Stable local volatility function calibration using spline kernel

Abstract

We propose an optimization formulation using the l ₁ norm to ensure accuracy and stability in calibrating a local volatility function for option pricing. Using a regularization parameter, the proposed objective function balances calibration accuracy with model complexity. Motivated by the support vector machine learning, the unknown local volatility function is represented by a spline kernel function and the model complexity is controlled by minimizing the 1-norm of the kernel coefficient vector. In the context of support vector regression for function estimation based on a finite set of observations, this corresponds to minimizing the number of support vectors for predictability. We illustrate the ability of the proposed approach to reconstruct the local volatility function in a synthetic market. In addition, based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is simple and resembles the observed implied volatility surface in shape. Stability is illustrated by calibrating local volatility functions using market option data from different dates.

Appendix: Kernels generating splines with an infinite number of knots

Here we briefly describe kernels generating splines with an infinite number of knots. The presentation follows from discussion in §11.6.2 in [24]. Suppose that we want to approximate a one-dimensional function of one variable s defined on the interval [−b,+∞), 0<b<∞, by splines of order d≥0 with infinite number of knots: {t _i }, 1≤i<∞. First the one-dimensional variable s is mapped into a vector u in the feature space of an infinite-dimension:

$$ s\rightarrow u=\bigl(1,s,\ldots,s^{d},(s-t_{1})^{d}_{+},\ldots,(s-t_{i})^{d}_{+},\ldots\bigr) $$

where

$$ (s-t_{k})^{d}_{+}= \begin{cases} 0 & \text{if $s\leq t_{k}$,} \\ (s-t_{k})^{d} &\text{if $s> t_{k}$.} \end{cases} $$

Then the spline has the form:

$$ g(s)=\sum^{d}_{i=0}a_{i}s^{i}+\int^{+\infty}_{-b}a(t)(s-t)^{d}_{+}dt, $$

where a _i , i=0,…,d and a(t) are coefficients of expansion. The kernel generating the spline can be obtained by determining the inner product as follows

$$ \mathcal{K}(s_j,s_{i})=\int^{+\infty }_{-b}(s_j-t)^{d}_{+}(s_{i}-t)^{d}_{+}dt+\sum^{d}_{k=0}s_j^k s^{k}_{i}. $$

For the linear spline with d=1 in particular, we have the following function representation for the kernel generating spline:

$$ \mathcal{K}(s_j,s_{i})=1+s_js_{i}+\frac{1}{2}| s_j-s_{i}|(s_j\wedge s_{i}+b)^{2}+\frac{(s_j\wedge s_{i}+b)^{3}}{3} $$

where s _j ,s _i are training data points in the interval [−b,+∞), and s _j ∧s _i denotes min(s _j ,s _i ). It can shown that the above kernel function is twice differentiable.