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Equity options strategies consist of a combination of options which are simultaneously entered into the market which enable one to achieve a financial return. A range of standard well-known strategies exist from which one can choose. The present paper looks beyond standard strategies and uses a memetic algorithm to choose from a myriad of option combinations in arriving at strategies which optimize specific fitness functions. The fitness function is formulated to find strategies that maximize return while, at the same time, limiting equity drawdown and achieving a desired rate of trade success. Over a decade of historical option data of the SPY, the S & P 500 Exchange Traded Fund, is used to choose strategies ranging from two to six option legs. Specific four- and six-leg option strategies were found to achieve optimum performance.

The great versatility of financial options has resulted in them becoming very popular in recent years as a means to achieve gains in the financial markets. Examples of the versatility can be seen by considering how profit can be made using options. This includes: 1) a correct prediction of the underlying price direction, 2) option time value decay, i.e. as an option is a time wasting asset then, for example, a seller of an option is able to buy back the option at a lower price at a later time, even though the underlying price may not have moved, thus achieving a profit, and 3) volatility prediction, e.g., if we again consider the case of selling an option when the underlying volatility is high (thus inflating the value of the option) one may buy it back at a lower price when volatility decreases thus achieving a profit. A typical example of this scenario is when an underlying has dropped in price typically resulting in volatility expansion which inflated the value of the option. At this point an option may be sold and subsequently if the underlying enters a period of price consolidation the volatility will contract, thus the value of the option will decrease at which point it may be bought back for a profit. These three different profit achieving mechanisms contrast, for example, with that of (non-dividend paying) equities where only price direction determines profitability. Further appreciation of the versatility of options can be obtained with the understanding that options can be used to mitigate risk, that is, limit the maximum loss that is achieved.

A single option position may be entered into the market, however, more generally a combination of option positions are entered. This combination is generally referred to as an option strategy. Well known option strategies include spreads, straddles and strangles for two-leg option combinations and condors for four-leg combinations. The profitability of certain option combinations has previously been examined in [

In the next section a brief overview of options is presented. This is followed in Section 3 with an introduction to basic evolutionary algorithms leading to the concept of the memetic algorithm. Section 4 provided a discussion on the implementation of the memetic algorithm used in this paper. The mechanism by which option strategies are described in terms of strike selection is presented in Section 5. In particular, we introduce the scaled, normalized strike mapping method. This method will be seen to have great performance advantages over the formerly used delta specified strike mapping method [

Financial (e.g. equity) options represent contracts between buyers and sellers and are traded on a public options exchange such as the CBOE (Chicago Board of Options Exchange). The contracts involve obligations and rights concerning buying and selling of the underlying equity associated with the option. There are a number of factors which determine the price of an option. Perhaps the foremost of these is the price of the underlying itself. Option contracts come in two types:

1) call.

2) put.

Both of these types can be bought or sold as an opening trade in an options transaction. Thus we can have the following four cases:

1) a bought call which is referred to as a long call.

2) a sold call which is referred to as a short call.

3) a bought put which is referred to as a long put.

4) a sold put which is referred to as a short put.

The payoff diagram shows how the profit and loss of an option varies with respect to the underlying price. The payoff diagrams (also referred to as the profit/loss (P/L) profiles) for the four cases mentioned above are shown in Figures 1(c)-(f). To put these in some context, the P/L profile of long and short stock is shown in

An option can be entered into the market anywhere in the moneyness range. Let’s consider the case of entering ATM, that is, the strike of the option chosen to be traded is at the current underlying price. For long calls and long puts a premium is paid to enter as indicated by the negative P/L value occurring at the ATM strike which is shown as 100 in

A number of options positions may be entered into the market simultaneously and represent an option strategy. Each strategy results in a characteristic P/L profile derived as an aggregate from its constituent P/L option profiles. The major aims of this paper are to derive an effective memetic algorithm which results in finding option combinations, i.e. strategies, which optimize specific performance metrics. This will be undertaken using historical option data for a time period of over a decade.

