Kostas Triantafyllopoulos University of Sheffield. Motivation / algorithmic pairs trading. Model set-up Detection of local mean-reversion

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1 Detecting Mean Reverted Patterns in Statistical Arbitrage Outline Kostas Triantafyllopoulos University of Sheffield Motivation / algorithmic pairs trading Model set-up Detection of local mean-reversion Adaptive estimation 1. RLS with gradient variable forgetting factor 2. RLS with Gauss-Newton variable forgetting factor 3. RLS with beta-bernoulli forgetting factor Trading strategy Pepsi and Coca Cola example

2 Introduction Statistical arbitrage. Algorithmic pairs trading market neutral trading. Buy low, sell high. Two assets with prices p At and p Bt At t: if p At < p Bt, buy low (A) and sell high (B). At t + 1: if pat > p Bt, buy low (B) and sell high (A)... In the long run, mean reversion of spread y t = p At p Bt. If y t goes up, y t will go down at t + 1. Take advantage of relative mispricings of A and B.

3 Introduction Share price stream Share price A,B Time Spread stream A B Time

4 Pepsi - Coca Cola date stream Share Prices (in USD) Mean of Spread = Pepsi Coca Cola Trading day

5 Model set-up (Elliott et al, 2005). y t is a noisy version of a mean-reverted process. y t = x t + ε t x t = α + βx t 1 + ζ t (Triantafyllopoulos and Montana, 2011). y t = α t + β t y t 1 + ϵ t = F T t θ t + ϵ t, θ t = Φθ t 1 + ω t, [ ] 1 F t = and θ y t = t 1 [ αt β t ]

6 Detecting mean reversion Define D t = (y 1,..., y t ) a data stream sample. (Elliott et al, 2005). If β < 1, then y t is mean-reverted. With D t, get online estimates ˆα and ˆβ. If ˆβ < 1, then mean-reversion. (Triantafyllopoulos and Montana, 2011). Under some assumptions y t is mean-reverted if β t < 1, for all t. We consider mean-reversion in segments or locally. Test again ˆβ t < 1. So we need online estimates ˆβ t.

7 Recursive least squares (RLS) (Haykin, 2001). Recursive Least Squares (RLS), if Φ = I. Find θ that minimizes the cost function t 1 λ j (y t j Ft jθ) T 2 j=0 0 < λ < 1 a forgetting factor. Past data are forgotten at a rate t 1 λ j = 1 + λ + + λ t λ j=0 [memory of the system] With λ = 1, ordinary regression, memory is With λ < 1, RLS, memory (1 λ) 1.

8 SS-RLS (state-space RLS) Variable forgetting factor (Haykin, 2001, Malik, 2006). Steepest descent: λ t λ t 1 = c(t) λ t = [λ t 1 + a λ (t)] λ+ λ, 1. λ (t) e t F T t Φψ t 1 2. ψ t = m t / λ = (I K t F T t )Φψ t 1 + S t F t e t 3. S t = P t / λ = λ 1 t P t + λ 1 t K t Kt T +λ 1 t (I K t Ft T )ΦS t 1 Φ T (I F t Kt T ). Need starting values m 1, P 1, ψ 1, S 1, λ 1.

9 GN-RLS (Gauss-Newton RLS) Song et al (2000) gave an approximate GN algorithm. λ t = [ λ t 1 + a ] λ+ λ(t) 2 λ (t), λ Here 2 λ (t) (F T t Φψ t 1 ) 2 e t F T t Φη t η t = ψ t 1 / λ = (I K t F T t )Φη t 1 + L t F t e t 2S t F t F T t Φψ t 1 where and L t = λ 1 t (I K t Ft T )ΦL t 1 Φ T (I F t Kt T ) +λ 2 t P t (I F t Kt T ) λ 1 t S t + M t + Mt T λ 2 t (I K t Ft T )ΦS t 1 Φ T (I F t Kt T ) M t = λ 1 t S t F t F T t {P t ΦS t 1 Φ T (I F t K T t )}.

10 GN-RLS cont. GN-RLS creates too abrupt jumps in λ t / too sensitive to changes. When smooth signal, we want SS-RLS and when noisy we want GN-RLS. We use λ t = [λ t 1 + a λ (t)] λ+ λ, if e2 t k [ λ t 1 + a λ(t) 2 λ (t) ] λ+ λ, if e2 t > k

11 BB-RLS (beta Bernoulli RLS) In the above we ask λ λ t λ +. If et 2 small, λ t λ + (smoothness) If et 2 large, λ t λ (adaptiveness) We set λ t = πλ + + (1 π)λ, π = Pr(et 2 k t ) Two events small prediction error / large prediction error { 1, if e 2 x t = t k t, with probability π 0, if et 2 > k t, with probability 1 π

12 Observation model x t Bernoulli(π). Prior for π is beta, π B(c 1, c 2 ) p(x t π) = π xt (1 x t ) 1 xt p(π) π c 1 1 (1 π) c 2 1 (bernoulli model) (prior beta model) p(π x t ) p(x t π)p(π) π c 1+x t 1 (1 π) c 2+1 x t 1 So π x t B(c 1 + x t, c 2 x t + 1). Sequentially: π x 1,..., x t π D t 1 Be(c 1t, c 2t ), where c 1t = c 1,t 1 + x t and c 2t = c 2,t 1 x t + 1. ˆπ t = E(π D t 1 ) = c 1t (c 1t + c 2t ) 1 ˆλ t = E(λ t D t 1 ) = ˆπ t λ + + (1 ˆπ t )λ From Pr(e 2 t k t ) = π, we have k t = q t F 1 χ 2 (π). We use k t q t F 1 χ 2 (ˆπ t 1 ).

