Quantitative trading

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An overview of optimal trading: taking into account the market microstructure to optimize your trades

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Quantitative trading

  1. 1. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day high frequency trading : from empirical evidences to quantitative optimization Charles-Albert Lehalle, PhD clehalle@cheuvreux.com Head of CA Cheuvreux Quantitative Research March 2008 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  2. 2. Practical aspects of auction markets Quantitative optimization of high freq trading Pre high-frequency world All seems to be smooth and regular, like a landscape covered by snow. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  3. 3. Practical aspects of auction markets Quantitative optimization of high freq trading Pre high-frequency world Now we discover some singularities: the world is not as linear as we thought. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  4. 4. Practical aspects of auction markets Quantitative optimization of high freq trading What fair value? When the fair value of a firm changes, the price does not immediately “jumps” from one value to another one: the process of price formation takes place. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  5. 5. Practical aspects of auction markets Quantitative optimization of high freq trading What fair value? When the fair value of a firm changes, the price does not immediately “jumps” from one value to another one: the process of price formation takes place. This exploration process is sometimes considered as the search for imbalance between agents (dynamic equilibrium search), as other authors consider it as the “trace” of a direct path to the “intrinsic fair value” of the firm into the market auction mechanisms. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  6. 6. Practical aspects of auction markets Quantitative optimization of high freq trading What fair value? When the fair value of a firm changes, the price does not immediately “jumps” from one value to another one: the process of price formation takes place. This exploration process is sometimes considered as the search for imbalance between agents (dynamic equilibrium search), as other authors consider it as the “trace” of a direct path to the “intrinsic fair value” of the firm into the market auction mechanisms. Here we will not focus on the fair value estimation process, but on how to deal with the price formation process. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  7. 7. Practical aspects of auction markets Quantitative optimization of high freq trading That is real stuff Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  8. 8. Practical aspects of auction markets Quantitative optimization of high freq trading That is real stuff This is the effect of the announcment by president Sarkozy that public TV channels will no more be allowed to sell advertising... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  9. 9. Practical aspects of auction markets Quantitative optimization of high freq trading That is real stuff This is the effect of the announcment by president Sarkozy that public TV channels will no more be allowed to sell advertising... The value of the main french private Channels jumped immediately. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  10. 10. Practical aspects of auction markets Quantitative optimization of high freq trading Effets of such a change 1. The fair values of private channels change first Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  11. 11. Practical aspects of auction markets Quantitative optimization of high freq trading Effets of such a change 1. The fair values of private channels change first 2. Propagation of the change to correlated stocks (Bouygues) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  12. 12. Practical aspects of auction markets Quantitative optimization of high freq trading Effets of such a change 1. The fair values of private channels change first 2. Propagation of the change to correlated stocks (Bouygues) 3. “Resistance” throught stocks of the same sector that are not affected by the event Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  13. 13. Practical aspects of auction markets Quantitative optimization of high freq trading Effets of such a change 1. The fair values of private channels change first 2. Propagation of the change to correlated stocks (Bouygues) 3. “Resistance” throught stocks of the same sector that are not affected by the event 4. Closed loops within the worldwide markets (and other classes of assets) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  14. 14. Practical aspects of auction markets Quantitative optimization of high freq trading Effets of such a change 1. The fair values of private channels change first 2. Propagation of the change to correlated stocks (Bouygues) 3. “Resistance” throught stocks of the same sector that are not affected by the event 4. Closed loops within the worldwide markets (and other classes of assets) This vision is the Capital Asset Pricing Model (CAPM) one: a worldwide market driven by few systematic factors. In this framework some temporary explorations are conducted by isolated stocks or sectors (specifics explorations). Extra day arbitrageurs focus on those aspects. At intra day scale, we try to understand and detect when and how auction mechanisms of market places are able to absorb such trajectories. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  15. 15. Practical aspects of auction markets Quantitative optimization of high freq trading Propagation - correlations Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  16. 16. Practical aspects of auction markets Quantitative optimization of high freq trading Propagation - zooming out: 30 minutes Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  17. 17. Practical aspects of auction markets Quantitative optimization of high freq trading Propagation - zooming out: 1 hour Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  18. 18. Practical aspects of auction markets Quantitative optimization of high freq trading Propagation - zooming out: daily Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  19. 