2016年-德勤全球_Interest_rate_derivatives_in_the_negative-rate_environment_15页_1mb

Summary of Interest Rate Derivatives in the Negative-Rate Environment: Pricing with a Shift

Core Content

This document discusses the challenges and solutions in pricing interest rate derivatives in a negative-rate environment, particularly focusing on the use of the shifted SABR model as a market standard tool. It outlines the rationale behind negative rates, the limitations of traditional models, and the methodology for calibrating and applying the shifted SABR model to accurately price derivatives such as caps, floors, and swaptions.

Main Views and Key Information

1. Rationale for Negative Rates

• The financial crisis (2007–2011) led to a loss of trust in financial transactions and prompted central banks, notably the ECB, to implement negative interest rates.
• Negative rates encourage investors to borrow money and invest in the economy, as holding cash incurs a cost.
• Central banks like the ECB, Switzerland, Denmark, and Sweden have introduced negative rates to stimulate economic activity.

2. Valuation Challenges in the Negative Rate Environment

• Traditional models like the Black model assume lognormality, which leads to undefined logarithms when forward rates are negative.
• The Hull-White model allows for negative rates but has issues with extreme negative values.
• The Bachelier model assumes normality and allows for negative rates but lacks the flexibility of the Black model.
• Caplet stripping is necessary to derive caplet volatilities from cap volatilities, especially in the negative rate context.
• Market quotes for volatilities at negative strikes are often unavailable, requiring extrapolation.

3. Market Standard Models for Negative Rates

• Hull-White model: Closed-form pricing, allows negative rates, but has issues with extreme negative values.
• Bachelier model: Closed-form pricing, allows negative rates, but less commonly used due to its lack of flexibility.
• Black model: Closed-form pricing, assumes non-negative rates, but breaks down when forward rates are negative.
• Shifted Black model: A modified version of the Black model that shifts forward rates to avoid negative values.
• SABR model: A stochastic volatility model that is widely used for volatility surface interpolation, but its implied volatility formula is an approximation.
• Shifted SABR model: Incorporates a shift parameter to handle negative rates, allowing the use of the Black model in such environments.

4. SABR Model and Its Application

• The SABR model is defined by four parameters: $\alpha$, $\beta$, $\rho$, and $\nu$.
• The model's implied volatility formula is an approximation and can lead to negative probability density functions near the zero strike.
• To address this, the shifted SABR model is used, which shifts the forward rate and strike by a parameter $b$, typically set at 1%.
• The calibration process involves:
  • Matching market-quoted cap prices with model outputs.
  • Using numerical methods like simulated annealing to find the optimal parameters.
  • Visualizing the error surface with heat maps to identify calibration accuracy.

5. Caplet Stripping and Calibration

• Caplet stripping is the process of deriving caplet volatilities from cap volatilities.
• The process involves:
  • Assuming equal caplet volatilities for the same cap (e.g., 1Y caplet volatility is assumed equal for all strikes).
  • Using the previous expiry's SABR model to interpolate caplet volatilities at the ATM strike of the current expiry.
• Calibration is crucial for accurate pricing, especially for negative strikes, and errors can accumulate across expiries.

6. Deloitte's Valuation Tool

• Deloitte uses the shifted SABR model calibrated with data from Bloomberg's BVOL CUBE.
• The tool allows for pricing of caps, floors, swaptions, and CMS options, even at negative strikes.
• It is used for Front Office pricing and as an independent validation tool for risk and validation teams.

7. Deloitte's Services

• Deloitte offers:
  • Managed services for independent valuations of interest rate derivatives.
  • Expert assistance in designing and implementing pricing engines.
  • Training on topics like the SABR model, volatility smiles, and stochastic modeling.
  • Stand-alone tools for valuation purposes.

Conclusion

In the context of negative interest rates, traditional models such as the Black model face significant limitations. The shifted SABR model is a robust and widely accepted solution that allows for accurate pricing of interest rate derivatives. Deloitte provides a comprehensive approach to valuation and calibration, ensuring reliable results for financial institutions in this challenging environment.