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Extreme Returns The Case of Currencies

This study explores the occurrence of extreme returns in the foreign exchange market and their impact on risk management and option pricing. It examines the factors contributing to extreme returns and the distribution of order flow. The study also discusses the concept of fat tails and kurtosis in exchange rate returns.

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Extreme Returns The Case of Currencies

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  1. Extreme Returns The Case of Currencies Carol Osler Brandeis University Tanseli Savaser Williams College QWAFAFEW July 20, 2010: Extreme Returns in FX

  2. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  3. 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”? QWAFAFEW July 20, 2010: Extreme Returns in FX

  4. Contributions • 4 Ways Price-Contingent Trading Increases Extreme Returns • Affect Distribution of Order-Flow Itself Three Ways • Distribution of trade sizes • Clustering of trades at times of day • 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  5. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  6. Kurtosis in Exchange-Rate Returns Link QWAFAFEW July 20, 2010: Extreme Returns in FX

  7. Kurtosis in Exchange-Rate Returns 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 Tall Skinny Middle Fat Tails QWAFAFEW July 20, 2010: Extreme Returns in FX

  8. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  9. Outline • Data • 3 Key Features of Price-Contingent Orders • Distribution of individual order sizes • Time-of-day clustering • Exchange-rate clustering • How much kurtosis? • 4th Factor: Feedback, Order Flow  Returns • How much kurtosis? • Linear feedback • Concave feedback • Summary QWAFAFEW July 20, 2010: Extreme Returns in FX

  10. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  11. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  12. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  13. Who Places Stop-Loss and Take-Profit Orders? QWAFAFEW July 20, 2010: Extreme Returns in FX

  14. Outline • Data • 3 Key Features of SL and TP Orders • Distribution of individual order sizes • Time-of-day clustering • Exchange-rate clustering • How much kurtosis? • 4th Factor: Feedback, Order Flow  Returns • How much kurtosis? • Linear feedback • Concave feedback • Summary QWAFAFEW July 20, 2010: Extreme Returns in FX

  15. 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

  16. Distribution of Order Sizes • High kurtosis in distribution of individual order sizes • EUR: 725! GBP: 21 JPY: 26 Tall Skinny Middle Fat Tails QWAFAFEW July 20, 2010: Extreme Returns in FX

  17. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  18. 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

  19. Distribution of Order Sizes • 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  20. Intraday Volatility Pattern and Kurtosis Exchange-Rate Levels Crossed per Half Hour New York London Asia QWAFAFEW July 20, 2010: Extreme Returns in FX

  21. 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

  22. 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 • Dailyorder 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  23. Exchange-Rate Preference and Kurtosis • People prefer to place orders at certain rates • Special preference for round numbers, for example $1.7600/£ QWAFAFEW July 20, 2010: Extreme Returns in FX

  24. 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 Stcrosses level ending in “00,” many orders (5 %) • If Stcrosses 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

  25. 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 • 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 Level Take-Prof Sell Stop-Loss Buy Time Link Exchange Rate QWAFAFEW July 20, 2010: Extreme Returns in FX

  26. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  27. Order Size Has Biggest Direct Impact • What if all three sources operate at once? QWAFAFEW July 20, 2010: Extreme Returns in FX

  28. Interactions Dominate QWAFAFEW July 20, 2010: Extreme Returns in FX

  29. Outline • Data • 3 Key Features of SL and TP Orders • Distribution of individual order sizes • Time-of-day clustering • 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

  30. 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? QWAFAFEW July 20, 2010: Extreme Returns in FX

  31. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  32. 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

  33. Simulated Rates Look Realistic • One simulated exchange-rate path Price Halts Price Cascades QWAFAFEW July 20, 2010: Extreme Returns in FX

  34. Calibration Actual Simulated QWAFAFEW July 20, 2010: Extreme Returns in FX

  35. Feedback Has Modest Direct Effect • Direct effect: Assume away order-flow factors • Size distribution, clustering … QWAFAFEW July 20, 2010: Extreme Returns in FX

  36. Feedback Has Huge Indirect Effects • Direct effect: Assume away order-flow factors • All effects: Restore order-flow factors QWAFAFEW July 20, 2010: Extreme Returns in FX

  37. 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

  38. Concave Feedback  Realistic Kurtosis • Simulations: OrderFlowt = F(St, St-1) St+1 - St Constant • OrderFlowt QWAFAFEW July 20, 2010: Extreme Returns in FX

  39. Concave Feedback  Realistic Kurtosis • Simulations: OrderFlowt = F(St, St-1) ln(St+1) – ln(St)=Constant • OrderFlowt QWAFAFEW July 20, 2010: Extreme Returns in FX

  40. Summary • Three properties of SL, TP orders generate kurtosis in returns • Order size distribution • Clustering in execution across trading day • Clustering across exchange-rate levels • 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

  41. Risk Management: Why Might Tails Get Fatter? • More kurtosis in order size distribution Greater use of barrier options • 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? • Stronger preference for round numbers • Stronger differences between stop-losses and take-profits QWAFAFEW July 20, 2010: Extreme Returns in FX

  42. Extensions • News? • Rising order flow? • The rest of order flow? QWAFAFEW July 20, 2010: Extreme Returns in FX

  43. 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 QWAFAFEW July 20, 2010: Extreme Returns in FX

  44. Influence From Rising Trading Volume? • Lowers kurtosis at shortest horizons • More orders, less fat tails • Raises kurtosis at longer horizons • More feedback effects QWAFAFEW July 20, 2010: Extreme Returns in FX

  45. Kurtosis From the Rest of Order Flow? • Kurtosis in size distribution of EBS (interdealer) trades: 99 • Time-of-day clustering in EBS trades? Yes QWAFAFEW July 20, 2010: Extreme Returns in FX

  46. 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

  47. Exchange-Rate Preference and Kurtosis • Stop-loss and take-profit orders cluster differently • Take-profit: Cluster BEFORE round numbers Exchange Rate Round Number Take-Prof Buy Time Take-Prof 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 Exchange Rate Stop-Loss Buy Round Number Time Time Stop-Loss Sell Exchange Rate QWAFAFEW July 20, 2010: Extreme Returns in FX

  49. 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

  50. 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 • k is fraction of observations included in Hill estimate Link Back QWAFAFEW July 20, 2010: Extreme Returns in FX

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