Volatility Surface Construction Models

Visit the Mathema Option Pricing System, supporting FX options and structured product pricing and valuation!

The volatility surface is a crucial tool in option pricing and risk management, depicting the distribution of implied volatility across different strike prices and maturities. In the foreign exchange (FX) options market, implied volatility often exhibits a "volatility smile" or "volatility skew," meaning that implied volatility is not uniform across all strike prices. To construct a complete volatility surface, interpolation or fitting methods are used to model the volatility smile.

Below are five commonly used models for constructing the volatility surface: Linear Interpolation, Cubic Spline Interpolation, SVI (Stochastic Volatility Inspired), SABR (Stochastic Alpha Beta Rho), and Vanna-Volga. Each model has its strengths and weaknesses, making them suitable for different market scenarios and requirements.

1. LINEAR (Linear Interpolation)

Formula:

Linear interpolation is a simple method that estimates implied volatility by linearly connecting two known data points:

$$IV(K) = IV(K_{\text{low}}) + \frac{IV(K_{\text{high}}) - IV(K_{\text{low}})}{K_{\text{high}} - K_{\text{low}}} \cdot (K - K_{\text{low}})$$

• Variable Definitions:
  • $IV(K)$: Implied volatility corresponding to strike price $K$.
  • $K_{\text{low}}, K_{\text{high}}$: The two nearest known strike prices to $K$.
  • $IV(K_{\text{low}}), IV(K_{\text{high}})$: Corresponding implied volatilities.

Characteristics:

1. Simple and Fast: High computational efficiency, suitable for real-time applications.
2. Lack of Smoothness: The curve may have "kinks" at data points, unable to capture complex volatility smile shapes.

Applications:

• Scenarios with sparse data or low precision requirements.
• Quick generation of preliminary estimates for the volatility surface.

---

2. CUBIC SPLINE (Cubic Spline Interpolation)

Formula:

Cubic spline interpolation fits a cubic polynomial between every two known data points, ensuring smooth and continuous interpolation across the entire surface. The formula is as follows:

For each segment ([K_i, K_{i+1}]):

$$IV(K) = a_i + b_i \cdot (K - K_i) + c_i \cdot (K - K_i)^2 + d_i \cdot (K - K_i)^3$$

• Variable Definitions:
  • $a_i, b_i, c_i, d_i$: Coefficients of the cubic polynomial for each segment, determined by boundary conditions and continuity constraints.
  • $K_i, K_{i+1}$: Adjacent known strike prices.

Boundary Conditions:

1. The interpolation function is continuous at each node:

   $$IV(K_i) = IV_{\text{data}}(K_i)$$

2. The first and second derivatives of the interpolation function are continuous at the nodes.

Characteristics:

1. Smoothness: The curve is smooth and continuous, without kinks.
2. Poor Extrapolation: The interpolation is only valid within the range of known data points.

Applications:

• Scenarios with a large number of evenly distributed data points.
• Situations requiring high smoothness for the volatility smile.

---

3. SVI (Stochastic Volatility Inspired)

Formula:

The SVI model, proposed by Jim Gatheral, models the implied volatility smile using total variance:

$$w(k) = a + b \cdot \left( \rho \cdot (k - m) + \sqrt{(k - m)^2 + \sigma^2} \right)$$

• Variable Definitions:
  • $w(k)$: Total variance, defined as $w(k) = \sigma^2 \cdot T$, where $\sigma$ is implied volatility and $T$ is time to maturity.
  • $k$: Log-moneyness, defined as $k = \ln(K / F)$, where $K$ is the strike price and $F$ is the forward rate.
  • $a$: Minimum total variance, controlling the baseline level of total variance.
  • $b$: Controls the slope of the total variance.
  • $\rho$: Controls the symmetry of the smile (typically negative, $\rho \in [-1, 1]$).
  • $m$: Center of the log-moneyness.
  • $\sigma$: Controls the width of the smile.

Implied Volatility:

Implied volatility is derived from total variance:

$$IV(K) = \sqrt{\frac{w(k)}{T}}$$

Characteristics:

1. High Flexibility: Capable of capturing complex smile shapes, including skewness and kurtosis.
2. Clear Parameter Interpretation: Each parameter has a clear financial meaning.

Applications:

• High-precision fitting of volatility smiles in FX or equity markets.

