Analysis of Financial Time Series, Third Edition

Appendix B: Approximation to Standard Normal Probability

The CDF Φ(x) of a standard normal random variable can be approximated by

where , k = 1/(1 + 0.2316419x), c1 = 0.319381530, c2 = − 0.356563782, c3 = 1.781477937, c4 = − 1.821255978, and c5 = 1.330274429.

For illustration, using the earlier approximation, we obtain Φ(1.96) = 0.975002, Φ(0.82) = 0.793892, and Φ( − 0.61) = 0.270931. These probabilities are very close to that obtained from a typical normal probability table.

Exercises

6.1 Assume that the log price pt = ln(Pt) follows a stochastic differential equation

where wt is a Wiener process. Derive the stochastic equation for the price Pt.

6.2 Considering the forward price F of a nondividend-paying stock, we have

where r is the risk-free interest rate, which is constant, and Pt is the current stock price. Suppose Pt follows the geometric Brownian motion dPt = μPt dt + σPt dwt. Derive a stochastic diffusion equation for Ft, T.

6.3 Assume that the price of IBM stock follows the Ito process

where μ and σ are constant and wt is a standard Brownian motion. Consider the daily log ...