19. Practical aspects of auction markets Quantitative optimization of high freq trading Propagation - sector: daily media Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  20. 20. Practical aspects of auction markets Quantitative optimization of high freq trading Contenu Practical aspects of auction markets Bid-ask spread, limit order books Intra day nomograms The two main enemies of intraday trading Alternate sources of liquidity Quantitative optimization of high freq trading High freq trading in equations In the heart of darkness: market impact models From simple to sophisticated optimization Two main evolutions of stock trading Non parametric approaches Misc Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  21. 21. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books The principles of an auction market Some agents wants to buy, others want to sell. Three main roles: Informed traders: they are convinced to have a clear idea of the fair value of the equity part of the firm Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  22. 22. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books The principles of an auction market Some agents wants to buy, others want to sell. Three main roles: Informed traders: they are convinced to have a clear idea of the fair value of the equity part of the firm Noise traders: they need to trade for another reason (their utility function is not directly related to the price of the equity) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  23. 23. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books The principles of an auction market Some agents wants to buy, others want to sell. Three main roles: Informed traders: they are convinced to have a clear idea of the fair value of the equity part of the firm Noise traders: they need to trade for another reason (their utility function is not directly related to the price of the equity) Arbitragers: they try to capture imperfections Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  24. 24. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books The principles of an auction market Some agents wants to buy, others want to sell. Three main roles: Informed traders: they are convinced to have a clear idea of the fair value of the equity part of the firm Noise traders: they need to trade for another reason (their utility function is not directly related to the price of the equity) Arbitragers: they try to capture imperfections Each agent place orders at given prices for given quantities, when they match: a transaction occurs Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  25. 25. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Bids and asks: from orders to LOB Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  26. 26. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Bids and asks: from orders to LOB Price is a function of quantity! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  27. 27. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Bids and asks: from orders to LOB Price is a function of quantity! ± ± Notations: fLOB for densities, FLOB for cumulated quantities Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  28. 28. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books From orders to LOB Statistical theory of the continuous double auction - Eric Smith , J Doyne Farmer, Laszlo Gillemot and Supriya Krishnamurthy - Santa Fe Institute - September 2003 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  29. 29. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Continuous auctions Actors Liquidity provider Wait to be executed (not executed for sure) Execution price is better than the last quoted one (sometimes) pays less fees Liquidity consumer Executed for sure Pays around half a bid-ask spread more than the last quoted price Pays Price Impact for large quantities Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  30. 30. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books The free option myth If I decide to place a Buy order (one share) at a price of S0 − δS as a liquidity provider during ∆T seconds, and then to buy at S∆T as a consumer (if I have not been executed): it’s as simple as a barrier option. A very special one: k.o. option with barrier level B = K the strike. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  31. 31. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books The free option myth If I decide to place a Buy order (one share) at a price of S0 − δS as a liquidity provider during ∆T seconds, and then to buy at S∆T as a consumer (if I have not been executed): it’s as simple as a barrier option. A very special one: k.o. option with barrier level B = K the strike. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  32. 32. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Should I stay or should I go? Let qT be the probability to cross the barrier between 0 and T (τB is the first time of crossing), bayes decomposition gives us: IE (S) = IE (S|τB < T ) I (τB < T ) +I (S|τB ≥ T )I (τB ≥ T ) P E P qT Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  33. 33. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Should I stay or should I go? Let qT be the probability to cross the barrier between 0 and T (τB is the first time of crossing), bayes decomposition gives us: IE (S) = IE (S|τB < T ) I (τB < T ) +I (S|τB ≥ T )I (τB ≥ T ) P E P qT S0 − B qT ⇒ IE (S|τB ≥ T ) = 1 − qT Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  34. 34. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Should I stay or should I go? Let qT be the probability to cross the barrier between 0 and T (τB is the first time of crossing), bayes decomposition gives us: IE (S) = IE (S|τB < T ) I (τB < T ) +I (S|τB ≥ T )I (τB ≥ T ) P E P qT S0 − B qT ⇒ IE (S|τB ≥ T ) = 1 − qT Finally, the payoff of such a strategy is (at this time scale, risk free interest rate can be neglected): IE (ST if τB < T , B otherwise) = S0 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  35. 35. