A Study on Option-Based Systematic Strategies

Designing All-Weather Overlays — A Study on Option-based Systematic Strategies

Ivan Guo, Grégoire Loeper

Monash CQFIS Working Paper 2020

Abstract

We perform an empirical analysis of systematic trading strategies on options. Namely, we focus on strategies which sell out of the money (OTM) call options to harvest the premium, and buy downside protection through OTM puts. We compare the risk adjusted performance across different choices of strike, maturity and option notional. In this paper we mostly focus on the S&P 500 index over the period 2007–2018. There is also a brief look of the performances of top strategies during the COVID-19 pandemic in early 2020.

https://www.monash.edu/science/quantitative-finance/publication Designing All-Weather Overlays — A Study on Option-based Systematic Strategies

Ivan Guo Gregoire Loeper

Centre for Quantitative Finance and Investment Strategies,∗ School of Mathematics, Monash University Clayton Campus, VIC, 3800, Australia

September 2, 2020

Abstract We perform an empirical analysis of systematic trading strategies on options. Namely, we focus on strategies which sell out of the money (OTM) call options to harvest the premium, and buy downside protection through OTM puts. We compare the risk adjusted performance across diﬀerent choices of strike, maturity and option notional. In this paper we mostly focus on the S&P 500 index over the period 2007–2018. There is also a brief look of the performances of top strategies during the COVID-19 pandemic in early 2020.

Keywords: systematic strategies, derivatives, volatility risk premium

Table of Contents

1 Introduction 2

2 Key takeaways 4

3 Basic Building Blocks5 3.1 Benchmark...... 5 3.2 Option strategies...... 5 3.3 Exercise Probabilities...... 9 3.4 Rebalancing/Compounding...... 11

4 Covered Calls with Downside Protection 13 4.1 Overall Results...... 14 4.2 Eﬃcient frontier...... 15 4.3 Top Strategy...... 17 4.4 Other VaRs...... 20 4.5 Without GFC...... 22 4.6 Covered Call Performance Contours...... 24

5 Put vs. Put Spread vs. Put Ratio 26 5.1 Covered calls with put spread...... 27 5.2 Top Strategies...... 30 5.3 Covered calls with put ratio...... 32 5.4 Top Strategy...... 35

∗The Centre is partially supported by BNP Paribas

1 6 Put Strike vs. Put Weight 37 6.1 Combining with index...... 37 6.2 Combining with covered call...... 39

7 Buying options vs. running delta hedge only 42 7.1 Single strategies...... 42 7.2 Covered calls...... 46

8 OTC vs. Listed 49 8.1 A bit of theory...... 49 8.2 Single strategies...... 49 8.3 Listed covered calls with downside protection...... 52 8.4 Top listed covered call vs. Top OTC covered call...... 55

9 Performances in 2020 57

10 Conclusion 61

A Maximising the Sharpe Ratio, a good idea? 62 A.1 A bit of theory...... 62

B Empirical Greeks via regression: theory 62

C Empirical Greeks via regression: results 63

1 Introduction

This work belongs to a series of papers aiming to explore empirically different types of systematic strategies on options, either listed or OTC1. There are of course many ways of building such strategies, due to the number of strikes and maturities available, which makes it a rich subject of research, and a difficult exercise when it turns to choosing the optimal strategy, with respect to some investment objectives. In our previous paper [5] we looked at strategies that harvest the so-called volatility risk premium (see, e.g., [1]). In that case, the investor usually assumes that the implied volatility is overpriced, and she might try to exploit the arbitrage by selling the option and executing the delta hedging strategy associated to it. In this paper we will pursue a different objective: the addition of options to a long market portfolio will be done in order to change the distribution of returns of the strategy. A common example is the covered call strategy [7], where for each unit of stock (or index) one sells a slightly out of the money call option with short maturity. This combination partly exchanges the returns from equity exposure with volatility exposure, which, in a sense, exchanges some beta with alpha. The performance from volatility is generated via the option premium, while equity return can still be generated up to the strike price. However, the strategy is susceptible to extremely large stock movements in a short period of time, as large stock increases do not provide extra returns, while losses from large stock price decreases are not mitigated. To mitigate the left tail of the distribution of returns, one can buy put options. From the point of view of the option seller, who will delta-hedge the option, a short option position presents a very asymmetric risk profile: everything else (in particular the implied volatility) being constant, the upside is limited to earning the decrease in time value of the option (the theta), while the downside is potentially unlimited. See SectionB and equation (1) for the definition of the various Greeks. The positive earning is described by the level of implied volatility, while the negative part depends on the realised volatility, multiplied by the level of convexity of the option (the gamma effect). The strategy tends to have steady performance in stationary market conditions

1Over the counter

2 but can suffer from heavy drawdowns in extreme conditions. This asymmetric, and negatively skewed profile is expected to be rewarded by a positive average expected return, the volatility risk premium. The difference between implied and realised volatilities has been well-established [8] and the existence of volatility risk premium is observable across many markets across the globe [3,9], including the S&P 500 index. It has also been extensively studied in our previous paper [5]. Therefore, a long option position is expected to have, in average, a negative carry, due to the “long convexity” it offers to the investor. When adding puts to the portfolio, one will have to be very careful to do it in the most cost-effective way, and the choice of the strike and maturity will be key. The type of strategy we study in this paper are those who consist of a long market position, plus an overlay of options. The overlay will consist of a short call / long put position. The short call position is supposed to generate alpha in bear to moderately bullish market conditions, while the long put position will offer downside protection, limiting the drawdown, at the expense of paying a negative carry. One sees immediately that answering the questions “which call (resp. put) should you sell (resp. buy) and in what quantity?” can only be done after answering a more difficult and profound ques- tion: how do you measure risk adjusted performance? The Sharpe-Ratio is an obvious candidate, but has known and well documented limitations, see for example [4], in particular given the non- gaussianity of the returns, therefore we will mostly focus on the VaR, over different quantiles and periods. When executing a systematic strategy on options, one has to be mindful of the potential liquidity risks, including transaction costs and market impact, see [2] and the references therein. We do not make any assumptions on the liquidity costs incurred or on the possible market impact. Instead, without doing assumptions on transactions cost, we can express the sensitivity of the performance to the mid-offer spread, in terms of volatility, as was done in our previous study. Our study is based on data that is made available to us through a partnership with BNP Paribas, obtained directly from settlement prices given by exchanges.

3 2 Key takeaways

• Our analysis is based on an extensive data set that comes from daily settlement prices.

• Our comparisons are based on an efficient frontier type of approach, where we look at risk adjusted performance for different measures of risk (variance and VaR over different quantiles and periods). • Our analysis shows that:

• When selling calls, better to go for a short maturity (15 business days is a good one) and slightly OTM (between 102% and 104% of the money). • Buying downside protection is expensive... • ... but drastically limits the drawdown if one takes into account events such as the GFC. • This is conﬁrmed by analysing the performance of the strategies during the recent market events due to the COVID-19 crisis. • When buying puts, it is better to go for longer maturity (longer than 6 months), and at the money or slightly OTM strikes (from 85% to 100% of the money).

• We tried to cheapen the cost of the protection by buying put spreads, or put ratios. Our result show that this strategy does not perform as well as buying puts. • We observe also that in this case, shorter put maturity (63bd) perform better.

• Running the strategies on unlisted OTC maturities with interpolated volatility improves the risk adjusted performance compared to using listed maturities only. • We interpret this as an eﬀect of the "pin risk", i.e. concentrating the options on small set of maturities, rather than spreading them every day.

• We include the recent period in our backtest at the end of the paper, with the COVID-19 crisis. Our analysis shows that the portfolios who are optimized over the period including 2008 perform much better in terms of limiting the drawdown, because of stronger downside protection.

4 3 Basic Building Blocks

3.1 Benchmark The Benchmark is constructed as the aggregate daily relative return of the BNPIFUS index. Recall that the BNPIFUS index is constructed by rolling the shortest maturity future contract on the S&P 500. The Benchmark has a constant dollar exposure (no compounding) to be consistent with the options strategies. Compounding can be implemented after combining strategies.

3.2 Option strategies The building block of our study consists of single strike and maturity strategy. Each strategy involves the systematic selling of constant strike and maturity option. Each day, one sells a given notional of options (e.g., in currency), of a given maturity (e.g., one month), and given moneyness (e.g. strike equals 102% of the current underlying spot). The options are kept until expiry and with no delta-hedging.

• Underlyings: S&P 500 Our current study will focus on options on the S&P 500 index. We chose this index because of its liquid options market, but the study can be reproduced on many indices.

• Option strikes and types: 80p, 85p, 90p, 95p, 100p, 100c, 101c, 102c, 102.5c, 103c, 104c, 105c We consider out-of-the-money (OTM) as well as at-the-money (ATM) options. For example, 80p corresponds to the put option struck at 80% of the underlying value today. • Option maturities: 15bd, 21bd, 42bd, 63bd, 84bd, 126bd, 189bd, 252bd The target maturity of the option is measured in business days, roughly corresponding to time periods of three weeks, one month, two months, three months, four months, six months, nine months and one year. • Notional

5 The notional of each option is chosen so that there is a constant exposure (in terms of currency) with respect to the spot of the underlying. So the payoﬀ of each option is of the + form N(ST /St − K) . • Hold to maturity, no delta hedge We consider options that are kept until expiry, and are not delta-hedged. • Compounding and non-compounding The basic strategies are also non-compounded. This means that the dollar exposure is constant. This is unlike a buy and hold strategy on an index, where the dollar exposure is equal to the value of the index. Compounding can be implemented after the strategies are combined, this will be discussed later.

• Over-the-counter (OTC) strategies The maturities of these options do not correspond to listed maturities. The pricing parameters (mostly implied volatility) are inferred by linear interpolation from listed markets prices (daily settlement prices provided by the exchange). Near the end of this paper, we will examine the effect of replacing them by options with listed maturities. A comparison between listed and OTC strategies will be done at the end of the paper, see section8. • Testing period Our study will be mostly carried out for the period 2007–2018, which includes the global financial crisis (GFC). There is also a brief section on the effect of removing the GFC by comparing with the period 2010–2018. • Risk measure In our study, we will focus on risk adjusted returns. Most of the time, we will use Value-at- Risk (VaR).

