Copywrite © 2021AnnaAronssonandElsaKjell´enAllrightsreservedPREDICTIONOFCURRENCYPAIRS,STATISTICALRELATIONSBETWEENFUTURESANDFORWARDCON-TRACTSSubmittedinfulﬁlmentoftherequirementsforthedegreeMasterofScienceinIndustrialEngineeringandManagementDepartmentofMat

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Master Thesis, 30 credits

PREDICTION OF CURRENCY PAIRS

Statistical relations between futures and forward contracts

Anna Aronsson and Elsa Kjellén

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PREDICTION OF CURRENCY PAIRS,

STATISTICAL RELATIONS BETWEEN FUTURES AND FORWARD CON- TRACTS

Submitted in fulfilment of the requirements for the degree Master of Science in Industrial Engineering and Management

Department of Mathematics and Mathematical Statistics Ume˚a University

SE - 907 87 Ume˚a, Sweden

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Abstract

Forecasting prices is a widely extended topic on the financial markets and is used by traders all over the world to make profitable trades. However, there exists limited amount of research regarding the relation between the price movements of futures and forward contracts. In this thesis work that relation has been investigated in order to see if it is possible to increase the efficiency of the pricing for two different currency pairs that are traded on the forex exchange market. The aim was to develop a statistical model that could find statistical relations so that an improvement in the predictions was seen. Throughout the project two different models were tested to find this relation, using time series data that included the trade dates, prices and delivery dates for the contracts.

The Random Forest algorithm performed best in this study with a prediction that generated low mean squared errors, and high out- of-bag scores. Even though the algorithm performed quite well, none of the results found, provided evidence of a useful statistical relation between futures and forward contracts.

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Sammanfattning

Prediktion av priser är ett väl undersökt omr˚ade p˚a de finansiella marknaderna. Det finns dock en begränsad mängd forskning p˚a förh˚allandet mellan tv˚a olika terminskontrakt. I detta examensar- bete har denna relation undersökts för att se om det är möjligt att

¨

oka effektiviteten i prissättningen för tv˚a olika valutapar som hand- las p˚a valutamarknaden. M˚alet var att utveckla en statistisk mod- ell som kunde hitta statistiska relationer s˚a att en förbättring av prediktioner s˚ags. Under projektets g˚ang testades tv˚a olika mod- eller för att hitta denna relation med hjälp av tidsseriedata som inkluderade handelsdatum, priser och leveransdatum för kontrak- ten. Random Forest-algoritmen fungerade bäst i denna studie med en förutsägelse som genererade l˚aga mean squared errors och höga out-of-bag scores. Trots att algoritmen fungerade ganska bra gav inget av de hittade resultaten bevis för en användbar statistisk relation mellan terminskontrakten.

.

PREDIKTION AV VALUTAPAR,

STATISTISKA F ¨ORH˚ALLANDEN MELLAN TERMINSKONTRAKT

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Acknowledgments

First of all, we would like to thank our supervisors at Swedbank FX Trading, Pauline Ingvarsson and P¨ar Hellstr¨om for their sup- port throughout this thesis work, always providing good feedback and creative solutions. We would also like to thank Jianfeng Wang for his positive feedback and great insight in the project throughout the entire semester. We are also very grateful to Markus ˚Adahl for his thoughts and efforts. Your statistical and risk oriented compe- tence helped us moving forward in the right direction. Lastly, we would like to acknowledge M˚ans Karlsson for his feedback, helpful suggestions and ideas when they were needed.

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1 Introduction

The introduction outlines the company description and the background to the project. It describes the aim for the thesis work and further the problem formulation, limitations and recommendations to the reader regarding disposition.

1.1 Company Description

Swedbank was founded in 1820 and their first office were positioned in Gothen- burg and enabled ordinary people to deposit and withdraw money at any time.

Today, Swedbank is one of the largest fullservice banks in Sweden and has over seven million customers worldwide with domestic markets in Sweden, Estonia, Latvia and Lithuania. It is coordinated as three main business areas, Swedish Banking, Baltic Banking, and Large Corporations and Institutions (LC&I).

The largest business area is Swedish Banking, which accounts for more than half of the overall profit and is the largest private bank in Sweden. Nearly one fifth of the operational profit is from the Baltic Banking. Their LC&I branch offers Trade Finance and other services to institutions, banks and corporations.

The Trade Finance unit operates with the purpose of minimizing the risk of import and export, where the transaction requires both a seller and buyer of different types of goods (Swedbank, n.d.).

1.2 Background

During the last couple of decades, technology has had a huge impact on financial markets. In trading, it is of high importance to be quick at adapting to new information, to enable making arbitrage. Algorithmic trading describes the phenomenon where trading is done automatically, instantly, when a couple of predetermined conditions are met. Over 30% of the trading made in the US is done with the help of algorithmic trading technology. These algorithms are based on statistical models and financial mathematics (Kolm and Maclin,

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System and Trade Finance, Swedbank provides solutions for businesses on the international market to make payments and deliveries. In currency trading, the margins are usually small. However, companies need to secure their cash flow preventing big losses due to fluctuations in the currency market.

The thesis project is centered around the trading of currency pairs in the futures and forwards market. With the use of statistical and financial tools, one wants to investigate if there are any statistical relations between the futures and forward FX markets that can be used to make profitable decisions when trading. Data of the currency pairs Japanese Yen (JPY)/US Dollars (USD) and Euros (EUR)/USD is used in this study.

1.3 Aim

The aim for this project is to investigate whether information in the futures market is valuable for predicting movements in the forward market, and vice versa. If so, the information in one of the two markets would be very useful for trading in the other, giving Swedbanks FX trading a competitive advantage.

One has succeeded in all cases where the results point in the direction whether if one instruments data is useful or not for predicting the other instrument.

1.4 Formulation of the Problem

With the aim established, the following formulation of the problem is given:

• Is it possible with help of futures contracts to predict the movement of forward contracts, and vice versa, so that it can generate more efficient pricing and statistical arbitrage when pricing currency pairs?

1.5 Project Scope

1.5.1 Confidentiality

The confidentiality in this thesis work regards trading data and the company’s business. The students have signed a confidentiality agreement and all information and results presented in the report is approved by the company before being published.