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Should I stay or should I go? Let qT be the probability to cross the barrier between 0 and T (τB is the first time of crossing), bayes decomposition gives us: IE (S) = IE (S|τB < T ) I (τB < T ) +I (S|τB ≥ T )I (τB ≥ T ) P E P qT S0 − B qT ⇒ IE (S|τB ≥ T ) = 1 − qT Finally, the payoff of such a strategy is (at this time scale, risk free interest rate can be neglected): IE (ST if τB < T , B otherwise) = S0 ⇒ All the complexity of intra day trading comes from market microstructure and price / volume dependencies. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  36. 36. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books My first continuous auctions formula − Let fLOB (s) be the volume available in the LOB by sellers (ask + prices) at price s (resp. fLOB (s) for bids quantities). When I want to buy a quantity V of shares of this instrument, I will generate a Market Impact of SPI implicitly defined by the equation (S0 is the last quoted price): SMI − (1) V= fLOB (s) ds s=S0 and my Volume Weightened Average Price (VWAP) will be: SMI 1 − s · fLOB (s) ds (2) VWAP = V s=S0 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  37. 37. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fondamental order book equation Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  38. 38. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fondamental order book equation Derivation of the market impact equation (1) gives: Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  39. 39. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fondamental order book equation Derivation of the market impact equation (1) gives: 1 − (3) fLOB (SPMI (v )) = ∂v SMI (v ) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  40. 40. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fondamental order book equation Derivation of the market impact equation (1) gives: 1 − (3) fLOB (SPMI (v )) = ∂v SMI (v ) Tell me your price impact function SMI (v ), I will tell you your implicit orderbook function. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  41. 41. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fondamental order book equation Derivation of the market impact equation (1) gives: 1 − (3) fLOB (SPMI (v )) = ∂v SMI (v ) Tell me your price impact function SMI (v ), I will tell you your implicit orderbook function. So if you think that price impact is constant, it means that you think that LOB is infinite. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  42. 42. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fondamental order book equation Derivation of the market impact equation (1) gives: 1 − (3) fLOB (SPMI (v )) = ∂v SMI (v ) Tell me your price impact function SMI (v ), I will tell you your implicit orderbook function. So if you think that price impact is constant, it means that you think that LOB is infinite. “The big secret...Quantitative finance is one of the easiest branches of mathematics” Paul Wilmott on his blog, April 2008... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  43. 43. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Fixing auctions The rules are different for a fixing auction. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  44. 44. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Fixing auctions The rules are different for a fixing auction. during the pre fixing phase, orders are not crossed to generate transactions Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  45. 45. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Fixing auctions The rules are different for a fixing auction. during the pre fixing phase, orders are not crossed to generate transactions a “theoretical price” is published Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  46. 46. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Fixing auctions The rules are different for a fixing auction. during the pre fixing phase, orders are not crossed to generate transactions a “theoretical price” is published at the end (5 minutes later): the fixed price is the natural Pareto equilibrium Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  47. 47. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Fixing price The fixing price S ∗ is such that FLOB (S ∗ ) = FLOB (S ∗ ) − + (4) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  48. 48. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books ˜ If I put a quantity δV into the fixing orderbook at a price S into − FLOB . Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  49. 49. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books ˜ If I put a quantity δV into the fixing orderbook at a price S into − FLOB . The new fixing price will be such that Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  50. 50. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books ˜ If I put a quantity δV into the fixing orderbook at a price S into − FLOB . The new fixing price will be such that −¯ +¯ FLOB (S) = FLOB (S) + δV · 1S<S ∗ (5) ˜ Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  51. 51. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fundamental Fixing equation When δV is small enough: Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  52. 52. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fundamental Fixing equation When δV is small enough: FLOB (S ∗ ) − fLOB (S ∗ ) · δs = FLOB (S ∗ ) + fLOB (S ∗ ) · δs + δV + o(δs) − − + + Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  53. 53. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fundamental Fixing equation When δV is small enough: FLOB (S ∗ ) − fLOB (S ∗ ) · δs = FLOB (S ∗ ) + fLOB (S ∗ ) · δs + δV + o(δs) − − + + and finally: − δV (6) δs = + o(δs) − + (S ∗ ) + fLOB (S ∗ ) fLOB Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  54. 54. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books One fundamental Fixing equation When δV is small enough: FLOB (S ∗ ) − fLOB (S ∗ ) · δs = FLOB (S ∗ ) + fLOB (S ∗ ) · δs + δV + o(δs) − − + + and finally: − δV (6) δs = + o(δs) − + (S ∗ ) + fLOB (S ∗ ) fLOB It is a matter of slopes. The effect of adding volume far from the previous equilibrium price S ∗ is mainly related to local properties of the orderbook around S ∗ . Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  55. 55. