Strategic Traders and Liquidity Crashes Alexander Remorov 6.254 Final Project December 7, 2013 Remorov Strategic Traders and Liquidity Crashes 1 / 21
Introduction Most of the time markets functioning well Prices don t move much Supply and demand for stock relatively balanced What happens if everyone wants to sell? Remorov Strategic Traders and Liquidity Crashes 2 / 21
Volume Price CREG Stock, Nov. 17, 2013 4.25 4.2 4.15 4.1 4.05 4 3.95 3.9 3.85 3.8 3.75 80000 60000 40000 20000 0 Data obtained from Google Finance Remorov Strategic Traders and Liquidity Crashes 3 / 21
Liquidity Crashes What happens if almost everyone wants to sell? Much lower price needed for someone to buy the stock Selling even a small position causes a large price decline This is caused an illiquid market Everyone selling at the same time causes a liquidity crash Remorov Strategic Traders and Liquidity Crashes 4 / 21
Strategic Traders Can view traders as playing a simultaneous game Need to decide on positions in the stock throughout the period Could wait it out since will price likely return to previous level However may be forced to sell when price is low: Due to an exogenous liquidity shock: Bernardo and Welch (2004) Loss limits: Morris and Shin (2004) Remorov Strategic Traders and Liquidity Crashes 5 / 21
Maket Participants Short-term strategic investors Risk-neutral; don t have price impact by their own trades May be forced to sell stock External liquidity shock: Bernardo and Welch (2004) Loss limit: Morris and Shin (2004) Market-makers/long-term investors Risk-averse Buy asset from short-term investors Set lower price if receive more sell orders Remorov Strategic Traders and Liquidity Crashes 6 / 21
Bernardo and Welch Model Three dates: t = 0: investors (possibly) trade t = 1: liquidity shock with prob. s, affected investors must trade t = 2: liquidation; investors are paid Investor behavior: α: fraction of investors selling at time 0 1 α: fraction selling at time 1 (assume correlated shocks) Prices p 0 (α) = v cα: price at time 0 p 1 (α) = v c(1 + α): price at time 1 v: expected value at time 2 Remorov Strategic Traders and Liquidity Crashes 7 / 21
Strategic investor s problem Decide on probability α to sell now Payoff to sell one share now: p 0 (α) Payoff to wait until date 2: { p 1 (α), if experiences shock u = v, if no shock Therefore will sell now if and only if: p 0 (α) s p 1 (α) + (1 s) v Remorov Strategic Traders and Liquidity Crashes 8 / 21
Deriving Equilibrium Define F (α) to be expected benefit to sell now: s 1 2 : mixed NE α = s > 1 2 : pure NE α = 1 F (α) = p 0 (α) s p 1 (α) (1 s) v s 1 s = c(s(1 + α) α) If probability of shock is high enough, everyone will sell right away before the shock even occurs Drawback: model too discrete; all investors acting the same Remorov Strategic Traders and Liquidity Crashes 9 / 21
Morris and Shin Model Two dates: t = 0: investors (possibly) trade, some get fired t = 1: liquidation; investors are paid Investor behavior: α: number of investors selling at time 0 loss limit q i ; if breached, investor is fired distribution: q i = θ + ω i, ω i IID Unif [ ɛ, ɛ] Prices Time 0 orders executed at an uncertain price v cu, U Unif [0, α] Price at the end of time 0 is v cα v: expected liquidation value at time 1 Remorov Strategic Traders and Liquidity Crashes 10 / 21
Strategic investor s problem Investor has stop limit q i. It is breached if ˆα i investors sell, where: q i = v c ˆα i Thus, if decides to hold the stock, expected payoff is: { v, if α ˆα i u(α) = 0, if α > ˆα i If decides to sell, then payoff is equal to expected sell price if loss limit not breached, and zero otherwise { v 1 w(α) = 2 cα, if α ˆα i ˆα i α (v 1 2 c ˆα i), if α > ˆα i Remorov Strategic Traders and Liquidity Crashes 11 / 21
Deriving Equilibrium Will look for threshold strategies: { sell, if q i > q (v) (v, q i ) hold, if q i q (v) In equilibrium, the fraction α of traders selling, conditional on your limit being q, is uniform [0, 1] Equilibirum condition: 1 Substituting formulas, becomes: 0 (u(α) w(α))dα = 0 v q i = c exp [ qi v ] 2(v + q i ) Remorov Strategic Traders and Liquidity Crashes 12 / 21
Loss Limit q Optimal Loss Limit Threshold Optimal Loss Limit Threshold, v = 100 100 90 80 70 60 50 40 30 20 10 0 Loss Limit Threshold Lowest Price 0 10 20 30 40 50 60 70 80 90 100 Price Sensitivity c Remorov Strategic Traders and Liquidity Crashes 13 / 21
Discussion and Modification Desired effect: trader sells as if most of other traders sell When c is small, act as if everyone else is selling Unrealistic! End up selling even if the price is 20% above limit... Remorov Strategic Traders and Liquidity Crashes 14 / 21
New Equilibrium Introduce transaction costs τ if successfully sell stock Furthermore if limit is breached, payoff is R > 0 Old equilibrium condition: [ qi v ] v q i = c exp 2(v + q i ) New condition: [ (qi v)(1 τ) 4vτ ] v q i = c exp 2(v + q i )(1 τ) 4R Remorov Strategic Traders and Liquidity Crashes 15 / 21
Loss Limit q Modified Model Results Optimal Loss Limit Threshold, v = 100 100 98 96 94 92 90 88 86 84 82 80 Lowest Price Tau = 0 %, R = 0 0 2 4 6 8 10 12 14 16 18 20 Price Sensitivity c Remorov Strategic Traders and Liquidity Crashes 16 / 21
Loss Limit q Modified Model Results Optimal Loss Limit Threshold, v = 100 100 98 96 94 92 90 88 86 84 82 80 Lowest Price Tau = 0 %, R = 0 Tau = 5 %, R = 20 0 2 4 6 8 10 12 14 16 18 20 Price Sensitivity c Remorov Strategic Traders and Liquidity Crashes 17 / 21
Loss Limit q Modified Model Results Optimal Loss Limit Threshold, v = 100 100 98 96 94 92 90 88 86 84 82 80 Lowest Price Tau = 0 %, R = 0 Tau = 5 %, R = 20 Tau = 5 %, R = 40 0 2 4 6 8 10 12 14 16 18 20 Price Sensitivity c Remorov Strategic Traders and Liquidity Crashes 18 / 21
Loss Limit q Modified Model Results Optimal Loss Limit Threshold, v = 100 100 98 96 94 92 90 88 86 84 82 80 Lowest Price Tau = 0 %, R = 0 Tau = 5 %, R = 20 Tau = 5 %, R = 40 Tau = 20%, R = 40 0 2 4 6 8 10 12 14 16 18 20 Price Sensitivity c Remorov Strategic Traders and Liquidity Crashes 19 / 21
Conclusion Nice framework for modeling liquidity crashes In equilibrium investors sell in fear that other investors may sell Due to a potential external shock Due to a loss limit When introduce transaction costs as well as positive payoffs if limit breached, results become more realistic Remorov Strategic Traders and Liquidity Crashes 20 / 21
Further extensions Want a multi-period model; things get more complicated Possible extension: three dates, two types of strategic investors: First type can sell at time 0 or 1 Second type can sell only at time 1 Then we get an optimal price drop for two periods more realistic Consider possibility of investors buying at the low price Example - Brunnermeier and Pedersen (2005) Large investor forced to sell others sell with him, then buy at the resulting very low price Remorov Strategic Traders and Liquidity Crashes 21 / 21