The typical option strategies considered here would have a long position in the index, a short position on a short maturity OTM call and a long position on a long maturity OTM put.

6 We begin with some basic plots of the OTM calls. In the following plots, each strategies is normalised by the 1-day 99% VaR.

Since each strategy consists of short positions in call options, we see a surge in performance during the GFC and a slow decline afterwards. Shorter maturities seem to perform better than longer maturties across the board.

7 Next, we have the OTM puts. Once again, each strategies is normalised by the 1-day 99% VaR.

Since each strategy consists of short positions in put options, we see a sharp drop in performance during the GFC as well as during other minor crises. Once again, shorter maturities tends to have better performances. This indicates that buying long maturity puts is a viable method of buying downside protection for extreme events.

8 3.3 Exercise Probabilities Here we compute and plot the exercise probabilities for various strikes and maturities.

100c 101c 102c 103c 104c 105c 110c 115c 120c 15bd 0.6237 0.4893 0.3360 0.1977 0.1100 0.0627 0.0073 0.0020 0.0003 21bd 0.6373 0.5317 0.3971 0.2766 0.1677 0.0989 0.0127 0.0023 0.0010 42bd 0.6650 0.5903 0.4860 0.4067 0.3202 0.2385 0.0357 0.0077 0.0030 63bd 0.6921 0.6386 0.5799 0.5054 0.4251 0.3388 0.0776 0.0176 0.0047 84bd 0.7066 0.6537 0.5977 0.5401 0.4780 0.4159 0.1481 0.0430 0.0147 126bd 0.7272 0.6996 0.6691 0.6296 0.5801 0.5286 0.2336 0.0935 0.0481 189bd 0.7403 0.7233 0.7021 0.6829 0.6628 0.6362 0.4441 0.2042 0.0824 252bd 0.7687 0.7474 0.7282 0.7021 0.6851 0.6649 0.5632 0.3467 0.1719

For calls, the exercise probabilities decrease as the strikes are further from the money, and the maturities decrease.

9 100p 99p 98p 97p 96p 95p 90p 85p 80p 15bd 0.3763 0.2740 0.1967 0.1450 0.1040 0.0760 0.0167 0.0070 0.0027 21bd 0.3627 0.2806 0.2204 0.1713 0.1326 0.0989 0.0224 0.0104 0.0043 42bd 0.3350 0.2721 0.2324 0.1941 0.1628 0.1342 0.0424 0.0185 0.0158 63bd 0.3079 0.2683 0.2304 0.2060 0.1806 0.1575 0.0640 0.0285 0.0207 84bd 0.2934 0.2590 0.2228 0.1924 0.1733 0.1498 0.0795 0.0396 0.0328 126bd 0.2728 0.2503 0.2309 0.2080 0.1841 0.1616 0.0855 0.0481 0.0440 189bd 0.2597 0.2420 0.2205 0.2042 0.1872 0.1706 0.1189 0.0853 0.0690 252bd 0.2313 0.2121 0.1929 0.1802 0.1734 0.1640 0.1194 0.1013 0.0865

For puts, the exercise probabilities decrease as the strikes are further from the money, and has no clear trend with respect to maturities.

10 3.4 Rebalancing/Compounding So far, the strategies presented are not compounded. We examine a few diﬀerent methods of compounding or rebalancing: • rebalancing periodically (252bd, 63bd or 1bd); • rebalancing whenever the value of the strategy deviates by a certain percentage (20% or 10%). We ﬁrst look at the examine the case of a covered call [Benchmark] + [103c,15bd].

Comparing diﬀerent methods of compounding. Next we have a covered call with downside protection: [Benchmark] + [103c,15bd] - [100p,126bd].

Comparing diﬀerent methods of compounding.

11 Naturally, compounding leads to higher performance in favourable market conditions. However, the diﬀerences between various compounding methods are not large. Going forward, we will simply rebalance every day whenever compounding is considered.

Takeaways

• The aggregate daily relative return of the BNPIFUS index is used as the Bench- mark. • It is a strategy that rolls SPX front month futures with a constant Dollar exposure.

• Each option strategy consists of selling an OTM option of a predetermined maturity and relative strike every day. The options are held to maturity without delta hedging. The notional is a constant Dollar notional. • This approach of having a constant Dollar exposure allows to combine strategies, compared to having a notional equal to the value of the strategy (compounding), which would imply to rerun the whole backtest for each new combination. • Daily rebalancing of the whole strategy is used whenever compounding is in- volved.

12 4 Covered Calls with Downside Protection

We examine a class of systematic strategies which longs the index, shorts a call option and longs a put option. Typically, a strategy that has a long position in the index while short call options is known as a covered call. It allows the investor to partially exchange equity exposure for volatility exposure, by foregoing gains from large sudden market surges and instead earning a consistent premium from the option. However, a covered call is still susceptible to market draw-downs. By incorporating an additional long position in put options, the investor can eﬀectively purchase downside protection while paying a small premium. Since our building block strategies involves selling options everyday, we look for

Strategy = Benchmark + aC − bP where

• Benchmark is deﬁned by the returns of the BNPIFUS index as explained in the previous section. We will use it as a proxy of the actual market index; • a, b are weights taken from the set {0, 0.25, 0.5, 0.75, 1}; • C is a strategy that sells an out-of-the-money, short to medium maturity call option everyday; • P is a strategy that sells an out-of-the-money, medium to long maturity put option everyday.

Several metrics are used to compare the performance of the strategies. The various headings used in the result table

• Perf denotes the performance, which is the total return of the strategy over the testing period. • Std denotes the standard deviation of non-compounded daily returns. • SR denotes the Sharpe-Ratio, i.e. the daily performance divided by the daily standard deviation. • NC and C refer to non-compounded/compounded versions of the same strategy. • Alpha and Beta are the regression coeﬃcients of the daily returns of the non-compounded strategy against the Benchmark daily returns, i.e.,

Strategy = Alpha + Beta × Benchmark + noise.

• VaR99_1 refers the 1-day 99% VaR. VaR95_1, VaR99_21 and VaR95_21 are deﬁned similarly.

In all following result tables, we rank all strategies according to Perf(NC)/VaR99_1, i.e., their non-compounded performance divided by the 1-day 99% VaR. Only the top twenty are displayed for each comparison.

13 4.1 Overall Results

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[102.5c,15bd]+[90p,189bd]*-1+[Bench] 44.099 1.175 0.83735 0.97991 0.69834 0.023517 0.85749 0.018988 0.011268 0.058344 0.046629 0.0063163 [103c,15bd]+[90p,189bd]*-1+[Bench] 44.002 1.2062 0.85634 0.96851 0.68761 0.022601 0.86753 0.019462 0.011666 0.060843 0.047418 0.0065605 [103c,15bd]+[95p,252bd]*-1+[Bench] 43.918 1.0646 0.77665 0.95869 0.69941 0.024135 0.83441 0.017684 0.010331 0.054495 0.037694 0.0058496 [102.5c,15bd]+[95p,252bd]*-1+[Bench] 43.863 1.0337 0.75766 0.96837 0.70979 0.025154 0.81908 0.017273 0.0099055 0.052221 0.036555 0.0056231 [103c,15bd]+[90p,252bd]*-1+[Bench] 43.666 1.2004 0.85301 0.96903 0.68862 0.022711 0.86558 0.019535 0.011359 0.059836 0.04447 0.0065254 [102.5c,15bd]+[95p,189bd]*-1+[Bench] 43.653 1.0085 0.74334 0.96351 0.7102 0.025321 0.81317 0.017028 0.009846 0.053311 0.036911 0.0055136 [103c,15bd]+[95p,189bd]*-1+[Bench] 43.492 1.039 0.76234 0.95304 0.69925 0.024263 0.82861 0.017528 0.010219 0.055585 0.0376 0.005743 [102.5c,15bd]+[90p,252bd]*-1+[Bench] 43.399 1.169 0.83401 0.97959 0.6989 0.023618 0.85473 0.019217 0.011162 0.057562 0.042581 0.0062862 [102c,15bd]+[95p,252bd]*-1+[Bench] 43.364 0.97438 0.72384 0.95613 0.71028 0.025619 0.80093 0.016692 0.0094657 0.050666 0.0348 0.0053684 [103c,15bd]+[100p,189bd]*-1+[Bench] 43.326 0.84221 0.64686 0.91039 0.69922 0.025713 0.7678 0.01493 0.0082311 0.04854 0.030614 0.0048733 [103c,15bd]+[100p,252bd]*-1+[Bench] 43.238 0.90912 0.6859 0.9398 0.70904 0.025848 0.78806 0.015863 0.0086744 0.046613 0.030874 0.0050959 [102c,15bd]+[90p,189bd]*-1+[Bench] 43.208 1.1135 0.80353 0.97114 0.7008 0.023944 0.84604 0.018597 0.010725 0.055814 0.04529 0.00604 [102.5c,15bd]+[85p,252bd]*-1+[Bench] 43.083 1.2709 0.8917 0.9736 0.68311 0.022008 0.8805 0.020697 0.012083 0.061468 0.050282 0.0068764 [103c,15bd]+[100p,126bd]*-1+[Bench] 42.933 0.74023 0.58661 0.83958 0.66534 0.024102 0.74589 0.013663 0.0076204 0.043627 0.029728 0.0046444 [102.5c,15bd]+[100p,126bd]*-1+[Bench] 42.932 0.71216 0.56761 0.84392 0.67263 0.025241 0.71739 0.013221 0.0073713 0.041561 0.028741 0.0044453 [102.5c,15bd]+[100p,252bd]*-1+[Bench] 42.905 0.87901 0.6669 0.94683 0.71836 0.026988 0.76489 0.015544 0.0083605 0.045276 0.029738 0.0048905 [102c,15bd]+[95p,189bd]*-1+[Bench] 42.819 0.94986 0.70952 0.9522 0.71126 0.025823 0.79496 0.01657 0.009391 0.050903 0.03567 0.0052549 [104c,15bd]+[100p,252bd]*-1+[Bench] 42.714 0.90456 0.68877 0.87781 0.6684 0.02246 0.82274 0.016125 0.0093535 0.049836 0.03301 0.0054283 [103c,15bd]+[85p,252bd]*-1+[Bench] 42.696 1.3024 0.91069 0.96323 0.67352 0.021209 0.88867 0.02133 0.012469 0.063743 0.051072 0.0071228 [102.5c,15bd]+[100p,189bd]*-1+[Bench] 42.68 0.81282 0.62787 0.91663 0.70805 0.026882 0.7421 0.014711 0.0077639 0.046266 0.028932 0.0046712 14 We can see from the table above that all of the top strategies favours selling short maturity calls (15bd) with strikes between 102c and 104c, while buying long maturity puts (189bd and 252bd). 4.2 Efficient frontier Here we present scatter plots of the strategies, plotting their performance versus VaR. The same plot is then colour-coded in various ways to distinguish between different features of the strategies. This representation is inspired by the efficient frontier approach in classical portfolio theory2, in which a portfolio is called efficient if it reaches the maximum performance for a given amount of risk, or reaches the minimum risk for a given level of performance. The capital market line represents the performance versus risk of the leveraged “optimal portfolio”, i.e. a constant leverage on the portfolio that achieves the best ratio of performance over risk. The intercept of this line is 0, since strategies we consider are already excess return. For reference, the Benchmark is added as a black point.

