Dimension-Reduced Modeling for Local Volatility Surface via Unsupervised Learning

Volatility is a key factor for option pricing. It displays skewness across different strike prices and maturity days when implied by the Black-Scholes formula. This phenomenon is called the volatility smile. The local volatility model is popular because it fits this smile. It assumes the volatilities a deterministic function of underlying asset and time. These volatilities form the local volatility surface (LVS). LVS evolves over time and this dynamics can be high-dimensional and fluctuating. In this research, we show that the LVS may be described by a small number of orthogonal factors. This is accomplished by studying the LVS dynamics with time series data on option prices and extracting their essences via principal component analysis (PCA) and multilinear PCA (MPCA). We aim at recognizing these dominant components. In this case, the dimensions of LVS are reduced, and these dominant components are used to reconstruct the LVS. Numerical results show that the reconstructed LVS retains the important characteristics while filtering out noise well. In particular, over 80% of observations are within 10% in the maximum absolute relative difference (MARD). Moreover, MPCA provides an extra degree of freedom for reconstruction as well as interpretation because it preserves the tensor structure.