Nature has devised a variety of methods for dealing with challenging situations. Among these is the notion of survival of the fittest. Parents reproduce to create offspring. These offspring contain genetic material from both parents and therefore inherent some of their parents’ physical characteristics. Fitness is a metric indicating likely survival. Highly fit offspring exhibit behavior that promotes survival. These offspring could survive until adulthood and have an opportunity to reproduce. Low fit offspring quickly die. At population levels, highly fit species persist whereas low fit species go extinct.

Evolutionary algorithms (EAs) first appeared more than 50 years ago. These algorithms mimic neo-Darwinistic evolution from Nature to search for solutions to difficult problems. Although there are several different EA versions, they all work using evolution via natural selection as the means to search a problem’s solution space. All EAs have the following components:

・ a genotype or individual that encodes a problem solution

The genome contains all of the genetic material associated with an organism. All of the information needed for the organism to function is contained in its genome. In EAs the genome is a data structure containing all of the parameters needed to create a solution to a given problem. In nature the term genotype refers to the complete set of genes describing an organism; different gene values (or alleles) describe individuals with different physical attributes. Similarly, different solution parameters sets define different genotypes or problem solutions in an EA.

・ a population of individuals that evolves over time

EAs are population-based stochastic search algorithms. As the population evolves, it contains new more fit candidate solutions. The initial population is randomly generated but over time converges to highly fit solutions.

・ a fitness function that determines if an individual survives

The fitness function maps a solution genotype onto the real-number line. Highly fit solutions have higher (positive) values. Fitness values are used during the selection process to decide which solutions are kept in the population and which ones disappear.

・ one or more stochastic reproduction operators

Individuals in the current population are chosen as parents that produce offspring for the next generation. In some EAs parents are chosen proportional to fitness (higher fitness implies higher probability of selection) whereas in other EAs parents are chosen randomly. Recombination operators take genetic material (solution parameters) from each parent to form the offspring. Mutation operators randomly perturb existing solution parameters in the parent to produce the offspring. All reproduction operators are purely stochastic.

・ a selection operator that chooses survivors

This operator determines which offspring survive to become the parents in the next generation. In some EA versions offspring must compete against their parents for survival. The selection operator keeps the population size constant. Individuals not selected are discarded.

・ a termination criteria

The EA terminates under any one of the following conditions: 1) a fixed number of generations have been processed, 2) the algorithm is assumed to have converged―i.e., no fitness improvement in 10 - 20 generations, or 3) a sufficiently good enough solution has been located.

Algorithm 1 shows the pseudo code for a generic EA. Each time through the while loop is considered a generation.

Any search algorithm has a limited computation budget. It must therefore tradeoff exploration and exploitation. Exploration refers to an ability to quickly find subsets in genotype space that contain highly fit solutions. Exploitation is the ability to find the best fit solutions in these regions. Too much time spent in exploitation limits the time available to explore. EAs are particularly good at exploration but not so good at exploitation. This has given rise to memetic algorithms [

The most common form of local search is Lamarckian local search. Let i be an arbitrary parent genotype. The neighborhood of individual i, denoted by N ( i ) , consists of those genotypes that are in some sense “close” to the parent. That is, a neighbor j ∈ N ( i ) has a genotype that differs only slightly from the genotype i. In a Lamarckian local search a neighbor j replaces the parent i if and only if f ( j ) ≥ f ( i ) .

Algorithm 2 shows how a Lamarckian local search is conducted. There are two types of termination criteria. In a greedy ascent the search terminates as soon as a higher fit neighbor is found whereas in a steepest ascent all neighbors are checked. An EA becomes a memetic algorithm by inserting a local search after line number 5 in Algorithm 1.

In this work, the genome for our memetic algorithm, which encodes an options strategy, is an N-bit binary string. This string may be partitioned into four equally sized sub-strings (see

A number of constraints were placed in determining the fitness of potential

solutions in order to lead to desired results. The constraints were:

1) Non-zero slope segments of P/L profiles were constrained to that of single option types, that is, a slope of plus or minus unity. Most currently used, popular option strategies have this feature. (In future work this constraint will be relaxed).

