Extreme Returns
The Case of Currencies
Carol Osler
Brandeis University
Tanseli Savaser
Williams College
QWAFAFEW July 20, 2010: Extreme Returns in FX
1
Extreme Returns in FX

Reality



October 7, 1998: Dollar-yen fell 11% … without news
October, November 2008: Frequent dollar moves of 2, 4, even 7%
High frequency of extreme moves


More frequent than normal distribution
But … reasons to expect returns distributed normally


Great variety of market shocks and Central Limit Theorem
Surprising to financial economists

In economic models, only information brings abrupt moves
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Extreme Returns Matter

Matter for risk management



Major market disruption: Funds go bankrupt
Value-At-Risk

How big IS tail risk?

Is it constant?
Matter for option pricing


What IS a “jump process,” anyhow?
What determines likelihood, size of “jumps”?
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Contributions


4 Ways Price-Contingent Trading Increases Extreme Returns

Affect Distribution of Order-Flow Itself Three Ways
1. Distribution of trade sizes
2. Clustering of trades at times of day
3. Clustering of trades at exchange-rate levels

Fourth Effect: Feedback from Order Flow to Returns
Evaluate Importance of Each Contribution



Most important single factor: Fat tails in order-size distribution
Interactions among factors also very important
Generalize? Algorithmic and Technical Trading in Equities
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Extreme Returns, Fat Tails, & Kurtosis

Fat tails: High frequency of extreme outcomes


Broader Concept: Kurtosis




Benchmark: Normal Distribution
Fat Tails
Tall Skinny Middle
Kurtosis of normal distribution = 3
 Kurtosis of financial returns >> 3
 Equities
 Bonds
 Forex
I (incorrectly) use “fat tails” and “kurtosis” interchangeably
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Kurtosis in Exchange-Rate Returns
EUR-USD
Kurtosis
15 Minutes
24
30 Minutes
19
1 Hour
14
2 Hours
12
6 Hours
7
12 Hours
5
24 Hours
4
48 Hours
5
Normal Distribution
3
QWAFAFEW July 20, 2010: Extreme Returns in FX
Link
6
Kurtosis in Exchange-Rate Returns
Share of (signed) 1/2-hour EUR returns by distance from mean
Ratio to share under normal distribution
Distance
from Mean
(Std. Devs.)
<
½
½
to
1½
1½
to
2½
2½
to
3½
3½
to
4½
4½
to
5½
5½
to
6½
Share Ratio
1.4
0.7
0.6
1.5
14
240
29,500
Tall Skinny Middle
Fat Tails
Example: 53 % of orders within 1/2 standard deviation of mean
38 % of observations within 1/2 std dev. for normal distribution
Ratio: 1.4 = 53/38
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Kurtosis in Exchange-Rate Returns

Earlier: Statistical description of return distribution



Normal Distribution ("Gaussian")? No
Student t distribution? Stable Paretian? Mixed evidence …
Mixture-of-normal distributions? (What’s that?)



Pick a group of random variables: X,Y,Z,A,B,C ….
All from normal distributions with same mean (say, 0)
 But different standard deviations
 Say: X,Y,Z have std.dev.= low; A,B,C have std.dev.=high
Distribution of the group X,Y,Z,A,B,C has fat tails

Little attempt at understanding

Assumes distribution is constant … which seems unlikely
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Outline

Data

3 Key Features of Price-Contingent Orders
1. Distribution of individual order sizes
2. Time-of-day clustering
3. Exchange-rate clustering

How much kurtosis?

4th Factor: Feedback, Order Flow  Returns

How much kurtosis?



Linear feedback
Concave feedback
Summary
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Data

Royal Bank of Scotland


Complete book of stop-loss, take-profit orders



Currently 5th largest FX dealing bank worldwide (Euromoney, 2007)
2 time periods
 1 September, 1999 - 11 April, 2000

1 June, 2001 through 9 September, 2002
3 major exchange rates
 Euro-dollar, Dollar-yen, Sterling-dollar
Contemporaneous exchange rates

Minute-by-minute indicative quotes Reuters FXFX
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Data

Basics:

