Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Risk Management for Whales Rama Cont and Lakshithe Wagalath Imperial College, London and IESEG School of Management, Paris Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Plan 1 Introduction and motivation 2 Modeling liquidation losses and liquidation-adjusted VaR 3 Taming the Whale 4 Conclusion Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Statistical risk measures for portfolios Current risk management frameworks focus on modeling/hedging the variation in Mark-to-Market P&L of portfolios. A common approach is to use a statistical model to estimate a risk measure (typically VaR or Expected Shortfall) for P&L distribution over a given risk horizon. Regardless of their complexity, such risk measures have a positive homogeneity property and hence scale linearly with portfolio size : if notionals are multiplied by 100, “risk” is multiplied by 100. Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Statistical risk measures for portfolios The computation of risk-measures enables to compute capital requirements, margin requirements, ..., to face losses in extreme risk scenarios . Examples: capital requirements for banks margin requirements for derivatives portfolios counterparty risk evaluation Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Liquidation risk vs market risk Such losses materialize in risk scenarios where one is forced to liquidate a sizable portion of the portfolio. This points to another, quite different, concept from market value: the liquidation value in a stress scenario. The purpose of margin and collateral requirements is to cover losses of the exchange/ clearinghouse in case a clearing member defaults. To fulfill this purpose, their level should cover liquidation costs in such scenarios. Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Liquidity risk: under the radar? Currently margin requirements, counterparty exposures, collateral requirements ... are computed based on a measure of market risk (VaR, ES,...) of the position. No consideration is given to the liquidity of instruments or the size of positions relative to market depth. How can the result reflect potential liquidation costs if it does not use any liquidity measure or market depth as input? The result is a serious, and massive, underestimation of portfolio exposures to liquidity risk. Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Taking liquidation risk seriously Several recent risk management fiascos have been associated with the miscalculation of risk for large positions. When the institutions tried to unwind their positions, the realized losses were much larger than what their risk models had anticipated. These considerations call for a comprehensive approach for integrating liquidation value into risk measures. Such an approach should: differentiate between positions across assets of varying liquidity account for limited market depth be scalable to complex, multi-asset portfolios with derivatives Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Example of endogenous liquidation Consider a fund with initial size V , equity E and subject to a V leverage constraint L max : E ≤ L max . If the fund’s portfolio experiences a percentage loss l , then its leverage ratio increases to V (1 − l ) E − lV . If l ∗ = 1 V then V (1 − l ) EL max − V L max − 1 < l < E E − lV > L max V The fund violates its leverage constraint. Two possibilities: increase equity → costly after a shock and not ideal due to the debt overhang problem decrease the size of assets → liquidation of a portion 1 − L max ( E − lV ) V (1 − l ) of its holdings Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Example of endogenous liquidation Fund liquidation 1,2 portion of the fund liquidated 1 portion of the fund liquidated 0,8 (smoothed) 0,6 0,4 0,2 Fund percentage loss 0 0,00% 0,30% 0,60% 0,90% 1,20% 1,50% 1,80% 2,10% 2,40% 2,70% 3,00% 3,30% 3,60% 3,90% 4,20% 4,50% 4,80% 5,10% 5,40% 5,70% 6,00% Figure: Portion of the fund liquidated as a function of the percentage loss for a fund with initial leverage ratio V E = 25 and leverage constraint L max = 33 Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Plan 1 Introduction and motivation 2 Modeling liquidation losses and liquidation-adjusted VaR 3 Taming the Whale 4 Conclusion Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Discrete-time setting: fundamental asset returns Time step ∆ t (typically one day) n assets: value of asset i at date k ∆ t denoted S i k Fundamental return of asset i at period k : S i k +1 − S i √ k = ǫ i ∆ t ξ i k +1 = k +1 + m i ∆ t S i k with m i the drift of asset i and ( ξ k +1 ) k ≥ 0 a sequence of iid centered random variables such that: Cov( ξ i k , ξ j k ) = Σ i , j Σ fundamental covariance matrix: captures a structural relationship between asset returns. Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Discrete-time setting: the fund Fund with positions α 1 , ..., α n n � α j S j Benchmark fund value: V k = k j =1 Buy-and-hold strategy and no price impact: n � α j S j k ǫ j V k +1 − V k = k +1 j =1 Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Discrete-time setting: fund liquidations Funds are often subject to a constraint: capital requirement, liquidity ratio, leverage constraint, performance constraint... Loss in asset values → breach of constraint → portfolio deleveraging Portion of the fund liquidated in response to price moves: � n � α i S i � V k � V k � ǫ i k f − f + k +1 V 0 V 0 V 0 i =1 Assumption of proportional liquidation and linear price impact ( D i market depth of asset i ) Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Discrete-time setting: price dynamics New price dynamics due to price impact from portfolio liquidation: � � �� � V k � n S i k +1 − S i α j S j V k � − α i k ǫ i V 0 ǫ j k = f − f V 0 + k +1 k +1 S i D i V 0 ���� k j =1 Fundamental return � �� � Feedback from liquidation When the fund value does not drop below a threshold → no fund liquidation and no price impact. In stress scenarios, the deleveraging process may generate significant deviations of price dynamics from fundamentals. Rama Cont and Lakshithe Wagalath Risk Management for Whales
Introduction and motivation Modeling liquidation losses and liquidation-adjusted VaR Taming the Whale Conclusion Continuous-time price dynamics Theorem 1 When ∆ t goes to 0 , the multi-period model converges weakly to a continuous-time limit described by a multi-asset diffusion model: dP i t = µ i t dt + ( σ t dW t ) i 1 ≤ i ≤ n P i t where the drift µ i t and the instantaneous covariance c t = σ t σ ′ t are given by � V t � < π t , Σ π t > t = m i + Λ i µ i 2 f ′′ V 2 V 0 0 � V s � V s � � � t = Σ + 1 s Σ + Σ π s Λ ′ ] + 1 � ( f ′ ) 2 c t = σ t σ ′ V 0 f ′ [Λ π ′ π ′ s Σ π s ΛΛ ′ V 2 V 0 V 0 0 � ′ is the dollar allocation of the portfolio and � α 1 P 1 t , ..., α n P n where π t = t � � ′ Λ = α 1 D 1 , ..., α n represents the positions expressed as a fraction of market D n depth. Rama Cont and Lakshithe Wagalath Risk Management for Whales
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