Large-scale simulations of synthetic markets

Transcription

1 Frac%onal Calculus, Probability and Non- local Operators: Applica%ons and Recent Developments Bilbao, 6-8 November 2013 A workshop on the occasion of the re%rement of Francesco Mainardi Large-scale simulations of synthetic markets Luca Gerardo-Giorda Joint work with: G. Germano (UC London) E. Scalas (U. Sussex)

2 High-frequency trading (HFT) HFT: Quantitative trading characterized by short portfolio holding periods. 1 Targets equities and foreign exchange trading. 2 HF traders move in and out of short-term positions aiming to capture sometimes just a fraction of a cent in profit on every trade. 3 All portfolio-allocation decisions are made by computerized quantitative models.

3 High-frequency trading (HFT) HFT: Quantitative trading characterized by short portfolio holding periods. 1 Targets equities and foreign exchange trading. 2 HF traders move in and out of short-term positions aiming to capture sometimes just a fraction of a cent in profit on every trade. 3 All portfolio-allocation decisions are made by computerized quantitative models. 4 As HFT strategies become more widely used, it can be more difficult to deploy them profitably. 5 Success of HFT strategies is largely driven by their ability to simultaneously process volumes of information.

4 High-frequency trading (HFT) HFT: Quantitative trading characterized by short portfolio holding periods. 1 Targets equities and foreign exchange trading. 2 HF traders move in and out of short-term positions aiming to capture sometimes just a fraction of a cent in profit on every trade. 3 All portfolio-allocation decisions are made by computerized quantitative models. 4 As HFT strategies become more widely used, it can be more difficult to deploy them profitably. 5 Success of HFT strategies is largely driven by their ability to simultaneously process volumes of information. Need for fast and reliable models of a whole HF financial market

5 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

6 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

7 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

8 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

9 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

10 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

11 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

12 Short history of HFT 1998 U.S. Securities and Exchange Commission (SEC) authorizes electronic exchanges Execution time of several seconds European Union liberalises stock exchanges 2010 Execution time had decreased to milli- and even microseconds May 6: Wall Street Flash Crash DJ industrial average lost $1 trillion of market value in half an hour August 1: Knight Capital debacle a computer glitch caused losses for $440 millions in 45 minutes September: Several European countries have proposed curtailing or banning HFT due to concerns about volatility 2013 September 2: Italy introduces a tax specifically targeted at HFT, charging a levy of 0.002% on equity transactions lasting less than 0.5 seconds.

13 Syntethic market modeling 1 Selection Select your favorite intra-day market model with full specification of the market process. 2 Fitting Fitting parameters of the market process with the available historical data. 3 Simulation Run Monte Carlo simulations of the fitted synthetic market. 4 Assessment of quantities of interest Compute the quantities of interest out of the Monte Carlo simulations Final price S(T ) = S(0) exp(η X(T )) Payoff (European call) P(T ) = max (S(T ) Sstrike, 0)

14 Syntethic market modeling 1 Selection Select your favorite intra-day market model with full specification of the market process. 2 Fitting Fitting parameters of the market process with the available historical data. 3 Simulation Run Monte Carlo simulations of the fitted synthetic market. 4 Assessment of quantities of interest Compute the quantities of interest out of the Monte Carlo simulations Final price S(T ) = S(0) exp(η X(T )) Payoff (European call) P(T ) = max (S(T ) Sstrike, 0)

15 Syntethic market modeling 1 Selection Select your favorite intra-day market model with full specification of the market process. 2 Fitting Fitting parameters of the market process with the available historical data. 3 Simulation Run Monte Carlo simulations of the fitted synthetic market. 4 Assessment of quantities of interest Compute the quantities of interest out of the Monte Carlo simulations Final price S(T ) = S(0) exp(η X(T )) Payoff (European call) P(T ) = max (S(T ) Sstrike, 0)

16 Syntethic market modeling 1 Selection Select your favorite intra-day market model with full specification of the market process. 2 Fitting Fitting parameters of the market process with the available historical data. 3 Simulation Run Monte Carlo simulations of the fitted synthetic market. 4 Assessment of quantities of interest Compute the quantities of interest out of the Monte Carlo simulations Final price S(T ) = S(0) exp(η X(T )) Payoff (European call) P(T ) = max (S(T ) Sstrike, 0)