2) Potential strategies which feature offsetting options at the same strike were discarded. Thus, a strategy with a long put (call) as well as a short put (call) at the same strike was not carried forward in this early stage in the algorithm.

3) Moneyness of selected options was constrained to avoid the use of deep in the money (DITM) options. DITM options have a large margin requirements as well as unfavorably wide bid-ask spreads. The term margin refers to the amount of funds necessary to put on a position. Large margin requirements limit the returns of the option strategy. It was decided, somewhat arbitrarily, to limit ITM options to the first three ITM strikes.

Any potential solution which violated any of the above conditions had a fitness value of zero returned and thus were removed from further processing.

Before option strategy performance metrics were evaluated by proceeding through the list of historical option chain data a potential solution was assessed for its feasibility, based on the above constraints, by application on a single option chain. As mentioned above, should this fail a zero fitness resulted. If passed, however, then the option configuration and the individual option delta values or, alternatively, the scaled, normalized strike values of the strike prices were recorded. These values were then used to map the strategy to other historical option chains. More will be said about strike mapping in the next section. Mapping the strategy into other time periods enables the trade P/L’s for the full historical period to be determined. At the end of this procedure a time series array of trade P/L values for the total historical data period is made available which enables the determination of the equity curve (i.e. cumulative sum of profits) and maximum drawdown percentage.

Option strategy performance metrics involved the use of two different fitness functions. The first required determining the average P/L per trade and percentage of profitable trades obtained throughout the time period of the historical data. The percentage of profitable trades needed to meet a desired threshold (we adopted an 80% threshold) so that the average P/L per trade was returned as the fitness value, otherwise a zero fitness value was returned. The second fitness function used was an extension of the first. Now the extra threshold of requiring the equity drawdown to be within certain limits was imposed on top of the percentage profitability requirement before setting the fitness value to the average P/L value determined. We considered maximum drawdown limits at the various levels of 40%, 30%, 20% and 10%. It was instructive to see how the strategies are progressively modified to achieve the drawdown limit requirements.

The MA is run for 30 generations with a population size of 200. Each individual encodes a trading strategy with K options; thus each individual has exactly K bits, where K is an integer, K ∈ [ 2,6 ] , set to logic 1 and the N-K remaining bits are logic 0. These K bits generally need to be distributed beyond one segment for feasible solutions to exist.

A cursory glance indicates most solutions have zero fitness because they are infeasible i.e., they violate one or more constraints. This poses a problem when constructing the initial population of the MA since initial populations are randomly generated. The MA must begin the search with only feasible individuals in the population. To accomplish this task we used a simple, brute force technique: a solution was randomly generated. It is inserted into the initial population if and only if it was feasible. Otherwise it was discarded and another solution was randomly generated. This process was repeated until a population of the desired size was created.

In MAs the search is conducted by applying stochastic reproduction operators to parents (existing solutions) to create offspring (new solutions) which are then evaluated for fitness. The problem is most solutions in the search space are infeasible, so conventional reproduction operators, such as recombination, are likely to be ineffective i.e., they will generate infeasible offspring. Indeed, our early trials found this to be the case. We therefore took a slightly different approach. It is reasonable to assume if an individual is feasible, so will at least some of its neighbors in the search space also be feasible. We simply cloned the individual and then let a local neighborhood search, called a local refinement, generate the offspring.

