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?



1 Answer 1


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:

enter image description here

And comparative plot for put looks like this:

enter image description here


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