qwafafew july 20, 2010: extreme returns in fx 1 extreme returns the case of currencies carol osler...

QWAFAFEW July 20, 2010: Extreme Returns in FX 1

Extreme Returns The Case of Currencies

Carol OslerBrandeis University

Tanseli SavaserWilliams College


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


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”?


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


Extreme Returns, Fat Tails, & Kurtosis Fat tails: High frequency of extreme outcomes

Benchmark: Normal Distribution

Broader Concept: Kurtosis 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


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 Link


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 1 ½

1 ½ to 2 ½

2 ½ to 3 ½

3 ½to 4 ½

4 ½to 5 ½

5 ½to 6 ½

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


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


Outline Data

3 Key Features of Price-Contingent Orders1. 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


Data

Royal Bank of Scotland Currently 5th largest FX dealing bank worldwide (Euromoney, 2007)

Complete book of stop-loss, take-profit orders 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


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


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


Who Places Stop-Loss and Take-Profit Orders?

Value ($ billions) Share %Take-Profit

Customers

Levered Money (e.g., Soros) 12.6 6.4 64.1

Real Money (e.g., Fidelity) 7.8 3.7 60.2

Broker-Dealers (e.g., Bear Stearns) 13.7 6.5 33.5

Gov’t Agencies, Central Banks 3.9 1.5 72.6

Large Corporates (e.g., GM) 20.1 9.9 71.4

Middle-Market Corporates 4.5 4.9 81.4

Banks

Royal Bank 77.3 35.6 55.3

Global Liquidity Providers 19.8 11.3 35.6

Regional Liquidity Providers 7.5 3.7 54.8

Customer-Service Banks 26.5 16.6 62.2


Outline

Data 3 Key Features of SL and TP Orders

1. Distribution of individual order sizes


3. Exchange-rate clustering How much kurtosis?



Summary


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


Distribution of Order Sizes

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 1 ½

1 ½ to 2 ½

2 ½ to 3 ½

3 ½to 4 ½

4 ½to 5 ½

5 ½to 6 ½

> 6 ½

Share Ratio 2 0.3 0.4 0.4 13 192 23175 31 Mill.

High kurtosis in distribution of individual order sizes EUR: 725! GBP: 21 JPY: 26

Tall Skinny Middle Fat Tails


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


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


Distribution of order flow Normal as N How fast?

Answer from simulation: Picking order sizes at random

How many orders executed per 1/2-hour, in reality? Back-of-the-envelope: 3 or 4. We go with 4

Distribution of Order Sizes

Orders per

Period1 2 3 4 5 10 20 50 100

Order-Flow

Kurtosis513 252 173 130 105 55 29 13 8


Intraday Volatility Pattern and Kurtosis

Exchange-Rate Levels Crossed per Half Hour

Asia

London

New York


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


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


Exchange-Rate Preference and Kurtosis People prefer to place orders at certain rates

Special preference for round numbers, for example $1.7600/£

0

2

4

6

00 10 20 30 40 50 60 70 80 90

Final Two Digits of Exchange Rate

Per

cent

of

all e

xecu

ted

orde

rs


Exchange-Rate Preference and Kurtosis

People prefer to place orders at certain levels End digit 0 preferred to 5 ….. 5 preferred to 2,3,7,8 ….

….. 2,3,7,8 preferred to 1,4,6,9

Orders executed depend on specific rates (St) crossed 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


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

But orders cluster at different levels, so less offsetting Lots of take-profits or lots of stop-losses More big returns

Exchange-Rate Preference and Kurtosis

Level

Exchange Rate

Stop-Loss BuyTime

Take-Prof Sell

Level

Exchange Rate

Stop-Loss BuyTime

Take-Prof Sell

Link


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


Order Size Has Biggest Direct Impact

What if all three sources operate at once?

Source Half-Hour Kurtosis

EUR JPY GBP

Order Size Distribution 105.3 9.5 7.4

Intraday Volatility Pattern 4.0 3.8 4.4

Exchange-Rate Preferences 4.4 4.3 4.5


Interactions Dominate


EUR JPY GBP

Order Size Distribution 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


Outline Data 3 Key Features of SL and TP Orders

1. Distribution of individual order sizes


3. Exchange-rate clustering Interactions more powerful than individual factors in isolation



Summary


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?


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


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


Simulated Rates Look Realistic One simulated exchange-rate path

Price Cascades Price Halts


Calibration

Actual

Simulated


Feedback Has Modest Direct Effect

Direct effect: Assume away order-flow factors Size distribution, clustering …


EUR JPY GBP

Feedback Direct Effect 13 11 14

Order-Flow Factors Only 305 20 24

Reality 19 14 11


Feedback Has Huge Indirect Effects

Direct effect: Assume away order-flow factors All effects: Restore order-flow factors


EUR JPY GBP


Feedback All Effects (Linear) 946 99 157


Reality 19 14 11


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


Concave Feedback Realistic Kurtosis

Simulations: OrderFlowt = F(St, St-1)

St+1 - St Constant • OrderFlowt


EUR JPY GBP


Feedback All Effects (Linear) 946 99 157

Feedback All Effects (Concave) 8 5 5


Reality 19 14 11


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%


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


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


Extensions

News?

Rising order flow?

The rest of order flow?


Influence of News?

Add actual U.S. macro statistical releases, 2004-2009 8 significant items

The usual suspects

Effect very small But much news excluded

GBP Return Kurtosis With and Without News

(Non-linear Simulations)

23456789

0.5 1 2 6 12 24 48 72Time Horizon (Hours)

With News No News


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 (Linear Simulations)

0

200

400

600

800

1000

0.5 1 2 6 12 24 48 72

Time Horizon (hours)

Low Orders High Orders

Eur Return Kurtosis(Non-Linear Simulations)

0

24

6

810

12

0.5 1 2 6 12 24 48 72

Time Horizon (hours)

Low Orders High Orders


Kurtosis From the Rest of Order Flow?

Percent Hourly EBS Volume: EURUSDOct 01 - Oct 02

0%

2%

4%

6%

8%

10%

12%

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

Kurtosis in size distribution of EBS (interdealer) trades: 99 Time-of-day clustering in EBS trades? Yes


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

How Much Order-Flow Kurtosis?


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

Round Number

Exchange Rate

Take-Prof Sell

Exchange Rate

Take-Prof BuyTime

Time


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

Round Number

Exchange Rate

Stop-Loss Buy

Time

Exchange Rate

Stop-Loss Sell Time


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

LinkBack


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, Moment of order exists if > k is fraction of observations included in Hill estimateEUR JPY GBP

Left Right Left Right Left Right

½-hour

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

12 hours

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

LinkBack

qwafafew july 20, 2010: extreme returns in fx 1 extreme returns the case of currencies carol osler...

Documents

fx2 extreme returns

fx5 extreme returns

kurtosis of financial

exchangerate returns

distribution of order

normal distribution

equities slide

fx6 kurtosis