13 Key points of λ t λ t is stochastic. We can derive its distribution p(λ t D t 1 ) = c(λ t λ ) c 1t 1 (λ + λ t ) c 2t 1 We can evaluate the mode and the variance of λ t. We can show ˆλ t xλ + + (1 x)λ If for many points e 2 t < k t, followed by few large e 2 t > k t, then locally ˆλ t does not work well.

14 Solution: intervention Example. 1. Set λ = 0.8, λ + = e 2 t k t (t = 1,..., 95) 3. e 2 t > k t (t = 96, 97, 98, 99, 100) 4. ˆλ t = (closer to λ + = 0.99, than to λ = 0.8). If change in x t (from 0 to 1 or from 1 to 0), reset the priors c 1,t 1 and c 2,t 1 to the initial values (c 1,1 = c 2,1 = 0.5). ˆπ t = c 1t c 1t + c 2t = c 1,1 + x t c 1,1 + c 2,1 + 1 = 1 + 2x t 4 ˆλ t = { 0.75λ λ, if x t = λ λ, if x t = 0 In the example, ˆλ 96 = =

15 Simulated streams Simulated Spread Time Prediction of B t SS RLS GN RLS BB RLS Trading day

16 Trading strategy Spread y t = p A,t p B,t. If y t not mean-reverted do nothing. It is not predictable. If y t < y t+1, p A,t+1 > p A,t or p B,t+1 < p B,t. Buy A / sell B. If y t > y t+1, p A,t+1 < p A,t or p B,t+1 > p B,t. Buy B / sell A. If y t y t 1, do nothing. At time t, we don t know y t+1, so we predict it and use ŷ t+1.

17 Trading strategy Trading strategy p(a) > p(b) Tradable time spread y t 1 t Non tradable time p(a) < p(b) t+1

18 With observed spread y t at t: Close the position of t 1 (if opened). If ˆβ t+1 < 0.99, declare y t+1 as mean-reverted. If ŷ t+1 h > y t, buy A / short-sell B. If ŷ t+1 + h < y t, buy B / short-sell A. Example: At t: y t = 10 and we project ŷ t+1 = 11. y t+1 can be 12 or 9 (rules change if we apply y t <> ŷ t+1. y t+1 = 9 can give loss, if we adopt y t < ŷ t+1 rule (10 < 11). Take h = 10% of ŷ t+1 = 1.1, and ŷ t+1 h = 9.9 < 10 = y t, we do not open a position buy A and short sell B. As ŷ t+1 + h = 11.1 > 10 = y t, we do not open a position short sell A and buy B.

19 Pepsi - Coca Cola date stream Share Prices (in USD) Mean of Spread = Pepsi Coca Cola Trading day

20 Detection of mean reverted patterns Prediction of B t BB RLS SS RLS GN RLS Trading day

21 MSE over time Mean square error of the three algorithms SS RLS GN RLS BB RLS Trading day

22 Trading performance Mean 1% 3% 5% SS-RLS GN-RLS BB-RLS STD 1% 3% 5% SS-RLS GN-RLS BB-RLS FB 1% 3% 5% SS-RLS GN-RLS BB-RLS h

23 Trading performance Cumulative profit BB RLS GN RLS SS RLS Trading day

24 Closing remarks Algorithmic pairs trading / statistical arbitrage require online machine learning methods. Pattern recognition methods for mean reversion / segments of stationarity. We develop variable forgetting factors for online learning. Other methods include sequential Monte Carlo. Need to take into account the shape of the distribution of the data stream. Larger data streams / complex portfolios / many pairs to consider simultaneously. Trading strategy can be improved.

25 References Elliott, R., Van Der Hoek, J., and Malcolm, W. (2005). Pairs trading. Quantitative Finance, 5: Haykin, S. (2001). Adaptive Filter Theory. Prentice Hall, 4th edition. Malik, M. B. (2006). State-space recursive least-squares with adaptive memory. Signal Processing, 86: Song, S., Lim, J.-S., Baek, S., and Sung, K.-M. (2000). Gauss-Newton variable forgetting factor recursive least squares for time varying parameter tracking. Electronics Letters, 36: Triantafyllopoulos, K. and Montana, G. (2011). Dynamic modeling of mean-reverting spreads for statistical arbitrage. Computational Management Science, 8:23-49.