Calibration Process of the Dupire Model

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The calibration process of the Dupire Model links the market-implied volatility surface to the local volatility model by inferring the local volatility function $\sigma(t, S)$ from market option prices. Below is a detailed step-by-step guide to the calibration process, including market data preparation, partial derivative calculations, and numerical implementation.

1. Objective of Calibration

The goal of calibrating the Dupire Model is to infer the local volatility function $\sigma(K, T)$ from market-implied volatility data, ensuring that the option prices calculated by the local volatility model match the market prices.

Specifically, we aim to satisfy the following conditions:

• Minimize the error between theoretical option prices $C_{\text{model}}(K, T)$ and market option prices $C_{\text{market}}(K, T)$.
• The theoretical price formula is:

  $$C_{\text{model}}(K, T) = \mathbb{E}\left[ e^{-r^d T} \cdot \max(S(T) - K, 0) \right]$$

  where the dynamics of $S(T)$ are driven by the local volatility model:

  $$\frac{dS(t)}{S(t)} = \left( r^d(t) - r^f(t) \right) dt + \sigma(t, S(t)) dW(t)$$

Through calibration, we need to find a local volatility function $\sigma(K, T)$ that aligns with the market-implied volatility surface.

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2. Inputs for Calibration

The calibration of the Dupire Model requires the following market data as inputs:

1. Market-Implied Volatility Surface $\sigma_{\text{imp}}(K, T)$

   • Market-implied volatility data, typically derived from European option prices. Implied volatility is a function of strike price $K$ and maturity $T$.

2. Risk-Free Interest Rate Curves $r^d(t), r^f(t)$

   • Domestic $r^d(t)$ and foreign $r^f(t)$ risk-free interest rate curves, extracted from market interest rate derivatives (e.g., forward rate agreements, swaps).

3. European Option Price Surface $C_{\text{market}}(K, T)$

   • Market prices of European options for different strike prices $K$ and maturities $T$.

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3. Dupire Equation

The Dupire equation is the core of the calibration process, linking the partial derivatives of European option prices to the local volatility function $\sigma(K, T)$:

$$\sigma^{2}(K, T) = \frac{\frac{\partial C(K, T)}{\partial T} + \left( r^d(T) - r^f(T) \right) \frac{\partial C(K, T)}{\partial K} + r^f(T) \cdot C(K, T)}{\frac{1}{2} K^2 \cdot \frac{\partial^2 C(K, T)}{\partial K^2}}$$

• $C(K, T)$: European option price.
• $\frac{\partial C(K, T)}{\partial T}$: Partial derivative of the option price with respect to maturity (Theta).
• $\frac{\partial C(K, T)}{\partial K}$: Partial derivative of the option price with respect to strike price.
• $\frac{\partial^2 C(K, T)}{\partial K^2}$: Second partial derivative of the option price with respect to strike price (Gamma).

Formula Breakdown:

1. Numerator Explanation:

   • $\frac{\partial C(K, T)}{\partial T}$: Time decay of the option (Theta).
   • $(r^d(T) - r^f(T)) \frac{\partial C(K, T)}{\partial K}$: Impact of the term structure on the option price.
   • $r^f(T) \cdot C(K, T)$: Effect of the foreign discount factor.

2. Denominator Explanation:

   • $\frac{\partial^2 C(K, T)}{\partial K^2}$: Second-order sensitivity of the option price to the strike price (Gamma), which adjusts the volatility surface.

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4. Calibration Process

The calibration process consists of the following steps:

Step 1: Extract the Implied Volatility Surface from Market Data

1. Implied Volatility Calculation:

   • Use the Black-Scholes formula to extract implied volatility $\sigma_{\text{imp}}(K, T)$ from market option prices $C_{\text{market}}(K, T)$:

     $$C_{\text{market}}(K, T) = Black-Scholes(K, T, \sigma_{\text{imp}})$$

   • The implied volatility surface $\sigma_{\text{imp}}(K, T)$ is the initial input for calibration.

