Abstract
Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.
Highlights
A recent approach to option pricing involves reformulating the pricing problem as a surrogate modeling problem
We introduce half-variance bounds into the penalization to improve the overall fit in prices and stabilize the local volatility surface
We introduced three variations of deep learning methodology to enforce no-arbitrage interpolation of European vanilla put option prices: (i) modification of the network architecture to embed shape conditions, (ii) use of shape penalization to favor these conditions, and (iii) additional use of local half-variance bounds in the penalization via the Dupire formula
Summary
A recent approach to option pricing involves reformulating the pricing problem as a surrogate modeling problem. The problem is amenable to standard supervised machine learning methods such as Neural networks and Gaussian processes This is suitable in situations with a need for fast computations and a tolerance to approximation error. In their seminal paper, Hutchinson et al (1994) use a radial basis function neural network for delta-hedging. Hutchinson et al (1994) use a radial basis function neural network for delta-hedging Their network is trained to Black–Scholes prices, using the time-to-maturity and the moneyness as input variables, without ’shape constraints’, i.e., constraints on the derivatives of the learned pricing function. Dugas et al (2009) introduce the first neural network architecture guaranteeing arbitrage-free vanilla option pricing on out-of-sample contracts
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