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1.5.2 Limitations

Due to the projects scope being large there are some limitations to this project, they are as following:

• Data that the thesis project is based on will be from a limited time period, being from November 2^nd - December 1^st 2020.

• The thesis work will be investigating two currency pairs, JPYUSD and EURUSD.

1.6 Thesis Disposition

This report includes 8 sections where the thesis project is thorough explained.

Section 1 gives an insight in the background for the project and clarifies the aim, Section 2 describes theory necessary for the thesis work. Section 3 describes the data sets relevant for this project and further, in Section 4, an explanation for the work methodology and tools that are used is conferred. In Section 5 the results discovered are presented objectively and factual. Section 6 includes an examination of the results and discussion regarding accuracy, if the aim is reached and potential further development. The 7th Section covers the conclusions that can be drawn from the thesis project.

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2 Theory

The theory in this project is important since the project involves a lot of lit- erature studies within the area of foreign exchange trading, different contracts and markets, technology developments and statistical results. The following sections regarding financial instruments, terms and markets will give further insight in the theory of this project.

2.1 Important Terminologies on the FX Market

2.1.1 Currency Pairs

Currencies are traded on the FX market, two currencies traded make a currency pair. The currency being bought is referred to as the base currency, while the currency being used to pay with is called quoted currency. An example could be buying EUR and paying with USD, creating the currency pair EURUSD (Weithers, 2006, p.2-18).

2.1.2 Over-The-Counter and Marked-To-Market

Participants at the FX market can trade Over-the-counter (OTC) which means that the buyers and sellers negotiate deals with each other and settle on a price in private. Trades can also be Marked-To-Market (MTM), meaning that the prices of the contracts are listed on a market and trading is done without any negotiation (Weithers, 2006, p.35-36).

2.1.3 Bid-Ask Spread

The Bid-Ask Spread is explained as the difference between the bid and ask prices

Bid-Ask Spread = Bid Price − Ask Price.

It is the cost for instantaneously buying and selling a security. The spread usually increases during times of ambiguity and high volatility. Given this, the security needs to have a larger gain to cover the spread and transaction cost of the trade. The spread is not uniform throughout the day, it fluctuates and usually increases when different trading sessions overlap. The bid and ask price is determined by the financial institutions themselves and it is important for them to be as accurate as possible to be able to compete on the market

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(Aldridge, 2010, p.118-120).

For example, if the bid price for a EURUSD transaction is 1.2520, and the ask price is 1.2521. Then the spread is 0.0001, also known as one pip, which is the fourth decimal point.

2.1.4 Market Transparency

Market transparency is the measurement of information available for participants. Pre-trade transparency often refers to the information available about prices and current market trend of buying and selling. Post trade transparency refers to the information available about previous trades, such as the price and order size and what party that were involved (Kolm and Maclin, 2010, p.21- 27).

2.2 Futures and Forward Contracts

The instruments relevant in this thesis are futures and forward contracts.

These two are financial derivatives that allow investors to speculate.

2.2.1 Futures Contract

A future contract is described as an agreement to either buy or sell a security or asset, at a predetermined price, at a specified time in the future. Futures contracts are standardized and used by speculators and hedgers, and the participants are not known for each other. The purpose for this is to prevent losses due to price changes and this is done by hedging the price movement. The contract obligates the parties to make the transaction at the arranged price, no matter what the current market price looks like (CME Group, 2021).

2.2.2 Forward Contract

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2.2.3 Duration and Expiration of Contracts

A forward contract with a duration of one month expires in 30 days from the contract being traded, with the exception of the expiration date landing on a bank holiday. Futures contracts however, always expire on the third Wednesday of either March, June, September or December, independent of purchase date (Brooks et al, 2001, p.35).

2.3 Vector Autoregression

The aim of this thesis is to investigate relations between the two time series, F utures_t and F orward_t. Vector Autoregression (VAR) is a tool for under- standing how different time series relate to each other and is used in this thesis. In order to understand Vector Autoregression, the basic concepts of time series are needed.

A time series refers to observations, equally spaced out in time, during spe- cific time periods. Time series analysis is a statistical tool for handling time series data (Prado and West, 2010, p.1). Important concepts for the time series analysis are autocorrelation, seasonality and stationarity. Autocorrelation is described as the similarity between observations as a function of the time lag between them (Aldridge, 2010, p.95-96). Seasonality means that periods of fluctuation can be found in the data that identifies increases and decreases that repeat with the same number of, for example minutes, days or years. Weak stationarity means that the covariance is sovereign of time and the mean and variance are constant. Thereby, the statistical properties do not change de- pending on time (Aldridge, 2010, p.98).

In a VAR-model, each time series is expressed as a linear combination of its own previous observations, as well as the previous observations of the other time series present in the model. Let one assume there are two stationary time series, F utures_t and F orward_t. In a VAR-model, these two time series could be expressed as following,

F utures_t = β₁₀ + β₁₁F utures_t−1 + · · · + β_1pF utures_t−p + γ₁₁F orward_t−1 +

· · · + γ_1pF orward_t−p+ _1t

F orward_t = β₂₀ + β₂₁F utures_t−1+ · · · + β_2pF utures_t−p + γ₂₁F orward_t−1 +

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where F utures_t−1represents the first lag of time series F utures_t ,F orward_t−1 the first lag of time series F orward_t. β and γ are coefficients estimated by ordinary least square for each series and vector of independent and identically distributed error terms with mean zero.

If at any time lag t in a time series, the coefficient of the lagged term has a low p-value, that term is statistically significant for predicting the time series (Hanck et al, 2020, Vector Autoregressions, paragraph 1).

2.3.1 Granger Causality

In order to understand the causality between futures and forward contracts the Granger Causality test is introduced. The Granger Causality test uses a VAR-model when testing if one time series is significant for predicting another.

The hypothesis of the test are the following:

H₀: Time series F utures_t does not Granger cause F orward_t, or vice versa H₁: Time series F utures_t does Granger cause F orward_t, or vice versa

If H₀ can be rejected then F utures_t is said to Granger cause F orward_t (or vice versa) and F utures_t is said to be statistically significant for predicting F orward_t The test is made under the assumption that all time series in the model are stationary (L¨utkepohl, 2005, p.41-43).

