# What is option rho?

By Simon Gleadall, CEO of Volcube.

Option rho tells us the amount an option’s value will change for a change in the risk-free rate of interest. It is usually standardised to show the change in option value for a 1% change in interest rates.

One way to understand rho is to remember one of the basic assumptions of most option pricing models. It is usually assumed that a stock or other asset that is paid for up front with cash must offer a return at least equal to the risk-free rate of interest. So it is assumed that for say a stock option, the stock underlying the option will in 12 months increase in price by at least the amount of the risk-free rate of interest. Stocks must be riskier than the ‘risk-free’ rate of interest, so they must, at least in theory, offer a return that is at least as good as the risk-free rate.

Now, think about a call option with a strike of 100 that expires in 12 months time, on a spot product currently trading 100. Imagine the risk-free rate of interest is 5%. You might think this call option is an at-the-money option because its strike price equals the spot price. But this is not really the case. The option is really struck not on the current spot price but on the *forward* price of the spot. In other words, the value of this call which has 12 months to run relates to where the spot price is expected to be in twelve months time and not where it is currently trading. So the at-the-money calls in this case are really the 105 calls, because this is the forward price of the spot.

Let’s go back to the definition of rho. Rho is the change in option value for the change in the interest rates. Now it should be clear to you that, in our example, if the risk-free rate increases from 5% to 10%, this is going to increase the value of the calls. The increase in interest rates has the effect of pushing the forward spot price upwards and this pushes the calls further in-the-money. The 100 strike puts, in contrast, will become further out-of-the-money and therefore their value drops. In other words, we can see that calls have positive rho and puts have negative rho. In simple terms, if you are long calls and short puts, even of the same strike, you have a positive exposure to interest rates.

## Adding rho and hedging rho risk with interest rate futures

Rho risk can be added across any options that are exposed to precisely the same interest rates. But great care must be taken to match the interest rate by type and duration. Only options with the same expiration and in the same currency can truly be added, in terms of rho risk. This is because interest rates vary by duration and by currency. The yield curve of interest rates is variable and specific to each currency. Rho can be hedged using the appropriate interest rate futures contract. Firms with several different options books can reduce hedging cost by aggregating rho across the portfolios, rather than hedging each book uniquely.

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