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books From price impact to market impact The price formation process can be seen as a diffusion in an heterogeneous environment: Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  56. 56. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books From price impact to market impact The price formation process can be seen as a diffusion in an heterogeneous environment: the more volume are in front of the diffusion, the more slow it will be able to progress, when you really zoom out, those effects disappear. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  57. 57. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books From price impact to market impact The price formation process can be seen as a diffusion in an heterogeneous environment: the more volume are in front of the diffusion, the more slow it will be able to progress, when you really zoom out, those effects disappear. You do not want to optimise your trading strategy at an order-by-order scale, so you need to zoom out a little and capture some market impact properties. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  58. 58. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books From price impact to market impact The price formation process can be seen as a diffusion in an heterogeneous environment: the more volume are in front of the diffusion, the more slow it will be able to progress, when you really zoom out, those effects disappear. You do not want to optimise your trading strategy at an order-by-order scale, (your trading tactic will be optimised on an order-by-order basis) so you need to zoom out a little and capture some market impact properties. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  59. 59. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Common sense for Market Impact model Crossing the Bid-Ask spread. If the last trade was on the opposite side to my order (once over two trades...), I pay the spread: 2ψ. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  60. 60. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Common sense for Market Impact model Crossing the Bid-Ask spread. If the last trade was on the opposite side to my order (once over two trades...), I pay the spread: 2ψ. Dynamical market depth. The more the volatility of the price diffusion is high, the more the LOB is structurally empty: M.I. increases with σ. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  61. 61. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Common sense for Market Impact model Crossing the Bid-Ask spread. If the last trade was on the opposite side to my order (once over two trades...), I pay the spread: 2ψ. Dynamical market depth. The more the volatility of the price diffusion is high, the more the LOB is structurally empty: M.I. increases with σ. Usual quantities: the more my volume v is large compared to the “usual” traded volume V∆T (during this time interval ∆T ), the more my M.I. increases. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  62. 62. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Qualitative Market Impact model Basic Market Impact model v (7) MI(v ) = ψ + F σ, V∆T where F (·, ·) increases with respect to its two inputs. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  63. 63. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Qualitative Market Impact model Basic Market Impact model v (7) MI(v ) = ψ + F σ, V∆T where F (·, ·) increases with respect to its two inputs. Usually σ is on the risk side of the equation, because market impact is subtracted to your return, we already see that in a quantitative trading optimisation framework σ is (at least) on the return side... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  64. 64. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Capture statistical invariants needs datasets If you have a proprietary datasets with your own orders (with enough trades into it), use them. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  65. 65. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Capture statistical invariants needs datasets If you have a proprietary datasets with your own orders (with enough trades into it), use them. It is also possible to work with public datasets. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  66. 66. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Capture statistical invariants needs datasets If you have a proprietary datasets with your own orders (with enough trades into it), use them. It is also possible to work with public datasets. You will need a permanent market impact model to capture fair value changes. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  67. 67. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Capture statistical invariants needs datasets If you have a proprietary datasets with your own orders (with enough trades into it), use them. It is also possible to work with public datasets. You will need a permanent market impact model to capture fair value changes. The choice of a time scale is very important! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  68. 68. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books Capture statistical invariants needs datasets If you have a proprietary datasets with your own orders (with enough trades into it), use them. It is also possible to work with public datasets. You will need a permanent market impact model to capture fair value changes. The choice of a time scale is very important! What do we need? As seen is equation (7), we need to capture σ, usual volumes, and half bid-ask spread. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  69. 69. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books What means “capture statistical invariants” No risk-neutral measure available, Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  70. 70. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books What means “capture statistical invariants” No risk-neutral measure available, so we will work under historical measure, Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  71. 71. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books What means “capture statistical invariants” No risk-neutral measure available, so we will work under historical measure, any statistical knowledge is welcome! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  72. 