2.3.2 Types of Stationarity

There are different types of stationarity.

• Weak stationarity refers to when the expected value of a time series is independent of time and the covariance function is independent of time but dependent on timehorizon (Brockwell and Davis, 2016, p.13).

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2.3.3 Augmented Dickey Fuller Test

Augmented Dickey Fuller Test (ADF) determines whether a time series is difference stationary by testing if the series has a unit root. The hypothesis for ADF-test are

H₀: There is a unit root present for the time series, H1: There is no unit root present for the time series.

If H₀ is rejected, the series is assumed to be difference stationary (Singh,2018).

2.3.4 Kwiatkowski-Phillips-Schmidt-Shin Test

The Kwiatkowski-Phillips-Schmidt-Shin (KKPS) test also uses unit roots to determine trend stationary of a time series. The hypothesis of the test follows H₀: The series is trend stationary,

H₁: A unit root implying non-stationarity can be found.

If H₀ can not be rejected, the series is assumed to be trend stationary (Singh, 2018).

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2.4 Random Forest

In bagging, decision trees are made using bootstrap samples. These trees can be highly correlated if similar sets of predictors are used for multiple trees. In random forest, this problem is solved by only allowing a random sample of m predictors, m ≈ √

p to be used in each split, making the trees less correlated (James et al, 2013, p.318-319).

2.4.1 Decision Tree

Decision trees are a supervised learning tool used for prediction. From training on historical data (supervised learning), decision trees identify a set of decision rules for the features of the data set, that best split up the data in such way that the prediction error is minimized. Decision trees can be used for both classification and regression. In this report, regression trees are used.

A regression tree uses a greedy algorithm called recursive binary splitting for splitting the predictor space, containing all possible values for the predictors X₁, X₂..X_p, into regions. The recursive binary splitting algorithm starts out with all observations from the training set in one region (the root of the tree).

It then splits this region into two smaller regions (creating two branches).

When the leafs (the terminal nodes) of the tree have less than a predetermined number of observations, the splitting stops. The algorithm is greedy, meaning that for each step it will make the split resulting in the lowest residual sum of squares (RSS) for that step. The formula to calculate RSS is presented in Equation (1).

RSS =

J

X

j=1

X

i∈Rj

y_i− ˆy_R_j2

(1) where ˆy_R_j is the j:th regions mean response from the training data (James et al, 2013, p.303-309).

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variance of the set can be reduced by dividing the data set into B number of small training sets, make a prediction model (such as a regression tree) for each small training set and then take the average of the resulting predictions. The formula for calculating the average of the resulting predictions is presented in Equation (2)

fˆ_avg(x) = 1 B

B

X

b=1

fˆ^b(x). (2)

However, since one usually only has one training set, one can use bootstrap, taking repeated samples of the training data to generate B bootstrapped training sets and train the prediction method on each of those sets. Then take the average of predictions, the calculation for this is shown in Equation (3)

fˆ_bag(x) = 1 B

B

X

b=1

fˆ^∗b(x). (3)

Bagging is very useful for decision trees that by themselves are prone to over- fitting (James et al, 2013, p.316-317).

2.4.3 Out-of-Bag Score

When using bagging models, each bootstrapped training sample usually includes around two thirds of the training data, leaving one third of the training data as ”out of the bag” (OOB). The mean prediction error of the out-of-bag samples is called out-of-bag error. The out-of-bag score, is defined as 1 − OOB error.

The importance of each feature in a bagging model can be estimated by muting that feature and then compare the OOB-error for the bagging model with the feature to the OOB-error with the feature muted, as shown in Equation (4)

Importance of featurex_j = Err_OOB(x_j) − Err_OOB(x^muted_j ) (4) (James et al, 2013, p.318-319).

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3 Data Description

This section includes a description of how the data was collected for this thesis work and also how the data was considered. The data was provided by the company and is collected from large market data providers. It covers the period November 2^nd - December 1^st 2020 and regards the currency pairs JPYUSD and EURUSD for forward and futures contracts, and spot prices. The data provided for the forward contracts contains different maturity dates, contract periods of 2 weeks, 3 weeks, 1 month and 3 months will be considered. The data was provided in CVS files and connected to each other by the timestamps where the prices were settled.

In the following subsection, the description of the data is split into three parts.

One part regarding the futures contract data, one regarding the forward contract data and lastly the spot price data. Furthermore, down in the method section the data pre-processing is described. Note, all data displayed in this thesis is fictional, due to confidentiality.

3.1 Futures Contract Data

The futures contract data includes all data provided that regards the futures contracts. It includes multiple parameters that is collected from the futures market when buying or selling these contracts. The most relevant information in this data is the timestamps and prices since this is what will be compared to with the forward contracts. An example data set is provided in Table 1.

Table 1: An example data set of futures contract data.

Date and Time Delivery Date Trade Price Ask or Bid Ticker Symbol

2020-11-02 00:00:00 2020-12 ... 95000 B JY

2020-11-02 00:00:01 2020-12 ... 95010 B JY

2020-11-02 00:00:03 2020-12 ... 95220 A JY

2020-11-02 00:00:05 2020-12 ... 95000 B JY

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futures contracts are always delivered the third Wednesday of either March, June, September and December. Trade Price refers to the total price of the contract and can either be a bid price or an ask price. The Ticker Symbol shows the product code.

3.2 Forward Contract Data

The forward contracts data includes timestamps and price parameters for both the bid and ask prices of the forward contracts. An example for the forward data is provided in Table 2.

Table 2: An example data set of forward contract data.

Ask Bid

Date and Time Open Price Close Price Open Price Close Price

2020-11-02 00:01 3.30 ... 3.29 3.29 ... 3.27

2020-11-02 00:02 3.31 ... 3.32 3.30 ... 3.29

2020-11-02 00:03 3.32 ... 3.33 3.31 ... 3.31

2020-11-02 00:04 3.30 ... 3.29 3.29 ... 3.27

2020-11-02 00:05 3.31 ... 3.31 3.30 ... 3.30

2020-11-02 00:06 3.32 ... 3.32 3.30 ... -´3.31

... ... ... ... ... ... ...