47,312 orders placed worth $253 billion

27 percent executed
 Otherwise deleted or remained open

Most orders executed within one day
 In fact, most executed within a few hours

Mean order size: $5.4 million
 Max order size: €858 million
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Stop-Loss and Take-Profit Orders


“Price-contingent” market orders

Stop-loss orders: Positive-feedback trading
 If market falls to $1.30, sell €50 million (exactly) at market price
 If market rises to ¥125/$, buy $25 million (exactly) at market price

Take-profit orders: Negative-feedback trading
 If market falls to $1.30, buy €50 million (exactly) at market price
 If market rises to ¥125/$, sell $25 million (exactly) at market price
Unlike limit orders


These orders absorb liquidity (especially stop-loss orders)
These orders used in quote-driven markets
 Customers assign dealers to monitor the market for them
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Who Places Stop-Loss and Take-Profit
Orders?
Value ($ billions)
Share
%Take-Profit
12.6
6.4
64.1
7.8
3.7
60.2
13.7
6.5
33.5
3.9
1.5
72.6
20.1
9.9
71.4
4.5
4.9
81.4
Royal Bank
77.3
35.6
55.3
Global Liquidity Providers
19.8
11.3
35.6
7.5
3.7
54.8
26.5
16.6
62.2
Customers
Levered Money (e.g., Soros)
Real Money (e.g., Fidelity)
Broker-Dealers (e.g., Bear Stearns)
Gov’t Agencies, Central Banks
Large Corporates (e.g., GM)
Middle-Market Corporates
Banks
Regional Liquidity Providers
Customer-Service Banks
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Outline


Data

3 Key Features of SL and TP Orders
1. Distribution of individual order sizes
2. Time-of-day clustering
3. Exchange-rate clustering

How much kurtosis?

4th Factor: Feedback, Order Flow  Returns

How much kurtosis?



Linear feedback
Concave feedback
Summary
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SL, TP Create Kurtosis In Order Flow

Reminder: Order flow = Buy-initiated – Sell-initiated


E.g., Market buy orders – market sell orders
Why kurtosis of order flow … instead of kurtosis of returns?

Order flow drives returns


Crudely:
Exchange-rate return  Constant • OrderFlow
Return distribution isomorphic to order-flow distribution



If order-flow distribution : Normal, Mean=0, Stand.Dev.=1
And if “constant” = 2
Return distribution of : Normal, Mean=0, Stand.Dev.=2
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Distribution of Order Sizes

High kurtosis in distribution of individual order sizes

EUR: 725!
GBP: 21
JPY: 26
Share of (signed) EUR order sizes by distance from mean
As fraction of share under normal distribution
Distance
from Mean
(Std. Devs.)
<
½
½
to
1½
1½
to
2½
2½
to
3½
3½
to
4½
4½
to
5½
5½
to
6½
>
6½
Share Ratio
2
0.3
0.4
0.4
13
192
23175
31
Mill.
Tall Skinny Middle
QWAFAFEW July 20, 2010: Extreme Returns in FX
Fat Tails
16
Distribution of Order Sizes

Suppose 1 order executed per half-hour

Each period, random pick of one order size

Also, random sign (Buy = +, Sell = -)

 Maybe x = €2.3 million sold = - €2.3 million
Order flow across the day is sequence of X’s
All sampled from same distribution with high kurtosis


So kurtosis of order-flow  kurtosis of order-flow sizes:

EUR: 725
GBP: 21
JPY: 26
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Distribution of Order Sizes

If 1 order executed per 1/2-hour


Kurtosis order-flow same as kurtosis of order-flow sizes:
 EUR: 725
GBP: 21
JPY: 26
If N = 2 orders executed per 1/2-hour

Each period, random pick of two order sizes


Assign random sign (buy/sell)
Order flow = x1 + x2
 Maybe x1 = -€2.3 million and x2 = 1.0 million
 So order flow = - €1.3 million

With many orders/period, OF distribution loses fat tails

Distribution  xi  Normal (kurtosis = 3) as N  
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Distribution of Order Sizes

Distribution of order flow  Normal as N  

How fast?

