1. The Challenge of Arbitrage-Free Curve Construction
The Yield Curve is the bedrock of fixed-income analysis, defining the relationship between yield-to-maturity and a bond's maturity. However, zero-coupon bonds rarely trade in the market. Practitioners must impute a continuous curve from a sparse set of liquid instruments (bonds, swaps) through a process known as bootstrapping.
A fundamental requirement, often missed by simple methods, is that the constructed curve must be arbitrage-free. For this to hold true, two conditions are mandatory:
- The Discount Factor Curve ($Z(0, t)$) must be monotonically decreasing in time $t$.
- The derived Instantaneous Forward Rates must be positive.
The key insight (as stressed by Hagan and West) is that interpolation is not separate from the bootstrap process; the interpolation method is intimately connected to the bootstrap itself, as it completes the incomplete market information.
2. Why Traditional Splines Fail (The Non-Local Problem)
Initial attempts to build a smooth curve often involve simple polynomial splines, like Linear Interpolation or Cubic Splines. However, these methods introduce significant pathologies:
- Forward Rate Discontinuities: Linear methods often result in forward rates that jump at each node, violating the requirement of continuity and stability.
- Non-Arbitrage Violation: Simple methods like "Linear on Rates" can easily produce negative forward rates from arbitrage-free economies.
- Lack of Locality: Many common methods suffer from non-locality. A small change (a perturbation) in one input can cause a material variation in large sections of the curve, which compromises the stability of hedging instruments.
- Oscillation and Instability: Quadratic and Cubic splines, though smoother than linear methods, can still exhibit undesirable oscillatory behaviour (the "zig-zag instability").
3. The Solution: Monotone Convex Interpolation
To overcome these deficiencies, we implemented the Monotone Convex Method introduced by Hagan and West. This approach stands out because it explicitly resolves the core issues of stability and no-arbitrage by applying rigorous conditions to the forward rate curve rather than the yield curve itself.
Key Advantages of Monotone Convex:
This method guarantees several critical quality criteria simultaneously:
- Positivity of Forwards: It explicitly ensures that all instantaneous forward rates are positive (whenever the discrete inputs allow it), thereby guaranteeing the arbitrage-free condition.
- Locality: The method is designed to be highly local. Changes in input at one location do not materially affect the value of the curve at distant locations.
- Continuity and Stability: The instantaneous forward curve is typically continuous and stable, meaning forward rates change proportionately to changes in inputs.
- Exact Pricing: All input instruments used in the bootstrap are exactly reproduced as outputs.
This implementation confirms that for serious financial modeling and hedging, the Monotone Convex method is generally considered the method of choice. The final constructed curve is validated against the Monotone Convex Spline due to its superior stability and locality properties. This level of precision is non-negotiable for derivatives valuation.
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