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Option Pricing under ARMA Processes Theoretical and Empirical prospective. Chou-Wen Wang. Astract.
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(1) the TAIEX options are European-style
(2) both the TAIEX options and futures are cash settled
(3) the expiration days of the TAIEX futures coincide with those of the TAIEX options
(4) the TAIEX options and futures are traded side by side on the same exchange involving the same clearing house
=>Those evidence documents the predictability of financial asset returns
and when the time interval approaches zero.
p , q : the AR and MA orders
: AR coefficients
: MA coefficients with
Assume that the underlying stock price process S satisfies Equation (1). Given that , and where and , repeated substitution in Equation (1) for m times yields
(Conditioning on , is measurable )
-measurable. The mean of stock return during N time intervals conditional on is
the conditional variance which depends on the time-to-maturity N satisfies
The local risk-neutralized probability measure Q, which is defined over the period from 0 to a finite integer T, satisfies the local risk-neutral valuation relationship (LRNVR), that is,
(1) Q and P are mutually absolutely continuous;
(2) for i=1,…,N; and
for all i, almost surely with respect to P.
With respect to local risk-neutralized probability measure Q, the asset price process, conditional on , with ARMA relation obeys
, , n=1,…,N.
Assuming that the dynamics of the underlying stock prices are given by Equation (1), the closed-form solutions for the ARMA( p, q) -type Futures options are as follows:
is the BS implied volatility for an option of strike X and time to maturity .
Panel A reports the aggregate (across 2003,2004,2005 and 2006) in-sample percentage valuation errors for all options by various models from weekly (every week) estimation by minimizing the sum of squared errors between model option values and market option prices for Ad hoc BS, ARMA(1,1), ARMA(2,2) and Ad hoc ARMA(1,1) models. RMSE is the ratio of root mean squared out-of-sample valuation errors to the average option price. MAE is the ratio of the mean absolute error to the average option price. Average premium is
the average option price in the sample.
Table 2B Out-of-sample percentage valuation errors