17 Syntethic market modeling 1 Selection Select your favorite intra-day market model with full specification of the market process. 2 Fitting Fitting parameters of the market process with the available historical data. 3 Simulation Run Monte Carlo simulations of the fitted synthetic market. 4 Assessment of quantities of interest Compute the quantities of interest out of the Monte Carlo simulations Final price S(T ) = S(0) exp(η X(T )) Payoff (European call) P(T ) = max (S(T ) Sstrike, 0)

18 Syntethic market modeling 1 Selection Select your favorite intra-day market model with full specification of the market process. 2 Fitting Fitting parameters of the market process with the available historical data. 3 Simulation Run Monte Carlo simulations of the fitted synthetic market. 4 Assessment of quantities of interest Compute the quantities of interest out of the Monte Carlo simulations Final price S(T ) = S(0) exp(η X(T )) Payoff (European call) P(T ) = max (S(T ) Sstrike, 0)

19 Notations {J i } i=1 Inter-trade durations (positive real random variables). {T n } n=1, T n = n i=1 J i Trading epochs N(t) = max{n : T n t} Counting process (number of events up to t) {Y i } i=1 Tick-by-tick logarithmic returns (real random variables). X(t) = N(t) i=1 Y i = i=1 Y i 1 {Ti t} Logarithmic-price process. S(t) Synthetic equity (a positive real variable) S0 opening price (at epoch T 0 = 0) S(t) = S0 e X(t) synthetic equity process (price process)

20 Single equity modeling: The behavior of a single equity in time is a combination of Price variation at any given trade Inter-trade duration

21 Single equity modeling: CTRW The behavior of a single equity in time is a combination of Price variation at any given trade Logarithmic returns Y i draws of a random variable γ x from Inter-trade duration Independent and identically distributed Lévy α-stable AutoRegressive Conditional Heteroskedasticity [ ] (ARCH) Generalized AutoRegressive Conditional Heteroskedasticity [ ] (GARCH) Inter-trade duration draws of a random variable γ t from Mittag-Leffler J i AutoRegressive Conditional Duration (ACD) [ ] In statistics, a collection of random variables is heteroskedastic if there are sub-populations that have different variabilities from others.

22 Remarks 1. Processes with α-stable i.i.d. increments that are subordinated to the limit of a Mittag-Leffler process have a one-point probability density function converging to the solution of the fractional diffusion equation. 2. AutoRegressive Conditional Heteroskedasticity (ARCH) models are used whenever there is reason to believe that, at any point in a series, the terms will have a characteristic size, or variance. 3. ARCH and GARCH are very popular in modeling financial time series that exhibit time-varying volatility clustering.

23 Logarithmic returns I - Lévy α-stable 3 Independent and identically distributed Lévy α-stable: Y t L α where L α is a random variable with probability density function given by f Lα (x) = 1 2π + e κ α e iκx dκ. Remark Ideal for mathematical analysis, but unrealistic.

24 Logarithmic returns II - ARCH(q) 1 AutoRegressive Conditional Heteroskedasticity: The error (return residuals with respect to a mean process) is split into: Y t ɛ t = σ t z t a stochastic piece zt (strong white noise process) z t N (0, 1) a time-dependent standard deviation σt σt 2 = α 0 + where α 0 > 0, α j 0, (j 0). q α j (ɛ t j ) 2 j=1

25 Logarithmic returns III - GARCH(p,q) 2 Generalized AutoRegressive Conditional Heteroskedasticity: An autoregressive moving average model (ARMA model) is assumed for the error variance, and the error is split into: Y t ɛ t = σ t z t a stochastic piece zt (strong white noise process) z t N (0, 1) a time-dependent standard deviation σt q p σt 2 = α 0 + α j (ɛ t j ) 2 + β i σt i 2 j=1 i=1 where α j > 0 (j = 0,..., q), β i > 0 (i = 1,..., p). Remark: GARCH(p,q) is stable provided that α 1 + β 1 < 1.

26 Inter-trade duration epochs I - Mittag-Leffler 1 Mittag-Leffler distribution, characterized by the survival function Ψ(t) = P(J > t) = E β ( (γt) β ), where E β (z) is the one parameter Mittag-Leffler function E β (z) = n=0 z n Γ(βn + 1), z C Remarks Ideal for mathematical analysis, but unrealistic.

