), positive \(T\), and finite model inputs, \(e^{-\delta T}>0\) and \(N(d_2)>0\). Therefore the call value increases with the risk-free force of interest within these assumptions. The result is an exact model sensitivity, not merely a comparison of discounted strikes.

Common Mistake and Practical Interpretation

A tempting shortcut is to say that because the present value of the strike falls, the call must rise by exactly the same amount. That reasoning treats the option as if it were a certain payment of \(K\). A call pays only when the stock price exceeds \(K\), and its value depends on the distribution of possible payoffs as well as discounting.

For example, in the Black–Scholes comparison above, discounting $100 for one year gives \(100e^{-0.02}=\$98.02\) at 2% and \(100e^{-0.06}=\$94.18\) at 6%. A mistaken calculation would call the option-value increase the full difference, \(98.02-94.18=\$3.84\). The actual model values rise by $2.07, not $3.84. The error is using the change in the present value of a certain strike without accounting for the rate-driven changes in \(d_1\), \(d_2\), and the weighted option payoffs.

The useful conclusion is conditional but strong: under the stated non-dividend European-call assumptions, raising the risk-free rate while holding the other model inputs fixed raises the call value. The present-value argument supplies intuition; a consistent pricing model supplies the full comparison. This distinction is aligned with the SOA VEE Accounting and Finance syllabus.