Empirical Test - Empirical Performance of Covered-Call Strategy under Stochastic

Chapter 2 Empirical Performance of Covered-Call Strategy under Stochastic

2.3 Empirical Test

2.3.1 Under All Period Market Conditions

According to the models, Tables 2.1 to 2.3 show the fixed implementation price strategy, dynamic adjustment strategy in compliance probability, risk values, and descriptive statistics. From Feb-2004 to Nov-2011, the monthly

3In the original Heston’s model, the underlying asset is the stock price. We use the futures index to replace the stock index in order to consider the time consistency compared to options and the Heston model is comparable with the Black model.

average return for a simple buy-and-hold strategy is 0.32%.

In covered-call strategies, when the moneyness is deeper OTM, the short position of a call will receive fewer premiums. Generally, the short position of 3% to 6% OTM call options will increase the monthly total return. The risk will be smaller than the naked futures’ position. Finally, whether we use the Sharpe Ratio or the Sortino Ratio as the performance indicators, the short position of a 6% OTM call option can get the best performance, significantly enhance return, and reduce the risk.

Table 2.1 Overall monthly performance of different fixed moneyness for Feb-2004 to Jan-2012.

Moneyness (percentage of out-of-money ) Naked

Table 2.2 and Table 2.3 show the results of dynamic adjustment covered-call, respectively, under the Black (1976) model and under the Heston (1993) model.

Both implied volatilities are calculated by using all day-end information, excluding any volume under 100 contracts. In the Heston model, we use the method of exhaustion to obtain the optimal parameters in a given lower bound [0.01, 0.01, -1, 0.01, and 0.01] and upper bound [0.6, 0.6, 1, 5, and 0.6].

(representing the volatility of variance, current variance, rho, kappa, and the long-run mean of the variance, respectively.)

In the Black (1976) model, the rewards in all seven compliance probabilities

from 17% to 49% are less than the naked futures’ position. However, there is still an advantage to choose the exercised probability of around 30% in order to reduce any fluctuation in the return. The Heston (1993) model significantly improves the performance of the remuneration. Apparently, the dynamic covered-call strategies under Black (1976) perform less than the naked futures’

position. The Heston (1993) model is much closer to the actual cases than the Black (1976) model.

Table 2.2 Overall monthly performances of different exercise probabilities under the Black model for Feb-2004 to Jan-2012.

Exercise probabilities

49% 42% 36% 30% 25% 20% 17%

Return 0.13% 0.16% 0.23% 0.30% 0.24% 0.27% 0.25%

SD 4.23% 4.68% 5.06% 5.46% 5.68% 5.92% 6.06%

Call Premium Return 2.65% 2.05% 1.64% 1.23% 0.97% 0.71% 0.56%

Coefficient of Variation 32.62 28.78 21.83 18.10 23.83 22.13 24.60

Median 1.70% 2.06% 2.05% 2.05% 1.95% 1.95% 1.55%

Max 5.25% 5.98% 8.77% 10.39% 11.31% 10.49% 10.49%

Min -16.95% -17.43% -17.79% -18.07% -18.29% -18.42% -18.42%

Semi-SD 8.14% 7.89% 7.51% 6.96% 6.18% 5.18% 3.73%

Sharpe Ratio 3.07% 3.47% 4.58% 5.52% 4.20% 4.52% 4.07%

Sortino Ratio 1.59% 2.06% 3.09% 4.33% 3.86% 5.16% 6.60%

Table 2.3 Overall monthly performances of different exercise probabilities under the Heston model from Feb-2004 to Jan-2012. conventional covered-call and dynamic covered-call strategies. Obviously, the left upper quadrant based on the naked futures position outperforms the other quadrants. In conclusion, the conventional covered-call and dynamic covered-call strategies under the Heston model have more risk than the naked futures position at the same return level.

Figure 2.6 The tradeoff between return and SD for naked futures, and conventional covered-call and dynamic covered-call strategies.

4.00% 4.50% 5.00% 5.50% 6.00% 6.50% 7.00%

Return

Future Fixed Strike Dynamic (BS) Dynamic (Heston)

2.3.2 Under Different Market Conditions

Financial products in different market conditions tend to behave differently.

This study refers to the setting of Che and Fund (2011), dividing Taiwan’s market into four conditions: sharply falling, moderately falling, sharply rising, and moderately rising. Tables 2.4 to 2.6 show the results.

