Investigating Statistical Arbitrage in Commodity Markets
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1 Investigating Statistical Arbitrage in Commodity Markets MDEF, Urbino, September 20, 2014
2 Outline 1. Statistical Arbitrage Framework 2. Commodity Markets and Inefficiencies 3. StatArb Methodology
3 Outline 1. Statistical Arbitrage Framework 2. Commodity Markets and Inefficiencies 3. StatArb Methodology
4 Efficient Market Hypothesis Efficient Market Hypothesis: Fama 1970 Market Efficiency implies stock prices fully reflect all publicly available information instantaneously, thus no investment strategies can systematically earn abnormal returns. Stock market anomalies are violations of Efficient Market Hypothesis in which consistently abnormal returns can be earned by on some investment strategies that are constructed based on potential market inefficiencies. The Joint Hypothesis problem refers to that testing for market efficiency is problematic, or even impossible. Any attempts to test for market (in)efficiency must involve an equilibrium asset pricing models.
5 A bit of History A bit of History Statistical Arbitrage: long horizon trading opportunity that generates riskless profits. 1980: Pairs Trading: Nunzio Tartaglia. 1990: Statistical Arbitrage (StatArb): statistical-mathematical models are used to individuate pricing inefficiencies and to profit from mispricing. 2002: Losing interest in StatArb models. Bad performance and poor confidence. 2005: New interest in StatArb models.
6 Statistical Arbitrage Definition Definition: Bondarenko (2003) For Bondarenko (2003) the definition of statistical arbitrage opportunity derives from the concept of pure arbitrage opportunity. A pure arbitrage opportunity is a zero cost trading strategy by which gains are received with no possibility of losses. Instead, a statistical arbitrage opportunity is a zero-cost trading strategy for which the expected payoff is positive and the conditional expected payoff in each state of the economy is nonnegative, meaning that the strategy payoff can be negative in some elementary states, as long as the average payoff in each final state is nonnegative.
7 Statistical Arbitrage Definition Definition: Bondarenko (2003) A zero-cost trading strategy with a payoff Z T = Z(I T ) is called a statistical arbitrage opportunity if 1. E[Z T I 0 ] > 0, and 2. E[Z T I ξ T 0 ] 0 for all ξ T, where I ξ T t := (I t ; ξ T ) = (ξ 1,..., ξ t ; ξ T ) denotes the augmented information set which in addition to the market information at time-t also includes the knowledge of the final state of the economy.
8 Statistical Arbitrage Definition Definition: Jarrow et al. (2003, 2005) For Jarrow et al. (2003, 2005) a statistical arbitrage is a long horizon trading opportunity that generates riskless profits. It is a natural extension of the trading strategies utilized in the existing empirical literature on anomalies. Statistical arbitrage is defined without any reference to any equilibrium model, therefore, its existence is inconsistent with market equilibrium and, by inference, market efficiency. The notion of statistical arbitrage enables the rejection of market efficiency without invoking the joint hypothesis of an equilibrium model. The joint hypothesis of an equilibrium model is replaced with an assumed stochastic process for trading profit.
9 Statistical Arbitrage Definition Definition: Jarrow et al. (2003, 2005) A statistical arbitrage is a zero initial cost, self-financing trading strategy with cumulative discounted trading profits v(n) and incremental trading profits v(n) such that: v(0) = 0 lim n E P [v(n)] > 0 lim n P(v(n) < 0) = 0 and = 0 if P[v(n) < 0] > 0, n < [lim n Var[ v(n) v(n) < 0] = 0] Var[ v(n)] lim n n P is the statistical probability measure.
10 Statistical Arbitrage Definition Financial meaning Statistical fair-price long-term relationship between assets Exploiting predictability in price dynamics
11 Commodity Markets and Inefficiencies Intermarket Spreads Intermarket Spread: simultaneously buy and sell different, but connected commodities, that have a stable relationship There are intermarket spreads when commodities are substitute or whose prices are correlated Casual perturbations of supply and demand in the spot and futures markets give rise to intermarket spread opportunities when futures prices diverge from spot prices
12 Commodity Markets and Inefficiencies Intermarket Spreads
13 Commodity Markets and Inefficiencies Common Spreads Crack spread: the spread between crude oil price and refined products prices Frac spread: the spread between liquid gas and natural gas prices Spark spreads: the spread between electricity price and natural gas price
14 Assumptions Assumptions Let (Ω, F, (F t ) t 0, P) be a filtered probability space over an infinite horizon [0, ), satisfying the usual conditions. P is the statistical probability measure. Let the stochastic process (M t ) t 0 represent the discounted portfolio values on which a zero cost initial, self-financing trading strategy is built.
