A Study of the Price Distribution in Foreign Exchange

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U ^MEÅ U ^NIVERSITY

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ASTER

’

S

T

HESIS

A Study of the Price Distribution in Foreign Exchange

Authors:

Jacob NORRMAN

Joakim FERNANDER

Supervisors:

Dr. Niklas LUNDSTRÖM^UmU

Dr. Pär HELLSTRÖM^SEB

A Master’s Thesis submitted in fulfilment of the requirements for a degree in M.Sc. in Industrial Engineering and Management, with

specialization in Risk Management, Umeå Universityat

June 20, 2016 Master’s Thesis, 30 hp

M.Sc. in Industrial Engineering and Management, Risk Management 300 hp

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i

Declaration of Authorship

We, Jacob Norrman and Joakim Fernander, declare that this thesis titled, “A Study of the Price Distribution in Foreign Exchange” and the work presented in it are our own. We confirm that:

• This work was done wholly or mainly while in candidature for a degree at this University.

• Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.

• Where we have consulted the published work of others, this is always clearly attributed.

• Where we have quoted from the work of others, the source is always clearly given. With the exception of such quotations, this thesis is entirely our own work.

• We have acknowledged all main sources of help.

• Where the thesis is based on work done by ourselves jointly with others, we have made clear exactly what was done by others and what we have contributed ourselves.

Signed:

“Risk comes from not knowing what you’re doing.”

Warren Buffett

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UMEÅ UNIVERSITY

Abstract

Science and Technology

Department of Mathematics and Mathematical Statistics

M.Sc. in Industrial Engineering and Management

A Study of the Price Distribution in Foreign Exchange by

Jacob NORRMAN

Joakim FERNANDER

The aim of this thesis was to develop models that can indicate at what relative level orders should be placed to obtain a specific market share in foreign exchange (FX). This was conducted at the E-markets department at Skandinaviska Enskilda Banken (SEB). To understand how trades occur, the Skew-function was developed. It is a suitable model for visualizing the price distribution within the spread, and can be used as an indicator for order placements in relation to market share.

Another aim was to find at what level orders should be placed in the market to obtain a maximized revenue. To reach this goal, the Optimal-skew model was developed. The Optimal-skew model visualizes how many percentages one needs to decrease into the spread in order to obtain a maximized revenue, with the condition of obtaining a required market share within a specific time frame. The model also demonstrates the time horizon that probably will lead to the highest revenue, when the required market share is set.

By testing the models with error metrics and backtesting, it was possible to approximately predict and forecast how the FX markets would behave in the future. With the forecast ability, it was in advance possible to know how trades would occur and therefore possible to take advantage of a deterministic market behavior. Both models indicate how markets differ from each other, and the market which is the most beneficial one to act on.

The models can be applied to any currency pair and market, but have in this thesis been focused on the EUR/USD instrument. These models could, if im- plemented correctly, generate a more deterministic view of the market, and result in more efficient trades, as well as better placement of orders. This should, as a consequence, provide better conditions for high-frequency trading within foreign exchange.

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Sammanfattning

Målet med detta examensarbete var att utveckla modeller som kan indikera på vilken nivå orderplacering bör ske för att erhålla en specifik marknadsandel vid valutahandel. Detta genomfördes vid avdelningen E-markets vid Skandi- naviska Enskilda Banken (SEB). För att skapa förståelse för hur exekveringar in- träffar, utvecklades Skevhetsfunktionen (the Skew-function). Det är en lämplig modell för att visualisera fördelningen av priser inom spreaden, och kan använ- das som en indikator för orderplacering i förhållande till marknadsandel.

Ett annat mål var att bestämma på vilken nivå order bör placeras på i marknaden för att maximera intäkten. För att nå detta mål utvecklades Optimala skevhetsmodellen (the Optimal-skew model). Den Optimala skevhetsmodellen visualiserar hur många procentandelar orderplaceringen bör minska i spreaden för att erhålla maximal intäkt, med villkor av en begärd marknadsandel inom en bestämd tidshorisont. Modellen visar också vilken tidshorisont som leder till den högsta intäkten, när den begärda marknadsandelen är bestämd.

Genom att testa modellerna med feluppskattningar och backtesting, var det möjligt att ungefär förutse hur valutamarknaden skulle te sig i framtiden. Med en prognostiseringsförmåga var det i förväg möjligt att veta hur handeln kom- mer att bete sig och därför möjligt att dra nytta av ett deterministiskt marknads- beteende. Båda modellerna visade hur marknaderna skiljer sig från varandra, och vilken marknad som är mest fördelaktig att agera på.

Dessa modeller kan tillämpas för alla valutapar och marknader, men har i detta examensarbete använts för instrumentet EUR/USD. Dessa modeller kan, om de implementeras rätt, skapa en mer deterministisk bild av marknaden och leda till effektivare affärer samt bättre orderplacering. Detta bör följaktligen ge bättre förutsättningar för högfrekvenshandel inom valutahandeln.

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Acknowledgements

First, we would like to acknowledge Dr. Pär Hellström at SEB for the motiva- tion and guidance that he has given us during this work, and to thank him for providing us with market data that made this thesis possible.

We would also like to thank our supervisor Dr. Niklas Lundström at Umeå Uni- versity for the attention he has given our work. He has always been available for questions and has always been providing great answers to them. We would furthermore like to thank him for the useful and accurate comments that has steered us in the right direction.

Finally, we would like to thank all the great teachers at Umeå University for the invaluable knowledge they have given us; especially Markus Ådahl, for giving us knowledge about risk management and for teaching us how to develop models from financial data.

