The previous articles in this Greek textbook series explored Delta, Gamma, Theta and Vega, each representing a different sensitivity in option pricing. In this article, we turn our attention to Rho, the Greek that measures an option’s sensitivity to changes in the risk-free rate.

Rho quantifies the change in an option’s value given a change in the risk-free rate, assuming all other factors remain constant. It is especially relevant in interest rate-sensitive environments or when pricing long-dated options.

Rho’s impact is often negligible for short-dated options, where changes in interest rates have little time to significantly affect option value. However, for long-dated options, such as structured products or other derivatives with long maturities, Rho becomes a more influential factor in both pricing and risk management.

Comparing Rho across option types and structured products

Figure 1 shows Rho per 1% change in rates against the risk free rate for five instruments; a call option, a put option and three structured product payoffs: leveraged return, protected growth and a reverse convertible. For the basis of these calculations, all instruments have a maturity of six years and are linked to an underlying with a volatility of 20% and a dividend yield of 2%. For the at-risk products, we assume full downside risk from the strike level. The gearings and coupons for the product were calculated assuming a product price of 100% when the risk-free rate was 4%. The reverse convertible pays a single fixed coupon payment at maturity. These choices are simplified versions of typical products, without barriers or caps. They provide a good indication of the rate sensitivity of different types of structured products.

The chart shows that call and put options react in opposite ways to changes in interest rates. Call options typically have positive Rho: as interest rates rise, the present value of the strike price falls, making the option more valuable. In contrast, put options generally have a negative Rho. Higher interest rates also reduce the present value of the strike price, but this makes the right to sell at that fixed price less attractive, lowering the put's value. This is shown clearly in Figure 1.

Additionally, call Rho increases with higher interest rates, indicating growing sensitivity. For puts, the opposite occurs: rate sensitivity decreases as interest rates rise.

The protected growth and reverse convertible products both have a negative Rho. The protected growth product demonstrates very similar Rho behaviour to the put option. This is due to the fact the zero-bond component makes up a significant portion of the protected growth product. A zero-coupon bond’s price is based on the discounted value of its principal, while a long-dated put’s value reflects the discounted expected payoff at expiry. When interest rates rise, discount factors fall, reducing the present value of these future amounts. As a result, both instruments decline in value, leading to a negative Rho. The near identical values are also explained by put call parity. The protected growth product in this example is bond plus call, which is known to be equal to underlying plus put. The (present values) of the underlying has no interest rate sensitivity, hence the relation observed.

The similarity extends beyond direction to the shape of the response. Both long-dated puts and bonds tend to show a concave Rho curve, where the rate of value loss slows as interest rates rise, which is a feature known as convexity in fixed income. Despite differences in structure, the underlying mechanics of discounting link these instruments, resulting in similar sensitivity to interest rate changes.

The reverse convertible product has a flatter curve than the principal-protected product. For this payoff, Rho remains relatively constant across different interest rate levels. This product can be viewed as a combination of a large fixed payment due at maturity and a put option. Both elements have negative Rho, making the product highly sensitive to interest rate changes at any rate level.

The leveraged return product has a positive Rho. It consists of put, call and bond components. Due to its geared upside, its sensitivity is more like that of the call option than the other two products.

Rho in multi-currency products

Rho exposure applies per currency, unlike Delta for example which is stated per underlying.. If all underlyings and the product payoff itself are denominated in the same currency, there will be a single Rho value associated with that currency. However, if the underlying asset is denominated in a different currency from the product (for example a USD product linked to the Eurostoxx 50 Index) then the product will have two Rho values: one for each relevant currency.

Understanding how Rho behaves in these multi-currency contexts is essential for accurately measuring and hedging interest rate exposure, especially in multi-asset products with underlyings selected from multiple regions.

While Rho might not receive as much attention as other Greeks like Delta or Vega, it plays a critical role in certain market conditions. It is vital for managing the pricing and risk of long-dated or cross-currency options. Like the other Greeks, Rho represents just one aspect of an option’s risk profile, but provides important insight into interest rate sensitivity.