Sornette-Ide model for markets: Trader expectations as imaginary part

Numerous microscopic models [1] for price fluctuations on stock or currency markets have been invented within the last decade [2, 3] in the physics literature. An alternative are more phenomenological differential equations for the price itself [4, 5, 6, 7, 8], with some noise as in Langevin equations. The present note follows Sornette and Ide [5] but uses a complex instead of a real variable. We interpret the real part of this complex variable z as the price (more precisely, it is proportional to the logarithm of the price in units of the fundamental price), and the imaginary part can be the trader expectation. Mathematics relates the changes in the real and imaginary parts. We will try to see crashes arising from the intrinsic market forces, with (log-periodic [9]) precursors in the real part and strong increases in the imaginary part. The expectations of the traders in the combination of price and expectation can be defined as a two-dimensional point in the market “phase space”, which is denoted by the complex number z with Re(z) = price and Im(z) = expectation.