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List of Figures

1.1 An example of market data . . . 4

3.1 The Skew-function model . . . 8

3.2 Skew-functions for market A and B . . . 9

3.3 Daily differences for the Skew-function, A . . . 10

3.4 Daily differences for the Skew-function, B . . . 10

3.5 Daily errors of the Skew-function model, A . . . 14

3.6 Daily errors of the Skew-function model, B . . . 14

3.7 Daily accuracy for the Skew-function model . . . 15

3.8 The 90% two-sided confidence interval on the sorted error-vector for the last 20 days . . . 16

3.9 The confidence interval for the Skew-function . . . 17

3.10 The confidence interval for the Skew-function with reailized values 18 4.1 The Optimal-skew model . . . 25

4.2 The Optimal-skew, A . . . 26

4.3 The Optimal-skew, B . . . 27

4.4 The errors for the Optimal-skew, A . . . 28

4.5 The errors for the Optimal-skew, B . . . 29

A.1 Skew-functions for more currencies, A and B . . . 35

B.1 Skew-functions with different time intervals . . . 36

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List of Tables

3.1 Error measures for the Skew-function forecast, A . . . 13

3.2 Error measures for the Skew-function forecast, B . . . 13

3.3 Violation ratio for the Skew-function forecast, A . . . 19

3.4 Violation ratio for the Skew-function forecast, B . . . 19

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Abbreviations

FX Foreign Exchange

HFT High-Frequency Trading...

MAD Mean Absolute Deviation MAPE Mean Absolute Percentage Error MSE Mean Squared Error

OTC Over the Counter VaR Value-at-Risk VR Violation Ratio

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x

Dictionary

Ask The price where a seller is willing to sell.

Bid The price where a buyer is willing to buy.

Given An execution on the ask side.

Market data The data representing the trading data for a specific instrument.

Market maker A provider of liquidity.

Paid An execution on the bid side.

Pips The smallest quoted unit of the price.

Quote A quotation of a bid or ask price and the corresponding sizes for a currency pair.

Skew To distort the market by placing a buy or sell order within the spread.

Spread The distance between the lowest ask and the highest bid.

Top of book The highest bid or lowest ask.

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Chapter 1 Introduction

SEB Foreign Exchange (FX) act as a market maker, i.e. they provide bid and ask prices for other market participants to trade on. The market is in constant flux and the bids and asks are constantly changing. The FX markets is an Over The Counter (OTC) traded market, and hence most of the market connects di- rectly, or via various brokers, to each other. There are several FX markets, some of the largest are: Thomson Reuters Dealing Aggregator, Currenex, Hotspot, EBS and BGC. Within these markets there are many different market participants, such as: financial institutions, hedge funds, investors and corporations through financial institutions. The FX market is the market with the highest daily turnover of all markets in the world. This together with the rapid development of computers and computer aided models, have led to a market which is dominated by various levels of high-frequency trading.

A market maker benefits from the bid-ask spread, the difference between the best ask and the best bid prices. In an interconnected market, such as the FX market, market makers compete for market shares. This has led to a decrease in spread. At the same time, profit margins can decline, and rapid market movements make every trade riskier. There is a wide interest in analyzing what hap- pens when changing a market makers trading strategy and intentions in finding the optimal strategy.

The assignment is designed by Dr. Pär Hellström, Senior Quant Trader at SEB E-markets in Stockholm. SEB request analysis of market data and the development of a model that provides improved control over FX transactions. The model will be based on actual market data and able to specify where orders should be placed in the market. The aim of the thesis is to find the suitable, relative level at which bid and ask orders should be placed. The model should be able to consider different currencies, market conditions, market depth, amount of updates and other variables that may be relevant. The purpose of the model is an attempt to optimize trading and make it more effective. The effects will result in more favorable transactions and hence an increased profit for both SEB and its clients.

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Chapter 1. Introduction 2

1.1 High-Frequency Trading

High-frequency trading (HFT) has had a great impact on how trades are made today. The main reason for the evolution of high-frequency trading is its prof- itability compared with traditional low-frequency trading. The main difference between high- and low-frequency trading is a higher turnover of capital in fast computer-driven responses due to changing market conditions. HFT trades are based on algorithm taking a decision and not someone pressing a key. High- frequency trading generally has a lower average gain per trade, but a higher amount of trades [1, p. 1]. High-frequency trading has a lot of advantages:

it has no or little correlation with long-term buy and hold strategies, which makes high-frequency trading a great tool when diversifying long-term portfo- lios. Another advantage is that it doesn’t require long evaluation periods, due to the statistical properties. How credible a high-frequency model is can therefore be confirmed, statistically, in a shorter period. High-frequency trading also provides benefits for the society. The automation of trades has led to savings in reduced staff and less errors caused by human interference, such as emotions and hesitations. Other social benefits are added liquidity, increased market ef- ficiency, improved computer technology and stabilized market systems [1, p.

2].

1.2 Market Maker

In order to make it possible for anyone to buy or sell an asset there has to be a counterparty that is willing to make the opposite transaction. Therefore every transaction has a buyer and a seller. A market maker or liquidity provider can be a financial institution who provides prices on the selling and buying side, hence it might be possible for a market participant to buy or sell a specific asset.

A market maker leverage the bid-ask spread to generate profit. A perfect scenario for a market maker is a “two-sided-flow”. This means that many trades occur on both sides of the spread at the same time. A market maker doesn’t necessarily gain from the spread. If the market maker is selling without buying in an upward trending market, it may find itself buying at higher prices than selling prices during a specific time frame. Hence a market maker is exposed to the risk of market movements [2, pp. 38–39].

With multiple market makers, there is competition in the spread. The competition is about flows. This tends to narrow the spread and imply smaller difference between executed bid-ask orders. Market makers tries to leverage their knowledge and price according to their belief of where the market is going.

Since the "two-sided-flow" is important there is an intention to have a certain proportion of the trades in the market. If a market participant need to trade a certian amount during a specific time, the participant wants to know which level in the spread an order should be placed at, to create the corresponding market share. Therefore it is important with market shares, since there is a con- nection between market shares and flows.

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FX prices are quoted out to the fourth decimal place. The spread is usually only a few “pips” wide in these markets. However, in order to compare different currency pairs it’s advantageous to measure the spread in percentage, in order to make a legitimate comparison [2, p. 44].

Volatility greatly influences the way market makers quote an instrument. It can choose to widen its spread to compensate for shortfalls in the model or hardware equipment. Market makers tries to estimate the positions of other market participants. It is important to know what’s going on in other markets, such as commodities, stocks and bonds, in order to make oneself well informed about potential market changes [2, p. 44].

