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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 ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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. ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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) ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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 0 γ − α l ( p − v + γ ) , p ≤ v − γ ;  α 0 ( v − p ) , v − γ < p < v + γ ; α ( p ) = − α 0 γ − α u ( p − v − γ ) , p ≥ v + γ .  v ∈ (0 , 1), 0 < γ < min ( v , 1 − v ), α 0 ≥ 0, α l ≥ 0, α u ≥ 0. Assumption 1 α l ≥ α 0 , α u ≥ α 0 ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 4 / 28

Fundamentalists ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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 l ( p − v ) , p ≤ v − ǫ − ;  p ≥ v + ǫ − ; β ( p ) = β u ( p − v ) , 0 , o.w.  with p ∈ (0 , 1), 0 < ǫ − < min ( v , 1 − v ), β l ≥ 0, β u ≥ 0. ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 6 / 28

Chartists ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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 p t +1 = f ( p t ) = p t + α ( p t ; γ, α 0 , α l , α u , v ) + β ( p t , ǫ − ; β l , β u , v ) (1) with p 0 ∈ (0 , 1). Case: γ > ǫ − If γ > ǫ − then the price process is given by p t +1 = f ( p t ) with  f 1 ( p ) = (1 − α l + β l ) p + α 0 γ + α l v − α l γ − β l v , 0 ≤ p < v − γ ;    f 2 ( p ) = (1 − α 0 + β l ) p + α 0 v − β l v , v − γ ≤ p ≤ v − ǫ − ;     v − ǫ − < p < v + ǫ − ; f ( p ) = f 3 ( p ) = (1 − α 0 ) p + α 0 v , v + ǫ − ≤ p ≤ v + γ ;  f 4 ( p ) = (1 − α 0 + β r ) p + α 0 v − β r v ,      f 5 ( p ) = (1 − α r + β r ) p − α 0 γ + α r v + α r γ − β r v , v + γ < p ≤ 1 .  ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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 α r = α l = α and β r = β l = β . Result 1 � � α − v γ ( α − β ) , α − 1 − v Let δ = max γ ( α − β ) . Suppose Assumptions 2 and 3 hold. If γ < v and α 0 ∈ ( δ, β ) then the equilibria p 1 = v − γ ( α − α 0 ) , p 3 = v , p 5 = v + γ ( α − α 0 ) α − β α − β exist. The equilibria p 1 and p 5 are locally stable if β > α − 2 and p 3 is stable if 0 < α 0 < 2. ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 9 / 28

Immediate basins b 1 = v − ǫ − − ( α − α 0 ) γ b 4 = v + ǫ − − ( α − α 0 ) γ b 2 = v − ǫ − b 3 = v + ǫ − 1 − α + β 1 − α + β ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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 p t +1 = p t + α ( p t ; γ, α 0 , α, v ) + β ( p t , ǫ − ; β + πξ t , v ) + εξ t (2) with p 0 ∈ (0 , 1), ε, π ≥ 0, ξ t ∼ N (0 , 1). π = 0 , ε > 0 additive shocks π > 0 , ε = 0 parametric shocks π > 0 , ε > 0 mixture ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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 p t +1 = f ( p t ) + ε g ( p t ) ξ t (3) where g ( • ) denotes a smooth function. Assumption 4 For ε = 0 (3) has an exponentially stable equilibrium ¯ p . Let p t ( ε ) be the solution of (3) with p 0 ( ε ) = ¯ p + εν 0 then p t ( ε ) − ¯ p z t = lim ε ε → 0 characterizes the sensitivity of the price equilibrium to i.i.d. shocks. z t +1 = f ′ (¯ p ) z t + g (¯ p ) ξ t ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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: V t = E [ z 2 t ] p )] 2 V t + g (¯ V t +1 = [ f ′ (¯ p ) Assumption 4 ⇒ | f ′ (¯ p ) | < 1 g 2 (¯ p ) ω = lim t →∞ V t = p )] 2 1 − [ f ′ (¯ √ 2 ω where k = erf − 1 (0 . 99) Confidence interval: ¯ p ± k ε Remarks: ω and ε define the borders of the confidence interval for ¯ p . D = ε 2 ω 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 ε 2 . ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] 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  1 − α + β, 0 ≤ p < v − γ ;    1 − α 0 + β, v − γ ≤ p ≤ v − ǫ − ;     v − ǫ − < p < v + ǫ − ; f ′ ( p ) = 1 − α 0 , v + ǫ − ≤ p ≤ v + γ ;  1 − α 0 + β,      1 − α + β, v + γ < p ≤ 1 .  1 1 1 ω 1 = ω 3 = ω 5 = 1 − (1 − α + β ) 2 1 − (1 − α 0 ) 2 1 − (1 − α + β ) 2 ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 14 / 28

Graphs of SSFs Figure: ω 1 for α = 5 and α − 2 < β < α − 1 Figure : ω 3 for 0 < α 0 < 2 ( [1] University of Agder, School of Business and Law, Kristiansand S, Norway [2] Ural F Jochen Jungeilges [1 , 2] , Tatyana Ryazanova [2] The stochastic sensitivity of bull- and bear states in an asset market September 5, 2019 15 / 28