Under the martingale pricing measures [[pi].sup.*] and the assumption that the T-period returns are independently and identically distributed, the first four RNMs of log([R.sub.t,[tau]]), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are given by
The right-hand sides of (5)-(6) show that the RNMs are the integrals of option prices over a range of strike prices [0, [S.sub.t]) and [[S.sub.t], [infinity]) with two singular points 0 and [infinity].
We then incorporate the constraints of RNMs recovered from options into the entropic pricing framework.
where [[pi].sup.*.sub.i] denotes the risk-neutral (martingale) probability of the underlying asset's gross return from time t-(I - i + 1)[tau] to t - (I - i)[tau] and [R.sub.t-(I-i+1)t,[tau]] and [m.sub.t,[tau]](j) are the RNMs serving as constraints (10).
In addition to the above options to be priced, we also need to generate a sample of call options in order to estimate the RNMs. Apparently this sample is different from the sample of calls used above for the valuation purpose.
The technical details of calculating the RNMs are given in Appendix C.1 and also outlined in footnotes 7 and 8.
Through the procedure specified above, we use only 8 options that are usually available in a real market (16) to obtain the RNMs ([m.sub.t,[tau]](1) and [m.sub.t,[tau]](2)).
Since there are multiple underlying asset prices in the experiment, we extract the RNMs for each underlying price.
In addition to the estimated RNMs, we also need to simulate return series following (17) for recovering the RND according to (12) and 13).
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