---

4. SABR (Stochastic Alpha Beta Rho)

Formula:

The SABR model's approximate implied volatility formula is:

$$IV(K) = \frac{\alpha}{K^\beta} \cdot \left[ 1 + \frac{(1 - \beta)^2}{24} \cdot \left(\frac{\alpha}{K^{1 - \beta}}\right)^2 + \frac{\rho \cdot \nu \cdot \beta \cdot \alpha}{4 \cdot K^{1 - \beta}} + \frac{(2 - 3 \cdot \rho^2) \cdot \nu^2}{24} \right]$$

• Variable Definitions:
  • $IV(K)$: Implied volatility corresponding to strike price $K$.
  • $\alpha$: Initial volatility, controlling the volatility level.
  • $\beta$: Controls the sensitivity of volatility to the underlying price (range: $[0, 1]$).
  • $\rho$: Correlation between the underlying asset price and volatility.
  • $\nu$: Volatility of volatility (vol-of-vol).
  • $K$: Strike price.
  • $F$: Forward price.
  • $T$: Time to maturity.

Characteristics:

1. Flexibility: Capable of capturing skewness and curvature, suitable for complex volatility smiles.
2. Applications: FX options and interest rate options markets.

---

5. Vanna-Volga

Formula:

The Vanna-Volga method fits the volatility smile using a linear combination of ATM volatility, risk reversal, and butterfly spread:

$$IV(K) = IV_{ATM} + w_1 \cdot RR + w_2 \cdot BF$$

• Variable Definitions:
  • $IV(K)$: Implied volatility corresponding to strike price $K$.
  • $IV_{ATM}$: ATM volatility.
  • $RR$: Risk reversal, defined as:

    $$RR = IV(K_{25C}) - IV(K_{25P})$$

  • $BF$: Butterfly spread, defined as:

    $$BF = \frac{IV(K_{25C}) + IV(K_{25P})}{2} - IV_{ATM}$$

Weight Calculation:

The weights $w_1$ and $w_2$ are determined by the sensitivities of Vanna and Volga:

$$w_1 = \frac{\text{Vanna}(K)}{\text{Vanna}(K) + \text{Volga}(K)}$$

$$w_2 = \frac{\text{Volga}(K)}{\text{Vanna}(K) + \text{Volga}(K)}$$

• Vanna and Volga are sensitivities of implied volatility to market volatility parameters:

$$\text{Vanna} = \frac{\partial^2 (\text{Option Price})}{\partial (\text{Underlying}) \partial (\text{Volatility})}$$

$$\text{Volga} = \frac{\partial^2 (\text{Option Price})}{\partial (\text{Volatility})^2}$$

Characteristics:

1. Fast and Efficient: Simple calculations, suitable for real-time pricing.
2. Limitations: Unable to capture complex smile shapes, lower fitting accuracy.

---

Summary and Comparison

Model	Formula	Advantages	Disadvantages	Applications
Linear	$IV(K) = IV(K_{\text{low}}) + \frac{IV(K_{\text{high}}) - IV(K_{\text{low}})}{K_{\text{high}} - K_{\text{low}}} \cdot (K - K_{\text{low}})$	Simple and fast, computationally efficient.	Curve lacks smoothness, cannot capture complex shapes.	Sparse data or low-precision scenarios.
Cubic Spline	$IV(K) = a_i + b_i \cdot (K - K_i) + c_i \cdot (K - K_i)^2 + d_i \cdot (K - K_i)^3$	Smooth and continuous, suitable for high precision.	Poor extrapolation, unstable with sparse data.	Volatility surface construction with dense data.
SVI	$w(k) = a + b \cdot (\rho (k - m) + \sqrt{(k - m)^2 + \sigma^2})$	Highly flexible, suitable for complex smiles.	Multiple parameters, complex fitting process.	High-precision fitting in FX or equity markets.
SABR	$IV(K) = \frac{\alpha}{K^\beta} \cdot \left[ 1 + \text{correction terms} \right]$	Captures skewness and curvature, suitable for complex smiles.	Computationally intensive, slower.	FX options and interest rate options modeling.
Vanna-Volga	$IV(K) = IV_{ATM} + w_1 \cdot RR + w_2 \cdot BF$	Simple and efficient, parameters have clear market meanings.	Cannot capture complex smile shapes, lower accuracy.	Quick generation of volatility surfaces in FX markets.

---

By considering specific requirements (e.g., computational speed, fitting accuracy, and data characteristics), the most suitable model can be selected to construct the volatility surface.