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What Is Rho?
Let me explain what rho is in the world of options trading. Rho is the rate at which the price of a derivative changes relative to a change in the risk-free rate of interest. It measures the sensitivity of an option or an entire options portfolio to a change in interest rates. Rho can also refer to the aggregated risk exposure to interest rate changes for a book of several options positions.
For instance, if an option or portfolio has a rho of 1.0, then for every 1 percentage-point increase in interest rates, the value increases by 1 percent. You should know that options most sensitive to interest rate changes are those that are at-the-money and have the longest time to expiration.
In mathematical finance, these quantities that measure a derivative's price sensitivity to changes in underlying parameters are known as the 'Greeks.' The Greeks are essential tools in risk management because they let you, as a manager, trader, or investor, measure how an investment or portfolio's value changes with a small parameter shift. More importantly, this allows you to isolate the risk and rebalance the portfolio to achieve your desired level of risk relative to that parameter. The most common Greeks are delta, gamma, vega, theta, and rho.
Key Takeaways
- Rho measures the price change for a derivative relative to a change in the risk-free rate of interest.
- Rho is usually considered to be the least important of all option Greeks.
Rho Calculation and Rho in Practice
The exact formula for rho is complicated, but it's calculated as the first derivative of the option's value with respect to the risk-free rate. Rho measures the expected change in an option's price for a 1 percent change in a U.S. Treasury bill's risk-free rate.
Take this example: assume a call option is priced at $4 and has a rho of 0.25. If the risk-free rate rises 1 percent, say from 3 percent to 4 percent, the value of the call option would rise from $4 to $4.25.
Call options generally rise in price as interest rates increase, and put options generally decrease in price as interest rates increase. So, call options have positive rho, while put options have negative rho.
Assume a put option is priced at $9 and has a rho of -0.35. If interest rates decrease from 5 percent to 4 percent, the price of this put option would increase from $9 to $9.35. In the same scenario, the call option from before would see its price decrease from $4 to $3.75.
Rho is larger for options that are in-the-money and decreases steadily as the option becomes out-of-the-money. Also, rho increases as the time to expiration increases. Long-term equity anticipation securities (LEAPs), which are options with expiration dates greater than one year away, are far more sensitive to changes in the risk-free rate and thus have larger rho than shorter-term options.
Though rho is a primary input in the Black-Scholes options-pricing model, a change in interest rates generally has a minor overall impact on option pricing. Because of this, rho is usually considered the least important of all the option Greeks.
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