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 under Heterogeneous Expectations

Model of Volume and Volatility in FX market Paper consists of 4 parts; 3. Model of Volatility as Markov Transition Process 4. Model of Distribution of Heterogeneous Expectations

3 Model of Volume and Volatility in FX market We use following tools of mathematics; Poisson Process Continuous Time Two-State Markov Process Infinitesimal Operator

4 2. Order Flow Generations 2.1. Two Sources of Trading Volume 1. Dealers’ Revising Expectations 2. Retail Transactions

5 2. Order Flow Generations 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.

6 2. Order Flow Generations 2.2. Expectation Revision 2/5 1. Dealer picks his reservation price. 2. He hits bid / ask. State 1 He takes open position. He submits limit order and waits. entry

7 2. Order Flow Generations 2.2. Expectation Revision 3/5 1. He wants to close position. 2. He hits bid or ask. State 1 State 0He keeps position closed. exit

8 2. Order Flow Generations 2.2. Expectation Revision 4/5 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.

9 2. Order Flow Generations 2.2. Expectation Revision 5/5 It is continuous time two state Markov process. Sojourn times in state 0 and state 1 have exponential distributions.

10 2. Order Flow Generations 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.

11 2. Order Flow Generations 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.

12 2. Order Flow Generations 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 Dealers who counterbalance retail transactions

13 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 How to characterize heterogeneous expectations?

14 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 How to characterize heterogeneous expectations? 3. Price Determination...

15 3. Price Determination... 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 How to characterize heterogeneous expectations?

16 3. Price Determination Accounting Equations 1/6 Dealers’ positions and retail transactions satisfy the accounting relationships. N 1 is the number of dealers in state 1. It is random variable as follows;

Accounting Equations 2/6 The number of dealers with short positions Long positions Dealers are allowed to have one unit of open position. 3. Price Determination...

Accounting Equations 3/6 Long and short positions are equated. Case 1: R = 0 Retail Transactions are balanced. 3. Price Determination...

Accounting Equations 4/6 ID number of dealer who is quoting bid rate. Next, we have case 2; R >0 or R< 0. The Z b th becomes market’s bid. We renumber dealers from the smallest FLE. 3. Price Determination...

Accounting Equations 5/6 Excess demand R : net retail transactions We can identify who is quoting bid rate. 3. Price Determination...

Accounting Equations 6/6 Dealers as a whole take net short position to accommodate positive R. Bid rate is equal to the Z b th FLE from the smallest. 3. Price Determination... Price changes enough to absorb R.

22 4. Markov Transitions between “Market Inventory State” We define a pair of N 1 and R as “market inventory state.” Q: What is this for? 4.1. Market Inventory States 1/3

23 A: We describe all the activities in the market as transitions between market inventory states. Each inventory state has unique value for Z b. and hence, unique expected value for bid rate Market Inventory States 2/3 4. Markov Transitions...

24 To do so, we construct a matrix of transition intensities; infinitesimal operator. When N 1 or R changes, a transition occurs 4.1. Market Inventory States 3/3 4. Markov Transitions... Want to have transition probabilities and stationary probabilities of the states.

25 We can consider intensity of leaving a given state and moving into another state. N 1 : two exponential distributions R : exponential because Poisson Inter-arrival time of transition 4. Markov Transitions Infinitesimal Operator 1/10

26 Dealer in state 1 exits from state 1 with intensity. Dealer in state 0 enters state 1 with intensity N 1 : the number of dealers in state 1 Change in N 1 4. Markov Transitions Infinitesimal Operator 2/10

27 R increases at Poisson intensity of retail seller. R decreases at the arrival intensity of retail buyer. Change in R R: Excess demand 4. Markov Transitions Infinitesimal Operator 3/10

28 exit from state 1: entry into state 1: retail buyer / seller: Example of intensity of leaving inventory state: N 1 = n, R = r. n d : number of dealers 4. Markov Transitions Infinitesimal Operator 4/10