Answer from simulation: Picking order sizes at random
Orders
per
Period
1
2
3
4
5
10
20
50
100
OrderFlow
Kurtosis
513
252
173
130
105
55
29
13
8

How many orders executed per 1/2-hour, in reality?
 Back-of-the-envelope: 3 or 4. We go with 4
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Intraday Volatility Pattern and Kurtosis
Exchange-Rate Levels Crossed per Half Hour
New York
London
Asia
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Intraday Volatility Pattern and Kurtosis

Key: Number of orders depends on number of rates crossed



From 1.0010 to 1.0011
 Execute orders ending in 11
From 1.0010 to 1.0015
 Execute orders ending in 11, 12, 13, 14, and 15
If order sizes distributed normally


In each ½-hour, order flow distributed normally
 Sum of variables with same normal distribution is normally
distributed
Order flow standard deviation high if N is high
 Vice versa
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Intraday Volatility Pattern and Kurtosis

Key: N depends on number of exchange rates crossed

Suppose individual order sizes distributed normally



Order flow distributed normally in each 1/2-hour
Order flow std. dev. high if number of orders is high, vice versa
Strong intraday variation in volatility

Daily order flow includes order flow from every time of day


That is, mixes normal distributions with varying standard deviations
So: Overall order flow has fat tails

Currency returns will have fat tails
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Exchange-Rate Preference and Kurtosis
People prefer to place orders at certain rates
Special preference for round numbers, for example $1.7600/£

Percent of all executed orders

6
4
2
0
00
10
20
30
40
50
60
70
80
90
Final Two Digits of Exchange Rate
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Exchange-Rate Preference and Kurtosis

People prefer to place orders at certain levels


Orders executed depend on specific rates (St) crossed



End digit 0 preferred to 5 ….. 5 preferred to 2,3,7,8 ….
….. 2,3,7,8 preferred to 1,4,6,9
If St crosses level ending in “00,” many orders (5 %)
If St crosses level ending “39,” few orders (0.3 %)
Suppose individual order sizes normally distributed



Number of orders per period varies due to exchange-rate preferences
So … standard deviation of order flow varies across period
So … mixture of normals, order flow has high kurtosis unconditionally
 And currency returns have high kurtosis
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Exchange-Rate Preference and Kurtosis

Executed take-profits and stop-losses might tend to offset


Example: Rate rise triggers take-profit sells and stop-loss buys
If same amount of each, no effect on returns
Level
Take-Prof Sell
Stop-Loss Buy
Time
Exchange Rate

But orders cluster at different levels, so less offsetting


Lots of take-profits or lots of stop-losses
More big returns
Level
Take-Prof Sell
Stop-Loss Buy
Time
Exchange Rate
QWAFAFEW July 20, 2010: Extreme Returns in FX
Link
25
How Much Kurtosis?

Simulations isolate effect of each factor on order-flow kurtosis






5 years of trading days
Half-hour horizon, 24-hours per day
4 orders per half hour, on average
No other trades
Calibrated simulations match properties of original orders data
30 simulations per case

Standard errors calculated across simulations
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Order Size Has Biggest Direct Impact
Source
Half-Hour Kurtosis
EUR
JPY
GBP
105.3
9.5
7.4
Intraday Volatility Pattern
4.0
3.8
4.4
Exchange-Rate Preferences
4.4
4.3
4.5
Order Size Distribution

What if all three sources operate at once?
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Interactions Dominate
Source
Half-Hour Kurtosis
EUR
JPY
GBP
105.3
9.5
7.4
Intraday Volatility Pattern
4.0
3.8
4.4
Exchange-Rate Preferences
4.4
4.3
4.5
Sum
113.3
17.6
16.4
Simulation With All 3 Factors
305.3
20.3
23.8
Order Size Distribution
QWAFAFEW July 20, 2010: Extreme Returns in FX
28
Outline

Data


3 Key Features of SL and TP Orders

1. Distribution of individual order sizes
2. Time-of-day clustering
3. Exchange-rate clustering

Interactions more powerful than individual factors in isolation

4th Factor: Feedback, Order Flow  Returns

How much kurtosis?



