Changes in interest rates affect the overall economy, stock markets, bond markets, and other financial markets, as well as macroeconomic factors. Changes in interest rates can also affect option valuation, which is a complex task involving multiple factors, including the price of the underlying asset, exercise or strike price, expiration time, risk-free rate of return (interest rate), volatility, and dividend yield Rate. Except for the strike price, all other factors are unknown variables and may change before the option expires.

## What is the interest rate for the pricing option?

It is important to understand the correct maturity rate used in the pricing options. Most option valuation models (such as Black-Scholes) use annualized interest rates.

If the interest-bearing account pays 1% every month, you will get 1%*12 months = 12% interest per year. correct?

No!

The conversion of interest rates for different time periods is different from simple multiplication (or division) of scaling up (or down) duration.

Suppose your monthly interest rate is 1% per month. How to convert it to annual interest rate? In this case, the time multiple = 12 months/1 month = 12.

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

2. Add 1 (to get 1.01)

3. Multiply by the time multiple (ie 1.01^12 = 1.1268)

4. Subtract 1 (to get 0.1268)

5. Multiply by 100, the annual interest rate (12.68%)

This is the annualized interest rate used in any valuation model involving interest rates. For standard option pricing models like Black-Scholes, the risk-free one-year Treasury interest rate is used.

It should be noted that the changes in interest rates are not frequent and small (usually in increments of 0.25% or 25 basis points). Other factors used to determine option prices (such as the price of the underlying asset, expiration time, volatility, and dividend yield) change more frequently and with greater magnitude, and they have a relatively greater impact on option prices than changes in interest rates.

Key points

- Changes in interest rates directly affect option pricing, and its calculation consists of many complex factors.
- For standard option pricing models such as Black-Scholes, the risk-free annualized Treasury bond interest rate is used.
- When interest rates rise, call options benefit, while put option prices are negatively affected.

## How interest rates affect the price of call options and put options

In order to understand the theory behind the impact of changes in interest rates, a comparative analysis between stock purchases and equivalent option purchases will be useful. We assume that a professional trader uses interest-bearing loans to trade in long positions and collects interest-bearing money through short positions.

### Interest advantages of call options

Buying 100 shares of stocks traded at 100 USD will require 10,000 USD. Assuming that the trader borrows money to trade, this will result in the payment of interest on this capital. Buying a call option at a price of $12 in 100 contracts only costs $1,200. However, the profit potential will be the same as the profit potential of holding long stocks.

In fact, the difference of $8,800 will save interest on the loan amount. Or, deposit the saved capital of $8,800 in an interest-bearing account and generate interest income-5% of interest will generate $440 in one year. Therefore, an increase in interest rates will result in saving interest on loan payments or increasing interest income from savings accounts. Both are good for this bullish position + savings. In fact, the price of call options rises to reflect this benefit from rising interest rates.** **

### Interest disadvantage of put options

In theory, shorting stocks in order to profit from falling prices will bring cash to short sellers. Buying put options can also get similar benefits from falling prices, but at a price, because put options have to pay a premium. There are two different situations in this situation: the cash received from short selling stocks can earn interest for the trader, while the cash for buying put options is the interest payable (assuming the trader borrows money to buy put options).

As interest rates increase, short-selling stocks is more profitable than buying put options because the former generates income, while the latter is the opposite. Therefore, put option prices are negatively affected by rising interest rates.

## Rho Greek

Rho is a standard Greek word used to measure the impact of changes in interest rates on option prices. It represents the amount by which the option price will change for every 1% change in interest rates. Assume that the call option is currently priced at $5 and the rho value is 0.25. If the interest rate increases by 1%, the call option price will increase by $0.25 (to $5.25) or its rho value. Similarly, the price of a put option will reduce its rho value by the amount.

Since interest rate changes are not so frequent and usually increase in 0.25% increments, rho is not regarded as the main Greek language because it affects option prices compared to other factors (or Greek words such as delta, gamma) No significant impact, vega or theta).

## How does interest rate changes affect the price of call options and put options?

Take the European in-the-money (ITM) call option with the underlying transaction price of US$100 as an example, the exercise price is US$100, the expiration is one year, the volatility rate is 25%, the interest rate is 5%, and the call option using the Black-Scholes model The price is $12.3092, and the call option value is 0.5035. The price of a put option with similar parameters is $7.4828, and the value of the put option is -0.4482 (Case 1).

*Source: Chicago Board Options Exchange (CBOE)*

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

The call option price rose to 12.7977 US dollars (change 0.4885 US dollars), and the put option price fell to 7.0610 US dollars (change-0.4218 US dollars). The amount of change in call option price and put option price is almost the same as the previously calculated call option price (0.5035) and put option price (-0.4482). (*The difference in score is caused by the calculation method of the BS model and can be ignored. )*

In fact, interest rates usually only change in 0.25% increments. As a practical example, let’s change the interest rate from 5% to 5.25%. The other numbers are the same as in Case 1.

The call option price rose to 12.4309 US dollars, and the put option price fell to 7.3753 US dollars (the call option price changed slightly by 0.1217 US dollars, and the put option price changed slightly-0.1075 US dollars).

It can be seen that after the 0.25% interest rate change, the change in the price of the call option and the put option is negligible.

The interest rate may change four times in a year (4 * 0.25% = 1% increase), that is, until the expiry time. Nonetheless, the impact of this change in interest rates may be negligible (for an ITM call option price of $12 and an ITM put option price of $7, it is only about $0.5). During the year, other factors may change at a higher rate and may significantly affect option prices.

Similar calculations for out-of-the-money (OTM) and ITM options produce similar results, with only minor changes in option prices observed after interest rate changes.

## Arbitrage opportunities

Is it possible to benefit from the arbitrage of expected interest rate changes? Generally, the market is considered efficient and it has been assumed that the price of an option contract contains any such expected changes. In addition, changes in interest rates usually have an adverse effect on stock prices, and the latter has a much greater impact on option prices. In general, because the change in interest rates leads to small changes in the proportion of option prices, it is difficult to use arbitrage gains.

## Bottom line

Although popular models such as Black-Scholes have been used for decades, option pricing is a complex process and is still evolving. A variety of factors affect option valuations, which may cause very large fluctuations in option prices in the short term. As interest rates change, the premiums of call options and put options will be adversely affected. However, the impact on option prices is small; option pricing is more sensitive to changes in other input parameters, such as the underlying price, volatility, expiration time, and dividend yield.

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