underlie

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Interestingly, the large losses are confined to the short-dated 30-day calls, as longer-dated OTM calls do not exhibit this pattern with bull market underlier returns.
In other words, a high-volatility environment is required for skewness in the returns of the underlier to produce call returns which match the pattern documented empirically.
Put option returns also exhibit departures from the results of the previous section when the assumption of normal returns in the underlier is relaxed, as detailed in Table 5.
Finally, skewness in the returns of the underlier helps explain the very large losses which have been documented for OTM put options empirically.
In addition to including skewness in the returns of the underlier, this section studies the effect of also allowing variation in volatility by using the volatility index (VIX) as the [sigma] parameter in the option prices.
29) A dfferent theory is that risk-seeking investors might pay a premium for the positive skewness in the returns of the call option itself, even without consideration of the distribution of returns in the underlier.
These returns are obviously lower than the average returns on the underlier itself, which were 0.
Once again, this is contrary to theoretical expected put option returns derived under the assumption of the underlier having normally distributed returns, but not surprising with negatively skewed returns in the underlier.