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MSc International Finance

Master Thesis

Thesis

Supervisor: Simon Benninga

Trailing stop-loss order

strategies in

algorithmic trading

A high frequency trading approach

Sinziana-Andra Lichi

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Abstract

This paper analyses two trailing stop loss strategies in order to attain an optimal strategy for a retail or institutional investor. One strategy is based on the random walk model with absorbing barriers and the other trails the best price to date by a fixed distance. The aim of this paper is to test two types of trailing stop-loss algorithms using a high frequency trading approach and determine a viable strategy which can be used by a financial trader in the FOREX market. The importance of optimisation of stop-loss algorithms will be emphasised. After a quantitative and qualitative analysis, both algorithms perform well on certain currency pairs. Investor risk adversity and trading style should be considered in further assessment.

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2 Contents

1. Introduction ... 3

2. Literature review ... 4

3. Methodology and hypothesis ... 7

4. Results and analysis ... 10

4.1. Back-testing algorithms - training and optimisation ... 10

4.2. Validation... 18

5. Conclusion ... 24

6. Recommendations ... 25

7. Appendices ... 26

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1. Introduction

The portfolio optimisation framework pioneered by Markowitz (1952) has had a prodigious dominance in the finance and investments literature, both in academia and in the industry. This has focused on constructing well-diversified static portfolios using low-cost index funds (Kaminski and Lo, 2007). Furthermore, the framework has led to a preconceived concept that active trading is too risky and too costly. Hence, the “buy and hold strategy” entered and invaded the mutual fund and financial planning professions. The passive approach is, however, often contradicted by behavioural finance, investors being in the unfavourable position, especially in moments of market turmoil, to “sell at the low” and “buy at the high” (Kaminski and Lo,2007).

Trading strategies usually consider mainly how to speculate the market and entry a trade with the exit being a distant aspect. The ability to exit a trade well can distinguish, however, between a profitable and a non-profitable trade. The reason for the lack of emphasis on the strategy is the fact that most technical traders have adopted a static type of stop loss – exit at a specific price. Due to highly volatile financial markets, with foreign exchange being in the lead in terms of volatility, the trader may miss maximizing the profit potential of a trade if the trend continues beyond expectations (Miner R.C, 2009).

A highly important aspect in active trading and frequent rebalancing is the stop-loss strategy. For stop-loss orders, the security is to be sold if its price falls below a stipulated limit. The stop loss order is activated when the stock hits a price limit. As the name suggests, the order lets the stock be sold to stop further losses from accumulating (Bodie et al.,2011). The most basic stop-loss strategy is placing a stop loss at a fixed percentage below the purchase price at the time of entry. For instance, if an investor is willing to accept a 5% loss on a position and he entered a position at 60, then the stop loss will be placed at 57.

The trailing stop-loss is an order placed when the investor enters a position. Stop order levels are also trailed below (above) the market price on long (short) positions. Trailing means that stops are raised (lowered) when the asset price rises (falls) in a

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long (short) trade, but remain stationary when the price falls (rises). Trailing is done when the market price moves in favour of the trade generating a profit (Brown et al, 2010). Trailing stops are designed to lock-in profits while allowing for further gains on a trend-following strategy (Acar and Toffel, 2000).

What the Efficient Market Hypothesis (EMH) states is that returns from speculative assets cannot be forecasted (Timmermann and Granger, 2004). If the EMH holds true, a stop-loss strategy will not be enforceable. As the concept of efficient markets does not consider systematic trading to be a valid concept, the stop-loss algorithms could not be accounted for in this context.

According to behavioural finance literature, investors tend to be more reluctant to realize losses than gains or in other words they tend to sell losers too late and winners too early (Bensaid and de Bandt, 1998). This psychological bias can be overcome by putting in place a clear automated stop loss strategy.

The aim of this paper is to test two types of trailing stop-loss algorithms using a high frequency trading approach and determine a viable strategy which can be used by an institutional trader in the FOREX market.