27 time distribution of Liffe bondfutures 4 Inter-trade duration epochs I - Mittag-Leffler 10 0 "(!) Survival probability for BTP futures with delivery date June Mittag-Leffler (β = 0.96, γ = 1/13) stretched exponential power law real data Figure from [1] ! {s} Figure 1. Survival probability for BTP futures with delivery date: June The [1] M Raberto, E Scalas, R Gorenflo, F Mainardi, Mittag-Leffler function (solid line) of index β =0.96 and scale factor γ =1/13 is comparedthe to thewaiting-time stretched exponential distribution (dash-dottedof line) LIFFE and thebond power (dashed futures line) functions. arxiv preprint cond-mat/ , 2000 [2] M Raberto, E Scalas, F Mainardi, rket, as it also takes into account both the non-markovian and the noners of the time evolution in financial time series. Waiting-times and returns in high-frequency financial data: an empirical study ing procedure Physica used in A: this Statistical paper differs Mechanics from that in and [4]. its Indeed Applications, we found 2002 of the function Ψ(τ) andthefittingproceduresfarfromtrivial. These

28 Inter-trade duration epochs II - ACD(p,q) 2 Autoregressive Conditional Duration The durations are given by J t = Θ t Z t, where Z t are positive, independent and identically distributed random variables, with E(Z t ) = 1. The time series Θ t is given by q p Θ t = α 0 + α j J t j + β i Θ t i j=1 i=1 where α 0 > 0, α j 0 (j = 1,..., q), β i 0 (i = 1,..., p).

29 Numerical simulations Full market with 2,000 independent equities. At the money (ITM) european call S strike = S 0 Monte Carlo simulation of the CTRW with 10,000 runs. Different scenarios Scenario 1 Scenario 2 Scenario 3 Scenario 4 Inter-trade durations J t Mittag-Leffler ACD(1,1) Log-returns Y t Lévy α-stable ARCH(1) GARCH(1,1) Parameters generated from normal distributions.

30 Results: scenario Scenario 1: final price 7 Scenario 1: realization Scenario Scenario 1: payoff log return Time Shares 15 20

31 Results: scenario Scenario 2: final price Scenario 2: realization Scenario Scenario 2: payoff log return Time Shares 15 20

32 Results: scenario Scenario 3: realization Scenario 3: final price Scenario Scenario 3: payoff log return Time Shares 15 20

33 Results: scenario Scenario 4: realization 80 Scenario 4: final price Scenario Scenario 4: payoff log return Time Shares 15 20

34 Summary We propose models to simulate a whole HF synthetic market: Efficient: Monte Carlo simulations can naturally run in parallel.

35 Summary We propose models to simulate a whole HF synthetic market: Efficient: Monte Carlo simulations can naturally run in parallel. We do not provide a winning recipe (model choice depends on the goal). No statistics performed so far. Simulation of the whole market is based on independent equities.

36 Summary and outlook We propose models to simulate a whole HF synthetic market: Efficient: Monte Carlo simulations can naturally run in parallel. We do not provide a winning recipe (model choice depends on the goal). No statistics performed so far. Simulation of the whole market is based on independent equities. Outlook Maximal likelihood estimations for the parameters of the different models.

37 Summary and outlook We propose models to simulate a whole HF synthetic market: Efficient: Monte Carlo simulations can naturally run in parallel. We do not provide a winning recipe (model choice depends on the goal). No statistics performed so far. Simulation of the whole market is based on independent equities. Outlook Maximal likelihood estimations for the parameters of the different models. Use of information criterions such as Akaike (AIC) to discriminate among models (in collaboration with T. Radivojevic and E. Akhmatskaya)

38 Summary and outlook We propose models to simulate a whole HF synthetic market: Efficient: Monte Carlo simulations can naturally run in parallel. We do not provide a winning recipe (model choice depends on the goal). No statistics performed so far. Simulation of the whole market is based on independent equities. Outlook Maximal likelihood estimations for the parameters of the different models. Use of information criterions such as Akaike (AIC) to discriminate among models (in collaboration with T. Radivojevic and E. Akhmatskaya) Introduce correlation among equities.

39 Summary and outlook We propose models to simulate a whole HF synthetic market: Efficient: Monte Carlo simulations can naturally run in parallel. We do not provide a winning recipe (model choice depends on the goal). No statistics performed so far. Simulation of the whole market is based on independent equities. Outlook Maximal likelihood estimations for the parameters of the different models. Use of information criterions such as Akaike (AIC) to discriminate among models (in collaboration with T. Radivojevic and E. Akhmatskaya) Introduce correlation among equities. A product that can be supportive for HFT firms and need to evaluate scenarios.

40 Commercials