Decoding Rho: understanding interest rate risk in structured products

Interest rates play a varying role in all derivatives and that’s sometimes difficult to analyse. It starts with the critical role of the non-existent risk-free rate.

The risk-free rate is a fundamental component in various financial models, including the Black-Scholes model for derivatives pricing

We will explore that in our series of articles dedicated to structured products and risk management with the Greeks (and beyond).

The concept of risk-free rate in finance

The risk-free rate is a fundamental component in various financial models, including the Black-Scholes model for derivatives pricing. However, in reality, a truly risk-free rate does not exist, as all interest rates carry some degree of market and credit risk (not forgetting the associated operational risk).

Common proxies for the risk-free rate

For USD‑denominated assets, the yields on U.S. Treasury securities (Treasury bills, notes, or bonds, depending on the horizon) are most often used as a proxy for the risk‑free rate. These instruments are perceived to carry almost no credit risk because the U.S. government can, in principle, meet its obligations by printing money and raising taxes. In practice, analysts also use rates that embed credit and liquidity premiums, such as swap rates, because these higher rates more accurately reflect the financing costs.

Intuitive analysis for Rho

It starts with the Forward, simplified the Forward = Costs (with financing costs in particular) -Benefits (for equity, the dividends). This is advantageous for a plain vanilla call option, as it has a positive Rho, meaning its value increases with higher interest rates. Conversely, it is disadvantageous for a long put option, which has a negative Rho, meaning its value decreases with higher interest rates. Notably, Rho is typically not the dominant factor in option pricing.

Interest rate risk in a Barrier Reverse Convertible

Our analysis will focus on the short down-and-in put option embedded in a Barrier Reverse Convertible (BRC) with the S&P 500 as the underlying asset. For the risk-free interest rate over the two-year horizon, we will use the yield on the two-year U.S. Treasury Note as a proxy—reflecting the market-standard practice of anchoring to Treasury yields for USD-denominated structured products with maturities aligned to on-the-run Treasury tenors.

The option’s parameters on the 6 February 2026 are – the option is embedded in the BRC with the index value (end of day): 6,932.30, Strike: 6,932.30, Barrier at 70%: 4,852.61, Expiration: two-years, Risk-free: 3.50% (U.S. Government Note yield), Volatility: 22.50 % (one-month at 17.76%, estimated). Dividend yield: 1.16% (estimated/accounted for).

The table below shows the option interest rate risk called Rho - the change in option value for a one percentage point (100 basis points) increase in the risk-free rate - at different interest rate and volatility levels, all else remaining the same: our own valuations. The forward is at 7,264.4432.

Instrument	Volatility	Risk-free	Value	Rho
Short down-and-in put	22.5	3.50%	500.428	54.7881
Short down-and-in put	22.5	3.00%	528.3912	57.1533
Short down-and-in put	22.5	4.00%	473.6328	52.4824
Short down-and-in put	25	3.50%	621.681	59.4896
Short down-and-in put	25	3.00%	651.963	61.7086
Short down-and-in put	25	4.00%	592.4969	57.3176

_{Source: Evolids Finance}

Let’s see if it works out well. At a volatility of 22.5%, we consider an increase in the risk-free interest rate from 3% to 4%:

The forward goes up from 7,192.1608 to 7,337.4520 (or Forward = 6,932.30 * EXP[(00% - 1.16%) * 2] = 7,337.4520, continuous compounding being applied. That means that a long call is more likely and a long-put position is less likely to be exercised and therefore, is worth more respectively worth less than before the increase in interest rate. Now the position is not a long but a short down-and-in put one: The Profit and Loss (P&L) of a short position is the exact inverse of the value change of the corresponding long option. Because a long put has a negative Rho, the short down-and-in put carries a positive position Rho. Therefore, an increase in interest rates results in a gain for the short position.

Numerically, the option's premium is estimated to decrease from 528.3912 to 528.3912 - 57.1533 (Rho of the option when the risk-free rate is at 3%) = 471.2379, the correct value being 473.6328. This difference shows that Rho provides a very good linear approximation, even given the large magnitude of the rate shock. Ultimately, the short down-and-in put captures this drop in premium, realizing a gain of approximately 57.1533.

An important remark is that Rho is a first‑order sensitivity. It captures the immediate impact of a rate change but ignores the curvature introduced by the barrier’s activation probability and the non‑linear discount factor.

Now, what is the behaviour close and very close to the barrier, with a near expiration? Barrier at 70%; 4,852.61 and expiration at 0.15 year:

Instrument	Volatility	Risk-free	Spot	Value	Delta	Rho
Short down-and-in put	22.5	3.50%	4,950	1,685.28	3.6719	14.1057
Short down-and-in put	22.5	3.50%	4,950	1,871.62	3.7738	12.453
Short down-and-in put	22.5	3.50%	4,950	2,046.52	3.8227	10.4133
Short down-and-in put	22.5	3.50%	4,950	2,054.44	0.9982	10.3438
Short down-and-in put	22.5	3.50%	4,950	2,104.35	0.9982	10.3439

 _{Source: Evolids Finance}

The table suggests that, near expiration and for constant volatility and risk-free rate, the (already small) Rho of a short down-and-in put continues to decrease in magnitude as the spot approaches the barrier from above. Once the barrier is hit and the option has knocked in, further changes in Rho appear relatively moderate in this example. For Delta, as we have seen, the very high value close to the barrier drops sharply after the barrier has been hit.

In a barrier reverse convertible (BRC), the interest rate sensitivity of the embedded short down-and-in put option typically opposes that of the long zero-coupon bond component. Investors holding such a position may therefore seek to hedge this offsetting interest rate risk to stabilize overall portfolio duration.

Interest rate risk in a plain vanilla long-term call option, a call-warrant

In the context of a call warrant with the following parameters:

Stock price:	EUR10
Strike:	EUR13
Expiration in:	3-years
Risk-free:	3%
Volatility:	25%

The forward is at: 10.9417 (at 3% risk-free):

Instrument	Volatility	Risk-free	Value	Rho
(Plain vanilla) Long-term call	25	3.00%	1.0787	0.096031
Long-term call	25	2.50%	1.0313	0.093387
Long-term call	25	4.00%	1.1773	0.101272
Long-term call	20	3.00%	0.744	0.089529
Long-term call	20	2.50%	0.7001	0.085964
Long-term call	20	4.00%	0.8371	0.096696

  _{Source: Evolids Finance}

The table confirms the positive link between risk-free rates and both value and Rho for long-dated calls at fixed volatility: at 25% volatility, rates from 2.5% to 4.0% lift the option value from 1.0313 to 1.1773 and Rho from 0.0934 to 0.1013.

Higher volatility pushes up the option’s time value but only softly impacts Rho, underscoring that rate sensitivity stems mainly from maturity and moneyness. A 1% rate shift alters a long-term call by about 9 - 10 cents, so large portfolios build material Rho exposure: that’s critical during sharp moves like 2022's hikes, which pressured equities due to rising rates.

In the next article, we will do a completely different exercise that consists in explaining some of the key topics addressed so far — especially the Greeks — in simple words, for the non-quants.

It will be a refreshing shift in perspective and a useful reminder that even the most technical ideas can be communicated clearly when we step back from the formulas and focus on the intuition behind them.

_{Image: Iamchamp/Adobe Stock}

This article is based on data and analysis provided by the SRP Greeks product. Find out more about SRP Greeks here

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_{Disclaimer: This content is not intended as a solicitation or an offer; it is provided solely for informational purposes to professional investors. The information presented herein has been prepared with great care; however, errors may still occur.}