Call Options, Semi-Static Arbitrage and Markov Chain Models

A presentation of the paper with abstract:

Under the assumption of the absence of arbitrage, European call prices on a given asset must satisfy well-known inequalities, which have been described in the landmark paper Merton (1973). If we further assume that there is no interest rate volatility and that the underlying pays continuously deterministic dividends, cross maturity inequalities must also be satisfied by the call prices.

In this paper, we show that there exists an arbitrage free model, which is consistent with the call prices, if these inequalities are satisfied. Furthermore, we describe an algorithm to obtain a realistic Markov chain model. The latter is a solution of the problem addressed by local volatility models since it is calibrated to all available call quotes but does not require the number of available quotes to be infinite or an interpolation of the implied volatility surface. Moreover some freedom in the model allows to have a direct control of the forward implied volatilities. The impact of the latter on exotics options, e.g. locally capped, globally floored compounding cliquet options, is investigated numerically.