## Interest rate - the puppetmaster behind option prices

The interest rate is the last variable we need to discuss. The dependence of the call price on the interest rate that emerges from the Black-Scholes formula is depicted in Figure 1. The dependence is positive, right? Not so fast. This is the case when common sense should be used instead of mathematical models. One economic factor can influence another through many channels, often leading to contradicting results.

John Hull offers two explanations.

- As interest rates in the economy increase, the expected return required by investors from the stock tends to increase. This suggests a positive dependence of the call price on the interest rate.
- On the other hand, when the interest rate rises, the present value of any future cash flow received by the long call holder decreases. In particular, this reduces the payoff if at expiration the option is in the money.

The combined impact of these two effects embedded in the Black-Scholes formula is to increase the value of the call options.

## However, experience tells the opposite

The Fed changes the interest rate at discrete times, not continually. Two moments matter: when the rumor about the upcoming interest rate change hits the market and when the actual change takes place. The market reaction to the rumor depends on the investors' mood - bullish or bearish. A bullish market tends to shrug off most bad news. In a bearish market, even a slight threat may have drastic consequences. By the time the actual change occurs, it is usually priced in.

In 2017, the Fed raised the rate three times: on March 15 (no reaction, judging by S&P 500 SPDR SPY), June 14 (again no reaction) and December 13 (a slight fall). The huge fall in the beginning of February 2018 was not caused by any actual change. It was an accumulated result of cautiousness ("This market has been bullish for too long!") and fears that the Fed would increase the rates in 2018 by more than had been anticipated. Many investors started selling stocks and buying bonds and other less risky assets. The total value of US bonds is about $20 trillion, while the total market capitalization of US stocks is around $30 trillion. The two markets are comparable in size, which means there is enough room to move from one to another and the total portfolio reshuffling can be considerable. Thus far, the mere expectation that the interest rate will increase has been able to substantially reduce stock prices and, consequently, call prices.

All this I summarized to my students as follows. When interest rates rise, bonds become more attractive. This is a substitution effect: investors switch from one asset to another all the time. Therefore stock prices and call prices fall. Thus the dependence of call prices on interest rates is negative.

The first explanation suggested by Hull neglects the substitution effect. The second explanation is not credible either, for the following reason. As I explained, stock volatility has a very strong influence on options. Options themselves have an even higher volatility. A change in interest rates by a couple percent is nothing in comparison with this volatility. Most investors would not care about the resulting reduction in the present value of future cash flows.

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