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.
Dynamically hedged at what frequency, with what cost? What jump risk are you assuming?
Continuously dynamically hedged short OTM puts with GBM assumptions are naturally going to look good, but that’s not real life. They are priced high because continuous hedging is too costly, and jumps are real, and very often those two are wrong-way risky (in that when the jumps happen are precisely when it’s very very expensive to hedge ex post).
Valid points. I used GBM without jumps. To compensate for the fact that GBM assumptions are not realistic, I priced the options at realized vol, i.e. without gap and skew risk premia. I also tried a stochastic vol model without jumps, and conclusions are about the same. Agree that for more accurate results, jumps (and stoch vol) should be used, especially for short dated options. Thanks for commenting.