Next few plots will zoom in around the optimal portfolio to get a closer look.

2The classical portfolio theory by Markowitz analyses portfolio choices based on their returns versus standard deviations. We use a variation here which uses VaR instead of standard deviation.

15 In the next plot, instead of using colours, we will label zoomed scatter plot with the composition of the optimal strategies.

Generally speaking, a strategy is more preferable if it is higher (more performance) and more to the left (less risk). According to the plots, the Benchmark (black point) has more risk than all the tested strategies, but not necessarily oﬀer the best performance. If we analyse the strategies according to each of their six parameters, we can see the following:

• Call maturity: shorter maturities clearly perform better than longer maturities, with 15bd being the best. • Call strike: while the overall result is mixed, the optimal strategies tend to have mid-level strikes, between 102c and 104c.

16 • Put maturity: longer maturities perform better than shorter maturities, the best ones are 252bd and 189bd. • Put strike: as we move more out-of-the-money, the performance increases but so does the risk levels. This is expected since out-of-the-money puts have lower premiums but oﬀer less downside protection. • Call weight: larger call weights (i.e, selling more calls) tend to have better performances. • Put weight: more negative put weights (i.e, buying more puts) have lower performances but has less risk.

The optimal strategies involves selling 102.5c or 103c at 15bd, while buying 90p or 95p at 189bd or 252bd.

4.3 Best Strategy Next, let us plot the top four strategies against the Benchmark. First, we plot the compounded performance without any scaling.

While the top strategies have only slightly better total return than the Benchmark across the entire testing period, they perform much more consistently and are more robust with respect to market draw-downs. This becomes clearer as we plot the compounded performance scaled by the 1-day 99% VaR.

17 In this scaled comparison, we see that the top strategies clearly outperform the Benchmark. Finally, we plot the non-compounded performance scaled by the 1-day 99% VaR. This performance criteria is equivalent to our ranking metric in the earlier table as well as the capital market line in our scatter plots

Evidently, the top strategies are signiﬁcantly better than the Benchmark.

18 Another way to visualise the performance is via histograms of daily returns. In the plots below, we show both non-smoothed and smoothed versions of the histogram. We can clearly see that the top stratigies oﬀer much more consistent performance compared to the Benchmark.

19 4.4 Other VaRs For the sake of completeness, we repeat the colour-coded scatter plots with capital market lines but with diﬀerent measures of risk. Instead of using the 1-day 99% VaR, we also examine the 21-day 99% VaR, the 1-day 95% VaR, and the 21-day 95 VaR. The overall results are still similar to the 1-day 99% VaR case.

20 Across all measures, we see that the best strategies still prefer selling one unit of short maturity (15bd), mid-level strike (102c to 103c) call, while buying one unit of long maturity (189bd or 252bd) put. Aside from that, the 1-day 95% VaR and the 21-day 95 VaR measures prefer a put strike that is closer to the money (100p or 95p), while the at-the-money call strike (100c) is also competitive in the case of the 21-day 95 VaR.

21 4.5 Without GFC A key features of the strategy is the presence of the put option to protect from extreme market draw-downs. Under normal market conditions without large draw-downs, one would expect the put to become less relevant. We can test this by repeating the same exercise, but starting in 2010 and excluding the GFC. We see that in these market conditions, it is preferable to buy little or no put options. It is worth noting that the following ranking is an in-sample comparison, for a period without extreme market downturns.

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[103c,15bd]+[100p,126bd]*-0.25+[Bench] 41.873 1.2275 0.85824 1.2141 0.84886 0.0062729 0.9821 0.020496 0.012169 0.064947 0.038674 0.0071096 [103c,15bd]+[100p,189bd]*-0.25+[Bench] 41.784 1.2442 0.86681 1.2189 0.84916 0.0063009 0.98192 0.020745 0.012273 0.066053 0.039526 0.0071781 [104c,21bd]+[90p,252bd]*-0.25+[Bench] 41.603 1.3868 0.94053 1.2365 0.83857 0.0053123 0.98861 0.022607 0.013442 0.072172 0.046739 0.0078869 [103c,15bd]+[95p,63bd]*-0.25+[Bench] 41.48 1.3105 0.90096 1.2319 0.84694 0.0059982 0.9853 0.02172 0.012825 0.067502 0.043226 0.0074805 [103c,15bd]+[95p,126bd]*-0.25+[Bench] 41.454 1.3019 0.89662 1.2298 0.84694 0.0060641 0.98393 0.021629 0.012723 0.067887 0.041332 0.0074444 [103c,15bd]+[100p,84bd]*-0.25+[Bench] 41.422 1.2103 0.84961 1.2063 0.84677 0.006104 0.98288 0.020511 0.011915 0.063668 0.039091 0.0070555 [103c,15bd]+[95p,84bd]*-0.25+[Bench] 41.415 1.3028 0.89726 1.228 0.84573 0.0059595 0.98453 0.021665 0.012821 0.067885 0.042533 0.0074603 [104c,15bd]+[Bench] 41.402 1.5932 1.0369 1.3022 0.84749 0.0056557 0.99316 0.025045 0.014312 0.07628 0.051121 0.0086036 [103c,15bd]+[95p,252bd]*-0.25+[Bench] 41.38 1.3136 0.90246 1.2335 0.84739 0.0060732 0.98433 0.021809 0.012585 0.067825 0.042306 0.0074889 [104c,21bd]+[85p,252bd]*-0.25+[Bench] 41.35 1.4293 0.96078 1.2516 0.84133 0.0054557 0.98924 0.023235 0.013741 0.073089 0.047503 0.0080303 [104c,21bd]+[95p,252bd]*-0.25+[Bench] 41.327 1.3346 0.91507 1.219 0.83585 0.0051835 0.98772 0.022142 0.013166 0.070957 0.045783 0.0076984 [104c,21bd]+[90p,189bd]*-0.25+[Bench] 41.288 1.3816 0.93839 1.2315 0.83646 0.005182 0.98855 0.022728 0.013408 0.072373 0.0468 0.0078888 [103c,15bd]+[100p,252bd]*-0.25+[Bench] 41.279 1.2555 0.87302 1.2176 0.84667 0.0060984 0.98287 0.021149 0.012267 0.066182 0.040217 0.0072508 22 [102.5c,15bd]+[100p,189bd]*-0.25+[Bench] 41.255 1.1848 0.83635 1.1989 0.84625 0.0063386 0.97731 0.020273 0.011986 0.064115 0.03734 0.0069497 [104c,21bd]+[80p,252bd]*-0.25+[Bench] 41.233 1.4621 0.9762 1.263 0.8433 0.0055588 0.98968 0.023675 0.013906 0.07379 0.048106 0.0081401 [104c,21bd]+[100p,63bd]*0+[Bench] 41.224 1.5625 1.0223 1.2982 0.84938 0.0058893 0.99077 0.024798 0.014259 0.076219 0.049791 0.0084635 [105c,63bd]+[95p,63bd]*-0.25+[Bench] 41.22 1.2251 0.85781 1.2045 0.84342 0.0058075 0.98465 0.020811 0.01219 0.065905 0.047471 0.0071519 [103c,15bd]+[90p,252bd]*-0.25+[Bench] 41.217 1.3656 0.92792 1.2509 0.85 0.0061844 0.98543 0.022513 0.012998 0.06904 0.044219 0.0076765 [104c,15bd]+[95p,126bd]*-0.25+[Bench] 41.197 1.3511 0.92385 1.2186 0.8332 0.0049228 0.98968 0.022425 0.013202 0.071081 0.046461 0.007797 [104c,21bd]+[85p,189bd]*-0.25+[Bench] 41.19 1.4302 0.96151 1.2491 0.83977 0.0053524 0.98935 0.023343 0.01379 0.073394 0.047689 0.0080512

We see that the top strategies prefer to buys little to no puts, while the optimal call option is still have mid-level strikes (103c to 104c) and short maturities (15bd). Again we provide a scatter plots of all strategies, plotting their performance against the 1-day 99% VaR.

We see that the overall shape of these plots are a lot “ﬂatter” than the previous plots. Without extreme market draw-downs, the range of available returns for any given level of risk is a lot narrower.

23 4.6 Covered Call Performance Contours In this subsection, we examine the eﬀect of changing maturities or strikes. First, we ﬁx the strikes and compare strategies of the form Benchmark + [x, 100c] − [y, 90p], with x and y varying across a range of maturities. A contour of the return/VaR ratios is given below.

The overall performance clearly increases as the call maturity shortens, with the best at 15bd. On the other hand, longer maturity puts tend to do better, with the optimal being between 189bd and 252bd. Next, we ﬁx the maturities and compare strategies of the form Benchmark + [x, 15bd] − [y, 189bd], where x is a call and y is a put, varying across a range of out-of-the-money strikes. A contour of the return/VaR ratios is given below.

