## Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case

This post is the continuation of the previous one on the riskiness of OTM vs. ATM short put options and the effect of leverage on the risk measures. In this installment we’re going to perform similar studies with the only exception that from inception until maturity the short options are dynamically hedged. The simulation methodology and parameters are the same as in the previous study.

As a reference, results for the static case are replicated here:

 ATM  (K=100) OTM (K=90) Margin Return Variance VaR Return Variance VaR 100% 0.0171 0.0075 0.1940 0.0118 0.0031 0.1303 50% 0.0370 0.0292 0.3844 0.0206 0.0133 0.2783 15% 0.1317 0.3155 1.2589 0.0679 0.1502 0.9339

Table below summarizes the results for the dynamically hedged case

 ATM  (K=100) OTM (K=90) Margin Return Variance VaR Return Variance VaR 100% -0.0100 1.9171E-05 0.0073 -0.0059 1.4510E-05 0.0062 50% -0.0199 7.6201E-05 0.0145 -0.0118 5.8016E-05 0.0121 15% -0.0660 8.7943E-04 0.0480 -0.0400 6.5201E-04 0.0424

From the Table above, we observe that:

• Similar to the static case, delta-hedged OTM put options are less risky than the ATM counterparts. However, the reduction in risk is less significant. This is probably due to the fact that delta hedging itself already reduces the risks considerably (see below).
• Leverage also increases risks.

It is important to note that given the same notional amount, a delta-hedged position is less risky than a static position. For example, the VaR of a static, cash-secured (m=100%) short put position is 0.194, while the VaR of the corresponding dynamically-hedged position is only 0.0073. This explains why proprietary trading firms and hedge funds often engage in the practice of dynamic hedging.

Finally, we note that while Value at Risk takes into account the tail risks to some degree, it’s probably not the best measure of tail risks. Using other risk measures that better incorporate the tail risks can alter the results and lead to different conclusions.