The Impact of Short-Selling Constraints on Financial Market Stability

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1 The Impact of Short-Selling Constraints on Financial Market Stability Mikhail Anufriev Jan Tuinstra 5th International Conference on Computing in Economics and Finance University of Technology, Sydney 6 July 29

2 Definition If a mean-variance investor, who demands A i,t (p) = E i,t[p t+ + y t+ ] ( + r f )p a i V i,t [p t+ + y t+ ], expects positive return then A i,t >, i.e. investor has long position expects negative return then A i,t <, i.e. investor has short position

3 Price Correction A i,t (p) = E i,t[p t+ + y t+ ] ( + r f )p a i V i,t [p t+ + y t+ ] If price change is not expected A i,t > iff ȳ > pr f, i.e., when asset is undervalued A i,t < iff ȳ < pr f, i.e., when asset is overvalued A i,t = iff ȳ = pr f, i.e., when price is on the fundamental value Notice that if price responds to the change in demand/supply, then strategy buy low, sell high is self-reinforcing and leads to price correction.,

4 Mechanism. investor s broker locates stocks stock is borrowed stock is actually not borrowed 2. security is sold and delivered to the buyer 3. investor closes ( covers ) his position, buying shares back 4. investor return the shares

5 Costs and risks of the short-selling strategy profit is limited, but loss are unlimited borrowing a stock might be difficult in an absence of a market for it a borrowed stock can be recalled at any moment by the lender legal restrictions hostility from society

6 Short Selling increases liquidity and informational efficiency, and eliminates mis-pricing Theory: Miller (JF, 977), Harrison and Kreps (QJE, 978), Diamond and Verrecchia (JFE, 987), Gallmeyer and Hollifield (JF, 28) Empirics: Jones and Lamont (JFE, 22), Lamont and Thaler (JPE, 23), Diether, Lee and Werner (RFS, 28) increases volatility and may lead to market crashes Lecce, Lepone and Segara (WP, 26), Setzu and Marchesi (WP, 28)

7 This Paper Take a model with heterogeneous agents (Brock and Hommes, JEDC, 998) Introduce the short-selling constraints Ā > : ( A i,t (p) = max Ā, E ) i,t[p t+ ] + ȳ ( + r f )p aσ 2 Analyse stability of the fundamental steady-state and amplitude of oscillations

8 Dynamical model of financial market. two assets riskless: risk-free interest rate r f risky: price p t and i.i.d. dividend y t with mean ȳ supply per investor S fundamental price p f = (ȳ aσ 2 S)/r f 2. mean-variance demand for the risky asset z h,t = E h,t [ pt+ + y t+ ( + r f ) p t ] / a σ 2 3. heterogeneous expectations of agents fundamentalists: Ef,t [p t+ ] = p f trend-followers: E c,t [p t+ ] = p f + g (p t p f ), g

9 Dynamical model of financial market 4. market clears, price p t is determined p t p f = + r f H n h,t E h,t [p t+ p f ] = h= g + r f n 2,t (p t p f ) 5. performances are computed ( Eh,t [x t ] ( + r f )x t A h,t r t = a σ 2 ) ( + S 2 S) x t (+r f )x t +aσ

10 Evolutionary updating of types 6. agents choose a new type for the next period past profits of two types U f,t = π f,t C U c,t = π c,t fraction of type h is computed as / n h,t+ = exp[β U h,t ] Z t, with Z t = exp[β U h,t] h β is the intensity of choice β = : equal distribution n f,t+ = n c,t+ =.5 β = + : all traders use the optimal strategy

11 Two regimes: stable and volatile Zero Supply Positive Supply

12 Two regimes: stable and volatile β < β : all agents have assets β < β < β : optimistic type is long, pessimistic is short β > β : fluctuations

13 Two attractors: overvaluation and undervaluation 2 2 Price Price 98 Return Return.5.5 Fractions.5 Fractions.5 2 fundamentalists chartists 2 fundamentalists chartists Positions Positions

14 Short-Sell Constraints Assume Ā > and impose a restriction: { A i,t (p t ) = max Ā, E i,t[p t+ ] + ȳ ( + r f )p } t a σ 2.

15 Short-selling constraints: Ā = primary bifurcation is not affected asymmetry between upper and lower attractors emerges the mispricing (amplitude of oscillations) increases

16 Adjusted demand and supply 4 3 Fundamentalists Chartists Aggregate 4 3 Price Deviation x c 2 E U Price Deviation x c 2 E C E U Fundamentalists Chartists Aggregate Adjusted Quantity -2 A+S Adjusted Quantity

17 Effect of short-selling constraints on upper trend 2 2 Price Price Return -.5 Return Fractions.5 Fractions.5 2 fundamentalists chartists 2 fundamentalists chartists Positions Positions

18 Short-selling constraints vs. No constraints When the short sell constraints are binding: level of price becomes higher smaller liquidity level of return is higher (smaller in absolute value) capital gain fundamentalists performance worsens w.r. to chartists (A f,t A c,t )r t fraction of fundamentalists is lower

19 Effect of short-selling constraints on crash 2 6 Price Price Return Return -3-5 Fractions.5 Fractions.5 2 fundamentalists chartists 6 fundamentalists chartists Positions Positions

20 Short-selling constraints vs. No constraints When the crash takes place under short-sell constraints: level of price is higher return is extremely low fractions of fundamentalists is much higher

21 Recall Lower Attractor vs. Upper Attractor 2 2 Price Price 98 Return.5.5 Return Fractions.5 Fractions.5 2 fundamentalists chartists 2 fundamentalists chartists Positions Positions

22 Lower Attractor without and with Crash 2 2 Price Price Return.5 Return.5 Fractions.5 Fractions.5 2 fundamentalists chartists 2 fundamentalists chartists Positions Positions

23 Summary Under short-sell constraints primary bifurcation (of the fundamental steady-state) is not affected (local stability is a local property, and the restrictions at the fundamental steady-state are not binding) there is an asymmetry between upper and lower attractors (constrained investors are present there in different proportions) amplitude of oscillations on the upper attractor increases (investors who try to eliminate mis-pricing are short)

24 Dependence on Ā for zero and positive supply

25 Fundamentalists vs. Contrarians

26 Fundamentalists vs. Sophisticated Trend Followers

27 Fundamentalists vs. Sophisticated Trend Followers 3 25 contsrained at t = 4, A = 7 no constraints

28 Fundamentalists vs. Sophisticated Trend Followers 3 contsrained at t = 4, A = 7 no constraints 3 contsrained at t = 4, A = 7 constrained at t = 5, A = no constraints

29 Conclusion Short-sell constraints affect the amplitude of cycle and drive price up liquidity effect composition of the ecology effect Short-sell constraints do not affect the local stability properties of the fundamental steady-state

30 Fundamentalists vs. Sophisticated Trend Followers Price Price Return Return Fractions.5 Fractions.5 6 fundamentalists chartists 6 fundamentalists chartists Positions 3-3 Positions

31 Fundamentalists vs. Contrarians Price Price Return -.5 Return Fractions.5 Fractions.5 fundamentalists chartists fundamentalists chartists.5.5 Positions Positions

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