24 In this case, the best call strikes are between 102.5c and 103c, while the best put strikes are between 90p and 95p.

Takeaways

• The covered call with downside protection strategy consists of a long position in the index, a short position in call options and a long position in put options. • Our principal performance measure is the ratio of non-compounded return divided by the 1-day 99% VaR, over the period 2007 to 2018. • For the short position in the call option, shorter maturities (15bd) are preferred, with a strike between 102c and 104c. • For the long position in the put option, longer maturities (189bd and 252bd) are preferred, with a strike between 90p and 100p. • If we exclude the GFC from the study, then it is preferable to buy a smaller amount of put options (e.g., 0.25 instead of 1), and shorter maturity puts (63bd or 126bd) also become competitive.

25 5 Put vs. Put Spread vs. Put Ratio

Instead of buying a put for downside protection, one could also consider replacing it with a put spread or a put ratio. The goal is to exchange some downside protection with some performance (via lower premia).

• Buying x put spreads refers to simultaneously buying x close to the money put options while selling x out of the money put options of the same maturity. • Buying x put ratios refers to simultaneously buying x close to the money put options while selling y out of the money put options of the same maturity. The ratio between x and y is equal to the inverse ratio of the two respective strikes.

The typical payoﬀs of put spreads and put ratios are show in the following diagrams.

Put spread payoﬀ Put ratio payoﬀ

In the following plot, we brieﬂy compare the eﬀect of: buying no downside protection, buying put option, buying put spread and buying put ratio.

In this particular case, the put spread and put ratio perform similarly. They both oﬀer an in- between option between the high-risk, high-return option of not buying puts, and the low-risk, low-return option of buying a single put. In order to properly study the use of put spreads and put ratios, we will perform a more comprehensive comparison in the next two subsections.

26 5.1 Covered calls with put spread In this section, we consider protecting the covered call with a put spread in the following way: by buying 100p, 95p or 90p while selling the 85p of the same maturity at the same quantity.

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[102.5c,15bd]+[100p-85p,63bd]*-0.75+[Bench] 36.977 0.97607 0.7578 0.72084 0.55964 0.012811 0.94003 0.020494 0.010376 0.06472 0.037982 0.007133 [102.5c,15bd]+[100p-85p,63bd]*-1+[Bench] 36.671 0.81737 0.65694 0.68468 0.55029 0.013185 0.89969 0.017914 0.0086946 0.051927 0.029028 0.0062887 [102.5c,15bd]+[95p-85p,63bd]*-1+[Bench] 36.621 1.0871 0.82653 0.7383 0.56133 0.012631 0.952 0.02257 0.012033 0.067347 0.04956 0.0077566 [103c,15bd]+[100p-85p,63bd]*-0.75+[Bench] 36.377 1.0039 0.77679 0.71838 0.55585 0.012377 0.94822 0.021354 0.010802 0.065676 0.038944 0.0073616 [103c,15bd]+[100p-85p,63bd]*-1+[Bench] 36.126 0.84463 0.67593 0.68433 0.54765 0.012712 0.91254 0.01871 0.0091724 0.052883 0.030542 0.0065017 [102c,15bd]+[100p-85p,63bd]*-0.75+[Bench] 36.091 0.92046 0.72398 0.70438 0.55402 0.012706 0.9296 0.02006 0.010034 0.063747 0.036311 0.0068837 [103c,15bd]+[95p-85p,63bd]*-1+[Bench] 36.019 1.1149 0.84553 0.73426 0.55684 0.012216 0.95758 0.023474 0.012335 0.069947 0.050788 0.0079988 [102c,15bd]+[100p-85p,63bd]*-1+[Bench] 35.905 0.76445 0.62312 0.66459 0.54172 0.013037 0.88323 0.017354 0.0082534 0.050954 0.027482 0.0060593 [102c,15bd]+[95p-85p,63bd]*-1+[Bench] 35.556 1.0302 0.79271 0.72465 0.55761 0.01256 0.94517 0.022295 0.011664 0.064502 0.048108 0.0074888 [104c,15bd]+[100p-85p,63bd]*-1+[Bench] 35.303 0.83672 0.67881 0.64269 0.52139 0.010623 0.93075 0.019228 0.0095956 0.054571 0.033069 0.0068582 [102.5c,15bd]+[100p-85p,84bd]*-0.75+[Bench] 35.272 0.9884 0.7699 0.7036 0.54806 0.012015 0.94281 0.021828 0.010763 0.069888 0.043307 0.0074001 [103c,15bd]+[100p-85p,84bd]*-0.75+[Bench] 35.2 1.0161 0.7889 0.70178 0.54485 0.011622 0.95081 0.022412 0.011198 0.070844 0.044186 0.0076274 [103c,15bd]+[100p-85p,189bd]*-1+[Bench] 35.161 1.0421 0.80522 0.70588 0.54543 0.011725 0.94802 0.022901 0.011427 0.07415 0.044227 0.0077769 [102.5c,15bd]+[100p-85p,189bd]*-1+[Bench] 35.047 1.0141 0.78623 0.70722 0.54832 0.012107 0.93964 0.022434 0.01111 0.073194 0.041942 0.0075534 [104c,15bd]+[100p-85p,63bd]*-0.75+[Bench] 35.042 0.99246 0.77967 0.67542 0.53061 0.010506 0.95998 0.02225 0.011562 0.067364 0.040539 0.0077405 [103c,15bd]+[100p-85p,84bd]*-1+[Bench] 34.939 0.86206 0.69207 0.66439 0.53338 0.01161 0.92104 0.019808 0.0096935 0.059773 0.037823 0.0068351 [103c,15bd]+[95p-85p,84bd]*-1+[Bench] 34.916 1.1177 0.85038 0.72297 0.55007 0.011755 0.95898 0.024355 0.012657 0.070654 0.051633 0.0081438 27 [102.5c,15bd]+[95p-85p,84bd]*-1+[Bench] 34.861 1.0899 0.83138 0.72634 0.55405 0.012144 0.95319 0.023848 0.012289 0.069698 0.050805 0.0079046 [103c,15bd]+[100p-85p,63bd]*-0.5+[Bench] 34.801 1.1678 0.87766 0.74116 0.55701 0.011997 0.96717 0.02522 0.012677 0.078469 0.046414 0.0083003 [103c,15bd]+[100p-85p,189bd]*-0.75+[Bench] 34.638 1.1555 0.87376 0.72726 0.54993 0.011651 0.963 0.025226 0.012763 0.081626 0.05017 0.0083698

The ranking metric is once again the return/VaR ratio. For reference, the original best performance ratio (where only a put is used for downside protection) was approximately 44. So according to this particular metric, put spreads do not perform as well as simply buying puts for downside protection. Next, we again compare the eﬀect of diﬀerent parameters using colour-coded scatter plots. Recall that the black point is the Benchmark. We examine all four risk measures: the 1-day 99% VaR, the 21-day 99% VaR, the 1-day 95% VaR, and the 21-day 95 VaR.

28 We make the following observations in the case of covered calls with put spreads: • Call maturity: short maturities (15bd) are clearly the best. For the 21-day 95 VaR, there are one or two mid-level maturity call option strategies that are also competitive in terms of performance ratio, despite having very low returns. • Call strike: mid-level strikes (102c to 103c) are generally the best. However for the 21-day 95 VaR, at-the-money calls are preferred. • Put spread maturity: shorter maturities (63bd) are preferred. This is in contrast with the previous results for pure puts. • Put spread strike: buying at the money put (100p) is preferred. • Call weight: larger call weights (i.e, selling more calls) are preferred. • Put spread weight: more negative put weights (i.e, buying more put spreads) are preferred. For the 21-day 95 VaR, a put weight of -0.75 is also competitive.

29 5.2 Best Strategies Let us plot the top four covered calls strategies with put spreads against the Benchmark. The following three plots respectively contain: compounded performance without scaling, compounded performance scaled by the 1-day 99% VaR, and non-compounded performance scaled by the 1-day 99% VaR.

30 We can see that, while the returns are similar to or worse than the Benchmark, the variance of daily returns of the strategies are much lower. So according our performance ratio, the covered calls strategies with put spreads outperform the Benchmark.

31 5.3 Covered calls with put ratio In this section, we consider protecting the covered call with a put ratio in the following way: by buying 100p, 95p or 90p while selling the 85p of the same maturity at the ratios of 100/85, 95/85 and 90/85 respective.

