This paper is about model of volume and volatility in FX market Our model consists of 4 parts; 1. M odel of Order Flow Generation 2. Model of Price Determination

This paper is about model of volume and volatility in FX

marketOur model consists of 4 parts;

1. Model of Order Flow Generation

2. Model of Price Determination

under Heterogeneous Expectations

Model of Volume and Volatility in FX market

Paper consists of 4 parts;

3. Model of Volatility as Markov

Transition Process4. Model of Distribution of

　 Heterogeneous Expectations

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Model of Volume and Volatility in FX market

We use following tools of mathematics;

Poisson Process

Continuous Time Two-State Markov Process

Infinitesimal Operator

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2. Order Flow Generations

• 2.1. Two Sources of Trading Volume

1. Dealers’ Revising Expectations

2. Retail Transactions

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• 2.2. Expectation Revision 1/5

Dealers randomly move between two states of expectation.

State 1

State 0

He wants to maintain closed position.

Dealer is confident enough with his expectation and wants to take position.

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1. Dealer picks his reservation

price.

2. He hits bid / ask. State 1

He takes open position. He submits limit order and waits.

entry

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1. He wants to close position.

2. He hits bid or ask.

State 1

State 0 He keeps position closed.

exit

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When one of the dealers revises expectation, he hits bid or ask.

It results in an asynchronous arrival of buyer or seller.

Expectation revisions thus generate trading volume.

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It is continuous time two state Markov process.

Sojourn times in state 0 and state 1 have exponential distributions.

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• 2.3. Retail Transactions

Retail customers arrive at dealer as buyers or sellers randomly; two Poisson processes.

Dealer counterbalances each retail transaction in the market.

Dealer is risk neutral.Maximum open position is one unit.

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• 2.3. Retail Transactions

Retail transactions go through to the market as Poisson arrivals.

Retail transactions thus generate trading volume in wholesale market.

In order to counterbalance, dealer hits bid or ask right away.

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• 2.4. Who initiate transactions?

Revision of expectation and retail transactions generate order flow.

Order Flow: signed trading volume

1. Dealers who enter or exit from state 1.2. Dealers who counterbalance retail transactions

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3. Price Determination under Heterogeneous Expectations

A: We use First Local Extremum, FLE.

It is first peak or trough of expected transaction price. It determines dealer’s present action.

3. 1. How to characterize heterogeneous expectations?

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Expectations are different with regard to FLE.

FLE: First Local Extremum on the expected time path of the transaction prices.

FLE is random variable among dealers.


3. Price Determination ...

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Distribution of FLE is that of long and short positions.

Dealers submit FLE as limit order prices.

Dealer is bullish and he has long position,

if FLE > current bid/ask .


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• 3.2. Accounting Equations 1/6

Dealers’ positions and retail transactions satisfy the accounting relationships.N1 is the number of dealers in state 1. It is random variable as follows;

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The number of dealers with short positions

Long positions

Dealers are allowed to have one unit of open position.


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Long and short positions are equated.

Case 1: R = 0Retail Transactions are balanced.


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ID number of dealer who is quoting bid rate.

Next, we have case 2; R >0 or R< 0.

The Zb th becomes market’s bid.

We renumber dealers from the smallest FLE.


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Excess demand R : net retail transactions

We can identify who is quoting bid rate.


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Dealers as a whole take net short position to accommodate positive R.

Bid rate is equal to the Zb th FLE from the smallest.


Price changes enough to absorb R.

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4. Markov Transitions between “Market Inventory State”

We define a pair of N1 and R as “market inventory state.”

Q: What is this for?

4.1. Market Inventory States 1/3

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A: We describe all the activities in the market as transitions between market inventory states.

Each inventory state has unique value for Zb. and hence, unique expected value for bid rate.


4. Markov Transitions ...

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To do so, we construct a matrix of transition intensities; infinitesimal operator.

When N1 or R changes, a transition occurs



Want to have transition probabilities and stationary probabilities of the states.

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We can consider intensity of leaving a given state and moving into another state.

N1 : two exponential distributionsR : exponential because Poisson

Inter-arrival time of transition


4.2. Infinitesimal Operator 1/10

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Dealer in state 1 exits from state 1 with intensity 　　 .

Dealer in state 0 enters state 1 with intensity

N1: the number of dealers in state 1

Change in N1



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R increases at Poisson intensity of retail seller.

R decreases at the arrival intensity of retail buyer.

Change in R

R: Excess demand



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exit from state 1:

entry into state 1:

retail buyer / seller:

Example of intensity of leaving inventory state: N1= n, R = r.nd: number of dealers



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Intensity to leave inventory state ( N1= n, R = r )

Meanwhile, the market may move into this inventory state from other.



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Each state has intensities of exiting from it and entering from others.

We can construct a list of intensities in a matrix form; infinitesimal operator Q.