The outline of the paper is as follows. Section 2 discusses the relevant literature on stop-loss and trailing stop-loss algorithms. Section 3 presents the methodology used in the paper along with the presentation of the algorithms being tested. Section 4 presents the results of the two trailing stop-loss algorithms and it analyses the performance using risk-adjusted measures. Section 5 concludes and section 6 provides a recommendation.

2. Literature review

The aim of the algorithms is to enhance order execution by strategically placing orders using computer-based pattern recognition programs with potential (not necessary) subsequent human judgment or intervention (Brownlees et al., 2011).

The fundamental building block for any trading strategy is the trading orders. Orders represent execution instructions. There are two main order types: market and limit orders. In addition, to those two, the standard orders, there are many other types

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such as: conditional, hidden, discretional, and routed and order-contingent (Johnson, 2010).This paper will mainly treat a conditional type of order: Trailing Stop. Trailing stops are often used in stock trading to limit the maximum of a possible loss and to lock in a profit.

A stop-loss order is an order placed with a broker to sell once the stock drops to a certain price level. A stop loss is designed to limit the investor’s loss on a security position (Yin et al., 2010).

Stop-loss orders are used in controlling losses. Stop loss orders are activated once an asset’s price hits a sufficiently unprofitable level and the trader exits the market with a market order. A trailing stop-loss strategy is a combination strategy. In this type of strategy a stop-loss order is periodically adjusted to lock in profits as the market moves in a favourable direction (Warburton et al., 2004). For instance, given a long trading position, a trader might place an order with a stop-loss 10% below the price of the security. In case the price goes up by 5% without the stop-loss being triggered, the stop-loss order is raised to 10% below the current price. This process continues until the stop-loss order is triggered.

There is not extensive research done on potential trade exit strategies. Trailing stop strategies have been considered by Glynn and Iglehart (1995), Warburton and Zhang (2006), Yin et al. (2010).

Warburton and Zhang (2006) have considered strategies in which the stop-loss order is raised only if a predetermined price target has been achieved or if minimally acceptable price performance has been achieved over a planning horizon.

While Glynn and Iglehart (1995) consider strategies in which a stop-loss order trails the best price to date by a fixed distance. In their model, each time the time price process reaches a new high, the stop level is raised by the difference between the new high and the previous high.

Warburton and Zhang (2006) discuss a computational method of trailing stop-loss in their paper. The price movements they model have a finite horizon and are done using the random walk model with absorbing barriers. The upper and lower

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absorbing barriers are defined as the stop loss and profit targets. The model is illustrated by the authors using a discrete time recombining trinomial tree in which the price moves up one level, stays constant or moves down one level at each time period. According to Warburton & Zhang(2006) their algorithm is readily implemented in practice and it has more flexibility than Glynn and Iglehart’s algorithm.

Acar and Toffel (2000) have also discussed the trailing stop-loss concept which they defined as an eclectic mix of a take profit and stop loss strategy. They explain that trailing stops are designed to lock-in profits while allowing for further gains on a trend-following strategy.

Yin et al (2008) developed stochastic approximation algorithms to estimate the optimal trailing stop percentage in a continuous case. They verified the derived optimal trailing-stop strategy by looking at real data and comparing it with a moving-average strategy. They found that the moving-average return from a trailing-stop was 71.45%, while the average return from a moving average was only 11.45%; the trailing stop therefore outperformed significantly. Abramov et al. (2008) assumed a binomial distribution of prices and described the features of an optimal trailing-stop strategy in a discrete setup. They analysed the relations among important statistics under certain conditions and claimed that there may not exist a trading strategy generating positive discounted gains.

There are other ways to obtain similar pay-offs to the stop-loss and profit taking strategies. One way is to trade a barrier option. A barrier option can be defined as an option where the payoff depends on whether the underlying asset’s price reaches a certain level during a specific period of time (Hull, 2012). These options can either cease to exist when the underlying asset price reaches a certain barrier (knock-out option), either begin their existence once a barrier was hit (knock-in option). This type of derivative is, however, only traded on over the counter markets and is not, therefore, easily accessible to a retail investor (Warburton, 2006).

