Ito Excursion Theory. Calum G. Turvey Cornell University

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1 Ito Excursion Theory Calum G. Turvey Cornell University

2 Problem Overview Times series and dynamics have been the mainstay of agricultural economic and agricultural finance for over 20 years. Much of the literature makes natural assumptions about dynamics and spends a great deal of time searching for unit roots without much further remark on what is implied by acceptance or rejections of a unit root. Rejection of a unit root simply means that variance of time series is not a function with time Not the same as the variance ratio property that the variance increases linearly in time for a Brownian motion

3 Quantitative Finance in Agriculture There is a need to better understand dynamics in agricultural economics Financial engineering and Ito s lemma is critical for modeling random variables over time From basic pricing of options to valuing farm programs (Turvey et al 2014) and the structure of markets (Assa 2015) Particularly useful for Monte Carl simulation But patterns in time series are also important to understand Mean regression or ergodic systems Persistent or long-memory systems Brownian vs Fractional Brownian motion.

4 Ito Excursion Theory we investigate how Itô s excursion theory can be usefully applied to economic time series data (Itô 2007) We relate excursion theory to geometric and fractional Brownian motion and the Hurst coefficient (also important for valuing and modeling exotic path dependant op[tions) We then calculate the Hurst coefficient for all stocks on the DOW 30, S&P 500 and Russell 2000, showing the distribution of Hurst measures and relating them statistically to excursions. Have also examined ag commodities provide a nice and intuitive link between Brownian motion and excursions, an application and consequence that we have not seen before

5 Ito Excursion Theory Itô s excursion theory is a probabilistic tool that can be applied to any continuous-time (near) Markov process with a recurrent state recurrent state -at some unknown and random point (time) in the future it will return to its original point (state). Excursion theory allows one to evaluate, measure, and quantify certain characteristic of the stochastic process and can be developed with great generality

6 A Simple Understanding of Excursion (Rogers 1989) game with a fair coin repeatedly tossed, with 1 assigned to a score each time the coin falls heads and -1 each time the coin falls tails. TT 0 = 0, TT 1 = 6, TT 2 = 10, TT 3 = 12, TT 4 = 18, TT 5 = 20 ξξ 1 = YY 0, YY 1, YY 2, YY 3, YY 4, YY 5, YY 6 ξξ 2 = YY 6, YY 7, YY 8, YY 9, YY 10 ξξ 3 = YY 10, YY 11, YY 12 ξξ 4 = {YY 12, YY 13, YY 14, YY 15, YY 16, YY 17, YY 18 } ξξ 5 = YY 18, YY 19, YY 20

7 Purpose of Ito Excursion Theory purpose of Itô s excursion theory is to describe the evolution of a (Markov) process in terms of its behavior between visits to a particular point m in the state space. E.g. probability distribution about the number of days that the security price returns to reference price. local time LL mm tt = ll is defined as the number of visits to point m up to time t. In the coin toss example, time is equal to 20 and LL 0 20 = 5 excluding the origin. Stopping Time: counts the number of excursions of type γ that occurs before the local time reaches l. It is a simple Poisson process

8 Simulated Random Walks Example Paths of Random Walk Canola Cash Price Cash Random 1 Random 2 Random Day

9 Binomial (Poisson) Distribution of first 10 Excursions in Simulated Random Walk (iter=30k)

10 Some Propositions about the Relationship between excursion theory and fractional Brownian motion A stationary autoregressive process AR(q>=1) may have multiple roots. If at least one root is a unit root it is a stationary process (extended proof from Assa and Turvey 2015 forthcoming) A time series with AR(q>1) is a (quasi) fractional Brownian motion with excursion patterns determined by nature of lags Most time series can be modeled using an autoregressive process with econometric (regression) techniques If Y ay ay ay ε q i= 1 t= 1 t 1+ 2 t q t q+ t a i = 1 at least one real root must be a unit root Then if q i= 1 a i 1 it has no unit root and the time series will converge to zero or infinite

11 The fractional Brownian motion dx αxdt xσ 2H = + dz dz =ε t H=0.5 : geometric Brownian motion H>0.5 : Persistence/long memory (Black noise) H<0.5 : Antipersistence/mean-reverting (Pink to White noise) H is the Hurst Coefficient (estimated from Variance ratio) [ ] = σ E xt ( ) xt ( ) ( t t) H Variance and Covariance [ ][ ] x { } = σ + E xt ( ) x(0) xt ( t) xt ( ) 0.5 ( t t) H t H t H 11

12 Estimated H, from simulation and from R/S calculation, with null Hypothesis Ho= 0.5 (GBM) Days/Contract (R/S) Alberta Barley price coffee price a,b,c cocoa price a,b,c a,b,c corn price a,b a a a,b Feeder Cattle price a a,b,c a,b,c Fluid Milk price Lean Hogs price a a,b,c a,b,c live cattle price a,b,c a,b,c a,b,c a,b,c a,b,c oats price a,b a,b a,b a,b,c orange juice price a,b,c a,b,c Pork Bellies price a a a,b,c a,b Rapeseed canola price a a a,b,c a,b Soybeans price a,b a,b a,b a,b a,b Sugar price a,b,c a,b,c a,b,c a,b,c wheat price a,b,c a,b,c a,b,c a,b,c a,b,c Winnipeg oats price a,b,c Winnipeg Wheat price a,b a a a a,b

13 The following graphs Illustrate by simulation how AR(q>1) processes of various lags generate non-unique quasi-fractional Brownian Motion Not how the excursion patterns for each process is unique Hurst Coefficients <0.5 reverse quickly with small excursions Hurst coefficients > 0.5 are persistence and with positive memory in the system show larger excursion patterns as H increases.