2020-12-01 22:59 3.34 ... 3.33 3.32 ... 3.32

The Open Price data shows the dots for the forward contract at beginning of each minute, and the Close Price at the end of each minute. A dot means 1/10000, for example in this case it would for the first timestamp be 3.30/10000.

3.3 Spot Price Data

The spot price data looks similar to the forward contract data. It consists of the prices and timestamps for both bid and ask prices. An example of the spot price data is provided in Table 3.

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Table 3: An example data set of spot price data.

Ask Bid

Date and Time Open Price Close Price Open Price Close Price

2020-11-02 00:01 105.9 ... 105.9 105.1 ... 105.2

2020-11-02 00:02 105.8 ... 105.9 105.2 ... 105.2

2020-11-02 00:03 105.9 ... 105.8 105.4 ... 105.3

2020-11-02 00:04 105.7 ... 105.7 105.3 ... 105.4

2020-11-02 00:05 105.6 ... 105.7 105.3 ... 105.3

2020-11-02 00:06 105.8 ... 105.7 105.4 ... 105.5

... ... ... ... ... ... ...

2020-12-01 22:59 105.6 ... 105.8 105.2 ... 105.3

The open and close prices in this data represent the price at the beginning versus the end of each minute.

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4 Method

This section contains a description of the methods implemented during the project, divided into four sub-parts. The pre-processing of data is presented first. Thereafter the data set up is explained. The pre-prosessing and data set up are the same for both the VAR-models and the Random Forest model.

Lastly the modelling is presented, the first section describing the modelling for the VAR-models, the second part describing the modelling of the Random Forrest-model.

4.1 Pre-Processing Data

Large data files of historical data were provided. As it is shown in the section Data, the original data for the two types of contracts were of two different time frequencies, inverted currency pairs and some prices provided with dots. In this section, the processing of the futures, forward and spot data are provided.

4.1.1 Processing Futures Contract Data

Through linear interpolation one can increase or decrease the frequency of data points. Assume a data set with points at x_i and x_i+1. A data point y(x), where x_i < x < x_i+1, can be created through the average of the two nearest points:

ˆ

y(x) = y_i+ (y_i+1− y_i) + (x_i+1− x_i) x_i+i− x_i

(Bayen and Siauw, 2015, p.212).

The data for the futures contracts was given with a frequency of seconds.

To downsize the data set and make it comparable with the time frequency for the forward contract data, the futures data was converted to minute frequency through linear interpolation.

The prices for the contracts were calculated with the Trade Price and the Trade Price Decimal by following formula:

F uturesP rice = T radeP rice 10T radeP riceDecimal

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These results in the final futures contract data that is presented in Table 4.

Table 4: An example data set of final futures contract data.

Date and Time Delivery Date Futures Price Ask or Bid Ticker Symbol

2020-11-02 00:01 2020-12 ... 0.009500 B JY

2020-11-02 00:02 2020-12 ... 0.009501 B JY

2020-11-02 00:03 2020-12 ... 0.009522 A JY

2020-11-02 00:04 2020-12 ... 0.009500 B JY

2020-11-02 00:05 2020-12 ... 0.009550 JY

2020-11-02 00:04 2020-12 ... 0.009590 A JY

... ... ... ... ... ...

2020-12-01 23:59 2020-12 ... 0.009600 A JY

4.1.2 Processing Forward Contract Data

As it can be seen in the Data Description section, the data for the forward contracts was given with dots and not the total price of the contract. In order to compare the different contract data, the forward dots were subtracted from the spot prices, resulting in the total price of the forward contracts. In addition, the currency pair given in the forward and spot data was inverted compared to the currency pair given in the futures data, meaning that the base currency in the forward contracts was the quoted currency in the futures contracts data. Due to this, the data had to be processed in the sense so that all the data can be comparable.

The method used to implement this was to add the forward data to the spot data. To posses the same currency as the product versus payment, the ask and bid, high and low data must be inverted. An explanatory example for this implementation is given in Tables 5, 5a, 5b and 5c.

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Table 5: Spot price and forward contract data with the same timestamps added together and inverted resulting in comparable forward contract data.

Ask Open High Price Price ...

105.9 105.9 ...

105.8 105.9 ...

105.9 106.0 ...

105.7 105.8 ...

105.6 105.7 ...

105.8 105.9 ...

... ... ... 105.6 105.8 ...

(a) Spot price example data.

Ask Open High

Price Price ...

3.30 3.29 ...

3.31 3.30 ...

3.32 3.31 ...

3.30 3.29 ...

3.31 3.31 ...

3.32 3.32 ...

... ... ... 3.34 3.33 ...

(b) Forward contract example data, expressed in dots.

Bid

Open Low

Price Price ...

0.009442 0.009442 ...

0.009451 0.009442 ...

0.009442 0.009433 ...

0.009460 0.009451 ...

0.009469 0.009460 ...

0.009451 0.009442 ...

... ... ... 0.009469 0.009451 ...

(c) Comparable forward contract example data.

4.2 Data Setup

4.2.1 Inverse Distance Weighted Interpolation

As mentioned in the theory section (See Theory 2.2.3), forward contracts can expire on any day apart from bank holidays, while futures contracts always expire on one out of four fixed dates. All the futures contracts in the futures data set expire on December 16^th, 2020, while the forward contracts expire on many different dates. It would be incorrect to compare the prices of a futures contract and forward contract bought at the same time, if they do not also expire the same time.

By performing inverse distance weighted interpolation (IDW) on forward contracts bought the same time t0 but with different maturity dates t1...tn, one can estimate the price of a fictional forward contract bought at t₀ expiring at any chosen time between t₁ and t_n by formula:

ˆ

y = Σⁿ_i=1_d¹

i ∗ yi

Σⁿ_i=1_d¹

i

,

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where ˆy is the estimated price of a forward contracts bought at t₀ expiring at chosen time t∗, t₁ < t∗ < t_n. y_i...y_n are the observed prices of contracts bought at t₀ expiring between t₁...t_n. d₁...d_n are the distances in time between the known observations y₁...y_n expires and the expiry of the estimated value ˆy (Bartier and Keller, 1996, p.795-796).

