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Abstract
Three main volatility models have been used so far in the finance industry: constant volatility, local volatility (LV) and stochastic volatility (SV). The first two models are complete. For their part, SV models are incomplete: the volatility is driven by one of several extra Brownian motions, and as a result perfect replication and price uniqueness are lost. To allow SV models to perfectly calibrate to the market smile, one can use stochastic local volatility (SLV) models. This article will show that path-dependent volatility (PDV) models, which are complete, can produce rich spot-vol dynamics and, furthermore, can perfectly fit the market smile. The two main benefits of model completeness are price uniqueness and parsimony: it is remarkable that so many popular properties of SLV models can be captured using a single Brownian motion. This paper first introduce the class of PDV models and then explain how the author calibrate them to the market smile. Subsequently, it investigates how to pick a particular PDV.