1.3 Market Data

Every trade in the foreign exchange market involves one currency pair (two currencies), such as the EUR/USD instrument. In this case the EUR is the base currency which is the currency a buyer would like to buy. The USD is the quote currency which is the currency you trade for one unit of the base currency [3, p.

3].

Every event in the foreign exchange market is discrete in time, and the data is called quote data. The quote data consist of:

• An indicator stating the currency pair

• A timestamp (date and time when the quote was registered)

• An ask price

• A bid price

• An ask volume

• A bid volume

The data representing an actual execution consist of:

• An indicator stating the currency pair

• A timestamp (date and time when the execution was registered)

• A price

• A volume

• An indicator stating if the price was given or paid (Indicating if the trade was done on the ask or bid side).

Figure 1.1 illustrates how the market can behave over time. It shows three levels of ask prices and three levels of bid prices, it also shows executions as given and paid.

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0 100 200 300 400 500 600 700 800 900 1000

Events 1.0851

1.08515 1.0852 1.08525 1.0853 1.08535 1.0854

Price EUR/USD

Ask level

Bid level

Given Paid

FIGURE 1.1: An example of market data for the EUR/USD instrument on market A. The different levels can be quoted by one or several market participants. There can be several levels outside and inside the spread. The executions not on a visible line, implies that the trade occurred on a quote not visible to the

trader.

1.4 Aim and Purpose

The aim is to analyze and develop models that can indicate at what relative level orders should be placed to obtain a specific market share. Moreover, the information about order placement should be used to optimize potential revenue. Such optimization will be based on the condition that a certain number of executions needs to occur during a specific time frame. The purpose of the models is to streamline trading and make it more deterministic. The aim of the model is to provide better control and understanding of order placements when attaining revenues and market share. The models will indicate differences in markets, which contribute to a better understanding in different behaviour between various markets.

1.5 Boundaries

The analysis and the models in this thesis will be based on market data for the currency pair: EUR/USD, during February. The market data is received from

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two different markets, which, in the following, will be referred to as market A and market B, due to confidentiality reasons.

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Chapter 2 Method

Literature studies have been done in order to evaluate existing models. Find- ings are that few studies seem to explain how to set up efficient trading strategies for market makers. This comes to no surprise. Most research, even though not academic, is done by trading firms and banks. If someone would come up with an optimal market maker, they are probably going to keep the method for themselves. This thesis has therefore led to practical testing of ideas and cre- ation of new models. These models are supposed to be used as tools when comparing and analyzing FX markets. The two models, the Skew-function model and the Optimal-skew model, originates from empirical studies of actual trading data. Furthermore the theoretical impact on revenue and price when executed orders are changed due to a different spread.

Market data has been studied to visualize the distribution of executed orders relative to the current spread. The distribution indicated how a market maker could attain a specific market share, by lowering a theoretical ask level. A clear difference was discovered between markets when studied during the same time period. This implied that it is more beneficial to attain market shares from one market than the other. In order to have practical use of such a model, it needs to be stable over time. This led to a study of how the distributions varies over time. The distributions obtained for the different markets appears to be very stable. To verify such claims, model tests have been performed. Such as error estimations between time periods. The accuracy of the model has been veri- fied by backtesting. These specific tests have been selected since they are often used as powerful tools in financial areas, when confirming the reliability of new models where they are recommended for testing high-frequency models. These tests also clarify the risk associated with the models.

The process of this method should be carried out, whenever a new set of data is to be tested on the models, which later on are presented in this thesis. The reason for doing the whole process becomes clear if there is an incentive to act on this information and to some extent, improve certainty of the outcome. The Skew-function model and the Optimal-skew model, which are the developed models, are presented in Chapter 3 and 4, respectively.

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Chapter 3 Skew-function

3.1 Introduction

The idea behind the development of a Skew-function is to obtain information at what level ask prices should be placed in order to obtain a specific market share. In other words, the Skew-function describes the distribution of prices within the spread. This means that the Skew-function will tell how much the ask level need to be lowered in order to obtain a desirable market share from a selling point of view. This can be achieved by creating a function from actual trading data. The general idea is to measure the proportions of executions at certain levels in the spread. Creating several Skew-functions from different markets and currency pairs, also gives an indication on how they differ from each other. This can be used as a basis when deciding on where there are possible opportunities when comparing currency pairs and markets between each other. The model has been developed by the authors of this thesis together with internal and external supervisors.

3.2 Definition of the Skew-function Model

To define the Skew-function we first note that in this thesis, vectors will be de- noted by bold letters. In particular, a vector x consisting of n elements is written as:

x_n= [x₁, x₂, ..., x_n]

Throughout this thesis the function |(x > y) will denote an indicator function declaring that the statement before the indicator function is true if the condition x > yis true, otherwise the statement is 0.

Let:

• nbe the number of events

• tbe the position in a vector, t 2 [1, ...n]

• a_nbe a vector of length n consisting of ask prices

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Chapter 3. Skew-function 8

• bnbe a vector of length n consisting of bid prices

• p_gn be a price vector of length n consisting of given prices, can include zeros

• p_pn be a price vector of length n consisting of paid prices, can include zeros

• i = i(an bn),8i 2 (0, 1], be a vector of length n consisting of the proportional distance in the spread

• Hitsi = Xn

t=1

1|(pgt at i),8i 2 (0, 1], be the number of times pgt at i

• Hits_tot = Xn

t=1

1|(pgt b_t), be the number of times pgt b_t

f (i) = Hits_i

Hits_tot,8i 2 (0, 1], be the Skew-function (3.1) Figure 3.1 is an illustration of how the model for the Skew-function works. It shows how the ask price is lowered and how it successively strikes prices at different levels. The steps in i has been selected to be 0.1 since the pips is of that size, approximately, relative to the spread.

0 100 200 300 400 500 600 700 800 900 1000

Events 1.08514

1.08516 1.08518 1.0852 1.08522 1.08524 1.08526 1.08528 1.0853 1.08532 1.08534

Price EUR/USD

Ask Bid

Lowered ask level

Given

FIGURE 3.1: Construction of the Skew-function. The dashed lines are the lowered ask level. The figure is fictional.