An IDW, centered around the futures contracts expiry date, 16^th of December 2020, was carried out using forward contracts of duration two and three weeks, one and three months, generating estimated prices for forward contracts traded from November 2^nd to December 1^st.

4.2.2 Stationary Time Series

When using statistical models to analyse data, it is highly preferable that the data is stationary (see Theory 2.3.2). The series of price data for futures contracts, and the series of generated price data for the forward contracts, were tested for stationarity using the ADF test and the KKPS test. Both tests were used since the ADF-test mainly determines a series difference stationary while the KKPS-test determines trend-stationarity (See Theory 2.3.3).

Returns on futures contracts and forward contracts were calculated using the price series data. This in order to create a stationary series, more suitable for modeling and further analysis.

4.3 Modelling VAR

Four data frames containing the relevant data from the pre-processing was created, two for each currency pair. Each data frame contains of futures and forward contacts, for either the ask-data or the bid-data, this since the bid and ask for a contract occurs at different times. In Table 6 an explanatory data frame for the ask data is shown.

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Table 6: An example data frame for Ask-data

Date and Time Forward Price Futures Price Forward Return Futures Return 2020-11-02 00:01 0.009510 0.009500 0.000002 0.000002 2020-11-02 00:02 0.009510 0.009501 0.000000 0.000002 2020-11-02 00:03 0.009508 0.009522 0.000001 0.000006

... ... ... ... ...

2020-12-01 23:59 0.009620 0.009601 0.000002 0.000003

The aim of the thesis is to investigate the relations between the price movements of futures and forwards contracts. As mentioned earlier (see Theory 2.3.2), it is suitable to use stationary data when analysing statistical relations of time series, Therefore the returns on futures and forwards were used when looking at correlation. A cross-correlation model was implemented, calculating the cross-correlation of the of the two time series

x_t: Time series of returns on futures contracts.

y_t: Time series of returns on forward contracts.

By formula

r_xy(T ) = σ_xy(T ) pσ_x²σ²_y,

where r_xy is the cross-correlation, X_tthe time series for returns on forwards, y_t the time series for returns on futures and T is the time lag that x_tis delayed by.

Furthermore, the two time series x_t and y_t were tested for Granger Causality.

Second order bivariate VAR(2)-models for the bid and the ask time series were implemented:

x_t y_t

=a₁ a₂

+ b¹₁₁ b¹₁₂ b¹₂₁ b¹₂₂

x_t−1 y_t−1

+b²₁₁ b²₁₂ b²₂₁ b²₂₂

x_t−2 y_t−2

+ ε_x,t ε_y,t

, where a₁ and a₂ are the intercepts,

¯1 coefficients for the first order lag terms and¯2 for the second order lag terms, vector of independent and identically distributed error terms with mean zero.

The amount of time lags in the model were selected by inspecting signifi-

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the Akaike Information Criterion (AIC) were also used as an indicator, AIC(p) = ln| ˆΣ| + 2k²p

T ,

where Σ is the models estimated covariance matrix, T represents the sample size and k the number of time series in the model and p the number of lags.

AIC penalizes complexity and favors model fit. The lower the AIC score the better.

After selecting number of lags, the models were tested for both long term and short term forecasting, with the help of pythons inbuilt function forecast.

The same method of testing futures and forwards for cross-correlation, fol- lowed by creating VAR models generating forecasts was done for both curren- ciy pairs, JPYUSD and EURUSD.

Thereafter, the bivariate VAR-models were evaluated, using the the forecasts and the coefficient of determination R² for each model

R² = 1 − Pn

i=1(y_i− ˆy_i)² Pn

i=1(y_i− ¯y)²,

where y is the average, R² shows the percentage of of original variation that has been explained by the model (James et al, 2013, p.69-70).

4.3.1 VAR-model with Two Currency Pairs

The bivariate VAR-models were used to investigate the relation between futures and forwards of the same currency pair. However, in order to see how the relation between futures and forwards is effected by the currency pair itself, a model taking currency into consideration was needed. A VAR(2)-model with all four time series was made and then used for forecasting,

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



 x_t y_t wt

z_t







=





 a₁ a₂ a3

a₄





 +







b¹₁₁ b¹₁₂b¹₁₃ b¹₁₄ b¹₂₁ b¹₂₂b¹₂₃ b¹₂₄ b¹₃₁ b¹₃₂b¹₃₃ b¹₃₄ b¹₄₁ b¹₄₂b¹₄₃ b¹₄₄











 x_t−1 y_t−1 wt−1

z_t−1





 +







b²₁₁ b²₁₂b²₁₃ b²₁₄ b²₂₁ b²₂₂b²₂₃ b²₂₄ b²₃₁ b²₃₂b²₃₃ b²₃₄ b²₄₁ b²₄₂b²₄₃ b²₄₄











 x_t−2 y_t−2 wt−2

z_t−2





 +





 ε_x,t ε_y,t ε_w,t ε_z,t





 ,

where a₁-a₄ are intercepts, b¹ coefficients for the first order lag terms and b² for the second order lag terms, independent and identically distributed error terms with mean zero.

Furthermore, the same steps as those are used for the bivariate time series were done for the VAR-model with two currency pairs.

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4.4 Modelling Random Forest

When using machine learning models such as Random Forest, it is preferred to have multiple variables in the model. The new varibles Rolling Mean, EWMA, MADQ, Hour and Spot Price were added to the data set.

Rolling mean is the same as a moving average (MA):

Rolling Mean = A₁, A₂...A_n n

where A is the average response for each time period n. EWMA is the Exponen- tially Weighted Moving Average:

EW M A_t= α ∗ r_t+ (1 − α) ∗ EW M A_t−1

where 0 ≤ α ≤ 1 is the weight given to the current observation, and r_t is the series value at time t. The higher the α, the more weight is put on the current observation compared to earlier observations (Danielsson, 2011, p.33).

