BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

202 ACTA WASAENSIA f˙1−c1 = v B f 1 −c 2 −f 1 −f 21 c 1 +c 2 +f 1 +f 2e α B(2(c 1 +c 2 +f 1 +f 2 )−1) − e −α B(2(c 1 +c 2 +f 1 +f 2 )−1) + v (f 1 + c 1 ) tanh(α(F 1 − C 1 )) − f 1 − c 1cosh(α(F 1 − C 1 ))f 1 + c1+(f 1 + c 2 ) tanh(α(F 1 − C 2 )) − f 1 − c 2cosh(α(F 1 − C 2 ))f 1 + c2+(f 1 + f 2 ) tanh(α(F 1 − F 2 )) − f 1 − f 2cosh(α(F 1 − F 2 ))f 1 + f 2 f˙2 = v B f 1−c1 −c 2 −f 1 −f 22 c 1 +c 2 +f 1 +f 2e α B(2(c 1 +c 2 +f 1 +f 2 )−1) − e −α B(2(c 1 +c 2 +f 1 +f 2 )−1) + v (f 2 + c 1 ) tanh(α(F 2 − C 1 )) − f 2 − c 1cosh(α(F 2 − C 1 ))f 2 + c1+(f 2 + c 2 ) tanh(α(F 2 − C 2 )) − f 2 − c 2cosh(α(F 2 − C 2 ))f 2 + c2+(f 2 + f 1 ) tanh(α(F 2 − F 1 )) − f 2 − f 1cosh(α(F 2 − F 1 ))f 2 + f 1(A9.2c). (A9.2d)The price dynamics (5.3) may be rewritten in terms of (A9.1) asp˙i = 1 lc ˙ i +(p fi − p i )f ˙f i , i =1, 2, (A9.3)iwherewehaveusedtheleverageparameterl := t c /t f introduced in proposition 2 inorder to express the relation of chartist relative to fundamentalist target holdings. Theentries of the Jakobian matrix⎛J =⎜⎝∂ċ 1 ∂ċ 1 ∂ċ 1 ∂ċ 1 ∂ċ 1 ∂ċ 1⎞∂c 1 ∂c 2 ∂f 1 ∂f 2 ∂p 1 ∂p 2∂ċ 2 ∂ċ 2 ∂ċ 2 ∂ċ 2 ∂ċ 2 ∂ċ 2∂c 1 ∂c 2 ∂f 1 ∂f 2 ∂p 1 ∂p 2∂f ˙ 1 ∂f ˙ 1 ∂f ˙ 1 ∂f ˙ 1 ∂f ˙ 1 ∂f ˙ 1∂c 1 ∂c 2 ∂f 1 ∂f 2 ∂p 1 ∂p 2∂f ˙ 2 ∂f ˙ 2 ∂f ˙ 2 ∂f ˙ 2 ∂f ˙ 2 ∂f ˙ 2∂c 1 ∂c 2 ∂f 1 ∂f 2 ∂p 1 ∂p 2∂ṗ 1 ∂ṗ 1 ∂ṗ 1 ∂ṗ 1 ∂ṗ 1 ∂ṗ 1⎟∂c 1 ∂c 2 ∂f 1 ∂f 2 ∂p 1 ∂p 2⎠∂ṗ 2 ∂ṗ 2 ∂ṗ 2 ∂ṗ 2 ∂ṗ 2 ∂ṗ 2∂c 1 ∂c 2 ∂f 1 ∂f 2 ∂p 1 ∂p 2of the coupled system (A9.2) and (A9.3) evaluated at the fundamental equilibrium(A9.4)c 1 = c 2 = f 1 = f 2 =1/8(A9.5)