📈 economics

On the Representation of Option Prices with Stochastic Interest Rates

This paper proposes a general representation formula for pricing vanilla options under stochastic, correlated interest rates that serves as an alternative to the Change of Numéraire method, providing semi-analytical approximations for affine models and a variance-reduced Monte Carlo estimator validated by numerical experiments.

Original authors: First Fabio Antonelli, Second Alessandro Ramponi

Published 2026-10-27

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: First Fabio Antonelli, Second Alessandro Ramponi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of finance, predicting the future value of an investment is a constant balancing act. Investors rely on mathematical models to determine the fair price of options, which are contracts giving the right to buy or sell an asset at a specific price by a certain date. These models must account for two moving targets: the price of the asset itself and the cost of borrowing money, known as the interest rate. For decades, the standard approach assumed that interest rates were either fixed or followed a simple, predictable path. However, in the real world, interest rates fluctuate unpredictably, and they often move in tandem with the prices of stocks or other assets. When these two forces are linked, calculating the price of an option becomes a notoriously difficult puzzle, often requiring computers to run millions of simulations just to get a rough estimate. This computational heaviness slows down trading strategies and makes it hard to adjust prices quickly when market conditions change.

Two researchers, Fabio Antonelli and Alessandro Ramponi, have proposed a new way to solve this puzzle. Instead of relying on the traditional, heavy-handed methods of simulation, they developed a general formula that breaks down the price of an option into simpler, more manageable pieces. Their approach treats the relationship between the asset price and the interest rate not as a tangled knot, but as a set of distinct contributions that can be calculated separately. By doing this, they created a method that is both highly accurate and significantly faster than existing techniques. Their work offers a fresh perspective on how to handle the complex dance between asset prices and interest rates, providing a tool that is precise enough for professional use while being efficient enough for real-time decision-making.

The core of their discovery lies in a new mathematical representation of the option price. Imagine trying to weigh a heavy, irregular object. The old way was to put it on a scale and wait for the needle to settle, a process that could be slow and prone to wobbling. The new method, developed by Antonelli and Ramponi, is like taking that object apart, weighing each piece individually with high precision, and then adding the weights back together. They found that the total price of an option can be expressed as a sum of specific parts: a base price, a term that accounts for the average interest rate over time, and specific adjustments for how the asset and the interest rate move together. This decomposition is powerful because it isolates the impact of the correlation between the two variables, making it clear exactly how much that relationship adds to or subtracts from the final price.

To test their theory, the researchers applied their formula to two well-known models of how interest rates behave. The first model, known as the Vasicek model, assumes that interest rates move in a smooth, bell-curve pattern. The second, the Cox-Ingersoll-Ross model, is more complex, assuming that rates cannot drop below zero and tend to fluctuate more wildly when they are low. In both cases, the researchers compared their new method against the standard, brute-force computer simulations that are currently the industry benchmark. The results were striking. When using their formula to estimate the price, the new method produced results that were nearly identical to the expensive, time-consuming simulations. In fact, the difference between the two was so small it was often invisible to the naked eye, even for options that were set to mature a full year in the future.

Beyond just finding the right number, the researchers discovered that their formula could also make the simulation process itself much faster. In the world of computer modeling, "variance" is a measure of how much the results jump around from one calculation to the next. High variance means you need to run the simulation thousands of times to get a stable average. The new formula acts as a built-in stabilizer. By using the decomposition they derived, the researchers showed that the random fluctuations in the calculation were drastically reduced. In their tests, the new method reduced the noise in the results by a factor of thousands compared to the standard approach. This means that to achieve the same level of accuracy, a computer using their method needs to run far fewer simulations, saving a tremendous amount of processing time and energy.

The team also explored how well their formula worked when the interest rates were allowed to change more drastically. They found that even when the volatility of interest rates was high, their method remained robust. The accuracy of their semi-analytical approximation—essentially a smart, simplified version of their full formula—stayed very high. For the simpler model, the error in their price estimate was less than one-tenth of one percent, even for long-term options. For the more complex model, the error remained similarly low. This suggests that their approach is not just a theoretical curiosity but a practical tool that can handle the messy, unpredictable reality of financial markets.