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[102.5c,15bd]+[100p-85p/0.85,63bd]*-1+[Bench] 36.252 0.86336 0.69193 0.669 0.53616 0.012255 0.90134 0.019087 0.009001 0.0617 0.029473 0.0067982 [103c,15bd]+[100p-85p/0.85,63bd]*-1+[Bench] 36.14 0.89086 0.71093 0.66974 0.53447 0.011852 0.91376 0.019672 0.0094445 0.062656 0.031217 0.007007 [102.5c,15bd]+[95p-85p/0.895,63bd]*-1+[Bench] 35.887 1.1203 0.84986 0.73098 0.55451 0.012168 0.9534 0.023681 0.012381 0.070056 0.050136 0.0080736 [102.5c,15bd]+[100p-85p/0.85,63bd]*-0.75+[Bench] 35.811 1.0119 0.78405 0.70975 0.54995 0.012207 0.93973 0.021894 0.01072 0.07205 0.038517 0.0075101 [103c,15bd]+[100p-85p/0.85,63bd]*-0.75+[Bench] 35.599 1.0399 0.80304 0.70816 0.54688 0.011818 0.94796 0.022558 0.011025 0.073006 0.039346 0.0077352 [103c,15bd]+[95p-85p/0.895,63bd]*-1+[Bench] 35.585 1.1482 0.86885 0.72763 0.55059 0.011786 0.95906 0.024416 0.012696 0.072656 0.051364 0.0083128 [102c,15bd]+[95p-85p/0.895,63bd]*-1+[Bench] 35.303 1.0628 0.81604 0.71689 0.55043 0.012078 0.94642 0.023115 0.011906 0.068686 0.048684 0.0078097 [102.5c,15bd]+[100p-85p/0.85,84bd]*-0.75+[Bench] 35.274 1.0181 0.79411 0.68671 0.53562 0.011209 0.94374 0.022513 0.011066 0.078074 0.043916 0.00781 [103c,15bd]+[100p-85p/0.85,84bd]*-0.75+[Bench] 35.113 1.0458 0.81311 0.68569 0.53312 0.010863 0.95164 0.023157 0.011594 0.07903 0.044745 0.0080344 [102c,15bd]+[100p-85p/0.85,63bd]*-0.75+[Bench] 35.094 0.95561 0.75022 0.69287 0.54395 0.01208 0.92925 0.021377 0.010208 0.071077 0.036854 0.0072654 [102c,15bd]+[100p-85p/0.85,63bd]*-1+[Bench] 35.037 0.80955 0.65811 0.64867 0.52732 0.012077 0.88549 0.018783 0.0084197 0.060727 0.027782 0.0065743 [104c,15bd]+[100p-85p/0.85,63bd]*-1+[Bench] 34.956 0.88198 0.7138 0.63148 0.51106 0.0099564 0.93142 0.02042 0.0098952 0.064344 0.033868 0.0073575 [104c,15bd]+[100p-85p/0.85,63bd]*-0.75+[Bench] 34.619 1.0276 0.80592 0.66752 0.52353 0.010065 0.95977 0.023279 0.011779 0.074694 0.04094 0.0081092 [103c,15bd]+[100p-85p/0.85,84bd]*-1+[Bench] 34.576 0.90081 0.72435 0.6428 0.51688 0.010503 0.92389 0.020949 0.0099697 0.070688 0.038822 0.0073822 [104c,15bd]+[95p-85p/0.895,63bd]*-1+[Bench] 34.31 1.1329 0.87173 0.68557 0.52753 0.01013 0.96759 0.025407 0.013392 0.077006 0.053066 0.0087049 [102.5c,15bd]+[100p-85p/0.85,84bd]*-1+[Bench] 34.183 0.87364 0.70535 0.64161 0.51802 0.010846 0.91249 0.020635 0.0096728 0.069732 0.037448 0.0071728

32 [102c,15bd]+[100p-85p/0.85,84bd]*-0.75+[Bench] 34.046 0.96214 0.76029 0.66997 0.52941 0.011055 0.93384 0.022331 0.010712 0.077101 0.042385 0.0075651 [104c,15bd]+[100p-85p/0.85,84bd]*-0.75+[Bench] 33.92 1.0329 0.81598 0.64718 0.51129 0.0092166 0.96301 0.024056 0.012265 0.080718 0.046339 0.0084071 [103c,15bd]+[100p-85p/0.85,63bd]*-0.5+[Bench] 33.894 1.1925 0.89515 0.73522 0.5519 0.011678 0.96706 0.02641 0.012812 0.083356 0.046682 0.0085441 [103c,15bd]+[95p-85p/0.895,189bd]*-1+[Bench] 33.863 1.2582 0.93933 0.72911 0.54431 0.011133 0.96987 0.027739 0.013938 0.089118 0.053556 0.0090907

The ranking metric is once again the return/VaR ratio. For reference, the original best performance ratio (where only a put is used for downside protection) was approximately 44. The top put spread performance approximately 36.98. So according to this particular metric, put ratios perform similar to put spreads, and do not perform as well as simply buying puts for downside protection. Next, we again compare the eﬀect of diﬀerent parameters using colour-coded scatter plots. Recall that the black point is the Benchmark. We examine all four risk measures: the 1-day 99% VaR, the 21-day 99% VaR, the 1-day 95% VaR, and the 21-day 95 VaR.

33 We make the following observations in the case of covered calls with put ratios:

• Call maturity: short maturities (15bd) are clearly the best. For the 21-day 95 VaR, there are one or two mid-level maturity call option strategies that are also competitive in terms of performance ratio, despite having very low returns. • Call strike: mid-level strikes (102c to 103c) are generally the best. However for the 21-day 95% VaR, at-the-money calls are preferred. • Put ratio maturity: shorter maturities (63bd) are preferred. This is in contrast with the previous results for pure puts. • Put ratio strike: buying at the money put (100p) is preferred. Buying 95p is also competitive for the 1-day 99% VaR. • Call weight: larger call weights (i.e, selling more calls) are preferred.

34 • Put spread weight: more negative put weights (i.e, buying more put spreads) are preferred.

The overall conclusions here are mostly similar to the put spreads.

5.4 Best Strategy Let us plot the top four covered calls strategies with put ratios against the Benchmark. The following three plots respectively contain: compounded performance without scaling, compounded performance scaled by the 1-day 99% VaR, and non-compounded performance scaled by the 1-day 99% VaR.

35 The conclusions are again more or less identical to the case with put spreads. While the returns are similar to or worse than the Benchmark, the variance of daily returns of the strategies are much lower. So according our performance ratio, the covered calls strategies with put spreads outperform the Benchmark.

Takeaways

• We investigate the possibility of using put spreads or put ratios instead of put options for downside protection. • Generally the best put spreads and put ratios tends to have shorter maturities (63bd). • While they save on the premium, put spreads and ratios oﬀer less downside protection. • Overall, put options still oﬀer better performance.

36 6 Put Strike vs. Put Weight

In this section, we further examine the eﬀect of buying puts for downside protection, and whether there are any equivalences between diﬀerent values of put strikes and put weights.

6.1 Combining with index First, let us look at the simpliﬁed case of buying puts to protect the index, and compare the eﬀect this has on the 1-day 99% VaR as well as the Perf/VaR ratio.

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[85p,126bd]*-1+[Bench] 27.105 0.81078 0.72164 0.46395 0.41294 0.002934 0.96671 0.026624 0.016225 0.08612 0.059735 0.0092059 [80p,126bd]*-1+[Bench] 26.99 0.88057 0.77826 0.47053 0.41586 0.0029034 0.9757 0.028835 0.017255 0.099332 0.065252 0.0098584 [85p,126bd]*-0.8+[Bench] 26.825 0.84006 0.75322 0.45391 0.40699 0.0022164 0.98107 0.028079 0.01696 0.094302 0.064512 0.0097492 [90p,126bd]*-1+[Bench] 26.573 0.68346 0.62576 0.43174 0.39529 0.0020933 0.95536 0.023549 0.014838 0.075127 0.05582 0.0083391 [95p,126bd]*-1+[Bench] 26.236 0.54189 0.51261 0.3929 0.37168 0.00096585 0.94028 0.019539 0.012524 0.064164 0.048636 0.0072653 [80p,126bd]*-0.8+[Bench] 26.22 0.89421 0.79851 0.45791 0.4089 0.0022259 0.98572 0.030454 0.017715 0.10487 0.071777 0.010287 [90p,126bd]*-0.8+[Bench] 26.036 0.739 0.67651 0.43078 0.39435 0.0015454 0.97585 0.025983 0.015805 0.081087 0.059332 0.0090368 [95p,126bd]*-0.8+[Bench] 25.96 0.62533 0.58599 0.40396 0.37855 0.00068842 0.97004 0.022573 0.013747 0.069581 0.054401 0.0081546 37 [90p,126bd]*-0.6+[Bench] 25.394 0.79115 0.72726 0.42597 0.39157 0.0010705 0.98839 0.02864 0.016691 0.092573 0.066346 0.0097838 [85p,126bd]*-0.6+[Bench] 25.389 0.86607 0.78479 0.44173 0.40028 0.0015691 0.99047 0.030911 0.017627 0.10248 0.072953 0.010328 [80p,126bd]*-0.6+[Bench] 25.125 0.90533 0.81876 0.444 0.40154 0.0015989 0.99258 0.032588 0.018161 0.11041 0.079022 0.010741 [100p,126bd]*-0.6+[Bench] 25.119 0.61363 0.58385 0.38668 0.36791 -0.00032794 0.98455 0.023243 0.013713 0.073783 0.055581 0.0083597 [100p,126bd]*-0.8+[Bench] 24.926 0.50357 0.48529 0.37008 0.35665 -0.00050995 0.96245 0.019469 0.011495 0.062479 0.049384 0.007168 [95p,126bd]*-0.6+[Bench] 24.683 0.70601 0.65937 0.40816 0.3812 0.00046207 0.98652 0.026714 0.015365 0.081776 0.061921 0.0091118 [95p,126bd]*-0.4+[Bench] 24.563 0.78309 0.73275 0.40771 0.3815 0.00027742 0.99506 0.029832 0.016905 0.096861 0.074048 0.010118 [100p,126bd]*-0.4+[Bench] 24.386 0.72127 0.6824 0.39525 0.37395 -0.00019013 0.99472 0.027983 0.01591 0.09119 0.06889 0.0096129 [85p,126bd]*-0.4+[Bench] 24.38 0.88867 0.81637 0.42802 0.39319 0.00098782 0.99613 0.033485 0.018184 0.11067 0.078991 0.010937 [80p,126bd]*-0.4+[Bench] 24.328 0.91387 0.83901 0.42915 0.394 0.0010206 0.99688 0.034487 0.018468 0.11595 0.080151 0.011218 [90p,126bd]*-0.4+[Bench] 24.296 0.83947 0.77801 0.41838 0.38775 0.00066063 0.99549 0.032023 0.017733 0.10406 0.07706 0.01057 [80p,126bd]*-0.2+[Bench] 23.968 0.91979 0.85926 0.41363 0.38641 0.00048869 0.99916 0.03585 0.018773 0.12149 0.082056 0.011714 In order to check whether there are equivalences between varying put strikes and put weights, it is easier to see the eﬀect via contour plots. We provide the contours for the 1-day 99% VaR and the Perf/VaR ratio below.

The equivalences are indicated by the level curves (or contour lines) in these plots. For VaR, we see the buying more puts (less negative weights) is equivalent to using more in-of-the-money strikes, as both oﬀer greater protection in exchange for higher premiums. For the Perf/VaR ratio, the relationship is less clear. Generally speaking, buying more puts (more negative weights) will lead to better results across all strikes until around the weight of -0.6, while the premium levels are still moderate. For weights of -0.8 and -1, due to the high premiums, it is preferable to buy more out-the-money puts.

38 6.2 Combining with covered call Next, let us look at the case of buying puts along with selling a covered-call, and compare the eﬀect this has on the 1-day 99% VaR as well as the Perf/VaR ratio.

