ABSTRACT
This chapter examines the most mainstream models of markets with continuous time. These models are based on the theory of stochastic integrals (stochastic calculus); stock prices are represented via stochastic integrals. Core concepts and results of mathematical finance are given (including self-financing strategies, replicating, arbitrage, risk-neutral measures, market completeness, and option price). The case of the market with a non-zero interest rate for borrowing can be described via the following bond-stock model. The chapter examines the problem of trading or choosing a strategy in a class of strategies that does not use future values of (S(t), r(t)). It presents the application of the Girsanov theorem and looks at the arbitrage possibilities and the arbitrage-free market. The chapter also reviews the option pricing for a complete market. The fair option price (Black-Scholes price) can be calculated explicitly for some cases. The corresponding explicit formula for the price of European put and call options is called the Black–Scholes formula.