29 Intensity to leave inventory state ( N 1 = n, R = r ) Meanwhile, the market may move into this inventory state from other. 4. Markov Transitions Infinitesimal Operator 5/10

30 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. 4. Markov Transitions Infinitesimal Operator 6/10

31 Assumption: Then Q matrix is 4. Markov Transitions Infinitesimal Operator 7/10

32 From infinitesimal operator, we can obtain transition probabilities between market inventory states. We solve Kolmogrov’s backward equation. 4. Markov Transitions Infinitesimal Operator 8/10

33 It gives a list of transition probabilities after time t elapses. Matrix form of exponential is defined as 4. Markov Transitions Infinitesimal Operator 9/10

34 Q: It is to find stationary probabilities. Then we use them to obtain expected variance of bid rate. A: What is transition probability for ? 4. Markov Transitions Infinitesimal Operator 10/10

35 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.

36 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 5. Transaction Price Volatility

37 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 Z b th from the smallest reservation price. The market moves between inventory states; (N1,R). X k = FLE = reservation price = limit order price 5. Transaction Price Volatility

38 Let X k for k=1… N 1,, be FLE values. If X k ～ U[0,1], then the Z b th value has a beta distribution. If FLE’s are uniformly distributed, then bid rate follows a beta distribution. 5.3 Bid Rate’s Distribution 5. Transaction Price Volatility

39 It makes difference in bid rate’s variance. However, FLE’s may have different distribution. Volatility depends on N1 and R. It also depends on heterogeneity of expectations. 5.3 Bid Rate’s Distribution 5. Transaction Price Volatility

40 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

41 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.

42 6. Effects on Volume and Volatility 6.2. Volume Effect 2/2 To absorb larger variance of R, transaction prices have to fluctuate more.

43 6. Effects on Volume and Volatility 6.3. Speed of Convergence Effect As revising expectation becomes more faster, number of transitions increases. The market reaches distant states faster. larger trading volume and larger short time volatility Faster expectation revising results in

44 6. Effects on Volume and Volatility 6.4. Thin Market Effect As the sojourn time in state 1 becomes relatively shorter, N 1 decreases. The market becomes thin. Volatility increases

45 6. Effects on Volume and Volatility 6.5 Heterogeneity Effect For the same value of Z b, 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.

46 7. Showing Heterogeneity Effect FLE’s distribution may not be uniform. The Z b th value’s distribution will complicates Distribution of Z b th value We approximate X k ’s distribution by a combination of two uniform distributions.

47 We define random variables FLE on three intervals. 1. X j ~ Uniform Distribution on [0,1] Then we introduce function J(x). J(x):piece ‐ wise linear and increasing. 2. Y j = J(X j ) 7.2 FLE as Three Random Variables 7. Showing Heterogeneity Effect

48 J(x): J(0)=0 and J(1)=1 An inverse of J(x) is (accumulative) distribution function of Y j. 3. C j = c 0 + c 1 Y j ; with support of [c 0,c 1 ] where 0<c 0 <c Showing Heterogeneity Effect 7.2 FLE as Three Random Variables

49 1. C j is random variable we observe. 2. Yj is standardized version of C j on [0,1]. 3. X j is uniformly distributed on [0,1]. X j = H( Yj ) where x = H(y)=Pr(Yj <y). 7. Showing Heterogeneity Effect In other words, X j Yj C j are random variables as follows; 7.2 FLE as Three Random Variables

50 Y: bid rate among Yj ’s X: bid rate among Xj ’s X ～ beta distribution J(x) describes heterogeneity of FLE Bid Rate Volatility We analyze Var[ Y ] as Var[ J(X) ]. 7. Showing Heterogeneity Effect

51 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 Bid Rate Volatility 7. Showing Heterogeneity Effect 2. It holds with all the inventory states.

52 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.

53 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.

54 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.

55 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.