MADQ stands for Moving Average Difference Quota and calculated the difference between long period MA and short period MA:

M ADQ = M A_tlong M Atshort

In our data set, the long period is 60 minutes and the short period is 30 minutes. The variable Hour indicates the hour of the day from 00:00 to 23:00.

Before time series data can be used in supervised learning models such as Random Forest it needs to be reformulated into a supervised learning problem. By creating an input and output column for each variable and put the values for each time step as input and then place the values for that variable one time step later in the output column. For example,

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After converting the data sets into learning problems, they were ready to be used in initial Random Forest models. The number dept and number of trees used in the Random Forest models were chosen through testing. Staring out small with low dept and fewer trees, increasing one parameter at the time, comparing results and complexity. As usual, there is a trade off between prediction accuracy and complexity of the model that needs to be taken into account.

In order to test whether the futures contract actually added any useful information for predicting the forward contract (and vice versa), the following predictions were made.

First, each contract type was predicted using the features of the other contract type as predictors but not including its own previous value as a predictor:

Response : Futures

Predictors :Rolling Mean_{f orward}, EWMA_{f orward} MADQ_{f orward}, Spot Price, Hour, Response : Forward

Predictors :Rolling Mean_{f utures}, EWMA_{f utures} MADQ_{f utures}, Spot Price, Hour.

Thereafter, the contract types were predicted using the features of the other contract as well as its own previous value:

Response :Futures

Predictors :Rolling Mean_{f orward}, EWMA_{f orward} MADQ_{f orward}, Spot Price, Hour, Futurest−1,

Response :Forward

Predictors :Rolling Mean_{f utures}, EWMAf utures MADQf utures Spot Price, Hour, Forwardt−1.

Lastly, predicting each time series without using any information from the other contract type was tested. After each prediction, the feature importance, OOB-score and MSE of the prediction was evaluated.

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5 Result

5.1 Results Vector Autoregression

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In Figure 1 it is shown that the cross correlation between futures and forwards for JPYUSD data is around 50%. For t = ±1 it is around 3%. For the EU- RUSD data set, the cross correlation at 0 time lags is 66% and around 3% in timelag t = ±1.

Furthermore, the Granger causality test found that all four time series Granger caused each other. The time series all have statistically significant relation- ships to each other. The time series of futures and forward contracts having high correlation and Granger causality, implying that one of the contract types could be useful in predicting the other. Following formula shows a representa- tion of the estimated VAR(2)-model

ˆx_t ˆ y_t

=ˆa₁ ˆ a₂

+

ˆb¹₁₁ ˆb¹₁₂ ˆb¹₂₁ ˆb¹₂₂

ˆx_t−1 ˆ y_t−1

+

ˆb²₁₁ ˆb²₁₂ ˆb²₂₁ ˆb²₂₂

ˆx_t−2 ˆ y_t−2

+ ε_x,t ε_y,t

.

Table 8: Estimates of VAR(2)-model for currency pair JPYUSD using ask data.

Data for JPYUSD Ask

Response Coefficients Estimates Standard Error P-value Futures Price, x_t aˆ₁ -0.0000 1.0000 0.896

ˆb¹₁₁ -0.0345 0.0064 0.000

ˆb¹₁₂ 0.0982 0.0130 0.000

ˆb²₂₁ -0.0301 0.0064 0.000

ˆb²₂₂ 0.0273 0.0130 0.036

Forward Price y_t aˆ₂ 0.0000 0.0000 0.771

ˆb¹₁₁ 0.0125 0.0032 0.000

ˆb¹₁₂ -0.0145 0.0064 0.024

ˆb²₂₁ -0.0015 0.0032 0.646

ˆb²₂₂ -0.0027 0.0064 0.679

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Table 9: Estimates of VAR(2)-model for currency pair JPYUSD using bid data.

Data for JPYUSD Bid

Response Coefficients Estimates Standard Error P-value Futures price, x_t aˆ₁ 0.000000 0.000000 0.897

ˆb¹₁₁ -0.021922 0.006389 0.001 ˆb¹₁₂ 0.073417 0.012848 0.000 ˆb²₂₁ -0.030082 0.006388 0.000 ˆb²₂₂ 0.025738 0.012851 0.045 Forward price, y_t aˆ₂ 0.000000 0.000000 0.770 ˆb¹₁₁ 0.014233 0.003178 0.000 ˆb¹₁₂ -0.019807 0.006391 0.002 ˆb²₂₁ –0.000198 0.003178 0.950 ˆb²₂₂ –0.008454 0.006393 0.186

Table 10: Estimates of VAR(2)-model for currency pair EURUSD using ask data.

Data for EURUSD Ask

ˆb¹₁₁ -0.025846 0.006865 0.000 ˆb¹₁₂ 0.044170 0.010287 0.000 ˆb²₂₁ -0.008260 0.006865 0.229 ˆb²₂₂ -0.001272 0.010286 0.902 Forward price, y_t aˆ₂ 0.000001 0.000001 0.159 ˆb¹₁₁ 0.025363 0.004579 0.000 ˆb¹₁₂ -0.037986 0.006862 0.000 ˆ

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Table 11: Estimates of VAR(2)-model for currency pair EURUSD using bid data.

Data for EURUSD Bid

ˆb¹₁₁ -0.019763 0.006882 0.0040 ˆb¹₁₂ 0.030716 0.010280 0.0030 ˆb²₂₁ -0.009356 0.006882 0.1740 ˆb²₂₂ -0.004404 0.010280 0.6680 Forward price, y_t aˆ₂ 0.000001 0.000001 0.1630 ˆb¹₁₁ 0.021491 0.004606 0.0000 ˆb¹₁₂ -0.041699 0.006881 0.0000 ˆb²₂₁ 0.003784 0.004606 0.4110 ˆb²₂₂ -0.023749 0.006881 0.0010

In the upper half in each of the Tables 8 - 11, one can see the estimated coefficients for predicting the futures price x_t, and in the lower half in each table the estimated coefficients for the forwards price y_t. There seems to be a pattern that the first and second lags of both futures and forwards are significant for predicting the futures contract, since this is the case for all data sets except the ask data for EURUSD in Table 9. Likewise, looking at the lower half of Tables 8 - 11, there seems to be a pattern of the first lags of futures and forwards being significant for predicting forward prices.