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[100p,126bd]*-1+[102.5c,15bd]+[Bench] 42.932 0.71216 0.56761 0.84392 0.67263 0.025241 0.71739 0.013221 0.0073713 0.041561 0.028741 0.0044453 [90p,126bd]*-1+[102.5c,15bd]+[Bench] 41.526 1.1039 0.80663 0.9021 0.65917 0.020823 0.86697 0.019425 0.011575 0.06021 0.048206 0.0064463 [95p,126bd]*-1+[102.5c,15bd]+[Bench] 41.495 0.91195 0.69349 0.87694 0.66686 0.022598 0.81293 0.016713 0.0098166 0.052395 0.037054 0.0054781 [100p,126bd]*-0.8+[102.5c,15bd]+[Bench] 41.235 0.86589 0.66617 0.86076 0.66222 0.021347 0.85309 0.016156 0.0091588 0.04832 0.033264 0.0052992 [95p,126bd]*-0.8+[102.5c,15bd]+[Bench] 41.029 1.0307 0.76687 0.87296 0.64949 0.019677 0.88944 0.018691 0.011041 0.057077 0.043795 0.0062198 [85p,126bd]*-1+[102.5c,15bd]+[Bench] 40.818 1.2776 0.90252 0.9289 0.6562 0.019846 0.90008 0.022111 0.012817 0.069286 0.052763 0.0072452 [90p,126bd]*-0.8+[102.5c,15bd]+[Bench] 39.928 1.1864 0.85738 0.88666 0.64076 0.018548 0.91371 0.021473 0.012256 0.06561 0.050926 0.0070487 [85p,126bd]*-0.8+[102.5c,15bd]+[Bench] 39.596 1.3254 0.93409 0.90397 0.6371 0.017917 0.93048 0.023591 0.01358 0.07381 0.054422 0.0077234 [80p,126bd]*-1+[102.5c,15bd]+[Bench] 39.494 1.3772 0.95913 0.92433 0.64373 0.018525 0.92251 0.024286 0.013866 0.07884 0.055736 0.0078489 [100p,126bd]*-0.6+[102.5c,15bd]+[Bench] 39.046 1.0211 0.76472 0.84607 0.63362 0.017936 0.92051 0.019585 0.010796 0.055906 0.040033 0.0063577 [95p,126bd]*-0.6+[102.5c,15bd]+[Bench] 38.63 1.1477 0.84025 0.85352 0.62486 0.01708 0.93326 0.021751 0.012031 0.062474 0.048846 0.0070836 [90p,126bd]*-0.6+[102.5c,15bd]+[Bench] 37.793 1.2655 0.90813 0.86246 0.6189 0.016462 0.94339 0.024029 0.013285 0.072082 0.053459 0.0077297 [80p,126bd]*-0.8+[102.5c,15bd]+[Bench] 37.46 1.4034 0.97938 0.89806 0.62673 0.016967 0.94291 0.026145 0.014289 0.08438 0.0568 0.0082319 [100p,126bd]*-0.4+[102.5c,15bd]+[Bench] 37.192 1.176 0.86328 0.82214 0.60351 0.015255 0.95339 0.023211 0.012312 0.070698 0.051224 0.0075352 [85p,126bd]*-0.6+[102.5c,15bd]+[Bench] 36.801 1.3694 0.96566 0.87392 0.61627 0.01611 0.95121 0.02624 0.014172 0.081993 0.056534 0.0082544 [80p,126bd]*-0.6+[102.5c,15bd]+[Bench] 36.134 1.4265 0.99963 0.86863 0.6087 0.015484 0.9575 0.027664 0.01483 0.08992 0.061541 0.008651

39 [95p,126bd]*-0.4+[102.5c,15bd]+[Bench] 35.623 1.2618 0.91363 0.82774 0.59933 0.014891 0.95758 0.025647 0.013387 0.076369 0.055961 0.0080303 [80p,126bd]*-0.4+[102.5c,15bd]+[Bench] 35.144 1.4465 1.0199 0.83724 0.59032 0.014089 0.96743 0.02902 0.01528 0.09546 0.062515 0.0091011 [85p,126bd]*-0.4+[102.5c,15bd]+[Bench] 35.07 1.4094 0.99724 0.84096 0.59501 0.01445 0.96467 0.028436 0.014903 0.090175 0.061408 0.0088288 [85p,126bd]*-0.2+[102.5c,15bd]+[Bench] 34.968 1.4453 1.0288 0.80663 0.57419 0.012944 0.97282 0.029421 0.015498 0.098357 0.062927 0.0094386 Once again we use contour plots to detect equivalences between varying put strikes and put weights. We provide the contours for the 1-day 99% VaR and the Perf/VaR ratio below.

The equivalences are indicated by the level curves (or contour lines) in these plots. For VaR, the results are similar to the previous case, where buying more puts (less negative weights) is equivalent to using more in-of-the-money strikes, as both oﬀer greater protection in exchange for higher premiums. For the Perf/VaR ratio, we see buying more puts (more negative weights) are better, all the way to the weight of -1. In contrast the the previous case, at the weight of -1, it is more preferable to buy in-the-money puts.

40 Takeaways

• When buying downside protection, in terms of Var adjusted performance, it is equivalent to buy more OTM puts, in greater quantity, or buy close to the money puts with a smaller gearing.

41 7 Buying options vs. running delta hedge only

Instead of buying or selling options, another possibility is to replace them by their corresponding delta-hedging strategy. Each day, one buys the delta-hedge computed with the implied vol of the option. In theory, the diﬀerence would mainly arise from the spread between the implied and realised volatilities. We will see how this aﬀects our strategies.

7.1 Single strategies As a ﬁrst example, let us look at the call option strategy [105c, 21bd]. In the following plot, the blue line is the original option, the red line is the result of running the corresponding delta hedge, and the yellow line is the diﬀerence between the two (or the option with delta-hedging). We also include a scatter plot of the option and the delta hedge against the Benchmark.

We see that there is a significant difference between the total return across the testing period, which corresponds to the spread between the implied and the realised volatility. The following is a scatter plot of the difference between pure option and delta-hedge, against the Benchmark.

42 Since this diﬀerence closely relates to the spread between the implied and the realised volatility, we see the shape of an upside-down volatility smile (since the strategies are shorting options).

43 In the next example, we look at the put option strategy [95p,126bd]. Again we use the same plots.

Once again, we see a difference between the option and the delta-hedge, which corresponds to the spread between the implied and the realised volatility. If we take into account of the scale, we see that the difference across the whole period is similar to the previous example with the call option, despite being at a complete different strike and maturity. To better illustrate this, we will plot this difference on its own, as well as provide a scatter plot of it against the Benchmark.

44 Similar to the case for the call, we once again see the shape of an upside-down volatility smile.

45 7.2 Covered calls We now combine the two as well as the Benchmark to construct a covered call with downside protection. In the following plot, the red strategy uses actual options, the yellow strategy uses delta-hedges, while the purple only uses option for the call and delta-hedge for the put.

Since actual options are usually more expensive than the delta hedge due to the volatility spread, we see that the best result is obtained by selling the call option while replacing the put by a long position in the corresponding delta-hedge. However, using a delta-hedge instead of actual put options poses signiﬁcant problems during periods of extremely high volatility. In particular, the delta-hedge is not able to oﬀer the same level of protection as pure put options. This can be illustrated by looking at the performance during the COVID-19 crisis in 2020, as shown below.

46 In the following ﬁgure, we have a plot of the top performing covered-call from Section4, [102.5c, 15bd] − [90p, 189bd] + [Bench], as well as the corresponding version which replaces the pure put option by its delta hedge.

Once again, we see the delta hedge seems to have better performance due to a lower cost-of-carry. But if we zoom into the respective performances in 2020, we see a diﬀerent story.

47 Finally, let us also plot the put leg of the respective strategies, as well as their diﬀerences.

It is clear that the pure option provides much better downside protection than the delta hedge. The ﬁrst quarter of 2020 is an exception period with extreme draw-downs, rebounds and volatilities. We will look at this period in more detail in Section9.

Takeaways

• Due to the spread between implied and realised volatility, there are some benefits to replace the put leg of the strategy with its corresponding delta hedge, as it reduces the cost-of-carry. • For the same reason, it is much more interesting to sell the call options rather than just doing the delta-hedge, as this benefit from a relatively high implied volatility. • However, doing so incurs additional risks during periods of extreme draw-downs and volatilities. For example, pure put options offered much better downside protection than its delta hedge counterpart in the first quarter of 2020.

48 8 OTC vs. Listed

Throughout the paper so far, we have only considered the use of OTC (over the counter) options, in which the prices are interpolated from the listed maturities. In this ﬁnal part, let us study the eﬀect of using listed options instead in the strategies.

8.1 A bit of theory We can give a general formula for the payoff of the elementary strategies (the ones that only trade one option): X P = f(Si,Sj(i)) − P (ti), i,j(i) where f is the payoff that depends on the strike level Si and on the maturity j(i), and P (ti) is the premium paid. Here we neglect again all effects due to compounding. The mapping j(i) that maps a trading date i to the option’s maturity date j(i) is either

i → i + T in the OTC case, where T is the maturity of the options, or

i → Matu(i) in the listed case, where Matu(i) is the listed maturity corresponding to the date i and the target maturity. It is therefore going to be a piecewise constant function, where exposure is accumulated on listed expiries. By elementary considerations, the variance of the listed version should be higher, as it benefits less from the diversification effect of spreading the expiries over time. Not only that, but it has also been widely documented in the literature that abnormal volatility occurs on expiry dates, see for example the book by Stoll and Whaley [10], therefore not only the risk is concentrated on some particular days, but those particular days are likely to be the most volatile.

8.2 Single strategies Let us begin by comparing the OTC and listed performances of a single option strategy. Fixing the strike at 102.5c, we compare calls with the listed maturities, 3fripb and 1m3fripb, with OTC calls with similar maturities. For this comparison, everything is scaled by their respective VaR.

49 The next ﬁgure compares 90p puts with longer maturities, 9m3fripb and 12m3frip, with their OTC counterparts.

From these plots, the overall evolutions of listed strategies appear to be similar to the corresponding OTC versions.