72. Practical aspects of auction markets Quantitative optimization of high freq trading Bid-ask spread, limit order books What means “capture statistical invariants” No risk-neutral measure available, so we will work under historical measure, any statistical knowledge is welcome! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  73. 73. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Volumes: do not use ”average” on log normal distributions Density of a log normal distribution: (ln(x) − µ)2 1 √ exp − (8) f (x) = 2σ 2 xσ 2π So (the median is not equal to the expectation): σ2 M = I (X ) = exp µ + E 2 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  74. 74. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Five months of FTE.PA, between 11h45 and 12h30 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  75. 75. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Log normality: want do we really need It is easy to see that IP (X > M) 30%. Do we want the mean value, or the median? Five months of FTE.PA (05/07 to 03/08) capital vs proportions Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  76. 76. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Log normality: want do we really need It is easy to see that IP (X > M) 30%. Do we want the mean value, or the median? Five months of FTE.PA (05/07 to 03/08) capital vs proportions If you really want the mean, use robust mean (expectation using median and truncated variance) instead of average! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  77. 77. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Log of proportions Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  78. 78. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Do not mix heterogeneous effects Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  79. 79. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Do not mix heterogeneous effects A large part of the variance comes from mixing Fridays with other days. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  80. 80. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Do not mix heterogeneous effects A large part of the variance comes from mixing Fridays with other days. You can use auto correlations to obtain more robust estimators. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  81. 81. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Volatility : do only add squared volatilities ! 2 2 We all know that N (0, σ1 ) + N (0, σ1 ) ∼ N (0, σ1 + σ2 ). Five months of FTE.PA (05/07 to 03/08) σGK vs proportions Volatility are not “as lognormal” as volumes are. Mainly, they are “far less i.i.d.”... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  82. 82. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility modelling Because of the Markovian aspect of σ intra day, we would like to model them as a diffusion process. A natural candidate is the C.I.R. model: d(σ 2 ) = γ · (m − σt ) dt + κσt dWt 2 (9) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  83. 83. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility modelling Because of the Markovian aspect of σ intra day, we would like to model them as a diffusion process. A natural candidate is the C.I.R. model: d(σ 2 ) = γ · (m − σt ) dt + κσt dWt 2 (9) The point is that m and κ should be time dependent here... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  84. 84. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility estimation Proportion of volatility seems to be more stable by volatility itself Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  85. 85. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility estimation Proportion of volatility seems to be more stable by volatility itself How could we use proportion of volatility? Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  86. 86. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility estimation Proportion of volatility seems to be more stable by volatility itself How could we use proportion of volatility? But measuring volatility itself is difficult Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  87. 87. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility estimation Proportion of volatility seems to be more stable by volatility itself How could we use proportion of volatility? But measuring volatility itself is difficult Our prices are discretized (rounded?) on a price grid Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  88. 88. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Intra day volatility estimation Proportion of volatility seems to be more stable by volatility itself How could we use proportion of volatility? But measuring volatility itself is difficult Our prices are discretized (rounded?) on a price grid A lot of interesting papers have been written on this subject: Jacod, Delattre, At-Sahalia, Zhang, Mykland, Shepard, Rosenbaum Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  89. 89. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms A “simple” model: √ (10) Xn+1 = Xn + σ δt ξn , Sn = Xn + ε Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  90. 90. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms A “simple” model: √ (10) Xn+1 = Xn + σ δt ξn , Sn = Xn + ε √ (Sn+1 − Sn )2 = 2nIE (ε2 ) + O( n) n Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  91. 91. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms A “simple” model: √ (10) Xn+1 = Xn + σ δt ξn , Sn = Xn + ε √ (Sn+1 − Sn )2 = 2nIE (ε2 ) + O( n) n thanks to Mathieu Rosenbaum Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  92. 92. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Bid-ask spread : when microstructure limits the use of diffusion models If only one microstructure effect should be kept, it is the Bid-Ask spread: Sell price = Buy price Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  93. 93. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Bid-ask spread : when microstructure limits the use of diffusion models If only one microstructure effect should be kept, it is the Bid-Ask spread: Sell price = Buy price Volume has an influence on the price Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  94. 94. