A new binomial approximation to the Black–Scholes model is introduced. It is shown that for digital options and vanilla European call and put options that a complete asymptotic expansion of the error in powers of n−1 exists. This is the first binomial tree for which such an asymptotic expansion has been shown to exist.

We present an introduction to mathematical Finance Theory for math ematicians The approach is to start with an abstract setting and then introduce hypotheses as needed to develop the theory We present the basics of European call and put options and we show the connection between American put options and backwards stochastic di erential equations

Since Black and Scholes published their seminal paper [2] in 1973, the pricing of options by means of deterministic partial differential equations or inequalities has become standard practise in computational finance. An option gives the right (but not the obligation) to buy (call option) or sell (put option) a share for a certain value (the exercise price K) at a certain time T (exercise date)...

This paper considered the notion of European option which is geared towards solving analytical and numerical solutions. In particular, we examined Black-Scholes closed form solution modified (MBS) partial differential equation using Crank-Nicolson finite difference method. These equations were approximated to obtain Call Put prices. The explicit price both options found accordingly. solutions c...