14 Geometric Brownian Motion, H=0.5, y = 1.0y t 1 + ε 14

15 Fractional Brownian Motion, H=0.4 y = 0.7 y + 0.2y + 0.1y + ε t 1 t 2 t 3 15

16 Fractional Brownian Motion, H=0.3, y = 0.4y + 0.2y + 0.4y + ε t 1 t 2 t 3 16

17 Fractional Brownian Motion, H=0.6, y = 1.4y 0.3y 0.1y + ε t 1 t 2 t 3 17

18 Fractional Brownian Motion, H=0.7, y = 1.5y 0.3y 0.2y + ε t 1 t 2 t 3 18

19 Time Path of Nonstationary Series,to Zero y = 1.0y 0.1y + ε t 1 t 2 19

20 Time Path of Nonstationary Series, to Infinite y = 1.0y + 0.1y + ε t 1 t 2 20

21 Dynamics of the excursion reference level. All simulations start at a base level of For all simulations, we computed the mean values and used these for the reference levels. The figure shows that the mean reference levels increase as H increases

22 Dynamics of the stopping natural time. The vertical axis measures the average time required for each process to complete 10 excursions across 30,000 replications. As expected the mean stopping natural time increases with Hurst. Mean Stopping Natural Time Stopping Natural Time H01 H02 H03 H04 H05 H06 H07 H08 H09 Hurst Exponent

23 Dynamics of the overall local time. The vertical axis measures the number of time the excursion crosses the reference point. As expected, the number of excursions in the simulated 2,150 step path is much higher for low Hurst than high Hurst. 250 Overall Local Time Mean Overall Local Time H01 H02 H03 H04 H05 H06 H07 H08 H09 Hurst Exponent

24 Relationship between Hurst exponent and mean excursion measure. Hurst exponent is measured on the vertical axis and excursion length is measured on the horizontal axis. The solid line is an exponential fit with R-squared of 0.78 showing that short excursion lengths are dominated by low Hurst, while high Hurst series are often associated with longer excursions 1 Corresponding Hurst Exponent with Highest Mean Count in Each Bin Hurst Exponent y = ln(x) R² = Bin (Excursion Length)

25 Further tests using stock market data.. Stock market gives many observations to test distributional assumptions Russel 2500, Standard and Poor, Dow Jones 2,023 time series with daily observations

26 Hurst coefficients and excursion measures for 2,023 stocks Local Time measures length of excursion Stopping time measures number of days for 10 excursions R2500 SPX DJIA Weighted Average Count 1, ,023 Hurst Coefficient Minimum Maximum Average Standard Deviations % > H > % > H > % > H > Overall Local Time Minimum Maximum Average Standard Deviations Stopping Natural Time (Days) Minimum Maximum 2,149 2,149 2,149 2, Average Standard Deviations

27 Frequency Distribution of Hurst Coefficients for DOW, S&P 500 and Russel Frequency DOW S&P Russel

28 Further Results For R2500, on average a 1% increase in Hurst value will decrease overall local time by %. For the SPX, this elasticity measure is % DJIA it is %. Hence, for a fixed sample size, the overall local time decreases as Hurst increases. We do not find strong a relationship for stopping time for the R2500 is and significant, which suggests that on average a 1% increase in Hurst will increase the stopping natural time by around 1%. We find very strong inverse relationship between stopping time and local time ( a tautology)

29 Conclusions and Discussion From an agricultural finance and risk management point of view advancing concepts in quantitative finance is important Understanding dynamics to understand price movements Modeling underlying processes Ito s lemma Recognizing that AR(q>1) models are quasi fractional is important in interpreting commodity pricing model. Part and parcel to this is drilling down to basic underlying structure Excursions Local time and stopping time Randomness is about things that are random and long excursions occur with lower frequency

30 Conclusions and Discussion Traders, insurers, farmers, researchers seem surprised when markets make a long term departure Long positive excursions are natural occurrences in a random walk More likely in persistent series (H> 0.5) Less likely in mean reverting processes (H<0.5) These things are natural and their occurrence says nothing about inefficient markets. A bubble is an excursion; it is special because it is rare, but the existence of a bubble says nothing about market efficiency. Bubbles are natural, periodic, excursions that are fully captured in a Brownian motion