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3. Methodology and hypothesis

The objective of the thesis is to find an optimal trailing stop loss strategy. This will be done by testing the methods proposed by Glynn and Iglehart (1995) and Warburton and Zhang (2006) against real time market data. The methods tested will be named: Trailing Stop-Loss 1 (TSL 1) and Trailing Stop-Loss 2(TSL 2) respectively. The data will consist of real time market data from a renowned market maker. The tests will be performed from the point of view of the buy side investor and only long trades will be considered.

Trailing Stop-Loss 1 is a strategy in which a stop-loss order trails the best price to date by a fixed distance. While in the second algorithm, we will back-test, the Trailing Stop-Loss strategy, the stop loss order is raised only if a predetermined price target has been achieved or if a minimally accepted price performance has been achieved over a planning horizon. Both algorithms are presented and adapted in the Appendices’ section.

In order to establish which stop loss is the most efficient one, the objectives and expectations of the investor should be considered. There are two main objectives investors may have: risk control and return enhancement. Firstly, a stop-loss strategy is expected to minimize the chances of losing significantly in a large market movement against a few positions opened. In this case, a stop-loss order should be able to limit losses to small amounts by avoiding a possible loss which could have undermined the capital base so much as to almost leading to a halt in the trading activity. This can also be defined as the risk control. Secondly, a placed stop-loss represents a discipline which forces traders to decrease loss on losing positions and increase revenue on winning ones. Trailing stop-losses especially are expected not to constrain profit potential and limit losses to small amounts. This would, thus, help an investor reach its second objective, to enhance revenues over time (Chan, 2013).

These are two investor objectives which will be considered when discussing the efficiency of the stop-losses throughout this paper.

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What we will do is to back-test the algorithms mentioned against the real time market data between 6th and 8th of May 2013 on three major currencies. For the first three days of trading of the week 6th-10th May, the algorithms will be back-tested and the main parameters of the algorithms will be optimised. Following an optimisation has been effectuated; a validation will be done on the next two days of trading. The tests were performed using the computer software programme Matlab.

We apply the methodology described above to representative series from the foreign exchange asset class. We use the EUR/USD, USD/JPY and GBP/USD. Our choice of data series is mainly due to availability, holistic coverage of the largest markets of foreign exchange and a combination of high volume and liquidity. The EUR/USD exchange rate is of prime interest to currency traders worldwide and its modelling has a significant relevance during the current turbulent times. This paper hopes to be able to provide

the buy side investor a practical solution to where to place the stop loss levels as an automated strategy.

Data used for testing

Data on the foreign exchange pairs will be taken from the data feed received from the financial broker. This will provide more accurate and realistic results. We will be testing on a short period of time- one week- using a high frequency approach. The sample on EUR/USD consists of 234,477 bid price observations, the GBP/USD sample 222,918 and the one on USD/JPY consists of 234,835 observations. This is the data for a full trading week. This specific week was chosen randomly out of the available data set.

Assumptions made for the algorithms

The ask and the bid prices have been used when computing the entries in a transaction and the values of the stop-losses, respectively. There is no reason, therefore, for extra transaction costs to be considered in the calculation of any profit measurements. The bid-ask spread is an accurate measurement of the cost an investor incurs.

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Optimisation

In order to assess the profitability of the two stop-loss strategies, entries in transactions had to be considered. In order to be unbiased in the assessment of the stop–loss values, the entries were generated randomly. The algorithms were programmed to enter a transaction roughly every ten minutes. For the TSL2 algorithm, a value for the target price level had to be assigned. The initial value given was 0.75% - this was based on (Stridsman ,2003) and optimised further on. Parameter optimisation is done during a back-testing procedure between 6th and 8th of May 2013 based on the Calmar ratio. The Calmar ratio is a risk-adjusted measure of performance that is given by the formula:

(Magdon-Ismail et al., 2004) The maximum cumulative loss from a market peak to a market through is called the maximum drawdown (MDD). This measures how sustained an investor’s losses can be (Magdon-Ismail et al, 2004).

