"Frontmatter". In: Analysis of Financial Time Series

EXERCISES 253which involves the CDF of a normal distribution with mean ln(P t )+(r +σ 2 /2)(T −t) and variance σ 2 (T − t). By using the same techniques as those of the secondintergation shown before, we havewhere h + is given by∫ ∞ln(K )e x f (x) dx = P t e r(T −t) (h + ),h + = ln(P t/K ) + (r + σ 2 /2)(T − t)σ √ .T − tPutting the two integration results together, we havec t = e −r(T −t) [P t e r(T −t) (h + ) − K (h − )]=P t (h + ) − Ke −r(T −t) (h − ).APPENDIX B. APPROXIMATION TO STANDARDNORMAL PROBABILITYThe CDF (x) of a standard normal random variable can be approximated by{1 − f (x)[c1 k + c(x) =2 k 2 + c 3 k 3 + c 4 k 4 + c 5 k 5 ] if x ≥ 01 − (−x) if x < 0,where f (x) = exp(−x 2 /2)/ √ 2π, k = 1/(1 + 0.2316419x), c 1 = 0.319381530,c 2 = −0.356563782, c 3 = 1.781477937, c 4 = −1.821255978, and c 5 =1.330274429.For illustration, using the earlier approximation, we obtain (1.96) = 0.975002,(0.82) = 0.793892, and (−0.61) = 0.270931. These probabilities are very closeto that obtained from a typical normal probability table.EXERCISES1. Assume that the log price p t = ln(P t ) follows a stochastic differential equationdp t = γ dt + σ dw t ,where w t is a Wiener process. Derive the stochastic equation for the price P t .2. Considering the forward price F of a nondividend-paying stock, we haveF t,T = P t e r(T −t) ,where r is the risk-free interest rate, which is constant and P t is the currentstock price. Suppose P t follows the geometric Brownian motion dP t = µP t dt +σ P t dw t . Derive a stochastic differential equation for F t,T .