The local refinement used follows the Larmarckian Learning paradigm [

The neighborhood size depends on the distance of the randomly chosen logic 1 to the closest set bit above and below in the same substring or in other substrings. Each substring represents the same range of strikes but for different option types and positions. Let us consider a genome that consists of N bits, then there are four substrings of length N/4 (where N/4 is an integer value). A randomly chosen logic 1 from amongst the N possible positions of the genome would translate to a certain position in the individual substring. Let us denote this position as i where 1 ≤ i ≤ N / 4 . The neighborhood is bounded from above by the closest set bit in the substring or by an identical position in any of the substrings. Let us denote this position as j where j > i . If there are no set bits, then j is set to the position value of the end of the substring, i.e. j = N / 4 . A similar discussion holds for finding the lower bound of the neighborhood. In this case we find the closest set bit below the randomly chosen bit in any of the substrings. Let us denote this position as k where k < i . If there are no set bits, then k is set to the position value of the start of the substring, i.e. k = 1 . Thus the neighborhood range is seen to be [k, j], which is a subset of the total range of a substring, i.e. [ 1, N / 4 ] . This neighborhood would next need to be translated to the appropriate position range in the genome which we will denote as N ( [ k , j ] ) . A neighbor is generated by clearing the initially chosen logic 1 bit and setting to logic 1 a bit m ∈ N ( [ k + 1, j − 1 ] ) , in the case where neither k or i represent a start or endpoint of a string, i.e. k ≠ 1 and j ≠ N / 4 , or if k = N / 4 , i.e. there is no set bit above the randomly chosen bit, then a bit m ∈ N ( [ k + 1, j ] ) is set to logic 1 (in the appropriate substring), or if k = 1 , i.e. there is no set bit below the randomly chosen bit, then a bit m ∈ N ( [ 1, j − 1 ] ) is set to logic 1 (in the appropriate substring). The neighbor replaces its parent in the population if and only if it has higher fitness. In our approach, we used a steepest ascent search so that all solutions in the neighborhood were tried with the best replacing the parent.

Let us illustrate this with an example. Consider a genome of length N = 160 . There will be four substrings each of length N / 4 = 40 . Thus in this case we see that there are 40 different strike values, which for the purpose of this example, range from 111 to 150. Suppose a random initialization results with the following four bits being set to 1 in the genome: 12, 64, 75 and 127. This translates to bit 12 in the long call option substring being set which represents a strike of 122, bits 24 and 35 in the short call substring being set which represents strikes of 134 and 145 and bit 7 in the short put option substring being set which represents a strike of 117. This initialization is shown in

Let us now consider that the randomly chosen set bit of the four is bit 64 of the genome, that is, the 24th strike in the short call options substring which has strike value of 134. The range of the neighborhood can be readily seen in

We also used elitism to help improve the search result―i.e., the best fit individual from the previous generation was cloned and replaced the worst fit individual in the current generation.

A strategy is initiated by noting the current underlying price and determining the appropriate option strikes in relation to this. Since clearly the underlying price moves around an effective relationship between the underlying price and the strategy option strikes needs to be found. That is, one does not have the luxury of specifying the option strikes directly but rather these need to be deduced indirectly using the underlying price at trade initiation.

In [

With the delta mapping method, given a specified delta value, the option whose strike is closest to the desired delta was chosen. Thus in this way the delta specification maps into a desired strike.

In this paper an alternative mapping is proposed and tested for its efficacy. Rather than specifying delta values, the alternative of specifying the scaled, normalized strike values is used. In this method, strikes are first normalized by dividing them by the current underlying price. So that, for example, using the data given in

After the strikes are normalized, they are then scaled in order to expand the strike selection. Strikes above the current underlying price are scaled by multiplying the normalized strike value by the scale factor, 1.03, i.e. increasing it by 3%. So in this case the scaled normalized strike value of a 120 strike option in

The scaling factors used in this paper were determined through a number of simulation runs by increasing/decreasing the scaling factor in 1% increments/decrements. The chosen scale factors gave the best results.

Thus with this method, the constituent option strikes for option strategies are specified by a scaled, normalized option strike value, so that when given the underlying price one searches for the option strike in the option chain that is closest to the desired value.

Row | Call Bid | Call Ask | Call Delta | Strike | Put Bid | Put Ask | Put Delta | Normalized Strike | Scaled Strike |
---|---|---|---|---|---|---|---|---|---|

1 | 25.69 | 25.97 | 0.9709 | 89 | 0.14 | 0.17 | −0.02538 | 0.777 | 0.753 |

2 | 24.71 | 24.99 | 0.96767 | 90 | 0.16 | 0.19 | −0.02868 | 0.785 | 0.762 |

3 | 23.73 | 24.02 | 0.96376 | 91 | 0.18 | 0.21 | −0.03211 | 0.794 | 0.770 |

4 | 22.76 | 23.04 | 0.95969 | 92 | 0.20 | 0.23 | −0.03568 | 0.803 | 0.779 |

5 | 21.79 | 22.07 | 0.95492 | 93 | 0.23 | 0.26 | −0.04056 | 0.811 | 0.787 |

6 | 20.80 | 21.14 | 0.94891 | 94 | 0.26 | 0.29 | −0.04561 | 0.820 | 0.796 |
---|---|---|---|---|---|---|---|---|---|