One of the most significant aspects of this work is that it does not require the entire financial system to be perfectly predictable to work. The researchers demonstrated that their formula holds true even when the relationship between the asset and the interest rate is complex and non-linear. This is a crucial distinction because many real-world financial models do not fit into neat, simple boxes. By showing that their method works for a broad class of interest rate models, including those with time-varying coefficients, they have provided a versatile framework that can be adapted to different market conditions without losing its precision.

The implications of this research extend beyond just calculating a single price. Because the formula breaks the problem down into clear, separate components, it allows risk managers to see exactly where the value of an option is coming from. They can instantly see how much of the price is driven by the correlation between the asset and the interest rate, and how much is driven by the volatility of the rates themselves. This transparency is invaluable for managing risk, as it allows financial institutions to understand the specific drivers of their exposure and adjust their strategies accordingly.

In their numerical experiments, the researchers used a standard set of parameters that are common in the industry, such as an interest rate of 3.5 percent and an asset volatility of 20 percent. They tested options with maturities ranging from three months to one year. Across all these scenarios, the representation-based method consistently outperformed the standard approach in terms of both speed and stability. The efficiency gains were particularly notable for shorter-term options, where the reduction in computational cost was most dramatic, but the benefits persisted even for the longest maturities.

Ultimately, Antonelli and Ramponi have offered a new lens through which to view option pricing. They have shown that by rethinking the structure of the problem, one can bypass the need for massive computational power while still achieving high precision. Their work bridges the gap between the theoretical elegance of mathematical finance and the practical demands of the trading floor. It provides a way to navigate the uncertainty of stochastic interest rates with a clarity that was previously difficult to achieve, turning a complex, noisy calculation into a streamlined, reliable process. For anyone involved in the valuation of financial derivatives, this represents a significant step forward in making the invisible mechanics of the market visible and manageable.

Technical Summary: On the Representation of Option Prices with Stochastic Interest Rates

1. Problem Statement

The pricing of financial derivatives, particularly plain vanilla options, in environments with stochastic interest rates remains a significant challenge in quantitative finance. While the classical "Change of Numéraire" technique provides a robust theoretical foundation, its practical implementation often relies on computationally intensive methods, such as standard Monte Carlo simulations. This is especially problematic for long-dated options or when rapid sensitivity analysis and real-time model calibration are required. Furthermore, existing literature often struggles to explicitly isolate the impact of the correlation between the underlying risky asset and the stochastic interest rate, or to provide efficient semi-analytical approximations for non-affine joint systems (e.g., when the asset and rate are correlated but the joint system is not affine).

2. Methodology

The authors propose a general representation formula for European option prices within a framework where the risky asset and the interest rate are correlated. The methodology proceeds through the following steps:

General Framework: The model assumes a risk-neutral measure $P$ with a stochastic short rate $r_t$ and an underlying asset log-price $X_t = \ln S_t$ . The Brownian motions driving the asset and the rate are correlated with coefficient $\rho$ .
Derivation of the Representation Formula: Building on previous works (specifically Antonelli et al. [2] and related literature), the authors derive a representation for the option price $V_t$ $V_{t}$ by applying Itô's formula to the product of the discount factor martingale $N(t, s)$ $N (t, s)$ and the Black-Scholes pricing function $P(s, X_s, R_s)$ $P (s, X_{s}, R_{s})$ , where $R_s$ $R_{s}$ is the optional projection of the average interest rate.
- The resulting formula (Equation 3.2) decomposes the option price into a current Black-Scholes term plus a series of integral terms involving the correlation parameter, interest rate volatility, and derivatives of the pricing function.
- Crucially, this decomposition separates the payoff into a "representation-based" component and a zero-mean martingale term.
Specialization to Affine Models: The general formula is specialized for affine interest rate models with time-dependent coefficients, specifically the Hull-White and CIR++ models. By utilizing the explicit dynamics of the martingales associated with these affine processes, the authors derive semi-explicit representation formulas (Equations 4.3 and 4.7).
Semi-Analytical Approximations: To make the representation computationally tractable, the authors propose approximations where certain expectation terms involving the Black-Scholes derivatives are "frozen" at their initial values. They provide an $L^2$ error estimate for these approximations, showing the error is of order $\sqrt{s-t}$ . A refined approximation is also introduced for the dominant first integral term to improve accuracy.
Variance Reduction: The pathwise decomposition allows the construction of a Monte Carlo estimator based on the representation formula. Since the difference between the standard discounted payoff and the representation-based estimator is a zero-mean martingale, the latter acts as a control variate, theoretically reducing the variance of the estimator.

