"Frontmatter". In: Analysis of Financial Time Series

232 CONTINUOUS-TIME MODELSThe prior distribution of stock price can be used to make inference. For example,suppose that the current price of Stock A is $50, the expected return of the stock is15% per annum, and the volatility is 40% per annum. Then the expected price ofStock A in 6-month (0.5 year) and the associated variance are given byE(P T ) = 50 exp(0.15 × 0.5) = 53.89,Var(P T ) = 2500 exp(0.3 × 0.5)[exp(0.16 × 0.5) − 1] =241.92.The standard deviation of the price 6 months from now is √ 241.92 = 15.55.Next, let r be the continuously compounded rate of return per annum from time tto T .ThenwehaveP T = P t exp[r(T − t)],where T and t are measured in years. Therefore,r = 1 ( )T − t ln PT.P tBy Eq. (6.9), we have( ) [( )]PTln ∼ N µ − σ 2(T − t), σ 2 (T − t) .P t 2Consequently, the distribution of the continuously compounded rate of return perannum is()r ∼ N µ − σ 22 , σ 2.T − tThe continuously compounded rate of return is, therefore, normally distributed withmean µ − σ 2 /2 and standard deviation σ/ √ T − t.Consider a stock with an expected rate of return of 15% per annum and a volatilityof 10% per annum. The distribution of the continuously compounded rate of returnof the stock over two years is normal with mean 0.15−0.01/2 = 0.145 or 14.5% perannum and standard deviation 0.1/ √ 2 = 0.071 or 7.1% per annum. These resultsallow us to construct confidence intervals (C.I.) for r. For instance, a 95% C.I. for ris 0.145±1.96 × 0.071 per annum (i.e., 0.6%, 28.4%).6.5 DERIVATION OF BLACK–SCHOLES DIFFERENTIAL EQUATIONIn this section, we use Ito’s lemma and assume no arbitrage to derive the Black–Scholes differential equation for the price of a derivative contingent to a stock valuedat P t . Assume that the price P t follows the geometric Brownian motion in Eq. (6.8)