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3.3 Result

The Skew-functions for two different markets, A and B, are presented in Fig- ure 3.2. The result of the Skew-function is based on a specific day, 24 hours. The circles are the actual values and to illustrate the Skew-functions, lines are fitted to the actual values with at least 99% accuracy. Lines are fitted by a built in application, Curve Fitting, in Matlab. The application uses polynomials, with the degree that represents the most accurate line. The lines are drawn for a visual purpose and not for further calculations.

0 10 20 30 40 50 60 70 80 90 100

Percentage down in spread (%) 0

10 20 30 40 50 60 70 80 90 100

Market share (%)

Actual A Fitted A Actual B Fitted B

FIGURE 3.2: The Skew-function for market A and B with currency pair EUR/USD, 24 hours.

At a first glance on Figure 3.2, there is a distinct difference between the two markets. The markets have the same currency pair and are constructed during the same time period. The result shows that the markets behave differently, which could be an opportunity for the market maker. In this case a market maker need to lower the ask level more in market B compared to market A, to obtain the same market share. From such a viewpoint it’s preferable to act on market A when a market maker intend to increase the market share by lowering the ask level.

To get an idea of how stable the Skew-function is over time, or if the Skew- function’s behavior is randomly constructed from day to day, a plot showing the Skew-functions for all the workdays in February was created, see Figure 3.3 and 3.4.

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0 20 20 40

15 100

Market share (%)

60

80

Day 10 80

60

5 40

20

0 0

FIGURE3.3: Daily differences for the Skew-function, A.

0 20 20 40

15 100

Market share (%)

60

80

Day 10 80

60

5 40

20

0 0

FIGURE3.4: Daily differences for the Skew-function, B.

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The Skew-functions appear to behave in a similar way day after day, see Fig- ure 3.3 and 3.4. When looking for opportunities between FX markets, this is a positive outcome. Since the Skew-function for a market appears similar from day to day, a market maker can, to some extent, predict how a market will look like in the future time period. The result shows that it is preferable to change the market maker in market A compared with market B. The purpose is, as stated before, to increase the market share to a desired level. Since the Skew-functions appear stable over time, it’s reliable to exploit one market before another. It may also be possible to predict the Skew-function’s shape. To strengthen the general idea and to evaluate the possibility of predictions, these statements must be evaluated by model tests. This will be carried out in the next section.

3.4 Model Test

In order to verify the reliability and accuracy of the model forecast, it’s necessary to perform model tests. This will be evaluated by comparing the model forecast and the realized values. The realized values are the actual values received in the future time period. These tests will give information on how accurate the forecast is. Information about differences gives an indication if there is any idea to take practical actions. These actions would be to change the trading strategy for a market maker. A positive outcome when testing the Skew-function would be that the forecast from one time period appear similar to the future realized values. This implies that the Skew-function on the specific FX market also behaves similar in the future time period. From such a result a market maker can skew the market, meaning changing trading strategies, that affect the spread. With the Skew-function, a market maker obtains information on how to change the ask prices in the spread, in order to reach a desired market share. If the results between time periods for a market are significantly different, the reliability for implementing a successful trading strategy decreases. The model will be tested using different error measurements between time periods.

These error measurements tell the difference in data between forecasted values and realized values. To measure the errors, the following error estimations were used.

The following methods are based on recommended metrics for backtesting errors [1, p. 221]. The lower the value is, on each of the following errors, the better the forecast performance is on the trading system.

Let:

• X_F,tbe the forecasted value for the Skew-function at some future time t

• X_R,tbe the realized value for the Skew-function at time t

• "_F,t= X_F,t X_R,tbe the forecast error for the given forecast of the Skew- function

Mean squared error (MSE) is calculated as the average of squared forecast errors over T estimation periods.

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M SE = 1 T

XT t=1

"²_F,t (3.2)

Mean absolute deviation (MAD) is calculated as the average of absolute values of the errors over T estimation periods.

M AD = 1 T

XT t=1

|"F,t| (3.3)

Mean absolute percentage error (MAPE) is calculated as the average of absolute values of the errors in relation to the realized values over T estimation periods.

M AP E = 1 T

XT t=1

"_F,t

X_R,t · 100% (3.4)

In order to apply the error estimations to the entire Skew-function, every forecasted value on each level of the percentage down in spread is measured against the respective realized value. The Skew-function has ten comparable measures between forecasted and realized values. These error measurements take into account that errors could be positive and negative, due to the quadratic or absolute computation of errors. At last the average of ten errors are calculated for each Skew-function.

The Skew-function is developed to estimate and predict a market in a future time period. In order to produce the most accurate forecast, it’s necessary to know which estimation window that minimize the errors. An estimation window is the length of the time period that is used in a model. In this case the estimation window is the number of days the Skew-function uses to forecast the following day. The second model test will tell how long the estimation window should be, in order to minimize the error between the forecast and the realized values. To compute the errors for each estimation window, the error calculations are done respectively for each time period. The estimation window which results in least errors is the one that should be applied in the model. One scenario could result in different recommendations for estimation windows, when looking at the error results between several different markets.

In such a scenario, one could for simplicity, choose the same estimation window for all the markets. If there is no big difference in errors between different estimation windows, one could, for computing simplicity, and for simplicity of the model, consider the shortest estimation window. The length of the estimation window should always be based on analysis of errors and the trade-off between accuracy and simplicity of the model.

The result from the error estimation is presented in Table 3.1 and 3.2 for markets A and B, respectively. The tables also present the error estimation for different estimation windows, used to predict the Skew-function. This will indicate

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which estimation window is more advantageous to use in the model. The results are based on 4 weeks of data, where the different estimation windows were tested on the same data.

TABLE 3.1: Error measures for forecasts by the Skew-function, EUR/USD, A.

Method\Est.wind. 1 day 2 days 3 days 4 days 5 days MSE 1.47e-04 1.56e-04 1.74e-04 1.95e-04 2.05e-04

MAD 0.0069 0.0075 0.0082 0.0091 0.0097

MAPE 0.81% 0.82% 0.94% 0.97% 1.00%

TABLE 3.2: Error measures for forecasts by the Skew-function, EUR/USD, B.