15 Assumptions The portfolio (M t ) t 0 Long-short Portfolio: it consists on a long(short) position on a target commodity and a short(long) position on a synthetic asset. It is a market neutral portfolio: the β of the portfolio is null, meaning that there is not correlation between the the return of the portfolio and the market return The assets are chosen by using the cointegration technique (M t ) t 0 follows a mean reverting dynamic (M t ) t 0 is called mispricing portfolio
16 The Mispricing portfolio The portfolio: the cointegration analysis If two time series (x n ) n N and (y n ) n N are first-order integrated, but some cointegrating vector of coefficients exists to form a stationary linear combination of them, then they are cointegrated. y n = β + αx n + ε n, ε n N(0, 1)
17 The Mispricing portfolio Cointegration: economic interpretation It can be seen as a long-period relationship: statistical fair-price relationship If x t and y t are linked by the same relationship, they can diverge in the short-term period, but the equilibrium will be restored There could be a statistical arbitrage
18 The Mispricing portfolio Description of the data We consider a set of commodities, crude oils: The West Texas Intermediate (WTI), traded on the New York Mercantile Exchange (Nymex) was launched in March, 1983, and it is now the most liquid futures contract. The WTI is deliverable to Cushing, Oklahoma which is accessible to the spot market via pipeline. The Brent crude oil, which is traded on the Intercontinental Exchange (ICE) was launched in July, The Dubai crude oil quote by Platt s Time series: 25/10/ /10/2009
19 The Mispricing portfolio Crude oil Mispricing Portfolio We consider an infinite number of trading date t i, with i = 0, 1,..., The following statistical fair-price relationship holds for a generic t i E[T ti F ti ] = Z ti = T ti, (1) representing a long-term relationship among variables, Z ti denotes the synthetic asset price at time t i, the portfolio consisting of the commodities Brent and Dubai, T ti is the target price, that is The WTI crude oil.
20 The Mispricing portfolio Crude oil Mispricing Portfolio The mispricing portfolio is obtained according to the following relationship M ti = T ti Z ti. (2) M ti 2 = T ti β 0 β j Ct j i, (3) j=1 where C 1 t i is the price of the Brent oil, C 2 t i is the price of the Dubai oil.
21 The Mispricing portfolio Crude oil Mispricing Portfolio Coefficients β 1, β 2 are estimated by using the cointegration regression: Variable Coefficient Std. Error t-statistic Prob. constant BRENT DUBAI R-squared Adjusted R-squared S.E. of regression Akaike info criterion Schwarz criterion F-statistic Prob(F-stat) Durbin-Watson statistic Table : Cointegration regression for crude oils.
22 The Mispricing portfolio Crude oil Mispricing Portfolio: Structural Break? The QLR test of Stock and Watson (2003) provides a way to check whether the long run relationship between the WTI, Brent and Dubai crude oil is stable. The QLR F-statistics tests the hyphothesis that the intercept and the coefficient in the equation (3) are constant against the alternative the break is in the central 70% of the sample. The critical value, F(3,455) = , indicates that the null hypothesis that these coefficients are stable is rejected at the 1% significant level. This is evidence of structural change in the sample. The breakpoint data is taken on Feb 14, Finally, applying the Johansen (1991) test we verify that the cointegration relation holds also in presence of structural break.
23 The Mispricing portfolio Crude oil Mispricing Portfolio Mt /12/ /04/ /09/ /01/ /05/ /10/ /02/ /07/ /11/2010 Time
24 The Mispricing portfolio Mean reverting investigation: Variance Ratio Function The Augmented Dickey-Fuller test for mispricings is with a p-value of ,2 1,0 Variance Ratio 0,8 0,6 0,4 0,2 0, τ (weeks) Figure : Variance Ratio Function
25 Mispricing Modelling Mean Reverting Model We use the Vasicek (1977) model under the statistical measure: dm t = α(θ M t )dt + σdw t, M 0 = M (4) where α, β, and σ are positive constants and W t is a standard Wiener process.
26 Mispricing Modelling Mean Reverting Model With 0 s t, we have: t M t = θ(1 e α(t s) ) + M s e α(t s) + σe α e αu dw u, (5) M t conditional to F s is normally distributed with mean and variance given respectively by E[M t F s ] = θ + (M s θ) e α(t s) (6) ( Var[M t F s ] = σ2 1 e 2α(t s)) (7) 2α s
27 Mispricing Modelling Mean Reverting Model: AR(1) The discrete time version of the mean reverting model, on a time grid 0 = t 0, t 1, t 2,... with (assume constant for simplicity) time step t = t i t i 1, is: where the coefficients are: M(t i ) = c + bm(t i 1 ) + δɛ(t i ), (8) c = Θ(1 e α t ), b = e α t, δ = σ t, and ɛ(t) is a Gaussian white noise ɛ N (0, 1).
