A Novel Option Pricing Model with Stochastic Interest Rates and Pure Jump Levy Processes

Martin Nicole^*
*Correspondence: Martin Nicole, Department of Electrical and Electronic Engineering, University of Chicago, Chicago, USA, Email:

Keywords

Quadrature • Derivatives • Stochastic

Introduction

The Black-Scholes model, introduced revolutionized the field of financial derivatives by providing a closed-form solution for pricing European options. However, its assumptions of constant volatility and interest rates, along with the absence of jumps in asset prices, often render it inadequate for real-world applications. Market conditions exhibit stochastic behaviors and sudden jumps, necessitating more robust models. In response, several extensions to the Black-Scholes model have been proposed. Stochastic volatility models, like the Heston model, account for changing volatility over time. Similarly, models incorporating stochastic interest rates, such as the Hull-White and Cox-Ingersoll-Ross (CIR) models, address the reality that interest rates fluctuate due to macroeconomic factors. Moreover, jump diffusion models, including the Merton and Kou models, introduce discontinuities in asset prices to capture market jumps [1].

Interest rates are influenced by a myriad of economic factors, making their behavior inherently stochastic. Traditional option pricing models often assume constant interest rates, which is a significant oversimplification. Stochastic interest rate models, such as the Hull-White and CIR models, provide a framework for modeling the dynamic nature of interest rates. These models allow for the incorporation of mean-reverting behavior and volatility, offering a more realistic representation of interest rate movements. In the context of option pricing, stochastic interest rates affect both the discounting of future cash flows and the dynamics of the underlying asset. By incorporating stochastic interest rates into the option pricing model, we can better capture the nuances of the financial environment and enhance the accuracy of option valuations [2].

Literature Review

Asset prices often exhibit sudden and significant changes, or jumps, which are not adequately captured by continuous-time models like Black- Scholes. Pure jump Lévy processes provide a mathematical framework for modeling these jumps. Lévy processes are characterized by their ability to capture both continuous and discontinuous movements in asset prices, making them well-suited for modeling the sudden shifts observed in financial markets. Pure jump Lévy processes focus exclusively on the jump component, ignoring the continuous part. This approach is particularly useful for capturing events such as market crashes, earnings announcements, or other significant financial news that can cause abrupt changes in asset prices. By integrating pure jump Lévy processes into option pricing models, we can account for these discontinuities and improve the model’s robustness [3].

Discussion

Our proposed model integrates stochastic interest rates and pure jump Lévy processes to address the limitations of traditional option pricing frameworks. This novel approach provides a more comprehensive and realistic method for valuing options in a dynamic financial environment. The model begins by integrating a stochastic interest rate component. We use the Hull-White model to capture the mean-reverting nature of interest rates. This component reflects the realistic behavior of interest rates over time, influenced by macroeconomic factors and central bank policies. The underlying asset price is modeled using a pure jump Lévy process. This process captures the discontinuities in asset prices, representing sudden jumps due to market events. The pure jump component is calibrated to reflect the historical jump behavior observed in the market, ensuring the model accurately represents real-world conditions. The final step involves integrating the stochastic interest rate and pure jump Lévy process components [4].

This combined model provides a comprehensive framework for option pricing, capturing both the dynamic nature of interest rates and the discontinuous movements in asset prices. The resulting option prices are more reflective of the true market conditions, offering enhanced accuracy for traders and risk managers. The integration of stochastic interest rates and pure jump Lévy processes into option pricing models has significant implications for the financial industry. By providing a more accurate representation of market dynamics, this model enhances the precision of option valuations, leading to better hedging strategies and risk management practices. Financial institutions can use the model to better assess and manage the risks associated with their option portfolios. The enhanced accuracy in option pricing allows for more effective hedging strategies, reducing the risk of significant losses due to market movements [5].

Traders can leverage the model to identify mispriced options and exploit arbitrage opportunities. The ability to accurately capture market jumps and stochastic interest rate behavior provides a competitive edge in developing profitable trading strategies. Regulatory bodies require financial institutions to accurately value their derivatives positions for reporting and compliance purposes. The novel option pricing model meets these requirements by providing a more realistic and robust framework for option valuation. By improving the accuracy of option pricing and risk management practices, the model contributes to overall market stability. It helps prevent the formation of asset bubbles and reduces the likelihood of systemic risks arising from mispriced derivatives [6].

While the novel option pricing model offers significant advancements, it also presents challenges. The complexity of integrating stochastic interest rates and pure jump Lévy processes requires sophisticated mathematical and computational techniques. Calibration of the model to real-world data is crucial and can be challenging due to the need for high-quality historical data and advanced statistical methods. Future research can focus on refining the model by exploring different stochastic interest rate models and Lévy processes. Additionally, the application of machine learning techniques to enhance model calibration and parameter estimation holds promise. Researchers can also investigate the model’s performance in various market conditions and its applicability to different types of options and derivatives.

Conclusion

The integration of stochastic interest rates and pure jump Lévy processes into option pricing models represents a significant advancement in financial theory and practice. This novel approach addresses the limitations of traditional models and provides a more comprehensive framework for valuing options in a dynamic and complex financial environment. By capturing the stochastic nature of interest rates and the discontinuous movements of asset prices, the model enhances the accuracy of option valuations, leading to improved risk management, trading strategies, and market stability. As the financial landscape continues to evolve, the development and refinement of such advanced models will be crucial for meeting the challenges and opportunities of the future.

Acknowledgement

None.

Conflict of Interest

None.

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