The predictability of ¯nancial asset returns are frequently observed in the
form of an autoregressive and moving-average (ARMA) model. We use the discrete-
time equivalent martingale measure and moment generating function for the stock
price process to derive a closed-form option formula under the ARMA(p; q) model for
the asset`s returns, which is the extended version of the trending O-U process. Its
formula is simply the Black-Scholes formula with an adjusted volatility, the condi-
tional standard deviation under the ARMA(p; q) for the return series. The Monte
Carlo simulation of the daily returns under the ARMA(1,1) model shows that the
selection of sampling period for the conditionally annualized volatility of the k-period
returns is important and the conditional volatility estimated from the maximum like-
lihood estimation is much better than the indirect estimation from the unconditional
autocorrelation as the input value of volatility.
