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Jump Telegraph Processes and a Volatility Smile

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We discuss a simple class of financial market models based on inhomogeneous telegraph processes. This model capture bullish and bearish trends as well as oversold/overbought market situations. The model under consideration is arbitrage-free if directions of jumps in stock prices are in a certain correspondence with their current velocity and interest rate behaviour. In the simplest case the model is complete. Diffusion rescaling of this model gives a natural representation of volatility. We provide explicit formulae for prices of standard European options are obtained, which permits to calculate directly implied volatilities with respect to various moneyness and maturity times of the option. volatility surface (see A.Jobert, L.C.G. Rogers, Option pricing with Markov-modulated dynamics). It gives an example of the implied volatility surface, which does not move by parallel shifts and which is based ona process different from exponential Levy (see L.C.G. Rogers, M.R.Tehranchi, The implied volatility surface does not move by parallel shifts.)

This model has a Markov-modulated implied

This talk is part of the Statistics series.

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