Financial Markets with Memory I: Dynamic Models

Abstract

Abstract. This is the first of two papers in which we consider a stock with price process defined by a stochastic differential equation driven by a process Y (·) different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process Y (·) is defined by a continuous-time AR(∞)-type equation and may have either short or long memory. We show that the process Y (·) has a good MA(∞)-type representation. The existence of such simultaneous good AR(∞) and MA(∞) representations enables us to apply a new method for the calculation of relevant conditional expectations, whence to obtain various explicit results for problems such as portfolio optimization. The financial market defined by the above stock price process is complete, and if the coefficients are constant, then the prices of European calls and puts are given by the Black-Scholes formulas as in the Black-Scholes model. Unlike the latter, however, the model allows for differences between the historical and implied volatilities. The model includes a special case in which only two additional parameters are introduced to describe the memory of the market, compared with the Black-Scholes model. Analysis based on real market data shows that this simple model with two additional parameters is more realistic in capturing the memory effect of the market, while retaining the simplicity and usefulness of the Black-Scholes model.

Bubbles and crashes in a Black–Scholes model with delay

“…In [1,2] the asymptotic behaviour of the autocovariance function of the solution of a linear stochastic integral equation is studied, and criteria for long memory established. In common with our work, it is found that the asymptotic behaviour of the autocovariance function depends on the asymptotic behaviour of the fundamental solution of an underlying deterministic linear functional equation.…”

Optimal Long-Term Investment Model with Memory

“…, n, where f (t; p, q) := q 2 (p + q) 2 + p(2q + p) (p + q) 3 · (1 − e −(p+q)t ) t , t > 0 (cf. [1], Examples 4.3 and 4.5). Notice that f (t; 0, q) = 1.…”

“…In the special case p j = 0, Y j (t) reduces to the Brownian motion W j (t). Driving noise processes with short or long memory of this kind are considered in [1], Anh et al [2] and Inoue et al [20], for the case n = 1. We define…”

“…, n (cf. Anh and Inoue [1]). Equivalently, we may define Y j (t) by the moving-average type representation…”

“…This filtration (F t ) t≥0 is the underlying information structure of the market model M. From (1.5), we can easily show that (Y (t)) t≥0 is a semimartingale with respect to (F t ) (cf. [1,Section 3]). In particular, we can interpret the stochastic differential equation (1.1) in the usual sense.…”