Table 12: Correlation of residuals for VAR(2)-model for JPYUSD ask data.

Futures contracts Forward contracts

Futures contracts 1.000 0.336

Forward contracts 0.336 1.000

Table 13: Correlation of residuals for VAR(2)-model for JPYUSD bid data.

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Table 14: Correlation of residuals for VAR(2)-model for EURUSD ask data.

Table 15: Correlation of residuals for VAR(2)-model for EURUSD bid data.

In Tables 12-15, one can see that the correlation of residuals for all time series are quite high. This indicates that there probably is an underlying factor not included in the model, that effects all time series in a similar way.

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Figure 3: Forecasts over a 200 minute time period for the JPYUSD bid data.

Figure 4: Forecasts over a 200 minute time period for the EURUSD ask data.

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Figure 5: Forecasts over a 200 minute time period for the EURUSD bid data.

Looking at Figures 2-5, none of the forecasts seem to be very accurate, even though there was strong statistical significance for many lag terms in the VAR- model.

Table 16: R² for the bivariate VAR(2)-model.

FuturesJ P Y U SD ForwardJ P Y U SD FuturesEU RU SD ForwardEU RU SD

Long Term Ask 0.0575 -0.0002 0.1332 -0.0651

Long Term Bid 0.0569 -0.0002 0.1037 -0.0474

Short Term Ask 0.0641 -0.0036 0.1412 -0.0828

Short Term Bid 0.0675 -0.0048 0.1765 -0.0623

Looking at Table 16 the R²values for the futures_{J P Y U SD}contracts are less than 0.07, meaning very little of the variation for futures_{J P Y U SD} is explained by the model. Futhermore, the long and short term forecasts for the forward_{J P Y U SD}

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in this section. The results for the bid data is found in Appendix A.1. Fur- thermore, since there are a total of 32 coefficients in the dual currency pair VAR(2)-model, the coefficients are written out in text form, rather than rep- resented by letters and indexes.

Table 17: Results for the VAR-model with both currency pairs, using ask data.

Response: Futures_{J P Y U SD}

Coefficients Standard Error P-value

Intercept 0.0000 0.0000 0.9050

Lag 1, Futures_{J P Y U SD} -0.0416 0.0070 0.0000 Lag 1, Forward_{J P Y U SD} 0.0980 0.0143 0.0000

Lag 1, FuturesEU RU SD 0.0002 0.0001 0.0030

Lag 1, Forward_{EU RU SD} -0.0001 0.0001 0.3340 Lag 2, Futures_{J P Y U SD} -0.0368 0.0070 0.0000 Lag 2, ForwardJ P Y U SD 0.0321 0.0143 0.0250

Lag 2, Futures_{EU RU SD} 0.0002 0.0001 0.0050

Lag 2, Forward_{EU RU SD} -0.0002 0.0001 0.0540 Response: Forward_{J P Y U SD}

Intercept 0.0000 0.0000 0.8230

Lag 1, Futures_{J P Y U SD} 0.0115 0.0033 0.0010 Lag 1, Forward_{J P Y U SD} -0.0317 0.0069 0.0000

Lag 1, Forward_{EU RU SD} 0.0003 0.0001 0.0000

Lag 2, Futures_{J P Y U SD} -0.0020 0.0033 0.5410 Lag 2, Forward_{J P Y U SD} -0.0076 0.0069 0.2650 Lag 2, Futures_{EU RU SD} -0.0000 0.0000 0.7410

Response : Futures_{EU RU SD}

Intercept 0.0000 0.0000 0.3610

Lag 1, Futures_{J P Y U SD} 0.5058 0.6801 0.4570 Lag 1, ForwardJ P Y U SD 7.8131 1.4042 0.0000 Lag 1, Futures_{EU RU SD} -0.0295 0.0072 0.0000

Lag 2, FuturesJ P Y U SD 0.0901 0.6799 0.8950 Lag 2, Forward_{J P Y U SD} 1.4721 1.4047 0.2950 Lag 2, Futures_{EU RU SD} -0.0088 0.0072 0.2190

Lag 2, Forward -0.0077 0.0109 0.4800

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Table 18: Continuation of Table 17.

Coefficients Standard Error P-value Response: Forward_{EU RU SD}

Intercept 0.0000 0.0000 0.1520

Lag 1, Futures_{J P Y U SD} 2.2008 0.4530 0.0000 Lag 1, Forward_{J P Y U SD} 6.5196 0.9352 0.0000

Lag 1, Forward_{EU RU SD} -0.0545 0.0073 0.0000 Lag 2, Futures_{J P Y U SD} -1.8690 0.4529 0.0000 Lag 2, Forward_{J P Y U SD} 2.8553 0.9356 0.0020 Lag 2, Futures_{EU RU SD} -0.0001 0.0048 0.9810 Lag 2, Forward_{EU RU SD} -0.0317 0.0073 0.0000 The result from the dual-currency VAR model shown In Table 17 do not show any obvious pattern.

Table 19: Correlation of residuals for VAR(2)-model with both currencies, using ask data.

FuturesJ P Y U SD 1.0000 0.3227 0.3957 0.1546

ForwardJ P Y U SD 0.3227 1.0000 0.2522 0.3912

FuturesEU RU SD 0.3957 0.2522 1.0000 0.4763

ForwardEU RU SD 0.1546 0.3912 0.4763 1.0000

The correlation of residuals shown in Table 19 are still relatively high, indicating that there is some underlying factor not included in the model, effecting all time series in a similar way. However, it is lower than in the bivariate VAR-models in Tables 12-15. Since there is lower correlation of residuals in the VAR-model with two currency pairs, it seems that the currency pair is a factor explaining some of the previously unexplained variation.

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Figure 6: Forecasts over a 200 minute time period for the currency pairs JPYUSD and EURUSD ask data.

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Similar to the results of the forecasts made by the bi-variate models, the forecasts made by the VAR-model using both currency pairs, seen in Figure 6, are not very accurate despite many lag-terms being significant in Table 17.

Table 20: R² for the VAR(2)-model with two currency pairs.