7 | 19.85 | 20.13 | 0.94476 | 95 | 0.29 | 0.32 | −0.05088 | 0.829 | 0.804 |

8 | 18.89 | 19.17 | 0.93825 | 96 | 0.33 | 0.36 | −0.05751 | 0.838 | 0.812 |

9 | 18.02 | 18.17 | 0.92893 | 97 | 0.37 | 0.41 | −0.06493 | 0.846 | 0.821 |

10 | 17.07 | 17.22 | 0.92083 | 98 | 0.42 | 0.46 | −0.07321 | 0.855 | 0.829 |

11 | 16.13 | 16.28 | 0.91137 | 99 | 0.48 | 0.51 | −0.08236 | 0.864 | 0.838 |

12 | 15.20 | 15.35 | 0.90054 | 100 | 0.56 | 0.57 | −0.09308 | 0.873 | 0.846 |

13 | 14.28 | 14.42 | 0.88878 | 101 | 0.62 | 0.65 | −0.10495 | 0.881 | 0.855 |

14 | 13.36 | 13.50 | 0.87600 | 102 | 0.70 | 0.74 | −0.11843 | 0.890 | 0.863 |

15 | 12.44 | 12.59 | 0.86211 | 103 | 0.79 | 0.83 | −0.13263 | 0.898 | 0.872 |

16 | 11.56 | 11.70 | 0.84499 | 104 | 0.90 | 0.93 | −0.14958 | 0.907 | 0.880 |

17 | 10.67 | 10.82 | 0.82705 | 105 | 1.03 | 1.06 | −0.16900 | 0.916 | 0.889 |

18 | 9.81 | 9.96 | 0.80629 | 106 | 1.16 | 1.19 | −0.18901 | 0.925 | 0.897 |

19 | 8.99 | 9.10 | 0.78312 | 107 | 1.31 | 1.35 | −0.21226 | 0.934 | 0.906 |

20 | 8.17 | 8.27 | 0.75779 | 108 | 1.51 | 1.53 | −0.23896 | 0.942 | 0.914 |

21 | 7.37 | 7.47 | 0.72966 | 109 | 1.69 | 1.73 | −0.26698 | 0.951 | 0.923 |

22 | 6.61 | 6.69 | 0.69850 | 110 | 1.92 | 1.96 | −0.29869 | 0.960 | 0.931 |

23 | 5.87 | 5.95 | 0.66440 | 111 | 2.18 | 2.22 | −0.33340 | 0.969 | 0.939 |

24 | 5.17 | 5.23 | 0.62741 | 112 | 2.49 | 2.51 | −0.37132 | 0.977 | 0.948 |

25 | 4.50 | 4.56 | 0.58739 | 113 | 2.81 | 2.84 | −0.41191 | 0.986 | 0.956 |

26 | 3.87 | 3.93 | 0.54459 | 114 | 3.17 | 3.21 | −0.45546 | 0.995 | 0.965 |

27 | 3.28 | 3.33 | 0.49921 | 115 | 3.58 | 3.63 | −0.50167 | 1.003 | 1.034 |

28 | 2.74 | 2.79 | 0.45171 | 116 | 4.03 | 4.07 | −0.55041 | 1.012 | 1.043 |

29 | 2.25 | 2.29 | 0.40253 | 117 | 4.54 | 4.59 | −0.60043 | 1.021 | 1.052 |

30 | 1.81 | 1.85 | 0.35261 | 118 | 5.10 | 5.15 | −0.65152 | 1.030 | 1.061 |

31 | 1.43 | 1.47 | 0.30331 | 119 | 5.71 | 5.80 | −0.70137 | 1.038 | 1.070 |

32 | 1.11 | 1.15 | 0.25603 | 120 | 6.37 | 6.49 | −0.75037 | 1.047 | 1.078 |

33 | 0.84 | 0.87 | 0.21069 | 121 | 7.10 | 7.22 | −0.79662 | 1.056 | 1.087 |

34 | 0.62 | 0.66 | 0.17032 | 122 | 7.86 | 8.00 | −0.84080 | 1.065 | 1.096 |

35 | 0.45 | 0.49 | 0.13473 | 123 | 8.67 | 8.94 | −0.87056 | 1.073 | 1.105 |

36 | 0.32 | 0.36 | 0.10453 | 124 | 9.53 | 9.80 | −0.90383 | 1.082 | 1.114 |

37 | 0.23 | 0.26 | 0.08014 | 125 | 10.42 | 10.70 | −0.93211 | 1.091 | 1.123 |

38 | 0.16 | 0.19 | 0.06061 | 126 | 11.34 | 11.63 | −0.95526 | 1.099 | 1.132 |

39 | 0.11 | 0.14 | 0.04552 | 127 | 12.28 | 12.57 | −0.97723 | 1.108 | 1.141 |

40 | 0.08 | 0.10 | 0.03419 | 128 | 13.25 | 13.53 | −0.96887 | 1.117 | 1.150 |

41 | 0.06 | 0.08 | 0.02708 | 129 | 14.22 | 14.51 | −0.97616 | 1.126 | 1.159 |