Method\Est.wind. 1 day 2 days 3 days 4 days 5 days MSE 4.69e-04 3.14e-04 3.34e-04 3.13e-04 3.28e-04

MAD 0.0101 0.0092 0.0094 0.0091 0.0096

MAPE 1.53% 1.31% 1.40% 1.38% 1.40%

In Table 3.1 and 3.2 all the errors are very low, and for market A the lowest errors can be found for an estimation window of one day. Market B has its lowest errors for an estimation window of four days, possibly two days, depending on what measurement to use. For example, the measurement MAPE indicates how off, on average, the forecast and the model is in percentage. For both A and B, MAPE is very low. MAPE is a relative measure of errors. MAPE and MAD doesn’t square errors, thus not giving more weight to large errors, which MSE does [4].

Based on the above result, and since all the errors are very low, regardless of estimation window or market, the estimation window to be used in this model is one day. This creates simplicity to the model, and a one day estimation window also gives the smallest errors for A. What this means is that only the day before is used to predict the following day.

The errors for a one day estimation window during a total time period of 20 days are presented in Figure 3.5 and 3.6, for the two markets.

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0 2 4 6 8 10 12 14 16 18 20

Day 0

0.005 0.01 0.015 0.02 0.025

Error

MAD MAPE MSE

FIGURE3.5: The daily errors, one day estimation window, A.

0 2 4 6 8 10 12 14 16 18 20

Day 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

Error

MAD MAPE MSE

FIGURE3.6: The daily errors, one day estimation window, B.

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As can be seen in Figure 3.6, the errors for market B is more extreme in comparison to the errors for market A. Therefore the model is more accurate for A, but it is still very good for B, since the maximum MAPE for B is below 4.5%, see Fig- ure 3.6. This means that the model only misses less than 4.5% of the prediction in the worst case. Worst case for A is approximately 2.5%, see Figure 3.5.

0 10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

Market share (%)

Actual A Fitted A Actual B Fitted B Next day Next day A Next day Next day B

FIGURE3.7: Illustrates how well the day before explains the following day, EUR/USD, A and B

The prediction of the Skew-function model with a one day estimation window is also illustrated in Figure 3.7, where one can see the fit between the solid lines (forecasted values) and the dashed lines (realized values). The prediction for A is better than the prediction for B since there is a higher level of visible cor- respondence between the dashed and solid lines for A than for the same lines representing B.

3.4.1 Value-at-Risk and Backtest

If the difference between forecasted values and realized values is small and if the error estimations are also small, then backtesting is a great tool to verify the model. With backtesting, individual models can be compared. Backtesting aims to take ex ante value-at-risk (VaR) forecasts from a model and compare them with ex post realized values [5, p. 142]. Value-at-risk is a measure that explains the probability of losing a certain amount [6, p. 183]. VaR has three elements, a time period, a certain confidence interval and an estimate of investment loss.

In this model the investment loss will be measured in percentage terms, which

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are based on the errors in changed market share. There are several methods to calculate the VaR. One is the Historical method, which will be used in this backtest [7, p. 36].

The idea behind VaR in the Skew-function, is that it estimates how big error the model could predict, at a confidence level. For simplicity it’s explained by an example. If a market maker want to have a 10% market share. Then forecasts the market, using the Skew-function and lower the ask level by 40% down in the spread. The VaR could with 90% probability, tell how many percent in market share (-5.1%, +6.1%) the model could predict wrong. This means that there is a 90% probability that the market share will end up in the interval (4.9%, 16.1%), during one day, when the ask level is lowered 40% down in the spread.

A confidence interval using the Historical method is calculated by simply sort the errors from largest to smallest, see Figure 3.8. In this model, negative and positive errors are considered equally bad, which leads to a two-sided confidence interval [7, p. 36].

…

–6.5 –5.1 6.1 8.2

Confidence interval 90%

–VaR VaR

FIGURE3.8: An illustration of the 90% two-sided confidence interval on the sorted errors for the last 20 days with the Historical

method

The VaR and the confidence interval will be different between each different percentage down in the spread level, see Figure 3.9. The reason is because the confidence intervals are constructed over the estimation window on each level (percentage down in spread). The 90%, two-sided, confidence interval for the Skew-function is constructed by taking the sorted errors for the last four weeks (20 days) and delete the first and the last values, as in Figure 3.8. These four weeks is the estimation window for the confidence interval. These errors will then be used as the 90% confidence interval for the Skew-function, during the next two weeks (10 days). These 10 days are the testing window for the confidence interval. An example of the Skew-function forecast for a specific day with a 90% confidence interval could be seen in Figure 3.9.

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0 10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

Market share (%)

Skew function with 90% CI

FIGURE3.9: An illustration of what the 90% confidence level for the Skew-function looks like

When the VaR is computed it’s possible to perform a backtest. This backtest will be based on the violation ratio (VR) [5, p. 145], that is defined below. VaR violations are infrequent events. With a 10% VaR, 10 violations are expected every 100 days. This results in a small sample size for statistical properties.

10 violations might be needed for a credible analysis. A higher level of VaR, require a longer estimation window [5, p. 147]. When applying a backtest to the Skew-function, it will be possible to see if the number of expected errors are more or less than what the confidence interval predicts. In other words, if a certain number of Skew-function values for the next day is outside of the confidence interval for the forecasted values, it will result in a certain number of violations. This result indicates if the Skew-function, with a certain confidence interval, is suitable for the used market data, see Figure 3.10.

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0 10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

Market share (%)

Predicted Skew function with 90% CI Realized Skew function

FIGURE 3.10: The predicted Skew-function, with a 90% confidence interval (red circles and errorbars), for one day (24 hours).

The blue circles are the realized values the next day. Violations have occurred for the percentage down in spread on level 10 and

90.