Linear feedback
Concave feedback
Summary
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Feedback from Order Flow to Returns

Price Cascade





Rate falls through 00 to 95
Triggers stop-loss sell orders
Rate falls further
More stop-loss sell orders
Rate falls even further …

Generates extreme returns, fat tails of return distribution

Common in FX


According to market participants
Once per week? Many times per week?
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Feedback from Order Flow to Returns

Price Halt






Rate falls through 110 to 105
Triggers take-profit buy orders
Buy orders impede rate from falling further
With stopped rate, no orders triggered next period
With no orders, rate stays put
Generates tiny returns, tall skinny middle of return
distribution
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Feedback Has Modest Direct Effect

Dynamic simulations
OrderFlowt = F(St, St-1)
ln(St+1) - ln(St ) = Constant • OrderFlowt

Simulations calibrated to match original RBS data




True order size distribution
True intraday exchange-rate volatility pattern
True exchange-rate preferences
Many other features of data
QWAFAFEW July 20, 2010: Extreme Returns in FX
32
Simulated Rates Look Realistic

One simulated exchange-rate path
Price Cascades
Price Halts
QWAFAFEW July 20, 2010: Extreme Returns in FX
33
Calibration
Actual
Simulated
QWAFAFEW July 20, 2010: Extreme Returns in FX
34
Feedback Has Modest Direct Effect

Direct effect: Assume away order-flow factors

Size distribution, clustering …
Source
Half-Hour Kurtosis
EUR
JPY
GBP
Feedback Direct Effect
13
11
14
Order-Flow Factors Only
305
20
24
Reality
19
14
11
QWAFAFEW July 20, 2010: Extreme Returns in FX
35
Feedback Has Huge Indirect Effects


Direct effect: Assume away order-flow factors
All effects: Restore order-flow factors

Source
Half-Hour Kurtosis
EUR
JPY
GBP
Feedback Direct Effect
13
11
14
Feedback All Effects (Linear)
946
99
157
Order-Flow Factors Only
305
20
24
Reality
19
14
11
QWAFAFEW July 20, 2010: Extreme Returns in FX
36
Feedback Has Huge Indirect Effects

Huge return kurtosis with all factors



For EUR, almost 1,000!
But: Exchange-rate kurtosis <<< 1,000!
Note: No linear relationship, order flow to returns


Large orders are managed, effect on returns is not proportionate
Next: Simulation where diminishing marginal effect of order flow
OrderFlowt = F(St, St-1)
ln(St+1) – ln(St ) = Constant • OrderFlowt
QWAFAFEW July 20, 2010: Extreme Returns in FX
37
Concave Feedback  Realistic Kurtosis

Simulations:
OrderFlowt = F(St, St-1)
St+1 - St  Constant • OrderFlowt
Source
Half-Hour Kurtosis
EUR
JPY
GBP
Feedback Direct Effect
13
11
14
Feedback All Effects (Linear)
946
99
157
8
5
5
Order-Flow Factors Only
305
20
24
Reality
19
14
11
Feedback All Effects (Concave)
QWAFAFEW July 20, 2010: Extreme Returns in FX
38
Concave Feedback  Realistic Kurtosis

Simulations:
OrderFlowt = F(St, St-1)
ln(St+1) – ln(St ) = Constant • OrderFlowt
Source
One-Hour Kurtosis
EUR
JPY
GBP
Feedback All Effects (Concave)
10.5
7.7
6.8
Reality
13.8
11.9
8.8
% of excess kurt. from SL & TP
69%
52%
65%
QWAFAFEW July 20, 2010: Extreme Returns in FX
39
Summary

Three properties of SL, TP orders generate kurtosis in returns
1. Order size distribution
2. Clustering in execution across trading day
3. Clustering across exchange-rate levels
4. Feedback with exchange-rate returns

SL, TPs produce substantial return kurtosis
 Accounts for ½ - 2/3 of excess kurtosis at one-hour horizon
Price-contingent order flow
important source of extreme returns
QWAFAFEW July 20, 2010: Extreme Returns in FX
40
Risk Management:
Why Might Tails Get Fatter?
1.
More kurtosis in order size distribution
Greater use of barrier options
2.
More extreme intraday volatility pattern
Much has to do with sleeping/waking patterns, and how many people
place orders at different hours
Rising international trade — More fat tails?
Bank consolidation — Less fat tails?
3.
Stronger preference for round numbers
4.
Stronger differences between stop-losses and take-profits
QWAFAFEW July 20, 2010: Extreme Returns in FX
41
Extensions

News?

