How and Why Interest Rates Affect Options

The US Federal Reserve is expected to raise the interest rates in coming months. Interest rate changes impact the overall economy, stock market, bond market, other financial markets and can influence macroeconomic factors. A change in interest rates also impacts option valuation, which is a complex task with multiple factors, including the price of the underlying asset, exercise or strike price, time to expiry, risk-free rate of return (interest rate), volatility, and dividend yield. Barring the exercise price, all other factors are unknown variables that can change until the time of an option's expiry.

It is important to understand the right maturity interest rates to be used in pricing options. Most option valuation models like Black-Scholes use the annualized interest rates.

If an interest-bearing account is paying 1% per month, you get 1%*12 months = 12% interest per annum. Correct?

No!

Interest rate conversions over different time periods work differently than a simple up- (or down-) scaling multiplication (or division) of the time durations.

Suppose you have a monthly interest rate of 1% per month. How can you convert it to annual rate? In this case, time multiple = 12 months/1 month = 12.

1.    Divide the monthly interest rate by 100 (to get 0.01)

2.    Add 1 to it (to get 1.01)

3.    Raise it to the power of the time multiple (i.e., 1.01^12 = 1.1268)

4.    Subtract 1 from it (to get 0.1268)

5.    Multiply it by 100, which is the annual rate of interest (12.68%)

This is the annualized interest rate to use in any valuation model involving interest rates. For a standard option pricing model like Black-Scholes, the risk-free one-year Treasury rates are used. (See related: The Black-Scholes Option Valuation Model.)

It is important to note that changes in interest rates are infrequent and in small magnitudes (usually in increments of 0.25%, or 25 basis points only). Other factors used in determining the option price (like the underlying asset price, time to expiry, volatility, and dividend yield) change more frequently and in larger magnitudes, which have a comparatively larger impact on option prices than changes in interest rates. (For a comparative analysis of how each factor impacts option prices, see Sensitivity Analysis For Black-Scholes Pricing Model.)

To understand the theory behind the impact of interest rate changes, a comparative analysis between stock purchase and the equivalent options purchase will be useful. We assume a professional trader trades with interest-bearing loaned money for long positions and receives interest-earning money for short positions.

• Interest Advantage in Call Option: Purchasing 100 shares of a stock trading at $100 will require $10,000, which, assuming a trader borrows money for trading, will lead to interest payments on this capital. Purchasing the call option at $12 in a lot of 100 contracts will cost only $1,200. Yet the profit potential will remain the same as that with a long stock position. Effectively, the differential of $8,800 will result in savings of outgoing interest payment on this loaned amount. Alternatively, the saved capital of $8,800 can be kept in an interest-bearing account and will result in interest income—a 5% interest will generate $440 in one year. Hence, an increase in interest rates will lead to either saving in outgoing interest on the loaned amount or an increase in the receipt of interest income on the saving account. Both will be positive for this call position + savings. Effectively, a call option’s price increases to reflect this benefit from increased interest rates. 
• Interest Disadvantage in Put Option: Theoretically, shorting a stock with an aim to benefit from a price decline will bring in cash to the short seller. Buying a put has similar benefit from price declines, but comes at a cost as the put option premium is to be paid. This case has two different scenarios: cash received by shorting stock can earn interest for the trader, while cash spent in buying puts is interest payable (assuming trader is borrowing money to buy puts). With an increase in interest rates, shorting stock becomes more profitable than buying puts, as the former generates income and the latter does the opposite. Hence, put option prices are impacted negatively by increasing interest rates.

Rho is a standard Greek (a computed quantitative parameter) that measures the impact of a change in interest rates on an option price. It indicates the amount by which the option price will change for every 1% change in interest rates. Assume that a call option is currently priced at $5 and has a rho value of 0.25. If the interest rates increase by 1%, then the call option price will increase by $0.25 (to $5.25) or by the amount of its rho value. Similarly, the put option price will decrease by the amount of its rho value.

Since interest rate changes don’t happen that frequently, and usually are in increments of 0.25%, rho is not considered a primary Greek in that it does not have as a major impact on option prices compared to other factors (or Greeks like delta, gamma, vega, or theta).

Taking the example of a European-style in-the-money (ITM) call option on an underlying trading at $100, with an exercise price of $100, one year to expiry, volatility of 25%, and an interest rate of 5%, the call price using Black-Scholes model comes to $12.3092 and call rho value comes to 0.5035. The price of a put option with similar parameters comes to $7.4828 and put rho value is -0.4482 (Case 1).

Source: Chicago Board Options Exchange (CBOE)

Now, let’s increase the interest rate from 5% to 6%, keeping other parameters same.

The call price has increased to $12.7977 (a change of $0.4885) and put price has gone down to $7.0610 (change of $-0.4218). The call price and put price has changed by almost the same amount as the earlier computed call rho (0.5035) and put rho (-0.4482) values computed earlier. (The fractional difference is due to BS model calculation methodology, and is negligible.)

In reality, interest rates usually change only in increments of 0.25%. To take a realistic example, let’s change the interest rate from 5% to 5.25% only. The other numbers are the same as in Case 1.

The call price has increased to $12.4309 and put price reduced to $7.3753 (a small change of $0.1217 for call price and of -$0.1075 for put price).

As can be observed, the changes in both call and put option prices are negligible after a 0.25% interest rate change.

It is possible that interest rates may change four times (4 * 0.25% = 1% increase) in one year, i.e. until the expiry time. Still, the impact of such interest rate changes may be negligible (only around $0.5 on an ITM call option price of $12 and ITM put option price of $7). Over the course of the year, other factors can vary with much higher magnitudes and can significantly impact the option prices.

Similar computations for out-of-the-money (OTM) and ITM options yield similar results with only fractional changes observed in option prices after interest rate changes.

Is it possible to benefit from arbitrage on expected rate changes? Usually, markets are considered to be efficient and the prices of options contracts are already assumed to be inclusive of any such expected changes. (See related: Efficient Market Hypothesis.) Also, a change in interest rates usually has an inverse impact on stock prices, which has a much larger impact on option prices. Overall, due to the small proportional change in option price due to interest rate changes, arbitrage benefits are difficult to capitalize upon.

Option pricing is a complex process and continues to evolve, despite popular models like Black-Scholes being used for decades. Multiple factors impact option valuation, which can lead to very high variations in option prices over the short-term. Call option and put option premiums are impacted inversely as interest rates change. However, the impact on option prices is fractional; option pricing is more sensitive to changes in other input parameters, such as underlying price, volatility, time to expiry, and dividend yield.