Reconstructing Smooth Local Volatility Surfaces for Cryptocurrency Options

Cryptocurrency markets play an important role in modern financial systems as they provide unique challenges and opportunities because of their high volatility, decentralized nature, and rapidly evolving market dynamics. We propose a numerical method for reconstructing smooth local volatility surfaces for cryptocurrency call options. The proposed method uses the generalized Black–Scholes (BS) equation, market option prices from cryptocurrency trading, and an optimization routine to reconstruct smooth local volatility surfaces. The generalized BS equation is computationally solved using a finite difference method. In the proposed algorithm, smooth local volatility surfaces are defined as bivariate polynomials. To validate the high performance of the proposed methodology, we conduct several computational experiments using real crypto call option prices from Bitcoin, Ethereum, Solana, and Ripple indices. The numerical results computed using the proposed bivariate polynomial local volatility surfaces successfully reproduce the real market option prices.