Rho (Options)

Rho, in options trading, is the measure of how sensitive the price of an option is to a change in the risk-free rate of interest. It is one of the option Greeks, the set of sensitivities that traders use to understand and hedge option positions. Rho is conventionally quoted as the change in the option's price for a rise of one percentage point in the interest rate, with all other inputs held constant.

Rho is positive for a call option and negative for a put option. For most short-dated options on shares it is the least important of the main Greeks, well behind delta, gamma, theta and vega. It becomes material for long-dated options and when interest rates are moving quickly.

A call gives the right to buy an asset later at a fixed strike price. Holding the call instead of the asset lets the buyer keep the cash until expiry and earn interest on it. The higher the rate, the more that deferral is worth, so call prices rise with rates. Put another way, a higher rate lowers the present value of the strike that the call holder will pay.

A put gives the right to sell at the strike later. A higher rate lowers the present value of the cash the put holder will receive, so put prices fall as rates rise. The same logic appears in the forward price of the underlying asset, which rises with the interest rate and so shifts value from puts to calls.

In the Black-Scholes-Merton model for a European option on an asset paying no dividends, rho for a call is:

and for a put:

• $K$ is the strike price.
• $T$ is the time to expiry in years.
• $r$ is the continuously compounded risk-free rate.
• $N(\cdot)$ is the cumulative standard normal distribution function.
• $d_2$ is the model term $d_1 - \sigma\sqrt{T}$, where $\sigma$ is the volatility of the underlying asset.

These expressions give the price change for a change of 1.00 in the rate, that is, 100 percentage points, so they are divided by 100 to obtain the usual quote per percentage point.

Take a one-year call with a share price of $100, a strike of $100, a rate of 5 percent and volatility of 20 percent. Then $d_1 = 0.35$ and $d_2 = 0.15$, with $N(0.15) = 0.5596$, and the discount factor $e^{-0.05}$ is 0.9512. Rho is $100 \times 1 \times 0.9512 \times 0.5596 = 53.23$, or about 0.53 per percentage point. The model price of the call is $10.45, so if the rate rose from 5 to 6 percent the call would be worth roughly $10.98. For the matching put, rho is $-100 \times 0.9512 \times 0.4404 = -41.89$, or about -0.42 per point.

• Time to expiry. Rho grows with the life of the option, because interest accrues over a longer period. Options with one or more years to option expiry carry far more rho than weekly or monthly contracts.
• Moneyness. An in-the-money option has larger rho in absolute terms, since it is more likely to be exercised and the strike payment is more certain. Deep out-of-the-money options have rho near zero.
• Strike and price level. Rho scales with the strike, so options on high-priced assets show larger figures in currency terms.

When policy rates sat close to zero for much of the period from 2009 to 2021, rho was widely ignored on equity option desks. The rapid rate rises by the Federal Reserve and other central banks in 2022 and 2023 brought it back into view, particularly for long-dated contracts and for positions that are neutral on other Greeks but large in notional size.

Rho also matters for positions that are close substitutes for borrowing or lending. A long call combined with a short put at the same strike behaves like a forward purchase of the asset, and its value depends directly on the financing rate. For currency options there are two rates, one for each currency, and so two rho figures. A related sensitivity to the dividend yield plays the same role for share options.

Rho assumes that a single interest rate moves while everything else stays fixed. In real markets a change in rates usually moves the price of the underlying asset and its implied volatility as well, and those effects are often larger than the direct rate effect. Rates also differ by maturity, so a single rho cannot describe exposure to changes in the shape of the yield curve. Interest rate derivatives, whose underlying is itself a rate, are managed with other measures and not with rho in this sense.