Long Term Ask 0.0581 -0.0005 0.1353 -0.0635

Short Term Ask 0.0798 -0.0088 0.1364 -0.0744

Looking at the R² values in Table 20, less than 8% of the variation for the futures_{J P Y U SD} contracts, and less than 14% of the futures_{EU RU SD} contracts is explained by the model. Furthermore, the long and short term forecasts for the forward contracts have negative R² values, indicating that using ¯y for predicting the forward contracts would explain more of the variance than the VAR-model does.

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5.2 Results Random Forest

After tuning and testing, the resulting Random Forest model had a width of 100 trees and a dept of 4 nodes. The data sets have been split up into six different training, validation and test sets.

5.2.1 Predictions made without the response in t − 1 as predictor The forecasts in Figures 7-8 presents six graphs each predicted only with the other contracts features. In other words, the futures contracts are predicted solely based on the forward contracts features. Likewise, the prediction of the futures contracts were predicted solely using the features of the forward contracts.

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Figure 7: The features of the futures contracts used for predicting the forward prices, and vice versa, for the JPYUSD Ask data.

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Figure 8: The features of the futures contracts used for predicting the forward prices, and vice versa, for the EURUSD Ask data.

Similar to the result for the JPYUSD ask futures predictions in Figure 7, there are two periods where the futures prediction deviates far from the actual values for the EURUSD ask data as well. One of them is found in the top left graph in Figure 8. The prediction results for JPYUSD and EURUSD bid is found in Appendix A.2 due to very similar forecasts as in the ask predictions.

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Figure 9: Feature importance in predicting the forward contracts for the JPYUSD ask data.

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Figure 10: Feature importance in predicting the futures contracts for the JPYUSD ask data.

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Figure 11: Feature importance in predicting the forward contracts for the EU- RUSD ask data.

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Figure 12: Feature importance in predicting the futures contracts for the EU- RUSD ask data.

It can be seen in Figures 9-12 that when predicting forward movements, the spot price is the most significant feature, while for the futures the features used varies. The feature importance for the JPYUSD and EURUSD bid were very similar to the feature importance for the JPYUSD and EURUSD ask, and are therefore presented in Appendix A.3.

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Table 21: Mean squared error for the prediction for JPYUSD and EURUSD.

The six splits represents the six test sets that were created for each data set.

Contract/Data Split 1 2 3 4 5 6

Forward_{J P Y U SD} Ask 5.767e-12 1.113e-11 8.439e-12 9.168e-12 4.707e-12 1.472e-11 Futures_{J P Y U SD} Ask 9.215e-11 4.085e-09 9.006e-11 1.847e-11 6.216e-10 1.491e-11 ForwardJ P Y U SD Bid 6.141e-12 1.122e-11 8.416e-12 9.412e-12 4.479e-12 1.635e-11 FuturesJ P Y U SD Bid 7.514e-11 3.655e-09 9.465e-11 1.854e-11 5.060e-10 1.401e-11 Forward_{EU RU SD}Ask 9.234e-08 2.084e-07 7.358e-08 5.912e-08 1.523e-07 1.011e-07 Futures_{EU RU SD}Ask 9.407e-06 3.525e-07 6.770e-08 4.301e-07 4.841e-07 1.506e-07 Forward_{EU RU SD}Bid 8.475e-08 2.023e-07 8.141e-08 5.577e-08 1.635e-07 9.219e-08 Futures_{EU RU SD}Bid 1.001e-05 3.012e-07 7.353e-08 4.097e-07 5.375e-07 4.704e-07

Table 22: Out of Bag Score for the prediction for JPYUSD and EURUSD. The six splits represents the six test sets that were created for each data set.

Contract/Data Split 1 2 3 4 5 6

Forward_{J P Y U SD} Ask 0.9843 0.9980 0.9924 0.9899 0.9948 0.9647 Futures_{J P Y U SD} Ask 0.8481 0.9169 0.9271 0.9762 0.8695 0.9423 Forward_{J P Y U SD} Bid 0.9841 0.9978 0.9924 0.9898 0.9950 0.9623 Futures_{J P Y U SD} Bid 0.8781 0.9198 0.9263 0.9767 0.8820 0.9450 Forward_{EU RU SD} Ask 0.9935 0.9881 0.9904 0.9777 0.9723 0.9863 Futures_{EU RU SD} Ask 0.8671 0.7780 0.9566 0.8817 0.9413 0.9308 Forward_{EU RU SD} Bid 0.9934 0.9875 0.9903 0.9772 0.9723 0.9866 Futures_{EU RU SD} Bid 0.8678 0.8326 0.9566 0.8799 0.9451 0.9137 In Table 21 the mean squared error is presented for the predictions. It can be seen that all the prediction for the currency pairs JPYUSD and EURUSD has low MSE. In Table 22 the out of bag score is presented, and it showed that the OOB score for the forward contracts is better than for the futures contracts.

5.2.2 Predictions made including the response in t − 1 as predictor

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Figure 13: The features of the futures contract, combined with one time lag historical price data from the own contract, used for predicting the forward prices, and vice versa, for the JPYUSD ask data.

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Figure 14: The features of the futures contract, combined with one time lag historical price data from the own contract, used for predicting the forward prices, and vice versa, for the EURUSD ask data.

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Due to similar prediction results for the JPYUSD and EURUSD bid data, these predictions are presented in Appendix A.4.

Figure 15: Feature importance in predicting the forward contracts for the JPYUSD ask data.

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Figure 16: Feature importance in predicting the futures contracts for the JPYUSD ask data.

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Figure 17: Feature importance in predicting the forward contracts for the EU- RUSD ask data.

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Figure 18: Feature importance in predicting the futures contracts for the EU- RUSD ask data.

Looking at Figure 16 it is very clear that F utures_t−1 is the most important feature for predicting F uturest. Likewise, in Figure 15 it is clear that the F orward_t−1is the most important feature for predicting F orward_t. The same pattern is shown for the EURUSD ask data in Figures 18 and 17. The results for the bid data sets were highly similar and can be found in the Appendix A.5. For each of the data sets time series the single most important predictor for time series in time t, is the time series itself in time t − 1.