In a classical

Time series search has received considerable attention from researchers in the recent past [

Time series search is categorized into min search and max search problems. In min search, the online player is searching for a minimum value, whereas in max search the online player is searching for a maximum price [

In the literature, algorithms are proposed for online k-min search problems using competitive analysis paradigm [

Rest of the paper is organized as follows. In Section 2, we present the preliminaries required to understand the work as well as the formal problem setting. Our proposed algorithm is presented in Section 3, along with derivation of the required formulae. Section 4 presents experimental settings based on various scenarios as well as the discussion on results. Section 5 concludes the work and present directions for future research.

In this section, we present the basic definitions forming the basis of this work, followed by the formal problem statement.

Generally, online algorithms are more challenging to design than offline algorithms. Online algorithms are evaluated using competitive analysis approach.

_{t} ∈ [_{t} and does not know about future prices. After observing a price quotation _{t}, the online player has to decide on the number of units to buy without knowledge of the future prices. The game ends when the player buys the required number of _{T}.

We present our proposed algorithm as shown in Algorithm 1.

According to the algorithm, the scheme is divided into two regimes; one when the forecast is true and second when it is false. When it is true the objective is to buy more than one unit. In fact,

Recall that the problem scope permits the online player to purchase multiple units of an asset. The determination of how many numbers of units to purchase is based on the extended price and tolerance level (

We consider a two-player game between an online player who wants to buy k units of an asset and a malicious adversary who wants to maximize the loss (competitive ratio) of the online player. We show that the malicious adversary has the ability impose a competitive ratio of at least c irrespective of the actions taken by the online player.

The online player has two options at each offered price. Either she refuses and look for a lower (better) price, or she accepts q_{t} and purchase some units. Remember the aim of the online player is to ensure the competitive ratio not worse than c. While the opponent objective is to enforce the competitive ratio not less than c. The opponent (adversary) provides a price

Similarly, if on a specific day the opponent offers the price

If the online player continues accepting all reservation prices, the adversary provides a price in a succession _{j} (

Solving

We consider the hybrid algorithm proposed by Iqbal and Ahmad (2015) and our proposed risk aware algorithm (Algorithm 1). Each algorithm is executed on DAX30 yearly data. Competitive ratio for both algorithms is calculated on yearly data. It will be helpful to distinguish if a scheme is performing well on the basis of competitive ratio or vice versa.

The experiments are performed on the real-world dataset of DAX30 from the year 1998 to 2017. DAX30 is the blue-chip stock market index consisting of 30 major German companies. We are considering daily closing prices.

We consider the following set of assumptions;

The prices are daily closing prices.

We assume that there is a forecast that in future the offered prices will reach a certain minimum level. It is not necessary that the forecast is always true, it can be false as well.

We assume a transaction fee of 2%.

We consider

We assume tolerance level

We assume Δ = {0.10, 0.20, 0.30} where Δ determines the price from where the forecast will come true. It means that forecast will be true if the offered price reaches m(1 + Δ).

In order to perform a meaningful comparison between the state-of-the-art algorithm and our proposed algorithm, we consider the following scenarios covering various aspects of the problem settings.

The experiments are performed on the real-world dataset for various scenarios as discussed in Section 4.3. Hybrid and our proposed risk aware algorithm are executed on the dataset. The experiments performed using Scenario 1 produces the results as presented in