Rising order flow?

The rest of order flow?
QWAFAFEW July 20, 2010: Extreme Returns in FX
42
Influence of News?

Add actual U.S. macro statistical releases, 2004-2009

8 significant items

GBP Return Kurtosis
With and Without News
The usual suspects
(Non-linear Simulations)

Effect very small

But much news excluded
9
8
7
6
5
4
3
2
0.5
1
2
6
12
24
48
72
Time Horizon (Hours)
With News
QWAFAFEW July 20, 2010: Extreme Returns in FX
No News
43
Influence From Rising Trading Volume?
Lowers kurtosis at shortest horizons

More orders, less fat tails

Raises kurtosis at longer horizons

More feedback effects

EUR Return Kurtosis
Eur Return Kurtosis
(Linear Sim ulations)
(Non-Linear Sim ulations)
1000
12
10
8
6
4
2
0
800
600
400
200
0
0.5
1
2
6
12
24
48
Time Horizon (hours)
Low Orders
High Orders
72
0.5
1
2
6
12
24
48
72
Time Horizon (hours)
Low Orders
QWAFAFEW July 20, 2010: Extreme Returns in FX
High Orders
44
Kurtosis From the Rest of Order Flow?


Kurtosis in size distribution of EBS (interdealer) trades: 99
Time-of-day clustering in EBS trades? Yes
Percent Hourly EBS Volume: EURUSD
Oct 01 - Oct 02
12%
10%
8%
6%
4%
2%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Global Trading Time
QWAFAFEW July 20, 2010: Extreme Returns in FX
45
How Much Order-Flow Kurtosis?

How do we get these numbers?

Calibrated simulations




E.g.: Contribution of intraday volatility pattern to kurtosis
Each period, choose number of exchange-rate levels to cross
Calibrate order execution frequency so average orders/half hour = 4
Pick order sizes from normal distribution, mean zero
QWAFAFEW July 20, 2010: Extreme Returns in FX
46
Exchange-Rate Preference and Kurtosis

Stop-loss and take-profit orders cluster differently

Take-profit: Cluster BEFORE round numbers
Take-Profit
Sell
Take-Profit
Buy
Above 00: 01-10
8
12
Below 00: 90-99
15
8
Exchange Rate
Round Number
Take-Prof Sell
Time
Take-Prof Buy
Time
Exchange Rate
QWAFAFEW July 20, 2010: Extreme Returns in FX
47
Exchange-rate Preferences and Kurtosis

Stop-loss and take-profit orders cluster differently


Take-profit: Cluster BEFORE round numbers
Stop-loss: Cluster AFTER round numbers
Take-Profit
Sell
Take-Profit
Buy
Stop-Loss
Sell
Stop-Loss
Buy
Above 00: 01-10
8
12
4
11
Below 00: 90-99
15
8
12
6
Exchange Rate
Stop-Loss Buy
Round Number
Time
Stop-Loss Sell
Time
Exchange Rate
QWAFAFEW July 20, 2010: Extreme Returns in FX
48
Exchange-rate Preferences and Kurtosis

Stop-loss and take-profit orders cluster differently



Take-profit: Cluster BEFORE round numbers
Stop-loss: Cluster AFTER round numbers
With different clustering, higher likelihood of order
clumps


Lots of take-profits, or lots of stop-losses
With more clumps, less offsetting, more big returns
Link
Back
QWAFAFEW July 20, 2010: Extreme Returns in FX
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Existence of 4th Moments?

Not an issue: For us, 4th moment just descriptive
device

But DO they exist? Maybe not at shortest horizons

Hill estimates of tail indexes, a; Moment of order q exists if q > a

EUR
JPY
k is fraction of observations
included in Hill
estimate
GBP
Left
Right
Left
Right
Left
Right
k = 0.1
3.58
3.38
3.6
3.47
3.68
3.76
k = 0.2
3.33
3.25
3.55
3.51
3.54
3.61
k = 0.1
5.34
5.07
5.2
4.39
4.16
5.38
k = 0.2
4.21
4.41
5.03
4.01
4.05
5.09
½-hour
12 hours
QWAFAFEW July 20, 2010: Extreme Returns in FX
Link
Back
50