Definition of violation ration (VR) [5, p. 145]:

Let:

• W_tbe the testing window (10 days)

• N be the number of values for 1 day (10 values, the values for 100% down in spread is not included)

• tot = N· Wtbe the total number of values within the testing window

• pbe the expected proportion of violations (10%)

• vbe the observed number of violations

V R = Observed number of violations Expected number of violations = v

p· tot (3.5)

If the violation ratio is greater than one, the model underforecasts the following Skew-function, and the confidence interval is too narrow. If the violation ratio is less than one, the model overforecasts the following Skew-function, and the

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confidence interval is too wide [5, p. 146]. When the violation ratio becomes greater than 1.5 or smaller than 0.5 one should consider to overhaul the model [5, p. 146].

The result for the backtest of the Skew-function with a 90% confidence interval was:

TABLE3.3: Violation ratio, A

Day 1 2 3 4 5 6 7 8 9 10 Tot

> VaR 7 2 4 0 0 0 0 1 1 2 17

< VaR 0 0 0 0 0 0 0 0 0 0 0

Tot 7 2 4 0 0 0 0 1 1 2 17

V R_A = 17 10 = 1.7

TABLE3.4: Violation ratio, B

Day 1 2 3 4 5 6 7 8 9 10 Tot

> VaR 1 1 0 6 0 0 1 0 0 0 9

< VaR 1 1 1 0 1 0 0 0 0 0 4

Tot 2 2 1 6 1 0 1 0 0 0 13

V R_B = 13 10 = 1.3

These results indicate that the Skew-function model with a 90% confidence interval, made over an estimation window of 20 days, tend to underforecast the prediction for both markets. The violation ratio for B is below 1.5 which implies that the model for the B-market is fairly accurate. The violation ratio for A is however some decimals greater than 1.5 which indicates some inaccuracy in the Skew-function model for the A-market with a 90% confidence interval, that is made with a 20 days estimation window. What can be observed in Ta- ble 3.3 is that the confidence interval constructed for the A-market is somehow distorted since all the violation occur above the prediction. This means that the confidence interval for the A-market is too narrow above the prediction and to wide below the prediction. For the B-market the confidence interval seems to be more accurately shaped since the violations are distributed more evenly above and below the predicted Skew-function, which can be seen in Tabel 3.4.

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3.5 Discussion and Conclusion

The idea behind the Skew-function was to obtain information at what level ask prices should be placed, in order to obtain a specific market share. It was also constructed to show how executed orders are distributed in the spread. The Skew-function fulfills these requirements and can be applied to high-frequency data. In order to verify the accuracy of the Skew-function, model tests were performed. The tests included error metrics and backtesting on realized values. The accuracy of the model will in the end be based on how the FX markets varies from day to day. This means that during some time periods, the Skew- function will be more or less suitable to use. A time period with low volatility in the Skew-function, followed by a high volatility period, will probably end up with a underforecasted Skew-function (a high violation ratio). The reason for this is because the confidence interval will narrow when the predicted Skew- functions are stable and widen during unstable time periods. During reverse circumstances the Skew-function would probably overforecast (a low violation ratio) the confidence interval in a backtest. The VaR was in this model constructed by the Historical method. If a higher certainty of backtesting is desired, the estimation window should be longer when constructing the VaR and the confidence interval. There is of course difficulties and more complexity in doing so. One should always consider that there is around 800 000 events of high-frequency data during 24 hours. And these events construct a one-day Skew-function prediction.

The Skew-function could, in some cases, not acquire every executed order in practice, even though executed orders are supposed to be captured through a lowered ask level. An executed order could be above the lowest ask, this means that an FX buyer has not bought at the lowest possible price in the spread. In practice a trader are not allowed to trade on every market or with every trading participator. The main reason for this is due to bilateral trading and limits. This means that trading participants have different trading contracts, legal obliga- tions and contacts on the market. They can therefore not get the lowest price. At last there is also a queue in the spread. This means that if there is several traders who have placed their orders at the same price, the order, which is placed first in time, gets executed first. This is of course when someone accepts and trade for this price in the market. The Skew-function doesn’t consider these events, which implies differences when using the model in theory and in practise.

When looking at how the Skew-function behaves from day to day, there is some- times a drift in the Skew-function’s direction. The reason for this could be that the supply and demand for the currency pair are changing towards a direction, but it’s in general hard to understand the specific reason for these occurrences.

Nevertheless, it’s good to know that these events exists, and that they could affect the Skew-function.

The Skew-function model was not suitable for including Sundays which could be explained by the limited amount of orders that occur on Sundays. For market A and market B there is very few trades on Sundays and the only operators are Middle Eastern countries, which makes market data from Sundays divergent in

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relation to data from Western working days (Monday to Friday). Therefore is the Skew-function and analysis made only for Western working days.

During the test of different, appropriate estimation windows for the Skew- function, shorter time periods (8 hours) were tested. The reason for testing shorter intervals was due to the fact that there is different trading patterns during 24 hours. This comes to no surprise since people trade more during normal working hours. The intention was to see if it was possible to predict a better Skew-function by dividing the day into smaller time intervals. The results from these error estimations responded differently. There was no improvement of the Skew-functions by using shorter time intervals. The fact is, that the 8-hour Skew-function is very similar to the 24-hour Skew-function, which can be observed in Figure B.1. This led to not using shorter time intervals than 24 hours (1 day). It’s also easier to apply the model to a one day time interval compared to shorter intervals.

The downside with the Historical method approach, is the recommended, but large sample size. In order to make the most accurate VaR measure, a general rule for the sample size is 3/p, where p is the VaR probability. A 1% VaR would require 300 days. The VaR in this thesis is based on a 10% probability of default.

The recommended estimation window should then have been 30 days (3/0.1).

The estimation window is based on 20 days, which is corresponding to 2/p. It has been shortened for simplicity since a 10% confidence interval was created for the Skew-function. An outcome of doing so, is that the VaR responds faster to volatility in the market, compared to a longer estimation window. A longer estimation window would though have been less sensitive to one-off extreme observations [5, pp. 97–98]. The implications of having a somewhat short estimation window resulted in a high violation ratio above one, in the backtest. As long as one understands the impact of shortened estimation windows, it’s very likely that the outcome will be reliable.

