Systemic Risk Illustrated

Transcription

1 Sysemic Risk Illusraed Jean-Pierre Fouque Li-Hsien Sun March 2, 22 Absrac We sudy he behavior of diffusions coupled hrough heir drifs in a way ha each componen mean-revers o he mean of he ensemble. In paricular, we are ineresed in he number of componens reaching a defaul level in a given ime. This coupling creaes sabiliy of he sysem in he sense ha here is a large probabiliy of nearly no defaul as opposed o he case of independen Brownian moions for which he disribuion of number of defauls is of binomial ype. However, we show ha his swarming behavior also creaes a small probabiliy ha a large number of componens defaul corresponding o a sysemic risk even. The goal of his work is o illusrae sysemic risk wih a oy model of lending and borrowing banks, using mean-field limi and large deviaion esimaes for a simple linear model. Inroducion In he oy model discussed below, he diffusion processes Y (i, i =,..., represen he log-moneary reserves of banks possibly lending and borrowing o each oher. The sysem is driven by independen sandard Brownian moions W (i, i =,..., and sars a ime = from Y (i = y (i, i = Deparmen of Saisics & Applied Probabiliy, Universiy of California, Sana Barbara, CA 936-3, fouque@psa.ucsb.edu. Work suppored by SF gran DMS Deparmen of Saisics & Applied Probabiliy, Universiy of California, Sana Barbara, CA 936-3, sun@psa.ucsb.edu.

2 ,...,. For simpliciy and wihou loss of generaliy for he purpose of his paper, we assume ha he diffusion coefficiens are consan and idenical, denoed by σ >. In he case of no lending or borrowing, Y (i, i =,..., are independen and simply given by drifless Brownian moions: dy (i = σdw (i i =,...,. ( Our oy model of lending and borrowing consiss in inroducing an ineracion hrough drif erms of he form (Y (j Y (i represening he rae a which bank i borrows from or lends o bank j. In his case, he raes are proporional o he difference in log-moneary reserves. Our model is: dy (i = α (Y (j Y (i d + σdw (i, i =,...,, (2 where he overall rae of mean-reversion α/ has been normalized by he number of banks and we assume α >. oe ha in he case α =, he sysem (2 reduces o he independen sysem (. In he spiri of srucural models of defaul, we inroduce a defaul level η < and say ha bank i defauls by ime T if is log-moneary reserve reached he level η before ime T (noe ha in his simplified model, bank i says in he sysem unil ime T. Here, we wan o commen on he difference beween sysemic risk which we will discuss below and credi risk. In he laer case, Y (i denoes he log-value of a firm (or is sock price as a proxy for insance, and dependency beween firms can be creaed by inroducing a correlaion srucure beween he Brownian moions W (i s (dependency can also be creaed hrough volailiies, see [2], bu for he sake of his commen we assume ha volailiies remain consan and idenical. In pricing credi derivaives he drifs are imposed by risk-neuraliy and do no play a role in he correlaion of defauls. In he independen case, as in sysem (, and assuming symmery (same iniial value, he loss disribuion (disribuion of he number of defauls is simply binomial. In he correlaed cases, for reasonable level of correlaion, he shape of he loss disribuion is roughly preserved wih some skewness and faer-ail effecs. We will show ha he shape of he loss disribuion generaed by he coupled sysem (2 is very differen wih mainly a large mass near zero (sabiliy of he sysem and a small (bu presen mass in he ail near (sysemic risk. 2

3 In he following secion, we illusrae he sabiliy of sysem (2 by simulaions for various values of he mean-reversion rae α and we compare wih he independen case α = as in (. As expeced, he possibiliy for a bank o borrow money from oher banks wih larger moneary reserves creaes his sabiliy of he sysem. In Secion 3, we derive he mean-field limi of sysem (2 as he number of banks becomes large. In his limi, banks become independen and heir log-moneary reserves follow OU processes. Ineresingly, before aking his limi, we observe ha each componen mean-revers o a common Brownian moion wih a small diffusion of order /. We exploi his fac in Secion 4, o explain sysemic risk as he small-probabiliy even where his mean level reaches he defaul barrier, wih a ypically large number of componens following he mean and defauling. Moreover, his small probabiliy of sysemic risk is independen of he mean-reversion rae α so ha a large α corresponds o more sabiliy bu a he same ime o (or a he price of a larger sysemic even. 2 Sabiliy Illusraed by Simulaions We firs compare he coupled diffusions (2 o he independen case ( by looking a ypical rajecories. For simpliciy of our simulaion, we assume y (i =, i =,...,. Also, we choose he common parameers σ =, η =.7, and =, and we used he Euler scheme wih a ime-sep = 4, up o ime T =. In Figures, 2 and 3, we show a ypical realizaion of he rajecories wih α =, α =, and α = respecively. We see ha he rajecories generaed by (2 are more grouped han he ones generaed by (. This is he swarming or flocking effec more pronounced for a larger α. Consequenly, less (or almos no rajecories will reach he defaul level η, creaing sabiliy of he sysem. ex, we compare he loss disribuions for he coupled and independen cases. We compue hese loss disribuions by Mone Carlo mehod using 4 simulaions, and wih he same parameers as previously. In he independen case, he loss disribuion is Binomial(, p wih pa- 3