2. Smooth the Implied Volatility Surface:

   • Use interpolation methods (e.g., linear interpolation, cubic splines, or the SVI model) to smooth the implied volatility data points, ensuring continuity in both $K$ and $T$ directions.

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Step 2: Calculate Partial Derivatives in the Dupire Equation

1. Calculate Partial Derivatives of Option Prices:

   • Use the market-implied volatility $\sigma_{\text{imp}}(K, T)$ and the Black-Scholes formula to compute option prices $C(K, T)$.
   • Use numerical methods to calculate partial derivatives:
     • Time derivative $\frac{\partial C(K, T)}{\partial T}$: Approximate using finite differences:

       $$\frac{\partial C(K, T)}{\partial T} \approx \frac{C(K, T+\Delta T) - C(K, T)}{\Delta T}$$

     • First and second derivatives with respect to strike price:

       $$\frac{\partial C(K, T)}{\partial K} \approx \frac{C(K+\Delta K, T) - C(K, T)}{\Delta K}$$

       $$\frac{\partial^2 C(K, T)}{\partial K^2} \approx \frac{C(K+\Delta K, T) - 2C(K, T) + C(K-\Delta K, T)}{\Delta K^2}$$

2. Numerical Optimization:

   • Partial derivative calculations may introduce errors, so smoothing or regularization techniques are often applied.

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Step 3: Infer Local Volatility

Substitute the calculated partial derivatives into the Dupire equation:

$$\sigma^{2}(K, T) = \frac{\frac{\partial C(K, T)}{\partial T} + \left( r^d(T) - r^f(T) \right) \frac{\partial C(K, T)}{\partial K} + r^f(T) \cdot C(K, T)}{\frac{1}{2} K^2 \cdot \frac{\partial^2 C(K, T)}{\partial K^2}}$$

• Key Considerations:
  • The denominator $\frac{\partial^2 C(K, T)}{\partial K^2}$ must be positive (otherwise, local volatility becomes infinite or unstable).
  • Additional regularization may be required for extreme cases (e.g., deep in-the-money or out-of-the-money options).

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Step 4: Validate Calibration Results

1. Numerical Validation:

   • Use the calibrated local volatility $\sigma(K, T)$ to recompute option prices $C_{\text{model}}(K, T)$ using Monte Carlo simulation or finite difference methods, and compare them with market prices $C_{\text{market}}(K, T)$.
   • If the error is too large, adjust data processing methods or regularization steps.

2. Error Metrics:

   • Use mean squared error (MSE) or relative error to evaluate the fit between model and market prices:

     $$\text{Error} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{C_{\text{model}, i} - C_{\text{market}, i}}{C_{\text{market}, i}} \right)^2$$

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5. Challenges in Calibration

1. Data Noise:

   • Market-implied volatility surfaces may contain noise, especially for deep in-the-money or out-of-the-money options, affecting the stability of local volatility.

2. Numerical Errors in Partial Derivatives:

   • Numerical differentiation methods are sensitive to data point distribution and spacing, potentially leading to instability.

3. Boundary Issues:

   • At extreme strike prices (e.g., deep in-the-money or out-of-the-money), the second derivative of option prices may approach zero, causing local volatility to become infinite.

4. Model Extrapolation:

   • The Dupire model assumes continuous paths and cannot capture jump behavior in the market.

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Conclusion

The calibration process of the Dupire Model is an inverse problem that maps market-implied volatility data to a local volatility function. Key steps include extracting the implied volatility surface from the market, numerically calculating partial derivatives of option prices, and inferring local volatility using the Dupire equation. Although the Dupire Model offers the advantage of precisely matching market prices, its calibration process is highly sensitive to data quality and numerical stability. In practice, interpolation, regularization, and numerical optimization techniques are often combined to ensure stable and accurate calibration results.