One of the main reasons the Sharpe ratio, the most common risk-adjustment method, was not used for optimisation purposes is that it penalises positive volatility. The measure does not distinguish between upside and downside risk and also it does not make a distinction between the irregular losses as opposed to the repeated ones. While an investor can have a large return on a position due to high volatility, the Sharpe ratio does not account for it (Gregoriou and Gueyie, 2003; Eling and Schuhmacher, 2007).

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4. Results and analysis

4.1. Back-testing algorithms - training and optimisation

Both strategies have been simulated for long strategies and therefore the stop-losses have been calculated on the bid price quoted by the financial broker. The data obtained was split in a six hour time frame in order to be presented and analysed in a more concise manner. This time frame manages to capture all three major FOREX market time zones. As it was mentioned in the previous section, the data used was chosen randomly. It can be observed right now, that the market was bearish on both EUR/USD and GBP/USD and exhibits a negative trend on those two currencies. This aspect should be considered within the analysis.

After testing the data for normality, the data set of returns on both TSL1 and TSL2 strategies showed a non-normal distribution. The testes performed for normality distribution were Anderson- Darling, where the null hypothesis- i.e. sample is following a normal distribution- is rejected if the p value is less than 0.05. In the case of our data of returns, the p value was significantly smaller than 0.05, hence the null hypothesis was rejected. For instance, in figure 1, the GBP/USD returns distribution on the TSL2 strategy is presented and shows non-normality.

The non-normal distribution of the returns puts in question whether the Sharpe Ratio is an adequate measure to assess the performance of the two trailing stop-loss strategies. According to Eling and Schuhmacher(2007), the Sharpe Ratio is a suitable measure of performance if the returns are normally distributed and also if the investor’s intention is to place all his risky assets in a single fund. According to academic sources (Mahdavi, 2004; Sharma, 2004), non-normally distributed returns cannot be adequately measured using the Sharpe Ratio. For instance, a covered call strategy produces a return distribution with a low standard deviation and a significantly negative skewness. The Sharpe ratio in this case will be misleading as it will have a larger value on the strategy than on the underlying asset. As this issue has been considered by the academia and the industry, other performance measures were developed (Mahdavi, 2004).

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Fig. 1. GBP/USD histogram of TSL2 returns distribution

Analysis of returns

Average return for three

days 6 h frame TSL1 TSL2

EUR/USD -0.0046% -0.0028%

GBP/USD -0.0030% -0.0017%

USD/JPY -0.0006% -0.0003%

Fig 2. Average return on the three-day training period six-hour time frame

Average return two-days 6 h

frame TSL1 TSL1

EUR/USD -0.0088% -0.0064%

GBP/USD -0.0079% -0.0047%

USD/JPY 0.0005% 0.0009%

Fig 3. Average return on the two-day validation period six-hour time frame -2 0 2 4 6 8 10 12 14 16 -0.03% -0.02% -0.01% 0.00% 0.01% 0.02% 0.03% Frequ enc y GBP/USD returns

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EUR/USD GBP/USD USD/JPY

Avg. bid ask spread 06-08/05/2013 0.0399% 0.0378% 0.0025% 09-10/05/2013 0.0402% 0.0351% 0.0027%

Fig 4. Average bid-ask spread on currency pairs tested in 6-10 May 2012

The profitability of the algorithms can be observed in the tables above. Following the optimisation process, both the TSL1 and TSL2 improved on the USD/JPY pair. Their profitability improved by 0.00108% and 0.00126% respectively and the profit turned to be slightly positive. The intention has been from the beginning to find an efficient stop-loss strategy which can prevent an investor from “running losses”. Furthermore, the losses increase on the other two pairs: EUR/USD and GBP/USD. From the profitability perspective, the algorithms may be better used in the original form, without the parameters being optimised, for the two pairs: EUR/USD and GBP/USD. This should be, however, investigated in addition to other risk management tools such as volatility, Sharpe Ratio.