We will soon see that this scaled, normalized strike mapping method performs better, under a variety of conditions, than both an unscaled, normalized strike mapping method as well the previously used delta strike mapping method.

In this section results are examined from searches performed by the MA. As mentioned previously, the MA is configured to maximize two different fitnesses:

1) Average profit per trade under the constraint that the probability of a profitable trade is greater than 80%.

2) The same as 1), but with the added constraint that the maximum percentage equity drawdown is limited to the various levels of 40%, 30%, 20% and 10%. These percentages reduce risk to the desired levels.

The underlying equity used in our study has ticker symbol SPY, which is the S & P 500 ETF (Exchange Traded Fund). The option data was obtained from IVolatility.com. The obtained data was sufficient to examine performance of 138 trades placed from January 10, 2005 to the exit of the final trade on July 15, 2016.

The trading protocol was as follows. Trades were entered on the close on the first trading day of the month and exited on the close on option expiration day which occurs on the 3^{rd} Friday of the following month. This allowed EOD (end of day) option data to be used. No adjustments were made to the trade during any trading period.

The memetic algorithm uses the data to discover option strategies which, over the time period of the historical data, maximizes the two performance metrics used. As will be seen below this produced a number of new and interesting option strategies. The strategies were categorized with respect to the number of option legs present. These were limited to two through six. Furthermore, the optimum number of option legs was thus also determined.

In [

First we consider the results of the delta strike mapping approach shown in

Target DD % | Option Type (P/C) | Long/ Short (L/S) | Delta (Abs. value) | Example ( | Average Profit/ Trade ($) | Total Profit ($) | Percent Profitable (%) | Max DD (%) |
---|---|---|---|---|---|---|---|---|

P | S | 0.212 | 107 | |||||

None | C | L | 0.403 | 117 | 71 | 9766 | 81.2 | 48.4 |

C | S | 0.211 | 121 | |||||

C | S | 0.135 | 123 | |||||

P | S | 0.212 | 107 | |||||

40 | C | L | 0.403 | 117 | 64 | 8835 | 81.2 | 39.7 |

C | S | 0.256 | 120 | |||||

C | S | 0.211 | 121 | |||||

P | S | 0.212 | 107 | |||||

30 | P | S | 0.239 | 108 | 48 | 6567 | 82.6 | 29.4 |

P | L | 0.333 | 111 | |||||

C | S | 0.135 | 123 | |||||

P | L | 0.025 | 89 | |||||

20 | P | L | 0.029 | 90 | 38 | 5257 | 81.9 | 14.7 |

P | S | 0.212 | 107 | |||||

C | S | 0.105 | 124 | |||||

P | L | 0.036 | 92 | |||||

10 | P | L | 0.041 | 93 | 36 | 4967 | 82.6 | 8.6 |

P | S | 0.212 | 107 | |||||

C | S | 0.105 | 124 |

As per the strike mapping technique discussed here the option strategies may be reproduced using the specified delta values. Examples using the data shown in