It is possible to distinguish markets from each other by looking at the Skew- function. Why markets behave differently could be explained by the different operators, that acts on the market. If different operators act on different markets, the demand that drives the market differs, which makes the markets behave dissimilar. In algorithmic trading, which constitutes a large part of foreign exchange, the computer program formulates the trading strategy and decides the executions. If the algorithms have different attributes on different markets, the markets will, as a consequence, behave differently, which could generate different Skew-functions. Why the Skew-functions then have predictable characteristics, with small prediction errors, is hard to elucidate. An explanation for the Skew-function’s predictability may be the similarity in the market’s com- position – that a market consists of specific, recurrent operators; and that the operators’ behaviour remains moderately constant over time. This could result in a Skew-function with small, daily changes, that is predictable.

Markets generates similar Skew-functions regardless of currency pair which can be seen in Figure A.1. What can be seen in the figure is that the Skew-functions from the A-market have resembling characteristics irrespective of currency pair.

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The same is true for the B-market. What’s also observable is that all Skew- functions for market A is above the Skew-functions for market B, and that the Skew-functions for market A has steeper slopes at the beginning of the function, while this behaviour is contrary for the B-market’s Skew-functions.

Finally, the Skew-function is a suitable tool for identifying a market’s price distribution within the spread and to compare different markets with respect to that characteristic. The Skew-function model is able to predict the next day’s Skew-function with small errors, see Figure 3.5, 3.6, and Table 3.1, 3.2. And by building a confidence interval with errors from the last 20 days as the Historical method instructs, the model receives a moderate 90% confidence interval that designates the uncertainty of the prediction.

3.6 Further Developments

When the current model is tested on several days one can see that the prediction many times underforecasts the actual Skew-function. To avoid this problem it could be beneficial to include drift in the model. By doing that, the model gets an “awareness” of the Skew-function’s vertical movements, and as a result of that it can make better predictions. By including drift the model can also construct a more accurate confidence interval and thus improve the problem with underforecasting.

Currently the model is made to find the price distribution within the spread based on how the number of trades are distributed. An other possible way to examine the price distribution is by studying how the volume is distributed within the spread. By doing that, one can get a better understanding of the market and where to place orders to fill a requested order size.

Other elaborations is of course to test and analyse other markets and other currencies to see how the model behaves with different inputs.

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23

Chapter 4 Optimal-skew

4.1 Introduction

A market maker can intend to have a specific proportion of the executions on the market. This can be achieved by an influence in the spread. By analyzing actual trading data, it’s possible to approximately determine how many executions there are under a specific time. When measuring the price for executed orders, the ask prices are reduced to the level which fulfills the revenue requirements. It’s important to know the purpose of the model, which is to reach a certain number of executions and find the level in the market that maximizes the revenue. It does not ensure increased profits. The method can force the market maker to sell on the bid side during certain conditions and circumstances.

With this model a market maker get a perception on which price, orders need to be issued, to achieve a desired number of executions. For example a market maker could state that they would like to get 10% of all given trades. The model will be able to tell at what level in the spread, orders should be placed, to maximize revenue. The model is not dependent on any specific FX market.

The purpose of an Optimal-skew model could be to find the level in the market that creates a desirable turnover, for managing the portfolio risk.

In this model and analysis, time is defined as events in the market [8, p. 186].

The reason for this is that the market activity is different in different time periods which could make 10 minutes at 14:00 more similar to 1 hour at 01:00.

Therefore could an event-based time definition be more equitable and more useful when comparing time series in the FX market.

4.2 Definition of the Optimal-skew Model

Let:

• m2 (0, 1] be the desired market share

• i2 (0, 1] be the proportional skew

• ⌧ 2 [1000, 10000] be the time frame

• nbe the number of events for one day

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Chapter 4. Optimal-skew 24

• tbe the position in a vector, t 2 [1, ...n]

• a_nbe a vector of length n consisting of ask prices

• bnbe a vector of length n consisting of bid prices

• p_gn be a price vector of length n consisting of given prices, can include zeros

• p_pn be a price vector of length n consisting of paid prices, can include zeros

• i = i(an bn),8i, be a vector of length n consisting of the proportional distance in the spread

• hit_{y(k,⌧ ),i} be the number of executions for a certain time frame y(k, ⌧) 2 [k, ⌧ + k],8⌧, that are above a certain ask level at i,8i

• tot_{y(k,⌧ )}be the total number of executions for a certain time frame y(k, ⌧) 2 [k, ⌧ + k],8⌧

• R(i, ⌧, m) be the average revenue under a time interval ⌧, when the desired market share is m and the skew is i, such that:

R(i, ⌧, m)

=

n ⌧X

k=1

"_{⌧ +k} X

t=k

⇥at i|(pgt at i^ hity(k,⌧ ),i toty(k,⌧ )m ] + pp⌧(tot_{y(k,⌧ )}m hit_{y(k,⌧ ),i})

#

n ⌧

8⌧, m, i

g(⌧, m) = i|(max(R(i, ⌧, m))), 8⌧, m, be the Optimal-skew (4.1) Figure 4.1 is an illustration of how the Optimal-skew model works. It shows how the ask price is lowered and how it successively strikes prices at different levels. If a specified number of given prices are not covered by the lowered ask, under a specific time, the rest has to be executed at the end of the bid serie, black asterisk. For an easier understanding, the definition is also explained with words. During each specific time interval (events), the total number of executed orders are summed up (given trades). Then a level of requested market share is set, followed by a lowered ask. Each followed and captured price execution is computed into the revenue, at the same price as the current ask level. This means that if the ask level captures a price execution that has a higher price than the current ask level, the price on the execution is assumed to be the same price as the current ask level. If the market share quota is not met at the end of the time interval, the model enforces to instantaneously sell on the bid side (paid). This is also included in the revenue. At last a mean of revenue is computed based on each revenue acquired from the followed time intervals. This is

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ultimately done for every ask level (percentage down in spread), each different time interval and requested market share, which results in a 3D plot, see Fig- ure 4.2 and 4.3. The steps in i has been selected to be 0.1 since the pips is of that size, approximately, relative to the spreads.

0 200 400 600 800 1000 1200

Events 1.08514

1.08516 1.08518 1.0852 1.08522 1.08524 1.08526 1.08528 1.0853 1.08532 1.08534

Price EUR/USD

Ask Bid

Lowered ask level

Given Paid

FIGURE 4.1: Construction of the Optimal-skew. The figure is fictional.