The bid-ask spread is already taken into account in the profitability of the algorithms. The value of the spread is on average in the period tested 0.04% on EUR/USD and GBP/USD while on USD/JPY is 0.0025%. This is an aspect which can vary from an investment perspective. Some investment entities may be able to trade with a lower spread. Hence, the stop-loss strategies may be profitable if implemented by those.

Risk-adjusted performance analysis

Training TSL1 Standard Deviation Sharpe Ratio Calmar Ratio EUR/USD 0.0005 -0.8909 -0.009% GBP/USD 0.0005 -0.8438 -0.006% USD/JPY 0.0004 -0.1376 -0.003%

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13 Training TSL2 Standard Deviation Sharpe Ratio Calmar Ratio EUR/USD 0.0008 -0.4913 -0.003% GBP/USD 0.0007 -0.4774 -0.003% USD/JPY 0.0006 -0.1232 -0.002%

Fig.6 Back-testing & training risk metrics for TSL2 optimised trailing stop loss

The Sharpe Ratio or Reward to Variability is calculated by dividing the excess return above the risk free rate by its standard deviation or variability.

Taking into account, the Sharpe Ratio does not make a distinction between the upside and downside risk and the returns are not normally distributed, another risk-adjusted measure was used in the analysis. The Calmar ratio (derived from California Managed Account Reports) suggested by Terry Young (1991) is a Sharpe type measure that uses maximum drawdown instead of the standard deviation to reflect the investor’s risk. In a hedge fund performance context it can be understood why it used more often by practitioners as the maximum possible loss from peak to valley as a more appropriate measure of risk. A drawdown is the cumulative loss from the last maximum to the next minimum of the price. Drawdowns are highly relevant because they measure directly the cumulative loss that an investment can incur. They also quantify the worst case scenario of an investor buying at the local high and selling at the next minimum (Eling and Schuhmacher, 2007).

Following the optimisation, the standard deviation on the EUR/USD and GBP/USD has lowered indicating that the risk was diminuated. However, the reward-to-volatility ratio is lowering as well. In theory, if investors are risk adverse they should be looking for high return and low variability in return (Bodie et al,2010). It can be argued, therefore, that the optimised version is more suitable for more risk adverse investors.

One way of improving the stop-loss profitability and efficiency is by changing the trade entries. For instance, an investor can take into account the informed opinion of a large

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investment firm for where to initiate a trade. The TSL2 algorithm on the USD/JPY pair meets both objectives of the investor: return enhancement as it does not constrain profit potential and limits losses to small amounts (this is done by TSL2 on more than one currency).

In the graphs below (Graphs 1-6), it can be seen how the trailing stop loss strategy behaved on the first three days of the week. An optimisation using the Monte Carlo simulation is effectuated, which will be validated on the following two trading days (9th and 10th of May). In order for the data to be presented in a clearer manner, an hour detail of the three day was extracted.

If the graphs in TSL1 and the ones in TSL2 are directly compared, there is one aspect which can be observed: the distance between the stop-loss and the current price. In the TSL2 strategy, the trailing stop-loss stays further throughout the whole period from the price than does the one from TSL1. This has a significant risk implication for the investor. It can be argued that the second algorithm should be used by a more risk adverse investor, while the first by a less risk adverse one. This situation has to be, however, analysed further along with a measure of the volatility such as the standard deviation.

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Graph 1. EUR/USD trailing stop-loss based upon TSL1 method on the bid price quote 1(one) trading hour on 6th May 2013

Graph 2. EUR/USD trailing stop-loss based upon TSL2 method on the bid price quote 1(one) trading hour on 6th May 2013

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Graph 3.GBP/USD trailing stop-loss based upon the TSL1 method– one trading hour on 6th May 2013

Graph 4. GBP/USD trailing stop-loss TSL2 method based upon the bid price - one trading hour on 6th May 2013

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Graph 5.USD/JPY trailing stop-loss based upon TSL1 method the bid price- one trading hour on 6th May 2013

Graph 6. USD/JPY trailing stop-loss TSL2 method based upon the bid price- one trading hour on 6th May 2013

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4.2. Validation

After the two algorithms have been back-tested on the sample of time randomly chosen (6th-8th May 2013), the main parameters of the algorithms have been optimised by maximising the Calmar ratio. The validation of optimisation has been done on the rest of the trading week – the next two days- 9th-10th of May 2013. The optimisation performed is a general, non-data dependent optimisation.