No maximum drawdown limitation: The P/L profile and equity curve for the option strategy discovered which imposes no maximum drawdown limitation are shown in

40% maximum drawdown limitation: Setting the maximum drawdown to a figure of 40% results in the P/L profile and equity curve shown in

30% maximum drawdown limitation: A 30% maximum drawdown target results in achieving an average profit per trade of $48 and an achieved maximum drawdown of 29.4%. The associated P/L profile and equity curve for the discovered option strategy are shown in

20% maximum drawdown limitation: With greater demands on reducing the drawdown the P/L profile further changes so that for very large underlying losses an appreciable profit can be achieved. The P/L profile and equity curve resulting from a target of 20% max drawdown are shown in

10% maximum drawdown limitation: The P/L profile and associated equity curve for a targeted 10% maximum drawdown changes little from the previous 20% drawdown limit case. These are shown in

We next consider a scaled, normalized strike as the basis by which the strikes of an option strategy are selected.

In

We will now look more closely at the results obtained by use of the scaled, normalized strike mapping method for each of the five option strategies with number of option legs varying from two to six. These are presented in

Two-leg option’s strategy: The option strategy chosen is one commonly known as a strangle, i.e. short call and short put. The delta mapping method also chose a strangle as the optimum two-leg option’s strategy [

Three-leg option’s strategy: The strategy chosen was a risk reversal with capped upside profitability, as before [

Four-leg option’s strategy: The P/L profile for this strategy is shown in

Five-leg option’s strategy: The P/L profile for this strategy is shown in

Number of Options | Delta Mapping Method ($ per Trade/DD %) | Normalized Strike Mapping Method ($ per Trade/DD %) | Scaled, Normalized Strike Method ($ per Trade/DD %) |
---|---|---|---|

2 | 55/34.5 | 66/38.6 | 75/33.8 |

3 | 52/60.4 | 56/61.5 | 69/55.0 |

4 | 68/48.3 | 72/44.9 | 87/42.8 |

5 | 56/58.4 | 58/51.4 | 78/47.1 |

6 | 65/48.9 | 69/50.9 | 87/35.0 |

No. of Opts. | Option Type (P/C) | Long/ Short (L/S) | Scaled Norm. Strike | Example ( | Average Profit/ Trade ($) | Total Profit ($) | Percent Profitable (%) | Max DD (%) |
---|---|---|---|---|---|---|---|---|

2 | C | S | 0.948 | 112 | 75 | 10,417 | 84.1 | 33.8 |

P | S | 1.061 | 118 | |||||

C | S | 0.948 | 112 | |||||

3 | C | L | 1.034 | 115 | 69 | 9508 | 85.5 | 55.0 |

P | S | 1.061 | 118 | |||||

C | S | 0.948 | 112 | |||||

4 | C | L | 1.034 | 115 | 87 | 12,005 | 85.5 | 42.8 |

P | S | 1.061 | 118 | |||||

C | S | 1.070 | 119 | |||||

C | S | 0.948 | 112 | |||||

C | L | 1.034 | 115 | |||||

5 | P | S | 1.061 | 118 | 78 | 10,773 | 86.2 | 47.1 |

C | S | 0.105 | 119 | |||||

C | L | 1.114 | 124 | |||||

C | S | 0.948 | 112 | |||||

C | S | 0.956 | 113 | |||||

6 | P | L | 0.965 | 114 | 87 | 12,001 | 87.0 | 35.0 |

C | L | 1.034 | 115 | |||||

P | S | 1.052 | 117 | |||||

P | S | 1.061 | 118 |

shown in

Six-leg option’s strategy: The P/L profile for this strategy is shown in

Conclusion from the results presented in

In the previous sections we have seen that the scaled, normalized strike mapping method is able to greatly improve the performance in terms of profitability and maximum drawdown. To further investigate this we will now look to imposing maximum drawdown limits at various levels. This was done earlier using the delta mapping method, and so a comparison with these previous results can be made.