4.3 Result

Figure 4.2 and 4.3 show the results of the Optimal-skew model for EUR/USD sample data based on a specific day, 24 hours. For the same day, different lengths of estimation windows are presented. The figures illustrates the optimal lowered ask level, with the goal of attaining a requested market share.

As Figure 4.2 and 4.3 show, the higher the requested market share is, the more the ask level need to be lowered. The plane itself indicate the maximum level the ask should be lowered, for each respective market share. Therefore it is unnecessary to lower the ask level (percentage down in spread) more than the plane. The reason is because the plane indicates the Optimal-skew, that maximizes the average revenue. It is important to know how the model has been developed, when trying to understand and compare the outcome of results. Es- pecially that every executed order is assumed to be lowered to the current ask level. This means that it can be advantageous and possible to get a higher revenue by selling on the bid side of the spread, at the end of the time interval.

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This is compared to have lowered the ask level. The reason for this event is the lost revenue from every executed order that was above the lowered ask level.

Naturally, an order is in theory assumed to be executed on the lowest ask. What Figure 4.2 and 4.3 also show is what time frame that will create the highest average revenue when a certain requested market share is set, these locations are presented with red asterisks in the figures.

So how can a market maker and a trader use this information in a practical way.

The model should be used to get an indication on how to place the ask level, according to a specified market share and time frame. The trader should then compute their and the total amount of orders that have been executed during the time period. If the market share quota is not met, the trader should instantly sell on the bidding side until the quota is fulfilled.

There is an apparent difference between the markets when comparing the percentage down in spread and the corresponding market share, see Figure 4.2 and 4.3. The ask level must be lowered more in market B compared to market A.

Therefore it is preferable to act on market A when the objective is to gain market share for the EUR/USD currency pair.

0 10000 5 10 15

8000 100

20

Percentage down in spread (%)

25

6000 80

30

Time interval (Events) 35

60

Requested market share (%) 4000

40

2000 20 40

0 0

FIGURE 4.2: The Optimal-skew for market A. Indicate how to lower the ask level during different time intervals. It optimizes the revenue for every requested market share. The red asterisks indicates the optimal time horizon for each requested market

share.

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0 10000 10 20

8000 100

30

Percentage down in spread (%)

6000 40

80

Time interval (Events) 50

60

Requested market share (%) 4000

60

2000 20 40

0 0

FIGURE 4.3: The Optimal-skew for market B. Indicate how to lower the ask level during different time intervals. It optimizes the revenue for every requested market share. The red asterisks indicates the optimal time horizon for each requested market

share.

4.4 Model Test

In order to verify how reliable and accurate this model is, the same error measurements used in Chapter 3 for the Skew-function, see page 12, were used for the Optimal-skew model as well. As a reminder, the error metrics were: Mean squared error (MSE), Mean absolute error (MAD), Mean absolute percentage error (MAPE).

The idea of the Optimal-skew model is that the day before could predict the following day. The error measurements below is therefore calculated with a window estimation of 1 day.

A 8>

<

>:

M SE = 0.0010 M AD = 0.0242 M AP E = 8.02%

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B 8>

<

>:

M SE = 0.0017 M AD = 0.0318 M AP E = 6.67%

By looking at the above measurements one can see that the errors for the Optimal- skew model is fairly low. MAPE for market A and B is 8.02% and 6.67%, respectively. This means that the model fails to predict 8.02% of the following day’s Optimal-skew for A, on average, and the model fails to predict 6.67% of the following day’s Optimal-skew for B, on average. However, the result also entails that the model is capable of predicting the following day’s Optimal-skew with an accuracy above 90%, on average, since:

100% M AP E_A^ 100% M AP E_B > 90%.

The errors for the Optimal-skew model with a one day estimation window during a time period of 20 days are presented in Figure 4.4 and 4.5, for the two markets.

0 2 4 6 8 10 12 14 16 18 20

Day 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Error

MAD MAPE MSE

FIGURE4.4: The daily errors for the Optimal-skew, A

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0 2 4 6 8 10 12 14 16 18 20

Day 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Error

MAD MAPE MSE

FIGURE4.5: The daily errors for the Optimal-skew, B

As can be seen in Figure 4.4 and 4.5, the most extreme errors for A and B is of similar magnitude when comparing the respective error metrics between the two markets. When observing Figure 4.4 and 4.5 one can see that the maximum MAPE-measure for A and B is approximately 16–17%, which means that the model only missed 16–17% of the prediction, in worst case, under the observed period.

4.5 Discussion and Conclusion

In practise it’s difficult to use this model and receive a maximized revenue, since the model could influence and skew the market. This means that the spread changes when the new ask level is set, which changes the market’s constitution as soon as one tries to set a new ask level. This does not occur in theory since the data the model is based on does not change when the model is constructed.

However, it’s favorable to know the ask level that optimizes revenue in theory.

It also gives an indication on which market it’s preferable to act on when one wants to request a specific market share, under a specific time interval, with the objective to maximize revenue.

The concept the model induce is to act on the market with the lowest Optimal- skew, since it could be beneficial to skew the market as little as possible. It’s in

A Study of the Price Distribution in Foreign Exchange

U MEÅ U NIVERSITY

M

’

T

A Study of the Price Distribution in Foreign Exchange

Declaration of Authorship

Abstract

Sammanfattning

Acknowledgements

Contents

List of Figures

List of Tables

Abbreviations

Dictionary

Chapter 1

Introduction

1.1 High-Frequency Trading

1.2 Market Maker

1.3 Market Data

1.4 Aim and Purpose

1.5 Boundaries

Chapter 2

Method

Chapter 3

Skew-function

3.1 Introduction

3.2 Definition of the Skew-function Model

3.3 Result

3.4 Model Test

…

3.5 Discussion and Conclusion

3.6 Further Developments

Chapter 4

Optimal-skew

4.1 Introduction

4.2 Definition of the Optimal-skew Model

4.3 Result

4.4 Model Test

4.5 Discussion and Conclusion

U ^MEÅ U ^NIVERSITY