28 Mispricing Modelling Implicit Statistical Arbitrage Rules Implicit statistical arbitrage (ISA) trading strategies are based on trading rules that rely implicitly on the mean-reverting behaviour of the mispricing. Burgess (1999). If in the long time the misprisings reduce as prices change, an operator, who has previously opened a position, can realize profits provided that the impact of transaction costs, supported to employ the strategy, turns out less than the impact of the mispricing gain component. ISA trading rules define the sign and the magnitude of the mispricing portfolio components, that is the target commodity and the replication portfolio.
29 Mispricing Modelling Implicit Statistical Arbitrage Rules We indicate with S(t i, M ti, k) the basic trading rule at date t i that depends on the sign and the level of the mispricing at the previous time and on the value of a sensitive parameter k according to the following formula S(t i, M ti, k) = sign(m(t i 1 )) M(t i 1 ) k, (9) When k = 0 we have a step function meaning that the entire holding is always invested in the mispricing portfolio. If k > 0, the size of portfolio increases as the magnitude of the mispricing enlarges and, in particular, a k > 1 corresponds to more aggressive strategies.
30 Mispricing Modelling Implicit Statistical Arbitrage Rules 8,00 6,00 4,00 2,00 0,00 Mispricing Trading Rule 2,00 4,00 6,00 06/12/ /04/ /09/ /01/ /05/ /10/ /02/ /07/ /11/2010
31 Mispricing Modelling Implicit Statistical Arbitrage Rules S h (t i, M ti, k, h) = S h t i = 1 h h S(t i j ) (10) j=1 S Θ (t i, M ti, k, Θ) = S Θ t i = (1 Θ)S ti + ΘS Θ (t i 1 ) (11) where h N is the moving average parameter and Θ R + 0 is an exponential smoothing parameter. By increasing the values of h and Θ, on one hand the number of transactions comes down so that the transaction costs decrease, on the other hand the accuracy of the smoothed trading signal diminishes.
32 Mispricing Modelling Implicit Statistical Arbitrage Rules S S^h S^O 2,50 2,00 1,50 1,00 0,50 0,00-0,50-1,00-1,50-2,00 05/02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/ /02/ /06/ /10/2009
33 Mispricing Modelling Measuring the Performance The return gotten over a generic trading time period [t i 1, t i ] : R ti = S ti M ti T ti + Z ti c S ti (12) where M ti = M ti M(t i 1 ), S ti = S ti S(t i 1 ), c is the percentage transaction costs and T ti + Z ti is the sum of the portfolio components. The total return or cumulative profit of a strategy: The Sharpe Ratio: Sh ti = cr ti = i R(t j ). (13) j=1 1 i i s=1 R(ts ) 1 i i s=1[r(t s ) 1 i i s=1 R(ts )]2
34 Mispricing Modelling Verifying StatArb according to Jarrow et al. (2003) Consider the discounted incremental cumulative profit of a strategy cr ti, and assume they satisfy cr ti = µt γ i + σt λ i Z i, (14) for i = 1,..., n, where Z i are i.i.d N (0, 1) random variables with Z 0 = 0. Note that cr(t 0 ) = 0
35 Mispricing Modelling Verifying StatArb according to Jarrow et al. (2003) Theorem A trading strategy generates a statistical arbitrage with 1 c percentage of confidence if the following conditions are satisfied: 1. µ > 0 2. λ < 0 or γ > λ 3. γ < max { λ 1 2 ; 1}. with the sum of the p values for the individual tests forming an upper bound for the type I error c
36 Mispricing Modelling Verifying StatArb according to Jarrow et al. (2003) Parametro Estimate Standard Error t Statistic p-value γ 0, , ,1384 0,8899 λ 0, , ,4838 0,0133 Mean of cr ti 0, St. Dev. of cr ti 0, Squared Residual Sum 0, Residual Standard Error (ˆσ) 0, R 2 0, Akaike criterion 3135,0 Schwarz criterion 3126,5
37 Out-of-Sample Analysis Out-of-Sample Analysis Year Total Return 7.76% 5.74% 4.76% 2.71% Sharpe Ratio Profitable weeks 45.45% 55.77% 42.31% 44.23% Year Total Return 2.09% 4.01% 13.89% Sharpe Ratio Profitable weeks 2.69% 50.00% 52.83% Table : In-sample performance Year Total Return 4.16% 3.16% 4.12% Sharpe Ratio Profitable weeks 42.31% 59.52% 50.00% Table : Out-of-sample performance
38 Out-of-Sample Analysis References Testing market efficiency using statistical arbitrage with applications to momentum and value strategies. S. Hogan, R. Jarrow, M. Teo, M. Warachka Journal of Financial economics, Elsevier Statistical arbitrage and securities prices. Bondarenko O.- Review of Financial Studies, 2003
Working Papers. Cointegration Based Trading Strategy For Soft Commodities Market. Piotr Arendarski Łukasz Postek. No. 2/2012 (68)
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