For the average return of seven probabilities, in a sharply falling market, selling options close to ATM can receive more premiums, and the dynamic adjustment covered-call (under both the Black and Heston models) strategies can effectively reduce losses. In a moderately falling market, the dynamic covered-call strategy is better than the conventional strategy under the Black model, and the performance of the Heston model is worse than the Black model.

However, in Taiwan there are only four months with a moderately falling situation during our study’s time period. Even so, the Heston model is still better than executing a naked futures’ position.

In a sharply rising market, all the covered-call strategies are worse than the naked futures strategy, however, if we evaluate the performance under the Sharpe ratio, the covered-call strategies are better than using naked futures, as they can effectively reduce risks. We are able to still point out that the Heston model is better than the Black model in the moderately rising market.

We note that the Black and Heston models choose the same probabilities of the closest ATM in a sharply falling market and those of the deeply OTM in a sharply rising market. The most interesting part is that the Heston model chooses to receive a higher premium than the Black model in the case of a moderately falling market and to receive fewer premiums than the Black model in the moderately rising situation in order to reduce the probability of exercise. Overall, this is the reason why the Heston model performs better than the Black model.

Table 2.4 Overall monthly performance of a fixed strike strategy under different market conditions.

Sharply Falling 1% 2% 3% 4% 5% 6%

Table 2.5 Overall monthly performance of a dynamic (Black) strike strategy under different market conditions.

Table 2.6 Overall monthly performance of a dynamic (Heston) strike strategy under different market conditions.

Coefficient of Variation -5.1728 -6.4412 -4.8511 -5.2111 -4.7019 -4.3909 -3.9669

Median 1.56% 0.94% 0.49% 0.15% -0.17% -0.17% -0.34%

Table 2.7 The performance under a conventional strategy and dynamic strategies (Black and Heston models).

We finally calculate the cumulative total return on the naked futures, a fixed strike strategy with OTM 3%, and both dynamic strategies with the exercise probability of around 30%. Although there are slightly different criterions between fixed and dynamic strategies, it is worth it to understand how the volatility influences the total return over the all periods. As shown as Figure 2.7, there are significant benefits to adopting the covered call strategy whether you use a fixed strategy or a dynamic strategy. Before the financial crisis of 2008, there are no obvious differences between the performances of the Black model and those of the Heston model. However, the fixed strike strategy outperforms all other strategies before year 2008. In the rapidly descending period in 2008, all strategies performed equally. After financial markets troughed out, the stock

market index bounced up quite fast. The call option positions suffered from buyers exercising them. The Heston model can properly adjust the moneyness, and the advantage can also be seen in Table 2.7. If we use other criterions with exercise probabilities, we find that the Heston model indeed performs better than the Black model.

Figure 2.7 The cumulative total return on the futures, a fixed strike strategy with the OTM 3%, a dynamic (BS) strategy, and a dynamic (Heston) strategy with exercise probability at 30%.

-40%

-30%

-20%

-10%

10%

20%

30%

40%

50%

20040218 20040519 20040818 20041117 20050216 20050518 20050817 20051116 20060215 20060517 20060816 20061115 20070226 20070516 20070815 20071121 20080220 20080521 20080820 20081119 20090218 20090520 20090819 20091118 20100222 20100519 20100818 20101117 20110216 20110518 20110817 20111116

Cumulative Total Return

Naked Futures Fixed Strike (3% OTM)

Dynamic (BS) (prob.=30%) Dynamic (Heston) (prob.=30%)

Figure 2.8 The cumulative total return on the futures, a fixed strike strategy with the OTM 1%, a dynamic (BS) strategy, and a dynamic (Heston) strategy with exercise probability at 20%.

Figure 2.9 The cumulative total return on the futures, a fixed strike strategy with the OTM 6%, a dynamic (BS) strategy, and a dynamic (Heston) strategy with exercise probability at 49%.

-40%

Cumulative Total Return

Naked Futures Fixed Strike (1% OTM)

Dynamic (BS) (prob.=20%) Dynamic (Heston) (prob.=20%)

-40%

Cumulative Total Return

Naked Futures Fixed Strike (6% OTM)

Dynamic (BS) (prob.=49%) Dynamic (Heston) (prob.=49%)

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