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 returns of IBM stock in 1997. The average return and the sample standard deviation are 0.00131 and 0.02215, respectively. Use the data to estimate the parameters μ and σ assuming that there were 252 trading days in 1997.
6.4 Suppose that the current price of a stock is $120 per share with volatility σ = 50% per annum. Suppose further that the risk-free interest rate is 7% per annum and the stock pays no dividend. (a) What is the price of a European call option contingent on the stock with a strike price of $125 that will expire in 3 months? (b) What is the price of a European put option on the same stock with a strike price of $118 that will expire in 3 months? If the volatility σ is increased to 80% per annum, then what are the prices of the two options?
6.5 Derive the limiting marginal effects of the five variables K, Pt, T − t, σ, and r on a European put option contingent on a stock.
6.6 A stock price is currently $60 per share and follows the geometric Brownian motion dPt = μPt dt + σPt dt. Assume that the expected return μ from the stock is 20% per annum and its volatility is 40% per annum. What is the probability distribution for the stock price in 2 years? Obtain the mean and standard deviation of the distribution and construct a 95% confidence interval for the stock price.
6.7 A stock price is currently $60 per share and follows the geometric Brownian motion dPt = μPt dt + σPt dt. Assume that the expected return μ from the stock is 20% per annum and its volatility is 40% per annum. What is the probability distribution for the continuously compounded rate of return of the stock over 2 years? Obtain the mean and standard deviation of the distribution.
6.8 Suppose that the current price of stock A is $70 per share and the price follows the jump diffusion model in Eq. (6.26). Assume that the risk-free interest rate is 8% per annum, the stock pays no dividend, and its volatility (σ) is 30% per annum. In addition, the price on average has about 15 jumps per year with average jump size − 2% and jump standard error 3%. What is the price of a European call option with strike price $75 that will expire in 3 months? What is the price of the corresponding European put option?
6.9 Consider the European call option of a nondividend-paying stock. Suppose that Pt = $20, K = $18, r = 6% per annum, and T − t = 0.5 year. If the price of a European call option of the stock is $2.10, what opportunities are there for an arbitrageur?
6.10 Consider the put option of a nondividend-paying stock. Suppose that Pt = $44, K = $47, r = 6% per annum, and T − t = 0.5 year. If the European put option of the stock is selling at $1.00, what opportunities are there for an arbitrageur?
Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, 10th ed. U.S. National Bureau of Standards, Washington, DC.
Ait-Sahalia, Y. (1996). Testing continuous-time models for the spot interest rate. Review of Financial Studies 9: 385–426.
Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approach. Econometrica 70: 223–262.
Bakshi, G., Cao, C., and Chen, Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance 52: 2003–2049.
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654.
Conley, T. G., Hansen, L. P., Luttmer, E. G. J., and Scheinkman, J. A. (1997). Short-term interest rates as subordinated diffusions. Review of Financial Studies 10: 525–577.
Cox, J. C. and Rubinstein, M. (1985). Options Markets. Prentice Hall, Englewood Cliffs, NJ.
Donsker, M. (1951). An invariance principle for certain probability limit theorems. Memoirs American Mathematical Society No. 6.
Duffie, D. (1995). Dynamic Asset Pricing Theory, 2nd ed. Princeton University Press, Princeton, NJ.
Elerian, O., Chib, S., and Shephard, N. (2001). Likelihood inference for discretely observed non-linear diffusions. Econometrica 69: 959–993.
Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. Journal of Business & Economic Statistics 19: 177–191.
Gallant, A. R. and Long, J. R. (1997). Estimating stochastic diffusion equations efficiently by minimum chi-squared. Biometrika 84: 125–141.
Gallant, A. R. and Tauchen, G. (1997). The relative efficiency of method of moments estimators. Working paper, Economics Department, University of North Carolina.
Hull, J. C. (2007). Options, Futures, and Other Derivatives with Derivagem CD, 7th ed. Prentice Hall, Upper Saddle River, NJ.
Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scandinavian Journal of Statistics 24: 1–19.
Kou, S. (2002). A jump diffusion model for option pricing. Management Science 48: 1086–1101.
Lo, A. W. (1988). Maximum likelihood estimation of generalized Ito's processes with discretely sampled data. Econometric Theory 4: 231–247.
Merton, R. C. (1976). Option pricing when the underlying stock returns are discontinuous. Journal of Financial Economics 5: 125–144.