K = T | ||||
---|---|---|---|---|

Δ | 0.1 | 0.2 | 0.3 | |

Year | Hybrid | Algorithm 1 | Algorithm 1 | Algorithm 1 |

1998 | 1.105 | 1.114 | 1.114 | 1.114 |

1999 | 1.127 | 1.127 | 1.133 | 1.133 |

2000 | 1.091 | 1.095 | 1.095 | 1.095 |

2001 | 1.248 | 1.245 | 1.249 | 1.252 |

2002 | 1.326 | 1.322 | 1.323 | 1.326 |

2003 | 1.258 | 1.256 | 1.257 | 1.259 |

2004 | 1.071 | 1.074 | 1.074 | 1.074 |

2005 | 1.046 | 1.048 | 1.048 | 1.048 |

2006 | 1.048 | 1.05 | 1.05 | 1.05 |

2007 | 1.053 | 1.057 | 1.057 | 1.057 |

2008 | 1.284 | 1.282 | 1.285 | 1.288 |

2009 | 1.214 | 1.214 | 1.214 | 1.217 |

2010 | 1.104 | 1.105 | 1.107 | 1.107 |

2011 | 1.16 | 1.159 | 1.163 | 1.167 |

2012 | 1.037 | 1.038 | 1.038 | 1.038 |

2013 | 1.059 | 1.063 | 1.063 | 1.063 |

2014 | 1.076 | 1.078 | 1.078 | 1.078 |

2015 | 1.052 | 1.056 | 1.056 | 1.056 |

2016 | 1.118 | 1.119 | 1.121 | 1.121 |

2017 | 1.028 | 1.028 | 1.028 | 1.028 |

During experiments, tolerance (τ) and the number of units

The experiment performed using the Scenario 2 produces the results presented in

k = 2 T | ||||
---|---|---|---|---|

Δ | 0.1 | 0.2 | 0.3 | |

Year | Hybrid | Algorithm 1 | Algorithm 1 | Algorithm 1 |

1998 | 1.105 | 1.114 | 1.114 | 1.114 |

1999 | 1.128 | 1.127 | 1.133 | 1.133 |

2000 | 1.091 | 1.094 | 1.094 | 1.094 |

2001 | 1.248 | 1.245 | 1.248 | 1.251 |

2002 | 1.326 | 1.32 | 1.322 | 1.324 |

2003 | 1.258 | 1.255 | 1.256 | 1.258 |

2004 | 1.071 | 1.074 | 1.074 | 1.074 |

2005 | 1.046 | 1.048 | 1.048 | 1.048 |

2006 | 1.048 | 1.05 | 1.05 | 1.05 |

2007 | 1.053 | 1.057 | 1.057 | 1.057 |

2008 | 1.285 | 1.281 | 1.284 | 1.287 |

2009 | 1.215 | 1.213 | 1.214 | 1.217 |

2010 | 1.105 | 1.105 | 1.106 | 1.106 |

2011 | 1.161 | 1.158 | 1.162 | 1.167 |

2012 | 1.037 | 1.038 | 1.038 | 1.038 |

2013 | 1.059 | 1.063 | 1.063 | 1.063 |

2014 | 1.076 | 1.077 | 1.077 | 1.077 |

2015 | 1.052 | 1.056 | 1.056 | 1.056 |

2016 | 1.118 | 1.118 | 1.121 | 1.121 |

2017 | 1.028 | 1.028 | 1.028 | 1.028 |

Tolerance (τ) and the number of units

The experiment performed using the Scenario 3 produces the results presented in

T/2 | T | 2 T | ||
---|---|---|---|---|

Year | Hybrid | Algorithm 1 | Algorithm 1 | Algorithm 1 |

1998 | 1.105 | 1.114 | 1.114 | 1.114 |

1999 | 1.128 | 1.127 | 1.127 | 1.127 |

2000 | 1.091 | 1.095 | 1.095 | 1.094 |

2001 | 1.248 | 1.248 | 1.245 | 1.245 |

2002 | 1.326 | 1.325 | 1.322 | 1.32 |

2003 | 1.258 | 1.258 | 1.256 | 1.255 |

2004 | 1.071 | 1.074 | 1.074 | 1.074 |

2005 | 1.046 | 1.048 | 1.048 | 1.048 |

2006 | 1.048 | 1.05 | 1.05 | 1.05 |

2007 | 1.053 | 1.057 | 1.057 | 1.057 |

2008 | 1.285 | 1.285 | 1.282 | 1.281 |

2009 | 1.215 | 1.216 | 1.214 | 1.213 |

2010 | 1.105 | 1.105 | 1.105 | 1.105 |

2011 | 1.161 | 1.159 | 1.159 | 1.158 |

2012 | 1.037 | 1.038 | 1.038 | 1.038 |

2013 | 1.059 | 1.063 | 1.063 | 1.063 |

2014 | 1.076 | 1.078 | 1.078 | 1.077 |

2015 | 1.052 | 1.056 | 1.056 | 1.056 |

2016 | 1.118 | 1.119 | 1.119 | 1.118 |

2017 | 1.028 | 1.028 | 1.028 | 1.028 |

It is evident from the previous two scenarios that the results of proposed risk aware algorithm did not improve much. In Scenario 3 better results are achieved. In Scenario 1 the average performance percentage was 18.33% while in Scenario 2 the average performance percentage achieved was 23.33%, whereas this percentage is improved and raised to 36.67% in Scenario 3.

In Scenario 4 (see

Scenario 5 relatively generates same results as we observed in Scenario 4. The only difference between the two scenarios are the number of units to buy such as

The outcomes of the experiments indicate that for several years’ hybrid provide better competitive ratio while for some the proposed risk aware algorithm attains high performance. Yet, the overall performance of the risk aware algorithm is inconsistent.

The risk aware algorithm shows good results when we have small value of Δ, higher value of τ and number of units k > T. The reason is that it is based on risk reward framework. The more the player takes risk the more she succeeds. It also happens that the more risk can lead to greater loss. For instance, It is observed from the results in year 2001, 2002 and 2003 that the competitive ratio is improved for risk aware algorithm over hybrid scheme and indeed reducing the total cost while in 2004 and 2007 the results show that the competitive ratio is at the higher side and risk aware lack in reducing the loss and indeed result in higher cost.

We presented an online k-min search scheme for the circumstance where an investor needs to purchase

The proposed scheme can be extended to other online algorithms for conversion problems with inter-related prices such as proposed by Schroeder et al. [