Whether cross-sectional option prices are consistent with the time-series properties of the underlying asset returns is probably the most fundamental of tests.

—David S. Bates

In the previous chapter, we discussed two different approaches for stochastic volatility parameter estimation: the cross-sectional one where we use a number of options prices for given strike prices (and possibly maturities); and the time-series approach where we use the stock prices over a certain period of time.

One natural question^{1} would therefore be the following: Will the theoretically invariant portion of the parameter-sets obtained by the two methods be the same?

More accurately, supposing we are at time and we use options with strikes and with maturity , we have

The above options could include calls or puts.

On the other hand, during the period , we can ...

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