3. Key Contributions

The paper makes two primary contributions:

General Representation Formula: It provides a new, general representation for option prices in correlated asset/rate frameworks. Unlike the Change of Numéraire approach, this method explicitly highlights the impact of the correlation parameter and interest rate volatility through a transparent decomposition of the price. It does not require the joint asset-rate system to be affine, only the interest rate process itself.
Dual Numerical Utility: The representation serves two distinct numerical purposes:
- Semi-Analytical Approximation: It yields accurate, semi-analytical pricing formulas for Hull-White and CIR++ models by replacing complex expectations with tractable quantities.
- Variance Reduction: It provides a basis for a Monte Carlo estimator with substantial variance-reduction properties compared to the standard discounted-payoff estimator.

4. Numerical Results

The authors validate their methodology using the Vasicek (Gaussian) and Cox-Ingersoll-Ross (CIR, square-root) models.

Variance Reduction Efficiency:
- In the Vasicek model, the representation-based estimator achieved variance reduction factors ( $VR$ ) ranging from $1.7 \times 10^3$ to $4.6 \times 10^4$ . The computational cost was approximately double that of the standard estimator, resulting in an overall efficiency gain ( $RE$ ) of up to $2.18 \times 10^4$ .
- In the CIR model, similar results were observed, with $VR$ ranging from $2.7 \times 10^3$ to $4.6 \times 10^4$ and efficiency gains up to $1.7 \times 10^4$ . The method proved effective even with non-Gaussian, square-root rate dynamics.
Approximation Accuracy:
- Vasicek: The semi-analytical approximation was benchmarked against the exact closed-form solution. For low volatility, the relative error remained below 0.02% even at one-year maturity. For high volatility, the maximum absolute error was approximately $8.6 \times 10^{-3}$ (relative error ~0.14%).
- CIR: Benchmarked against the representation-based Monte Carlo estimator, the approximation showed high accuracy. In the high-volatility scenario, the maximum error was approximately $3.1 \times 10^{-3}$ (relative error ~0.05%).
Error Analysis: In both models, the term-by-term error analysis confirmed that the primary source of approximation error is the first integral term (involving the correlation and rate dynamics), while terms weighted by volatility and correlation parameters contributed negligibly to the error.

5. Significance and Claims

The authors claim that their approach offers a viable alternative to the classical Change of Numéraire method, particularly in scenarios requiring computational efficiency or sensitivity analysis. The significance of the work lies in:

Transparency: The decomposition isolates the specific contributions of correlation and rate volatility to the option price, which is useful for risk management.
Efficiency: The representation-based Monte Carlo estimator offers substantial variance reduction without requiring the joint system to be affine, making it applicable to a broader class of models (like CIR with correlation) where closed-form solutions are unavailable.
Accuracy: The proposed semi-analytical approximations provide a high degree of accuracy for affine interest rate models, validated against exact benchmarks (Vasicek) and high-precision Monte Carlo estimates (CIR).

The paper concludes that the probabilistic structure of the pricing problem can be effectively exploited not only to derive approximations but also to significantly improve the statistical efficiency of simulation-based pricing.