Basic Pair Trading Study Using Linear Regression

linear_regression.q

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+// Alpha and beta functions 
+// These formulas are derived from Ordinary Least Squares (OLS) regression for one variable
+// OLS regression is a method used to estimate the relationship between a dependent variable (y) and one or more independent variables (x).
+
+// @kind function
+// @desc Function to calculate the beta coefficient (slope) using OLS
+//       The beta coefficient (slope) represents the change in the dependent variable for a one-unit change in the independent variable.
+//       This function computes the beta coefficient using the formula:
+//                beta = ((n * Σ(x*y)) - (Σx * Σy)) / ((n * Σ(x^2)) - (Σx)^2)
+// @param x {number[]} Independent variable
+// @param y {number[]} Dependent variable
+// @return {number} Beta (slope)   
+betaF:{dot:{sum x*y};                                      
+      ((n*dot[x;y])-(*/)(sum')(x;y))%                         
+      ((n:count[x])*dot[x;x])-sum[x]xexp 2};
+
+
+// @kind function
+// @desc Function to calculate the alpha coefficient (intercept) using OLS
+//       The alpha coefficient (intercept) represents the value of the dependent variable when the independent variable is zero.
+//       This function computes the alpha coefficient using the formula:
+//                alpha = Mean(y) - beta * Mean(x)
+// @param x {number[]} Independent variable
+// @param y {number[]} Dependent variable
+// @return {number} alpha (intercept)
+alphaF: {avg[y]-(betaF[x;y]*avg[x])};

streamPair.q

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+// import linear regression module
+\l linear_regression.q
+
+// load tables
+tab1: 1_ flip `dateTime`bid`ask`bidVol`askVol!("*FFFF";",") 0: `:data/USA500IDXUSD.csv;
+tab2: 1_ flip `dateTime`bid`ask`bidVol`askVol!("*FFFF";",") 0: `:data/USATECHIDXUSD.csv;
+tab3: flip `dateTime`spread`mean`up`low`ewma`up2`low2!("P"$();"F"$();"F"$();"F"$();"F"$();"F"$();"F"$();"F"$());
+historial_tab1: 1_ flip `open`high`low`close`adjClose`vol!("FFFFFF";",") 0: `:data/SP500_hist.csv;
+historial_tab2: 1_ flip `open`high`low`close`adjClose`vol!("FFFFFF";",") 0: `:data/NASDAQ100_hist.csv;
+
+// Fix data and take log(prices)
+priceX: 0!1_(update delta:0f^deltas dateTime from distinct select distinct dateTime, log bid, log ask from update dateTime:"P"$@[;19;:;"."] each dateTime from tab1);
+priceY: 0!1_(update delta:0f^deltas dateTime from distinct select distinct dateTime, log bid, log ask from update dateTime:"P"$@[;19;:;"."] each dateTime from tab2);
+
+// Calculate alpha and beta from historical values
+beta_lr: betaF[px:-100#log historial_tab1`close;py:-100#log historial_tab2`close]; // we only take most recent 100 values for the alpha and beta 
+alpha_lr: alphaF[px;py];
+// We calculate an historical standard deviation
+std_lr: dev[(1000#exec bid from priceY) - (1000#exec bid from priceX)];
+
+/ load and initialize kdb+tick 
+/ all tables in the top level namespace (.) become publish-able
+\l tick/u.q
+.u.init[];
+
+// Read and write on buffer functions
+.ringBuffer.read:{[t;i] $[i<=count t; i#t; i rotate t] }
+.ringBuffer.write:{[t;r;i] @[t;(i mod count value t)+til 1;:;r];}
+
+// Initialize index and empty tables (We will access directly to these objects from dashboards)
+.streamPair.i:-1;
+.streamPair.iEWMA:-1;
+.streamPair.priceX: 1000#tAux: 1_1#priceX;
+.streamPair.priceY: 1000#tAux;
+.streamPair.spreads: 1000#tab3;
+
+// Timer function
+timer:{t:.z.p;while[.z.p<t+x&abs x-16*1e6]}    / 16 <- timer variable
+
+.streamPair.genPair:{
+      // We wait some delta
+      d: `float$(priceX[.streamPair.i+:1][`delta]);
+      timer[d];
+      // We take the i element from our tables
+      resX: enlist priceX[.streamPair.i];
+      resY: enlist priceY[.streamPair.i];
+
+      // We calculate spreads for linear regression
+      s: priceY[.streamPair.i][`bid] - ((priceX[.streamPair.i][`bid] * beta_lr)+alpha_lr);
+      ewma: dev[ema[0.04; .streamPair.iEWMA#0f^(exec spread from .streamPair.spreads)]];
+      $[.streamPair.iEWMA>999;.streamPair.iEWMA:998;.streamPair.iEWMA+:1];
+      resSpread: enlist `dateTime`spread`mean`up`low`ewma`up2`low2!("p"$(priceX[.streamPair.i][`dateTime]);"f"$(s);"f"$(0);"f"$(1.96*std_lr);"f"$(-1.96*std_lr);"f"$0f^(ewma); "f"$(0f^(1.96*(last 1000 mdev (exec spread from .streamPair.spreads)))); "f"$0f^(-1.96*(last 1000 mdev (exec spread from .streamPair.spreads))));  
+
+      // We update our buffer tables with those values
+      .ringBuffer.write[`.streamPair.priceX;resX;.streamPair.i];
+      .ringBuffer.write[`.streamPair.priceY;resY;.streamPair.i];
+      .ringBuffer.write[`.streamPair.spreads;resSpread;.streamPair.i];
+      resX
+
+ }
+
+// Publish stream updates each milisecond
+.z.ts: {.streamPair.genPair[]]} 
+
+// Snapshot read from our buffer
+.u.snap:{[t] .ringBuffer.read[.streamPair.priceX;.streamPair.i]} // reqd. by dashboards
+
+\t 16

tick/u.q

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+/2019.06.17 ensure sym has g attr for schema returned to new subscriber
+/2008.09.09 .k -> .q
+/2006.05.08 add
+
+\d .u
+init:{w::t!(count t::tables`.)#()}
+
+del:{w[x]_:w[x;;0]?y};.z.pc:{del[;x]each t};
+
+sel:{$[`~y;x;select from x where sym in y]}
+
+pub:{[t;x]{[t;x;w]if[count x:sel[x]w 1;(neg first w)(`upd;t;x)]}[t;x]each w t}
+
+add:{$[(count w x)>i:w[x;;0]?.z.w;.[`.u.w;(x;i;1);union;y];w[x],:enlist(.z.w;y)];(x;$[99=type v:value x;sel[v]y;@[0#v;`sym;`g#]])}
+
+sub:{if[x~`;:sub[;y]each t];if[not x in t;'x];del[x].z.w;add[x;y]}
+
+end:{(neg union/[w[;;0]])@\:(`.u.end;x)}