Term Structures of Implied Volatilities: Absence of Arbitrage and Existence Results

Abstract

This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h. We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h. For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h, namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions.

Local volatility dynamic models

“…[30] and [8] are early examples of attempts to go beyond static models, but despite the fact that they consider only a cross section of the surface (say for K fixed), the works of Schönbucher [32] and Schweizer and Wissel [34] are more in the spirit of the market model approach which we advocate in this paper.…”

“…As we already mentioned, Schönbucher and Schweizer and Wissel have argued that the implied volatility was not the right code-book for equity market models (see [32] and [34]), and in the simpler case of a cross section they work instead with the term structure of volatility for a fixed option. Our point of view is to follow the spirit of the approach advocated by Schönbucher in the case of credit portfolios [33].…”

“…But again, isolating tractable conditions characterizing the absence of arbitrage is very technical and cumbersome. See for example [32] and [34] for a discussion of the particular case when K is fixed.…”

“…In so doing, they derive conditions very similar to our spot consistency and drift conditions. See also [34] for a discussion of some of the difficulties associated with the simultaneous dynamics of all the call prices.…”

A Note on Market Completeness with American Put Options

“…Indeed, in the recent years many papers have treated problems like absence of arbitrage, hedging, optimal portfolio choice in a financial market where investors are allowed to trade in the underlying assets as well as to assume static positions in some class of derivatives. Here, we recall only few of them: Campi [1] for no-arbitrage and completeness issues, the papers by Ilhan et al [11,12] and Carr et al [5] for optimal investment problems, and the more recent papers by Schweizer and Wissel [20,21] and by Jacod and Protter [14] where an HJM approach for European call options is developed.…”