In the discussion below reference is made to ITM options that were selected for the strategies that were discovered. As previously mentioned, due to margin requirements the ITM options were restricted to just the first three ITM strikes. With reference to

No maximum drawdown limitation: The P/L profile and equity curve for the option strategy discovered which imposes no maximum drawdown limitation are shown in

40% maximum drawdown limitation: Setting the maximum drawdown limit to a figure of 40% results in the P/L profile and equity curve shown in

30% maximum drawdown limitation: A 30% maximum drawdown target results in achieving an average profit per trade of $66 and an actual maximum drawdown of 29.6% as shown in

20% maximum drawdown limitation: With greater demands on reducing the drawdown the P/L profile further changes so that for very large underlying losses an appreciable profit can be achieved. The P/L profile and equity curve are shown in

Target DD % | Option Type (P/C) | Long/ Short (L/S) | Scaled Norm. Strike | Example ( | Average Profit/ Trade ($) | Total Profit ($) | Percent Profitable (%) | Max DD (%) |
---|---|---|---|---|---|---|---|---|

C | S | 0.948 | 112 | |||||

None | C | L | 1.034 | 115 | 87 | 12,005 | 85.5 | 42.8 |

P | S | 1.061 | 118 | |||||

C | S | 1.070 | 119 | |||||

C | S | 0.948 | 112 | |||||

40 | C | L | 1.034 | 115 | 83 | 11,457 | 86.2 | 38.5 |

C | S | 1.052 | 117 | |||||

P | S | 1.061 | 118 | |||||

P | S | 0.931 | 110 | |||||

30 | C | S | 0.948 | 112 | 66 | 9134 | 87.0 | 29.6 |

P | L | 0.965 | 114 | |||||

P | S | 1.061 | 118 | |||||

P | L | 0.889 | 105 | |||||

20 | C | S | 0.948 | 112 | 42 | 5728 | 80.4 | 9.7 |

P | S | 1.061 | 118 | |||||

C | L | 1.114 | 124 | |||||

P | L | 0.889 | 105 | |||||

10 | C | S | 0.948 | 112 | 43 | 5987 | 80.4 | 7.1 |

P | S | 1.061 | 118 | |||||

C | L | 1.123 | 125 |

10% maximum drawdown limitation: The P/L profile and associated equity curve for a targeted 10% maximum drawdown changes little from the previous 20% drawdown limit case. These are shown in

Conclusion from the results presented in

In this paper the work initiated in [

The results presented considered strategies ranging from two to six option legs such that an 80% winning trade percentage was achieved. Both a six- and four-leg strategy was found to be optimum. Further examination of four-leg strategies was made to see the effect on strategy structure as the maximum drawdown percentage limit was progressively decreased in 10% decrements from 40% to 10%. This was done using the scaled, normalized strike mapping as well the previously used strike mapping method using deltas. At all drawdown levels the scaled, normalized strike mapping approach was found to give superior results.

In summary, the main contributions to this work are threefold:

1) a memetic algorithm has been formulated to find effective option trading strategies,

2) a number of strategies have been uncovered, the most effective of which are four- and six-leg strategies, and

3) a method of selecting strikes was devised which optimizes performance in regards to both profit and drawdown. Thus not only has the basic form of optimal strategies been found, but also, an effective method to assign strikes has also been proposed.

Considering the degree of influence that the method of determining strikes has, further work will involve incorporating a volatility component into the scaled, normalized strike mapping approach as a possibility to further enhance the results. We anticipate that this enhancement will lead to higher profits as well as reduced drawdowns. Also the use of ITM options and their tradeoff with margin requirements should also be investigated.

The authors declare no conflicts of interest regarding the publication of this paper.

Tymerski, R. and Greenwood, G. (2018) Designing Equity Option Strategies Using Memetic Algorithms. Technology and Investment, 9, 179-202. https://doi.org/10.4236/ti.2018.94013