Laguerre path-dependent volatility model

Abstract

There has been much effort devoted to designing and engineering new and improved stochastic models for option pricing. Much attention in both the academia and the industry has been drawn to stochastic models that model the asset dynamics without regard to the historical path of the asset price leading up to the time when option pricing is performed.

A path-dependent volatility model is a stochastic model where the volatility dynamics are driven by the whole path of the asset price. The use of a path-dependent volatility model allows for the incorporation of the historical path of the asset price in modeling the volatility dynamics. It is perhaps intuitive that the price of an asset is driven by market factors which may not be adequately captured by financial variables measured at one instant; instead, there may be information regarding such factors that can be extracted from the historical path of the asset price. For instance, an upward sloping path in the recent history may signal a positive outlook, or even a market bloom; a downward sloping path may signify market distress; a trough or a crest in the recent times may signify a reversion of market conditions, to name a few - the apparent or hidden patterns in the historical path of asset prices may hint at how the market will continue to evolve. Therefore, the path-dependent volatility model is a natural extension of the prevailing stochastic models for option pricing.

The mainstream path-dependent volatility models take the approach of inventing path-dependent state variables that encode the path-dependent information of the historical path of asset prices and inventing a volatility function that extracts the path-dependent information from the path-dependent state variables; often the inventions are based on intuition or ad-hoc analysis. We propose an innovative formulation of a path-dependent volatility model called the Laguerre Path-Dependent Volatility (LPDV) model. We apply series expansion to a historical price path with Laguerre polynomials, turning a path into a sequence of coefficients of the series. This sequence can be interpreted as a representation of the path, and we select a finite subset of the sequence as the path-dependent state variables with the property that they approximately represent the path. Then, we choose a volatility function that is both sufficiently flexible and theoretically connected to the Laguerre series expansion. The theoretical analysis is supported by a sound theoretical framework that we develop.

We also provide a detailed account on model calibration. We discuss comprehensively the considerations and challenges that one might face in model calibration. In addition, we propose an innovative calibration procedure that is uncommon in the literature but is suitable for the LPDV model. Finally, we conduct a numerical experiment where we test the LPDV model in a simulated controled setting. We discuss the details in various aspects of the implementation of both model calibration and option pricing. We provide example cases to study the performance of the model, paying attention to how path-dependent volatility comes into play.