50 However, more can be revealed if we closely look at the sensitivities of the strategies. For instance, let us compare the Vega of the listed and OTC strategies.

In both cases, we see that the listed Vegas are much more volatile. Aside from the usual ﬂuctuations, it also appears to contain a periodic component which can be attributed to the monthly expiry time of options. This phenomenon can be attributed to what is known as pin risks. Since the expiries of options used in the listed strategies tend to concentrate on particular dates (e.g., 3rd friday of each month), as opposed to the OTC case where one option is expiry every day, the listed strategies are more susceptible to extreme events. On the other hand, OTC strategies would have relatively smoother price evolutions.

51 8.3 Listed covered calls with downside protection Let us now study the covered call strategies with downside protection, but using only listed options. The same metrics are used as before. The top strategies are listed below.

Perf(NC)/VaR99_1 Perf(C) Perf(NC) SR(C) SR(NC) Alpha Beta VaR99_1 VaR95_1 VaR99_21 VaR95_21 Std ______

[104c,3fri,pb]*1+[95p,9m,3fri,pb]*-1+[Bench] 41.88 0.96922 0.73526 0.82687 0.62727 0.020095 0.81328 0.017556 0.010814 0.055846 0.037622 0.0061747 [104c,3fri,pb]*1+[90p,9m,3fri,pb]*-1+[Bench] 41.556 1.1012 0.81439 0.84135 0.62219 0.019037 0.84421 0.019598 0.011932 0.062244 0.046804 0.006895 [104c,3fri,pb]*1+[95p,12m,3fri,pb]*-1+[Bench] 41.123 0.97375 0.73783 0.82873 0.62795 0.019811 0.82699 0.017942 0.010888 0.055285 0.03757 0.0061896 [104c,3fri,pb]*1+[90p,12m,3fri,pb]*-1+[Bench] 41.033 1.0841 0.80479 0.83645 0.62095 0.018689 0.85551 0.019613 0.011949 0.060471 0.04177 0.0068274 [105c,3fri,pb]*1+[95p,9m,3fri,pb]*-1+[Bench] 41.003 0.96658 0.73795 0.79713 0.60858 0.018336 0.83765 0.017997 0.011108 0.058099 0.038308 0.0063876 [103c,3fri,pb]*1+[90p,9m,3fri,pb]*-1+[Bench] 40.839 1.036 0.77647 0.82928 0.62153 0.019617 0.81818 0.019013 0.011596 0.058737 0.045065 0.0065811 [105c,3fri,pb]*1+[100p,9m,3fri,pb]*-1+[Bench] 40.704 0.83119 0.65224 0.78272 0.6142 0.019794 0.79144 0.016024 0.0094635 0.052432 0.032064 0.005594 [104c,3fri,pb]*1+[100p,9m,3fri,pb]*-1+[Bench] 40.658 0.83176 0.64955 0.80942 0.63211 0.02174 0.75713 0.015976 0.0092385 0.04983 0.031112 0.0054131 [103c,3fri,pb]*1+[95p,9m,3fri,pb]*-1+[Bench] 40.457 0.90588 0.69734 0.81046 0.62389 0.020706 0.77878 0.017237 0.010522 0.052339 0.035735 0.005888 [103c,3fri,pb]*1+[95p,12m,3fri,pb]*-1+[Bench] 40.364 0.91035 0.69991 0.81269 0.62483 0.020414 0.79351 0.01734 0.010384 0.052284 0.036228 0.0059008 [105c,3fri,pb]*1+[95p,12m,3fri,pb]*-1+[Bench] 40.101 0.97097 0.74052 0.79852 0.60899 0.018057 0.85045 0.018466 0.0112 0.057887 0.038248 0.0064055 [104c,3fri,pb]*1+[100p,12m,3fri,pb]*-1+[Bench] 40.072 0.83934 0.65498 0.80508 0.62825 0.020836 0.78482 0.016345 0.0094181 0.050395 0.03137 0.005492 [103c,3fri,pb]*1+[90p,12m,3fri,pb]*-1+[Bench] 40.006 1.0192 0.76687 0.82404 0.62003 0.019252 0.82951 0.019169 0.011502 0.056964 0.040428 0.0065154 [105c,3fri,pb]*1+[100p,12m,3fri,pb]*-1+[Bench] 39.683 0.83837 0.65767 0.77709 0.6096 0.018911 0.81627 0.016573 0.0097701 0.052998 0.032415 0.0056832 [105c,3fri,pb]*1+[90p,12m,3fri,pb]*-1+[Bench] 39.608 1.0796 0.80748 0.80566 0.60258 0.017091 0.87402 0.020387 0.012387 0.062572 0.043969 0.007059 [105c,3fri,pb]*1+[90p,9m,3fri,pb]*-1+[Bench] 39.574 1.0966 0.81708 0.81061 0.60397 0.017443 0.86292 0.020647 0.012336 0.064345 0.047715 0.0071266 [102.5c,3fri,pb]*1+[90p,9m,3fri,pb]*-1+[Bench] 39.308 0.95818 0.7338 0.78956 0.60467 0.018956 0.80139 0.018668 0.011418 0.056262 0.043692 0.0063928 52 [104c,3fri,pb]*1+[85p,12m,3fri,pb]*-1+[Bench] 39.279 1.1783 0.86132 0.83881 0.61316 0.017814 0.87161 0.021929 0.013133 0.066737 0.05065 0.0073999 [104c,3fri,pb]*1+[85p,9m,3fri,pb]*-1+[Bench] 39.245 1.207 0.87713 0.84548 0.61442 0.01801 0.86675 0.02235 0.013244 0.069726 0.051302 0.0075202 [104c,3fri,pb]*1+[100p,9m,3fri,pb]*-0.75+[Bench] 39.215 0.99094 0.7519 0.80647 0.61193 0.017416 0.88507 0.019174 0.010953 0.057589 0.03922 0.0064727 \end{matlaboutput}

Recall that the top performing OTC strategies had a Perf/VaR ratio of 44. As shown here, the top performing Listed strategies have a ratio of 41.88, which is noticeably lower. For the colour-coded scatter plots with capital market lines, we again use the following four measures of risk: the 1-day 99% VaR, the 21-day 99% VaR, the 1-day 95% VaR, and the 21-day 95 VaR.

53 The top listed strategies uses short maturity (3fri) calls with strikes between 103c and 105c. For puts, the 12m options are slightly more favoured, and the best strikes are 90p and 95p. It is also generally preferable to sell as much call and buy as much put as possible.

54 8.4 Best listed covered call vs. Best OTC covered call Let us plot the top listed covered call strategy against the top OTC covered call strategy. In particular, we plot the non-compounded performance scaled by the 1-day 99% VaR, which is equivalent to our ranking metric.

To better understand the eﬀect of switching from OTC to Listed options. we also compare the top OTC strategy, [102.5c, 15bd] + [90p, 189bd] ∗ −1 + [Bench], against its corresponding listed version [102.5c, 3fri] + [90p, 9m, 3fri] ∗ −1 + [Bench], and the resulting plot is shown below.

We see that the top OTC strategy clearly outperforms the corresponding listed strategy. Both of them also outperform the Benchmark after scaling by VaR.

55 Takeaways

• Strategies using OTC options outperform the corresponding strategies using listed options in terms of Var adjusted returns. • Listed option strategies inherit pin risks due to a large number of options in the strategy portfolio expiring simultaneously. The volatility exposure they oﬀer is much less stable than the OTC strategies.

56 9 Performances in 2020

Finally, let us brieﬂy examine the performance of our covered call strategies in the ﬁrst four months of 2020. In 2020, we saw unprecedented market drawdowns during February and March as a result of the COVID-19 outbreak, followed by a sharp rally in April. We will be paying close attention to the following two strategies:

• Strategy A: [102.5c, 15bd] + [90p, 189bd] ∗ −1 + [Bench], the top strategy when optimised over the period 2007–2018, which included the GFC; • Strategy B: [103c, 15bd] + [100p, 126bd] ∗ −0.25 + [Bench], the top strategy when optimised over the period 2010–2018, which did not include the GFC.

While the call legs of the two strategies are fairly similar, we see that Strategy A has a much larger position in puts for downside protection. First let us plot the evolution of these strategies, as well as the Benchmark, during the ﬁrst four months of 2020. Note that these are the non-compounded versions and we do not scale them by VaR since the VaR estimates in this short period would be extremely volatile.

We can see that Strategy A, due to a larger position in puts, suﬀered a much smaller drawdown than Strategy B in March. However, Strategy A beneﬁted less from the sharp market rally in May than Strategy B, which was optimised over a predominantly bull market.

57 Breaking it down further, let us plot separately the call leg and the put leg in both strategies. First, this is done for the ﬁrst four months of 2020. We see that the two call legs are almost identical. The put leg of Strategy A provided much greater protection than the put leg of Strategy B.

We can compare this with performances from the GFC, as shown in the following plot. Once again, Strategy A’s put leg offered greater protection during the crisis. However, there is one significant difference between GFC and the events of 2020. The call legs in 2020 performed worse due to the extremely sharp rebound in April, as much higher proportion of the 15bd call options were exercised when compared to the period following the GFC. Moreover, high levels of volatility lead to more expensive put premia during the same period. As a result, the total performance of the strategies in early 2020 was reduced, despite doing well in minimise the effect of the market drawdowns.

58 In contrast, if we make the same plot over a relative normal period (2012–2014) The performance of both call and puts are both fairly steady.

For completeness, we also plot the top 5 strategies for 2007-2018 as well as the top 5 strategies from 2010–2018. As shown in the ﬁgures, the evolutions within each of the two cases are quite similar. The more conservative strategies optimised over the period including the GFC performed better during the market drawdown.

59 Takeaways

• In an out-of-the-sample test, our top strategies manage to reduce the eﬀect of market drawdowns during the COVID-19 market turmoil. • The more conservative strategy, which was optimised over a period including the GFC, was better at minimising the drawdown, compared to the strategy optimised after the GFC. • We have examine the contribution of all legs of the strategy during 3 periods: GFC, COVID-19, and a bull market perdiod 2012-2014. • The sudden market rebound and extreme volatility in April 2020 reduced the performance of the strategies, since many 15bd calls were exercised and puts had unusually high premia, but the drawdown has been considerably reduced.

