The Black-Scholes model for pricing stock options is a revolutionary result in mathematical finance. Many of the pricing techniques and models used today in finance are rooted in the methods and ideas from the Black-Scholes model. In this chapter, we present the Black-Scholes model and relevant results.

**Definition 42.1** (Continuous Market). Let be an *n*-dimensional Brownian motion on some filtered probability space be an adapted interest rate process. A continuous market is a (*d* + 1)-dimensional stochastic process

where

and

Here the mean rate of return vector and the volatility matrix are also adapted processes.

**Definition 42.2** (Black-Scholes Market). Let {*B*_{t} : *t* ≥ 0} be a Brownian motion on some probability space ...

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