How Interest Rates Affect Equity and Futures Option Rho
Summary
The document explains rho as an option’s sensitivity to interest rates and compares how its direction depends on the underlying market and model assumptions. In the Black–Scholes framework for equity options, a higher rate raises call values and lowers put values. The discussion attributes this difference to the financing and cash flows associated with holding or shorting the stock. For options on futures, where entering the futures position requires no initial underlying purchase, it describes rates as lowering both call and put values.
The answer gives standard Black–Scholes expressions for call and put rho and interprets them in terms of discounted strike value, exercise probability, and time to maturity. It notes that rho often receives less attention because its price impact is small relative to other option risks. The explanation is tied to the stated model and assumptions; actual rate sensitivity can depend on contract conventions, the instrument, and how rates and carry are represented.
Key ideas
- Rho measures how an option’s value changes when interest rates change.
- For equity options in the Black–Scholes model, higher rates increase call values and decrease put values.
- The document describes rates as lowering both call and put values for options on futures under its assumptions.
- Black–Scholes rho depends on the discounted strike, the probability of exercise, and time to maturity.
- Rho may have a smaller price impact than other option sensitivities.
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Full text
# Can someone provide a good definitive explanation for rho in relation to option risks?
# Can someone provide a good definitive explanation for rho in relation to option risks?
I have a pretty good understanding of option risks except for one thing, rho. Unfortunately, interest rates tend to have a small effect on option prices, and thus most literature tend to just gloss over this stuff.
My current understanding of rho is that it really depends on the product you trade and the cash flows in and out. For example, if we are talking about the BSM model on equities, rho will have opposite effects on put and call prices. In such a model, we assume that trading underlying requires 100% cash outlay. Thus, when rates rise, call prices get more expensive, and put prices get cheaper (to compensate for the fact that it is more expensive to buy the underlying stock and you get greater interest for shorting a stock).
Meanwhile, if we are talking about the BSM model on futures where we assume that there is 0 cash outlay to take a position in futures, interest rates will have the same effect on options. An increase in rates will decrease the prices of both calls and puts as it is now "more expensive" to borrow money and buy and option.
So the answer to the question how do interest rates affect options really just depends. Is my rational correct?
Thanks!
## Answer by Magic is in the chain (score 2)
https://quant.stackexchange.com/a/47059
It is simpler than the other Greeks, and the reason you don't hear a lot about $\rho$ is because it has smaller impact in the scheme of things. Let's say we are in the BS world, then the rho formulae for a call or put are rather simple:
$\rho_{\mathrm{Call}} = K { e^{- r_{d} \tau} }\tau { N\left (d_{2} \right ) }$
$\rho_{\mathrm{Put}} = - K { e^{- r_{d} \tau} }\tau { N\left(-d_{2} \right ) }$
Now you know the terms containing the N of $d_2$ are just the exercise probabilities, so essentially rho is just the discounted value of the strike times the probability of exercise times maturity. So it is in line with your intuitions.
I have plotted the rho for an ATM call option for different levels of vol just to show how it looks like:
And comparative plot for put looks like this:Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.