60 10 Conclusion

In this study, we looked at the performance of a particular class of systematic option strategies: covered call with downside protection. They generally involve a long position in the index, a short position call options and a long position in put options. The main ﬁndings are as follows:

• For the call option, short maturities (15bd) with mid-level strikes (102.5c and 103c) have the best performance. • For the put option, long maturities (189bd and 252bd) are best. Strikes between 100p and 85p are all competitive, with a trade-off between return and risk protection. • The put options mainly contribute during periods of extreme market drawdowns. • Buying put options is better than put spreads or put ratios for the purpose of downside protection. • When using put spreads or put ratios, shorter maturity (63bd) perform better. • Most of the performance come from market gains (Delta), with some from the volatility spreads. • One can achieve cheaper cost-of-carry by replacing the put option by its corresponding delta- hedge. However, this exposes the strategy to additional risks since the delta hedge offers less reliable downside protection. • Strategies using OTC options perform better than the listed option counterparts by spreading the expiries over every day, instead of concentrating them on listed expiries. • In an out-of-the-sample test, the top strategies were able to significantly reduce the extreme drawdowns of the index in March 2020, while maintaining steady performance during the strong rebound in April.

61 A Maximising the Sharpe Ratio, a good idea?

A.1 A bit of theory The authors in [4] argue that, in a complete market where options are available, the payoff maximising the Sharpe Ratio will be typically a concave payoff. This is in accordance with the observation that selling calls increases the sharpe ratio in average, while buying puts decreases it. Actually, considering a market with a finite number of future states indexed i ∈ [1..N], each having an (objective) probability pi and a (risk-neutral) probability pî. The problem is to find the payoff xi for each state i that is consistent with the risk neutral probability, and maximises the Sharpe Ratio. It is equivalent to finding P 2 ( pixi) max P 2 pi(xi − x¯) under constraints X X pixi =x, ¯ pîxi =x. ˆ [4] shows that the optimal payoff will satisfy

pi − pˆi xi − x¯ = γ . pi

If instead, one chooses to maximise a utility function, say U, the optimal payoﬀ zi will satisfy

0 pˆi U (zi) = θ , pi where the constant θ ensures that the risk neutral expectation of the marginal utility U 0 is ﬁxed. Comparing those two expressions, one gets that

−1 0 xi − x¯ = γ(1 − θ U (zi)). If U 0 is convex, which would be the case for any U of the form CARA utility function given xγ U(x) = , γ ∈ (0, 1), γ then xi is a concave function of zi. Therefore, assuming that zi are optimal (i.e. the market is priced efficiently), the Optimal Sharpe ratio payoff is a concave function of the utility maximising payoff. The concavity implies that average returns give rise to higher payoff, while extreme returns (positive or negative) lead to higher losses/smaller gains. Somehow, the Sharpe Ratio is less risk averse than any CARA utility function.

B Empirical Greeks via regression: theory

Let us denote the current value of the strategy by u(t, s, σ), where t is time, s is the forward price of the underlying (at a fixed maturity) and σ is the current implied volatility (at a fixed relative strike). Neglecting the effect of interest rates, the evolution of the strategy is given by 1 1 du = ∂ udt + ∂ udS + ∂ udσ + ∂ u(dS )2 + ∂ u(dS )(dσ) + ∂ u(dσ)2 + ··· (1) t s t σ 2 ss t sσ t 2 σσ = T heta + Delta + V ega + Gamma + V anna + V olga + ··· . (2) In other words, the performance of the strategy can be broken down into its sensitivity or Greeks with respect to the different indicators. Since the portfolio composition of each strategy is dynamic, with old options expiring and new options entering every day at different absolute strike levels, it is difficult to analytically compute the contribution of each term. Instead, we aim to empirically estimate the Greeks via regression. Note that, modulo the effect of interest rate and dividends, it is equivalent to substitute the underlying value by its forward price when analysing the contributions from Greeks. The notations ∆fwd and ∆iv are used to denote the daily changes in the forward and the implied volatility, respectively.

62 • Delta, ∂sudSt — sensitivity with respect to the forward price of the underlying. Since the strategies are already delta-hedged (in terms of Black-Scholes delta), we normally would expect the delta to be zero. However, this is not the only way to compute delta in models with non-constant volatility (see, e.g., [6]). As discussed in the previous section, there may be exist some residual delta due to correlations between the underlying and the volatility. We will measure this by regressing the returns against ∆fwd.

• Vega, ∂σudσ — sensitivity with respect to the implied volatility. We will measure this by regressing the returns against ∆iv. Note that the implied volatility is only taken daily at a particular relative strike determined by the strategy. The dynamic nature of the portfolio composition means that the strategy is actually sensitive to multiple parts of the volatility surface at any time. Hence this term will only be an estimate of the full eﬀect.

1 2 • Gamma, 2 ∂ssu(dSt) — second order sensitivity with respect to the forward price of the underlying. 2 2 We will measure this by regressing the returns against (∆fwd) . Since (dSt) = 2 (σrealisedSt) dt also corresponds to the quadratic variation of the underlying and the realised volatility, we can interpret gamma as a measure of the performance generated by the realised volatility.

• Theta, ∂tudt — sensitivity with respect to time. Theta corresponds to the change of the value of the strategy over time, assuming all other parameters (forward, implied volatility) are held constant. We cannot measure this directly by regressing with respect to time, since that would give us the total (not partial) derivative. In order to properly measure this, let us ﬁrst recall that, neglecting the interest rate, the 1 2 2 Black-Scholes equation stipulates that ∂tu + 2 σ St ∂ssu = 0 where σ is the implied volatility. Hence we will measure this by regressing the returns against (iv)2. Thus theta can also be interpreted as a measure of the performance of the strategy generated by the implied volatility.

• Vanna, ∂sσu(dSt)(dσ) — sensitivity with respect to the leverage between underlying and implied volatility. We will measure this by regressing the returns against (∆fwd)(∆iv). This eﬀectively corresponds to the correlation between the underlying and the implied volatility.

1 2 • Volga 2 ∂σσu(dσ) — second order sensitivity with respect to the implied volatility. This term is measureed by regressing the returns against (∆iv)2.

Since the various indicators may be correlated, we expect to see some overlap between the contributions of diﬀerent Greeks. Even though the overlap cannot be explicitly assigned to any individual term, the regression will naturally take this into account when computing the total variance contributed by the Greeks. The complete regression is of the form

∆u = A∆fwd + B∆iv + C(∆fwd)2 + D(iv)2 + E(∆fwd)(∆iv) + F (∆iv)2 + residuals.

Note that A, B, C, D, E and F are not constants, instead they are functions of the form P5 i i=−2 αi(fwd) ﬁtted by the regression. These are used to allow the Greeks some local dependence on the forward price of the underlying.

C Empirical Greeks via regression: results

In this section, we regress the daily returns of the top covered call strategy, Benchmark+[102.5c,15bd]-[90p,189bd], against the Benchmark daily returns and the changes in implied volatilities, to estimate performance contributions from various sensitivities.

63 As we can see from the scatter plots and correlations, the performance of the strategy has strong correlation with the performance of the market, and has negative correlation with the implied volatilities. This is not surprising as the strategy is constructed to have long exposure to the market index, but have low exposure to volatile market movements via the mechanism of the covered call and the downside protection. Next, we perform a P&L explanation as a function of the diﬀerent Greeks of the portfolio. The methodology is explained in the Appendix, sectionB. In the following plot, we ﬁrst separate the contribution of the P&L in to each of the sensitivity components: Delta, Gamma, Vega, Theta, Vanna and Volga.

While the second order contributions (Gamma, Theta, Vanna and Volga) have large magnitudes, they mostly cancel out once aggregated. The following plot combines Gamma, Theta, Vanna and

64 Volga in to a single component.

As we can see, the Delta component captures the equity exposure of our strategy with respect to the index. The Vega is the exposure of the strategy to volatility, which only became significant during the GFC. The combined second order component represents the spread between implied and realised volatilities. Finally, there is a sizeable residual component which was not captured by our regression. Please note that our methodology has several limitations and should only taken as a rough approximation. For starters, the systematic strategy contains a large portfolio of options with a variety of absolute strikes and implied volatilities, so the terms affected by the implied volatility (such as Vega) are only crude empirical estimates of the actual components which are much more difficult to compute. Moreover, there would be correlations amongst the various components, so it is not possible our regression to properly distinguish the true contributions.

References

[1] Ang, I.-C., Israelov, R., Sullivan, R., and Tummala, H. Understanding the volatility risk premium. AQR research paper (2018).

[2] Ayed, A. B. H., Said, E., Bel, A., Ayed, H., Thillou, D., Rabeyrin, J.-J., and Abergel, F. Market impact: A systematic study of the high frequency options market. arXiv preprint arXiv:1902.05418 (2019).

[3] Fallon, W., Park, J., and Yu, D. Asset allocation implications of the global volatility premium. Financial Analysts Journal 71, 5 (2015), 38–56.

[4] Goetzmann, W., Ingersoll, J., Spiegel, M. I., and Welch, I. Sharpening sharpe ratio. NBER working paper (2002).

[5] Guo, I., and Loeper, G. The volatility risk premium: an empirical study on the s&p 500 index. CQFIS working paper (2018).

[6] Hull, J., and White, A. Optimal delta hedging for options. Journal of Banking & Finance 82 (2017), 180–190.

65 [7] Israelov, R., Klein, M., and Tummala, H. Covering the world: global evidence on covered calls. Journal of Risk, Forthcoming (2018).

[8] Jackwerth, J. C., and Rubinstein, M. Recovering probability distributions from option prices. The Journal of Finance 51, 5 (1996), 1611–1631.

[9] Londono, J. M. The variance risk premium around the world. Available at SSRN 2517020 (2015).

[10] Stoll, H. R., and Whaley, R. E. Expiration day eﬀects of index options and futures. No. 3. Salomon Brothers Center for the Study of Financial Institutions, Graduate . . . , 1987.