In the validation process, it can be observed how the optimised strategy works and if the optimisation improved or not the strategy.

The trailing stop-loss 1 did not improve on EUR/USD, nor did on GBP/USD, however on the USD/JPY pair; the TSL1 Sharpe ratio indicates a positive effect. The Calmar ratio on this pair is the same before and after optimisation.

For the second algorithm of trailing stop-loss, the Calmar ratio shows an increase of 0.016% on the GBP/USD. The Calmar ratio has a large importance in the analysis because the returns distribution is non-normal.

As a result, the optimisation was successful for both algorithms on two different currency pairs(USD/JPY and GBP/USD respectively).

Validation TSL1 Standard Deviation Sharpe Ratio Calmar Ratio EUR/USD 0.0004 -1.0139 -0.017% GBP/USD 0.0003 -1.3517 -0.015% USD/JPY 0.0006 0.0305 -0.003%

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19 Validation TSL2 Standard Deviation Sharpe Ratio Calmar Ratio EUR/USD 0.0005 -1.0455 -0.006% GBP/USD 0.0006 -0.8352 0.013% USD/JPY 0.001 0.0765 -0.010%

Fig.8 Validation risk metrics TSL2 optimised trailing stop loss

Graph 7. Optimised version of EUR/USD trailing stop-loss TSL1method based upon the bid price –one trading hour on 9th May 2013

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Graph 8. Optimised version of EUR/USD trailing stop-loss TSL2 method based upon the bid price – one trading hour on 9th May 2013

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Graph 10. Optimised version GBP/USD trailing stop-loss TSL2 method -the bid price -one trading hour on 9th May 2013

Graph 11. Optimised version of USD/JPY trailing stop-loss TSL1 method bid price 1(one) trading hour 9th May 2013

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Graph 12. Optimised version of USD/JPY trailing stop-loss TSL2 method based upon the bid price - one trading hour on 9th May 2013

The only strategy which changes its profitability after the optimisation is the trailing stop-loss calculated after the TSL2 method. This can be seen in the graph below. This is an indication of an improvement of the strategy following Calmar ratio or a profit over drawdown optimisation.

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Graph 13. Profit and cumulative profit based upon optimised version of USD/JPY trailing stop-loss TSL2 method for long positions in the period 9th-10th May 2013

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5. Conclusion

In this paper two trailing-stop loss algorithms were back-tested and validated (forward-tested) on real-time data provided by a well renowned financial broker. Despite the increase exposure of the buy and hold strategies in financial literature and industry in the past, the concept of active portfolio management and high frequency trading is gaining ground ( Kaminski and Lo,2007). The two algorithms were based on the trailing-stop loss strategies proposed by Warburton and Zhang (2006) and Glynn and Iglehart (1995). Following the back-testing, a generally applied optimisation was performed. Once the optimised parameters were obtained, the algorithms were validated on the rest of the data sample randomly chosen. In order to be used in practice an analysis of the trailing-stop loss strategies based on risk-adjusted performance measures was performed. As the returns distribution data is non-normally distributed, a drawdown based performance measure, the Calmar ratio, was used to provide a better basis to the analysis. Taken into account the downward movement of the currency pairs tested in the period chosen, the trailing stop-loss strategies proved to be very efficient in terms of both qualitative and quantitative aspects. In terms of the quantitative analysis, the variability and the performance adjusted to risk has improved on both algorithms: USD/JPY on TSL1 and GBP/USD on TSL2. Following an optimisation using the Calmar ratio, the strategies have turned to be less volatile. The optimisation proved efficient on the second TSL algorithm in terms of profitability and risk-adjusted returns on USD/JPY pair.

The risk control is significantly improved by the first algorithm as the stop-loss values stay closer to the price of the currency pair. Both algorithms penalise any significant change in the trend by exiting the entered transaction in a favourable point. For a high-frequency trading practitioner, who does not consider transactions cost a problem; this is a very useful aspect.

