What is Rho in Options?
Rho refers to the metric that is used for assessing the sensitivity of an option to the changes in the risk free interest rate. In other words, it shows the amount of money that an option would either gain or lose in case the risk free interest rate changes by 1%. In the US, the interest rate for US Treasury bills is used as the proxy for risk free interest rate. Typically, Rho is expressed in terms of dollar amount.
Please note that Rho is one of the least used Greek option metrics as the option price is not significantly impacted due to a change in interest rates.
How to Calculate Rho in Options?
The exact formula for Rho can be expressed in a very complicated way, wherein it is calculated as the first derivative of the value of the option relative to the risk-free interest rate. However, in a simpler way, the formula for Rho can also be expressed by using the spot priceSpot PriceA spot price is the current market price of a commodity, financial product, or derivative product, and it is the price at which an investor or trader can buy or sell an asset or security for immediate delivery., option strike priceStrike PriceExercise price or strike price refers to the price at which the underlying stock is purchased or sold by the persons trading in the options of calls & puts available in the derivative trading. Thus, the exercise price is a term used in the derivative market., normal cumulative distribution function, risk-free interest rateRisk-free Interest RateA risk-free rate is the minimum rate of return expected on investment with zero risks by the investor. It is the government bonds of well-developed countries, either US treasury bonds or German government bonds. Although, it does not exist because every investment has a certain amount of risk., standard deviationStandard DeviationStandard deviation (SD) is a popular statistical tool represented by the Greek letter 'σ' to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data's reliability., and time to option’s expiry.
Mathematically, it is represented as,
ρ = K * t * e−r*t * N(d2)
where, d1 = [ln(S/K) + (r + σ2/2) * t] σ√t
d2 = d1 − σ√t
- S = Spot price
- K = Option strike price
- N = Normal cumulative distribution function
- r = Risk-free interest rate
- σ = Standard deviation
- t = time to option’s expiry
Examples of Rho
Let us take a simple example to illustrate the concept of Rho. Imagine that there is a call option priced at $5.00, and it has a rho equivalent to $0.50. Now, if the risk-free interest rate increases by 0.5% (from 2.5% to 3.0%), then what will be the impact on the value of the call option.
Theoretically, every 1% increase in an interest rate should increase the value of the call option by $0.50. In this case, the interest rate increased by 0.5%, so the value of the call option should increase by $0.25 (= 0.5%/1% * $0.50). So, the new value of the option would be $5.25.
Let us take another example of a put option to explain the computation of Rho in more detail. In this case, the spot price of the underlying is $45, the strike price is $50, the risk-free interest rate is 1%, and the standard deviation is 0.25. Determine the Rho of the option is the time to option’s expiry is one year.
- Option strike price, K = $50
- Spot price, S = $45
- Risk free interest rate, r = 1%
- Standard deviation, σ = 0.25
- Time to option’s expiry, t = 1 year
Now, the value of d1 and d2 can be calculated as,
d1 = [ln(S/K) + (r + σ2/2) * t] σ√t
- = [ln($45/$50) + (1% + 0.252/2) * 1] 0.25√1
- = -0.2564
d2 = d1 − σ√t
- = -0.2564 – 0.25√1
- = -0.5064
Now, the Rho of the option can be calculated by using the above formula as,
- = $50 * 1 * e−1%*1 * N(-0.5064)
- Rho = $15.16
Therefore, for every 1% change in the interest rate, the value of the put option will increase by $15.16.
Option Conditions in Rho
The three major option conditions with respect to Rho are as follows –
- Out-of-the-Money (OTM) – An Out of the MoneyOut Of The Money”Out of the money” is the term used in options trading & can be described as an option contract that has no intrinsic value if exercised today. In simple terms, such options trade below the value of an underlying asset and therefore, only have time value. option can either be a put option for which the strike price is lower than the spot price or a call option for which the strike price is higher than the spot price. Typically, out-of-money options exhibit a very low value of Rho.
- At-the-Money (ATM) – The strike price of At the MoneyAt The MoneyATM refers to a situation in which the option holder's exercise of the option results in no loss or gain since the exercise price or strike price is equal to the current spot price of the underlying security. option is the same as the spot price of the underlying asset. If both call and put options are simultaneously at-the-money, then both may increase in value provided there is huge uncertainty about the underlying stock’s future price. In such cases, the value of Rho of call and put option decides which way the market perceives the underlying stock’s future price movement. Typically, at-the-money options exhibit a higher value of Rho.
- In-the-Money (ITM) – An In the MoneyIn The MoneyThe term "in the money" refers to an option that, if exercised, will result in a profit. It varies depending on whether the option is a call or a put. A call option is "in the money" when the strike price of the underlying asset is less than the market price. A put option is "in the money" when the strike price of the underlying asset is more than the market price. option can either be a call option for which the strike price is lower than the spot price or a put option for which the strike price is higher than the spot price. Typically, in-the-money options exhibit a higher value of Rho.
If all other factors remain the same, then the value of an option with positive Rho will increase with the increase in interest rates rise and decrease with the fall in interest rates fall.
If all other factors remain the same, then the value of an option with negative Rho will decrease with the increase in interest rates rise and increase with the fall in interest rates fall.
Although Rho is an indispensable part of the Black–Scholes options-pricing model, it is regarded as one of the least used Greek option metrics because for Rho to have a significant impact on the price of an option; the interest rate has to change drastically, which is usually not the case.
So, it can be seen that Rho is particularly useful only when the interest rate changes dramatically, and this is the reason that it is not part of the vast majority of options trading strategies.
This has been a guide to what is Rho in Options. Here we discuss examples, positive, negative Rho, and how to calculate it along with its option in conditions. You may learn more about financing from the following articles –