The stochastic sensitivity of bull- and bear states in an asset - - PowerPoint PPT Presentation

The stochastic sensitivity of bull- and bear states in an asset market

Jochen Jungeilges [1,2] Tatyana Ryazanova [2]

[1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural Federal University, Institute of Natural Sciences and Mathematics,

Ekaterinburg, Russia

September 5, 2019 NED 2019 Conference on Nonlinear Economic Dynamics Kyiv School of Economics Kiev, Ukraine, September 4-6

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 1 / 28

Background

Research area: Asset markets with heterogenous investors Seminal model: Day and Huang (1990), Huang and Day (1993). Tramontana et al. (2010) Tramontana et al. (2011) Tramontana et al. (2013) Tramontana et al. (2014) Tramontana et al. (2015) Sushko et al. (2015) Panchuk et al. (2018) Quest for future efforts:

1 Allow for asymmetric response around the fundamental value. 2 Diversify no-trade intervals. 3 Intensify the stochastic modelling effort. Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 2 / 28

Scope of the project

Model: A stochastic DH asset-price process Asymmetries in trading behavior within agent-type No-trade intervals of agent-types do not coincide Types of noise: additive and parametric Goal: Further our understanding of the asset price dynamics in spec- ulative markets

1 Study the dynamics of the deterministic map (5 linear

pieces map with 2 discontinuities).

2 Analyze the sensitivity of stochastic equilibria. 3 Identify different types of transitions. 4 Unravel the ”genesis” of the transitions.

Method: Indirect method, stochastic sensitivity function (SSF) Milstein and Ryashko (1995)

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 3 / 28

Fundamentalists

Definition 1

The excess demand of α-investors is given by α(p) =    α0γ − αl(p − v + γ), p ≤ v − γ; α0(v − p), v − γ < p < v + γ; −α0γ − αu(p − v − γ), p ≥ v + γ. v ∈ (0, 1), 0 < γ < min(v, 1 − v), α0 ≥ 0, αl ≥ 0, αu ≥ 0.

Assumption 1

αl ≥ α0, αu ≥ α0

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 4 / 28

Fundamentalists

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 5 / 28

Chartists

Definition 2

The excess demand of the β-investors is given by β(p) =    βl (p − v), p ≤ v − ǫ−; βu (p − v), p ≥ v + ǫ−; 0,

• .w.

with p ∈ (0, 1), 0 < ǫ− < min(v, 1 − v), βl ≥ 0, βu ≥ 0.

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 6 / 28

Chartists

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 7 / 28

Deterministic Price process

Price process

Relying on Definitions 1 and 2 the asset price process can be given as pt+1 = f (pt) = pt + α(pt; γ, α0, αl, αu, v) + β(pt, ǫ−; βl, βu, v) (1) with p0 ∈ (0, 1).

Case: γ > ǫ−

If γ > ǫ− then the price process is given by pt+1 = f (pt) with

f (p) =                f1(p) = (1 − αl + βl)p + α0γ + αlv − αlγ − βlv, 0 ≤ p < v − γ; f2(p) = (1 − α0 + βl)p + α0v − βlv, v − γ ≤ p ≤ v − ǫ−; f3(p) = (1 − α0)p + α0v, v − ǫ− < p < v + ǫ−; f4(p) = (1 − α0 + βr)p + α0v − βrv, v + ǫ− ≤ p ≤ v + γ; f5(p) = (1 − αr + βr)p − α0γ + αrv + αrγ − βrv, v + γ < p ≤ 1.

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 8 / 28

γ > ǫ−: Equilibria

Assumption 2

α > β + 1

Assumption 3

αr = αl = α and βr = βl = β.

Result 1

Let δ = max

• α − v

γ (α − β), α − 1−v γ (α − β)

• . Suppose Assumptions 2

and 3 hold. If γ < v and α0 ∈ (δ, β) then the equilibria p1 = v − γ(α − α0) α − β , p3 = v, p5 = v + γ(α − α0) α − β

• exist. The equilibria p1 and p5 are locally stable if β > α − 2 and p3 is

stable if 0 < α0 < 2.

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 9 / 28

Immediate basins

b1 = v − ǫ−−(α−α0)γ

1−α+β

b2 = v − ǫ− b3 = v + ǫ− b4 = v + ǫ−−(α−α0)γ

1−α+β Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 10 / 28

The stochastic model

Stochastic price process

Relying on Definitions 4 and 6 the asset price process can be given as pt+1 = pt + α(pt; γ, α0, α, v) + β(pt, ǫ−; β + πξt, v) + εξt (2) with p0 ∈ (0, 1), ε, π ≥ 0, ξt ∼ N(0, 1). π = 0, ε > 0 additive shocks π > 0, ε = 0 parametric shocks π > 0, ε > 0 mixture

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 11 / 28

Sensitivity analysis via SSF

We can represent (2) as pt+1 = f (pt) + εg(pt)ξt (3) where g(•) denotes a smooth function.