As an algorithmic and more likely as a high frequency trader, you would potentially have access to a better, thus lower, bid-ask spread. This could help you in gaining a significant advantage along with using the stop-loss strategies tested in this paper.

All these aspects should be considered in conjunction with the investor’s intentions and risk aversion before choosing any of the strategies.

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One of the limitations of the research is that the algorithms were optimised only on the basis of the Calmar ratio, a risk-adjusted performance measure based on the drawdown. In addition, for a more accurate performance of the algorithms and before using them in practice, a longer period of history should be used.

The tests and analysis in this paper are aimed at financial analysts, financial traders, hedge fund managers and any other financial specialist who manages an active portfolio on a high frequency basis. Both algorithms presented should be helping the financial industry in finding an optimal exit strategy, suitable to the investor’s needs, trading style and risk aversion.

Quantitative analysis Qualitative analysis

Profit Investor expectation and risk adversity Sharpe Ratio & Standard deviation

Return enhancement and risk control Calmar ratio - Profit/Drawdown

6. Recommendations

As it has been proven along the paper, both stop-loss strategies employed improved their variability and on certain currency pairs their risk-adjusted profitability, following an optimisation based on the Calmar ratio. An aspect which should be highlighted is that an optimisation must be performed for any algorithm tested. The optimisation using the Calmar ratio proved to be very useful on the USD/JPY pair as it clearly improved the strategy’s profitability. The strategies presented should be used in accordance to the investor’s profile and its risk aversion.

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7. Appendices

The algorithms tested

TSL1 as seen in Glynn and Iglehart (2006)

1. Stop-loss order trails the best price to date by a fixed distance

2. Consider the subdivision of the entire history, let’s say ), where n is the total length of the history ,and

3. The maximum price achieved by the security up to time n is given by :

, where

4. Let be a trailing stop which is lower than the maximum price reached by the “security” up to the current period of time .The trade stops at the end of period n if and only if

.

5. Let be the total amount the trailing stop has been raised up to the given period, Let , the amount the trade has lost in comparison to its best level to date.

6. Let , and . We can see that .We stated that , and that implies ,and that the queue is stable.We study the cycles that begin at a random time point n for which . We have that(the mean,

7. Every time the price makes a new high we raise our stop. The amount by which the stop is raised

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2. TSL2 as seen in Warburton and Zhang(2006)

Infinite horizon

 Assumptions: o infinite horizon

o security price moves to different levels in accordance to the trinomial model: the price can go up, stay the same ( no motion), the price can go down.

o T(total number of time intervals in the planning horizon) is unbounded (the waiting time until the price hits the Stop Loss) <- no requirement of growth rate -

o Or assuming the history has fixed length ,let’ s say , means the time in which the price at the moment .

o =the target price level in time interval t;

o Ω

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o Current price level= 0

o K(the target price level ); M=- L(the stop-loss price level);K,M,L>0

 If Ω (price level) =-M before it rises to K=> trade is stopped out at price level =-M => trade terminates

 Else Ω==K , before it hits level =-M=> we rest our stop loss to level K-M  Else Ω==2K, before it hits level =-M=> we reset our SL to level 2K-M  => As soon as the price level reached (j-1)K, j>=1 =>

SL reset to (j-1)K-M

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Graph 23. EUR/USD Profit and Cumulative profit based upon the bid price trailing stoploss results on TSL1 method in the period 6th -8th May 2013

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Graph 26. USD/JPY Profit and cumulative profit based upon trailing stop-loss TSL2 method based on the bid price in the period 6th -8th May 2013

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8. References

Abramov, V., Khan, M. K., Khan, R. A., 2008. A probabilistic analysis of the trading line strategy. Quantitative Finance, Vol. 8, No. 5, 499-512.

Acar, E., Toffel R., 2000. Stop-loss Investment Returns. Draft May 2000, Investment Conference, Faculty and Institute of Actuaries, Hatfield Heath, June 2000.

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