We mentioned in the previous sections that in finance, returns are assumed to follow a normal distribution, whereas prices follow a lognormal distribution. The stock price at time t+1 is a function of the stock price at t, mean, standard deviation, and the time interval, as shown in the following formula:
In this formula, St + 1 is the stock price at t+1, ˆ μ is the expected stock return, t _ is the time interval (T t n_= ), T is the time (in years), n is the number of steps, ε is the distribution term with a zero mean, and σ is the volatility of the underlying stock. With a simple manipulation, equation (4) can lead to the following equation that we will use in our programs:
In a risk-neutral work, no investors require compensation for bearing risk. In other words, in such a world, the expected return on any security (investment) is the risk-free rate. Thus, in a risk-neutral world, the previous equation becomes the following equation:
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