A gentle introduction to mathematical finance: The mathematical theory of arbitrage, pricing and hedging of derivative securities in a discrete, elementary setting.
The one-period model (Arbitrage, risk neutral measure, pricing, complete markets, optimal portfolios) The multiperiod model (self-financing portfolios, duality, Dalang/Morton/Willinger theorem) The Binomial Model, Cox-Ross-Rubinstein Model, distribution of the maximum Markov Models Taking limits in the Binomial Model, Black-Scholes Model American Options, Snell envelope, Doob decomposition Optimal portfolios
Hans Föllmer, Alexander Schied: Stochastic Finance. An Introduction in Discrete Time. De Gruyter, 3rd ed., 2011, ISBN: 978-3110218046.