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Assumption:

Then Q matrix is



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From infinitesimal operator, we can obtain transition probabilities between market inventory states.

We solve Kolmogrov’s backward equation.



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It gives a list of transition probabilities after time t elapses.

Matrix form of exponential is defined as



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Q: It is to find stationary probabilities. Then we use them to obtain expected variance of bid rate.

A: What is transition probability for ?



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5. Transaction Price Volatility

5.1 Two Definitions of Volatility

2. Short time valatility : variance of price difference in 10 minutes. Transition probabilities matter.

1. Long time volatility : variance of all the samples contained in three hour interval.

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Volatility consists of two elements: 1. bouncing between bid and ask

2. shift of their average

We approximate change in volatility of transaction price by that of bid rate.

5.2 Approximation by Bid Rate


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Each market inventory state has a respective distribution function for bid rate.

5.3 Bid Rate’s Distribution

For a given state, bid rate is the Zb th from the smallest reservation price.

The market moves between inventorystates; (N1,R) .

Xk = FLE = reservation price = limit order price


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Let Xk for k=1… N1 ,, be FLE values. If Xk ～ U[0,1], then the Zb th value has a beta distribution.

If FLE’s are uniformly distributed, then bid rate follows a beta distribution.



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It makes difference in bid rate’s variance.

However, FLE’s may have different distribution.

Volatility depends on N1 and R. It alsodepends on heterogeneity of expectations.



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6. Variation in Volume and Volatility

They summarize model’s conclusions on volume and volatility

6.1. Four Effects on Volume and Volatility

1. volume effect 2. speed of convergence effect 3. thin market effect 4. heterogeneity effect

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6. Effects on Volume and Volatility

• 6.2. Volume Effect 1/2

Volume Effect: Volume of retail transactions increases volatility.

Excess demand R is difference of two Poisson variables.

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• 6.2. Volume Effect 2/2

To absorb larger variance of R, transaction prices have to fluctuate more.

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• 6.3. Speed of Convergence Effect

As revising expectation becomes more faster, number of transitions increases. The market reaches distant states faster.

larger trading volumeand larger short time volatility

Faster expectation revising results in

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• 6.4. Thin Market Effect

As the sojourn time in state 1 becomes relatively shorter, N1 decreases. The market becomes thin. Volatility increases

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• 6.5 Heterogeneity Effect

For the same value of Zb , bid rate comes to have larger variance.

Difficulty: Showing this effect involves an inverse of distribution function of FLE.

More heterogeneous expectations make limit orders less concentrated.

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7. Showing Heterogeneity Effect

FLE’s distribution may not be uniform. The Zbth value’s distribution will complicates....

7.1. Distribution of Zbth value

We approximate Xk’s distribution by a combination of two uniform distributions.

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We define random variables FLE on three intervals.

1. Xj ~ Uniform Distribution on [0,1]Then we introduce function J(x). J(x):piece‐wise linear and increasing.

2. Yj = J(Xj)

7.2 FLE as Three Random Variables


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J(x): J(0)=0 and J(1)=1

An inverse of J(x) is (accumulative) distribution function of Yj.

3. Cj = c0 + c1Yj ; with support of [c0,c1] where 0<c0<c1.



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1. Cj is random variable we observe.

2. Yj is standardized version of Cj on [0,1]. 3. Xj is uniformly distributed on [0,1].

Xj = H( Yj ) where x = H(y)=Pr(Yj <y).


In other words, Xj Yj Cj are random variables as follows;


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Y: bid rate among Yj ’sX: bid rate among Xj ’s

X ～ beta distribution

J(x) describes heterogeneity of FLE.

7.3. Bid Rate Volatility

We analyze Var[ Y ] as Var[ J(X) ].


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1. For a given (N1,R), as the expectations become more heterogeneous, Var[J(X)] increases.

How to show Heterogeneity Effect

3. Long time and short time volatilities increase.

7.3. Bid Rate Volatility


2. It holds with all the inventory states.

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8. Conclusion• Issue of economics

Use of arrival intensity is useful approach to analyze continuous time auctions.

No truistic definition of equilibrium for continuous auctions exists. It is difficult to apply equilibrium approach.

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8. Conclusion

• Issue of economics

Arrival intensities of buyers and sellers is useful tool to analyze continuous time auctions.

No truistic definition of equilibrium for continuous auctions exists. It is difficult to apply equilibrium approach.

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• Issues of microstructure; 1/2

We provide a model which heterogeneous expectations interact with retail transactions.

8. Conclusion

In financial market, heterogeneous expectation is one of the reasons to trade.

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• Issues of microstructure; 1/2

It makes it possible to explain intra- and inter-day variations of volume and volatility.

8. Conclusion

Constructing infinitesimal operator provides new tools to analyze continuous auctions in financial markets.

Documents

This paper is about model of volume and volatility in FX market Our model consists of 4 parts; 1. M odel of Order Flow Generation 2. Model of Price Determination