Assumption 4

For ε = 0 (3) has an exponentially stable equilibrium ¯ p. Let pt(ε) be the solution of (3) with p0(ε) = ¯ p + εν0 then zt = lim

ε→0

pt(ε) − ¯ p ε characterizes the sensitivity of the price equilibrium to i.i.d. shocks. zt+1 = f ′(¯ p)zt + g(¯ p)ξt

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 12 / 28

Sensitivity continued

Focus on the dynamics of second moments: Vt = E[z2

t ]

Vt+1 = [f ′(¯ p)]2Vt + g(¯ p) Assumption 4 ⇒ | f ′(¯ p) |< 1 ω = lim

t→∞ Vt =

g2(¯ p) 1 − [f ′(¯ p)]2 Confidence interval: ¯ p ± kε √ 2ω where k = erf −1(0.99) Remarks: ω and ε define the borders of the confidence interval for ¯ p. D = ε2ω is related to the variance matrix of the stationary density. ω is the stochastic sensitivity function (SSF) for the attractor ¯ p. The SSF relates the intensity of stochastic signal ε2.

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 13 / 28

Sensitivity analysis for the stochastic price process

Case γ > ǫ− : additive noise, i.e. g(¯ p) = 1 f ′(p) =                1 − α + β, 0 ≤ p < v − γ; 1 − α0 + β, v − γ ≤ p ≤ v − ǫ−; 1 − α0, v − ǫ− < p < v + ǫ−; 1 − α0 + β, v + ǫ− ≤ p ≤ v + γ; 1 − α + β, v + γ < p ≤ 1. ω1 = 1 1 − (1 − α + β)2 ω3 = 1 1 − (1 − α0)2 ω5 = 1 1 − (1 − α + β)2

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 14 / 28

Graphs of SSFs

Figure: ω1 for α = 5 and α − 2 < β < α − 1 Figure : ω3 for 0 < α0 < 2 Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 15 / 28

Confidence intervals

p1 : v − γ(α − α0) α − β ± kε

• 2

1 − (1 − α + β)2 p3 : v ± kε

• 2

1 − (1 − α0)2 p5 : v + γ(α − α0) α − β ± kε

• 2

1 − (1 − α + β)2

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 16 / 28

Case 2: stable: p1, p5 unstable p3; ε = 0

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 17 / 28

Case 2: stable: p1, p5 unstable p3; p0 = p1(p5) , ε = 0.01

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 18 / 28

Case 2: stable: p1, p5 unstable p3; p0 = p5, ε = 0.03

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 19 / 28

Case 2: stable: p1, p5 unstable p3; p0 = p1, ε = 0.03

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 20 / 28

Case 3: stable p1, p3, p5, ε = 0

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 21 / 28

Case 3: stable p1, p3, p5, p0 = pi, ε = 0.01

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 22 / 28

Case 3: stable p1, p3, p5, p0 = pi, ε = 0.025

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 23 / 28

Case 3: stable p1, p3, p5, p0 = pi, i ∈ {1, 3, 5}, ε = 0.025

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 24 / 28

Conclusion

Key elements in the genesis of transition between boom and bust states:

1

immediate basins of attraction,

2

confidence ellipses of attractors

The noise levels at which transitions become likely depends on trading intensities. In the case of additive noise, transitions between equilibria might

• ccur even if all equilibria are stable.

Transitions are more likely to occur under parametric noise than under additive noise (work in progress).

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 25 / 28

References I

Day, R. H. and Huang, W. (1990). Bulls, bears and market sheep. Journal

• f Economic Behavior & Organization, 14(3):299–329.

Huang, W. and Day, R. (1993). Chaotically switching bear and bull markets: The derivation of stock price distributions from behavior

• rules. In Day, R. and Chen, P., editors, Nonlinear Dynamics and

Evolutionary Economics. Oxford University Press. Milstein, G. and Ryashko, L. (1995). The first approximation in the quasipotential problem of stability of non-degenerate systems with random perturbations. Applied Mathematics and Mechanics, 1(59):51–XX. in Russian. Panchuk, A., Sushko, I., and Westerhoff, F. (2018). A financial market model with two discontinuities: Bifurcation structures in the chaotic

• domain. Chaos, 28(055908).

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 26 / 28

References II

Sushko, I., Tramontana, F., Westerhoff, F., and Avrutin, V. (2015). Symmetry breaking in a bull and bear financial market model. Chaos, Solitons & Fractals, 79:57 – 72. Proceedings of the MDEF (Modelli Dinamici in Economia e Finanza Dynamic Models in Economics and Finance) Workshop, Urbino 18th20th September 2014. Tramontana, F., Gardini, L., and Westerhoff, F. (2011). Heterogeneous speculators and asset price dynamics: Further results from a

• ne-dimensional discontinuous piecewise-linear map. Computational

Economics, 38(3):329. Tramontana, F., Westerhoff, F., and Gardini, L. (2010). On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders. Journal of Economic Behavior & Organization, 74(3):187 – 205.

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 27 / 28

References III

Tramontana, F., Westerhoff, F., and Gardini, L. (2013). The bull and bear market model of huang and day: Some extensions and new results. Journal of Economic Dynamics and Control, 37(11):2351 – 2370. Tramontana, F., Westerhoff, F., and Gardini, L. (2014). One-dimensional maps with two discontinuity points and three linear branches: Mathematical lessons for understanding the dynamics of financial

• markets. Decisions in Economics and Finance, 37.

Tramontana, F., Westerhoff, F., and Gardini, L. (2015). A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities. Mathematics and Computers in Simulation, 108:16 – 40. Complex Dynamics in Economics and Finance - Papers

• f the MDEF 2012 Workshop.

Jochen Jungeilges [1,2], Tatyana Ryazanova [2] ([1] University of Agder, School of Business and Law, Kristiansand S, Norway

[2] Ural F

The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 28 / 28