Deterministic implied volatility models

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Abstract
In this paper, we characterize two deterministic implied volatility models, defined by assuming that either the per-delta or the per-strike implied volatility surface has a deterministic evolution. Practitioners have recently proposed these two models to describe two regimes of implied volatility (see Derman (1999 Risk 4 55–9)). In an arbitrage-free sticky-delta model, we show that the underlying asset price is the exponential of a process with independent increments under the unique risk neutral measure and that any square-integrable claim can be replicated up to a vanishing risk by trading portfolios of vanilla options. This latter result is similar in nature to the quasi-completeness result obtained by Bjork et al (1997 Finance Stochastics 1 141–74) for interest rate models driven by Levy processes. Finally, we show that the only arbitrage-free sticky-strike model is the standard Black-Scholes model.