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Bid-ask spread : when microstructure limits the use of diffusion models If only one microstructure effect should be kept, it is the Bid-Ask spread: Sell price = Buy price Volume has an influence on the price Volatility estimations are not so simple Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  95. 95. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms What about real statistics? “Spread effect” on three stocks Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  96. 96. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  97. 97. Practical aspects of auction markets Quantitative optimization of high freq trading Intra day nomograms Give me two of the three curves, I will give you the missing one... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  98. 98. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market impact (volume driven) demands to trade slowly As suggested by equation (7), a good way to reduce your Market Impact is to trade on a larger time interval ∆T . Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  99. 99. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market impact (volume driven) demands to trade slowly As suggested by equation (7), a good way to reduce your Market Impact is to trade on a larger time interval ∆T . v ≡ cst v ⇒ψ+F σ, with ∆T V∆T with ∆T V∆T Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  100. 100. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market impact (volume driven) demands to trade slowly As suggested by equation (7), a good way to reduce your Market Impact is to trade on a larger time interval ∆T . v ≡ cst v ⇒ψ+F σ, with ∆T V∆T with ∆T V∆T ⇒ trade as slowly as possible! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  101. 101. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market impact (volume driven) demands to trade slowly As suggested by equation (7), a good way to reduce your Market Impact is to trade on a larger time interval ∆T . v ≡ cst v ⇒ψ+F σ, with ∆T V∆T with ∆T V∆T ⇒ trade as slowly as possible! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  102. 102. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market risk (volatility driven) demands to trade fast The more you wait and the more you tale market risk. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  103. 103. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market risk (volatility driven) demands to trade fast The more you wait and the more you tale market risk. For an arithmetic Brownian diffusion, the amplitude of the risk is: √ α · σ ∆T Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  104. 104. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market risk (volatility driven) demands to trade fast The more you wait and the more you tale market risk. For an arithmetic Brownian diffusion, the amplitude of the risk is: √ α · σ ∆T ⇒ trade as fast as possible! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  105. 105. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Market risk (volatility driven) demands to trade fast The more you wait and the more you tale market risk. For an arithmetic Brownian diffusion, the amplitude of the risk is: √ α · σ ∆T ⇒ trade as fast as possible! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  106. 106. Practical aspects of auction markets Quantitative optimization of high freq trading The two main enemies of intraday trading Those two effects are mixed Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  107. 107. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Continuous auctions mechanisms Two main behaviours: Liquidity provider: provide orders to the orderbook (a quantity at a given price for a defined side), my orders are consolidated inside the “limit order book”. Liquidity consumer: give orders (at best, price limited, or more complex) which consume one or more orders of the LOB. Each market place (Dark Pools, Multilateral Trading Facilities, etc) has this set of specific orders, like iceberg, pegged or totally hidden). Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  108. 108. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Multilateral trading facilities (MTF) Chi-X, turquoise, etc... are here or are coming More order types: totally hidden, pegged, etc... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  109. 109. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity The more liquidity pools you can address, the less market impact your orders will have Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  110. 110. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity The more liquidity pools you can address, the less market impact your orders will have Beware of the latency issues! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  111. 111. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Dark pools Light pools garantee to executed the full quatity of your (agressive) order at a price to be definde, Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  112. 112. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Dark pools Light pools garantee to executed the full quatity of your (agressive) order at a price to be definde, Dark pools garantee your execution price (often in relationship with a perfect market) but not the executed quantity. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  113. 113. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Dark pools Light pools garantee to executed the full quatity of your (agressive) order at a price to be definde, Dark pools garantee your execution price (often in relationship with a perfect market) but not the executed quantity. Illuminating the new dark influence on trading and U.S. market structure, Carl Carrie, JOT Winter 2008 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  114. 114. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Smart Order Router take a snapshot of the available markets (best exec proof), Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  115. 115. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Smart Order Router take a snapshot of the available markets (best exec proof), take your decision (split) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  116. 116. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Smart Order Router take a snapshot of the available markets (best exec proof), take your decision (split) what about Dark pools (no market data)? Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  117. 117. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Smart Order Router take a snapshot of the available markets (best exec proof), take your decision (split) what about Dark pools (no market data)? need of statistical tables Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  118. 118. Practical aspects of auction markets Quantitative optimization of high freq trading Alternate sources of liquidity Smart Order Router take a snapshot of the available markets (best exec proof), take your decision (split) what about Dark pools (no market data)? need of statistical tables What does mean VWAP of an order in such a context? Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  119. 119. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Equations I have a volume V ∗ to buy from 0 to T ; I will assume a regular time grid of width δt (n from 1 to N = [T /δt]). Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  120. 120. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Equations I have a volume V ∗ to buy from 0 to T ; I will assume a regular time grid of width δt (n from 1 to N = [T /δt]). My volume will be split in N slices vn such that n vn = V ∗ . Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  121. 121. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Equations I have a volume V ∗ to buy from 0 to T ; I will assume a regular time grid of width δt (n from 1 to N = [T /δt]). My volume will be split in N slices vn such that n vn = V ∗ . My price follows an arithmetic Bronwian diffusion: √ (11) Sn+1 = Sn + α δt + σn+1 δt ξn+1 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  122. 122. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Equations I have a volume V ∗ to buy from 0 to T ; I will assume a regular time grid of width δt (n from 1 to N = [T /δt]). My volume will be split in N slices vn such that n vn = V ∗ . My price follows an arithmetic Bronwian diffusion: √ (11) Sn+1 = Sn + α δt + σn+1 δt ξn+1 My market impact is additive given by a ηn (vn ) function. Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  123. 123. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Equations I have a volume V ∗ to buy from 0 to T ; I will assume a regular time grid of width δt (n from 1 to N = [T /δt]). My volume will be split in N slices vn such that n vn = V ∗ . My price follows an arithmetic Bronwian diffusion: √ (11) Sn+1 = Sn + α δt + σn+1 δt ξn+1 My market impact is additive given by a ηn (vn ) function. So my total price is: N vn · (Sn + ηn (vn )) (12) W= n=1 ˜ Sn (vn ) Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  124. 124. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations The classical rewriting N xn = vn n Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  125. 125. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations The classical rewriting N xn = vn n N N √ (13) W = V · S0 + vn · ηn (vn ) + V · α δt xn σn δt ξn + n=1 n=1 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  126. 126. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Cost minimization Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  127. 127. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Cost minimization N IE (W |σ, V , α) = V · S0 + vn · ηn (vn ) + V · α δt n=1 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  128. 128. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Cost minimization N IE (W |σ, V , α) = V · S0 + vn · ηn (vn ) + V · α δt n=1 η(vn ) + vn · η (vn ) = λ Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  129. 129. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Cost minimization N IE (W |σ, V , α) = V · S0 + vn · ηn (vn ) + V · α δt n=1 η(vn ) + vn · η (vn ) = λ Linear market impact: η(vn ) = η · vn /Vn Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  130. 130. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Cost minimization N IE (W |σ, V , α) = V · S0 + vn · ηn (vn ) + V · α δt n=1 η(vn ) + vn · η (vn ) = λ Linear market impact: η(vn ) = η · vn /Vn Vn V∗ vn = N k=1 Vk Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  131. 131. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Cost minimization N IE (W |σ, V , α) = V · S0 + vn · ηn (vn ) + V · α δt n=1 η(vn ) + vn · η (vn ) = λ Linear market impact: η(vn ) = η · vn /Vn Vn V∗ vn = N k=1 Vk Trade regularly, follow the market... Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  132. 132. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Risk minimization Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  133. 133. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Risk minimization N 22 V(W |σ, V , α) = xn σn δt n=1 Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  134. 134. Practical aspects of auction markets Quantitative optimization of high freq trading High freq trading in equations Risk minimization N 22 V(W |σ, V , α) = xn σn δt n=1 Trade all your quantity during the first time interval! Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  135. 135. Practical aspects of auction markets Quantitative optimization of high freq trading In the heart of darkness: market impact models A market impact effect? Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  136. 136. Practical aspects of auction markets Quantitative optimization of high freq trading In the heart of darkness: market impact models Market Impact main model With ψ the half spread, σ the volatility (in currency), and V a constant homogeneous with a quantity of shares, The most common model is: v γ fm (v ) = ψ + κσ · (14) V Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization
  137. 137. Practical aspects of auction markets Quantitative optimization of high freq trading In the heart of darkness: market impact models This M.I. model put inside equation (3) gives: v γ−1 v γ − · κσγ fLOB S0 + ψ + κ σ =1 Vγ V Charles-Albert Lehalle Head of CA Cheuvreux Quantitative Research Intra day high frequency trading : from empirical evidences to quantitative optimization

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