4 Figure : One realizaion of he rajecories of he coupled diffusions (2 wih α = (lef plo and rajecories of he independen Brownian moions ( (righ plo using he same Gaussian incremens. The solid horizonal line represens he defaul level η =.7. Figure 2: One realizaion of he rajecories of he coupled diffusions (2 (lef plo wih α = and rajecories of he independen Brownian moions ( (righ plo using he same Gaussian incremens. The solid horizonal line represens he defaul level η =.7. 4

5 Figure 3: One realizaion of he rajecories of he coupled diffusions (2 (lef plo wih α = and rajecories of he independen Brownian moions ( (righ plo using he same Gaussian incremens. The solid horizonal line represens he defaul level η =.7. rameer p given by ( p = IP min (σw η T ( η = 2Φ σ, T where Φ denoes he (, -cdf, and we used he disribuion of he minimum of a Brownian moion (see [3] for insance. Wih our choice of parameers, we have p.5 and herefore he corresponding loss disribuion is almos symmeric as can be seen on he lef panels (dashed lines in Figures 4, 5, and 6. Observe ha in he independen case, he loss disribuion does no depend on α, and herefore is he same on hese hree figures (up o he Mone Carlo error esimae. ex, we compare he loss disribuion generaed by our coupled sysem (2 for increasing values of α (solid lines, α =, α =, and α = in Figures 4, 5, and 6, respecively. We see ha increasing α, ha is he rae of borrowing and lending, pushes mos of he mass o zero defaul, in oher words, i improves he sabiliy of he sysem by keeping he diffusions near zero (away from defaul mos of he ime. However, we also see ha here 5

6 .25.2 prob of # of defaul prob of # of defaul # of defaul 6 8 # of defaul Figure 4: On he lef, we show plos of he loss disribuion for he coupled diffusions wih α = (solid line and for he independen Brownian moions (dashed line. The plos on he righ show he corresponding ail probabiliies. is small bu non-negligible probabiliy, ha almos all diffusions reach he defaul level. On he righ panels of Figures 4, 5, and 6 we zoom on his ail probabiliy. In fac, we will see in he nex secion ha his ail corresponds o he small probabiliy of he ensemble average reaching he defaul level, and o almos all diffusions following his average due o flocking for large α. 3 Mean-field Limi In order o undersand he behavior of he coupled sysem (2, we rewrie is dynamics as: dy (i = α = α [( (Y (j Y (i d + σdw (i Y (j Y (i ] d + σdw (i. (3 6

7 .5.2 prob of # of defaul prob of # of defaul # of defaul 6 8 # of defaul Figure 5: On he lef, we show plos of he loss disribuion for he coupled diffusions wih α = (solid line and for he independen Brownian moions (dashed line. The plos on he righ show he corresponding ail probabiliies..2 prob of # of defaul prob of # of defaul # of defaul 6 8 # of defaul Figure 6: On he lef, we show plos of he loss disribuion for he coupled diffusions wih α = (solid line and for he independen Brownian moions (dashed line. The plos on he righ show he corresponding ail probabiliies. 7

8 In oher words, he processes Y (i s are OUs mean-revering o he ensemble average. ex, we observe ha his ensemble average saisfies ( ( d Y (i σ = d W (i, and assuming for insance ha y (i =, i =,...,, we obain Y (i = σ W (i, (4 and consequenly dy (i [( σ = α W (j Y (i ] d + σdw (i. (5 oe ha in fac he ensemble average is disribued as a Brownian moion wih diffusion coefficien σ/. In he limi, he srong law of large numbers gives W (j a.s., and herefore, he processes Y (i s converge o independen OU processes wih long-run mean zero. In order o make his resul precise, one can solve (5 Y (i = σ W (j + σe α e αs dw (i s σ (e α e αs dw s (j, and derive ha Y (i converges o σe α eαs dw s (i which are independen OU processes. This is in fac a simple example of a mean-field limi and propagaion of chaos sudied in general in [4]. oe ha he disribuions of hiing imes for OU processes have been sudied in []. Le us denoe p = IP (τ T, 8

9 τ being he hiing ime of he defaul level for an OU process wih long-run mean zero, given by dy = αy d + σdw. In he ineresing regime where p λ >, obained as and η appropriaely, he loss disribuion converges o a Poisson disribuion wih parameer λ. In his sable regime, he mass is mainly concenraed on a small number of defauls. In he nex secion, we invesigae he small probabiliy of a large number of defauls when he defaul level η is fixed. 4 Large Deviaion and Sysemic Risk In his secion, we focus on he even where he ensemble average given by (4 reaches he defaul level. The probabiliy of his even is small (when becomes large, and is given by he heory of Large Deviaion. In our simple example, his probabiliy can be compued explicily as follows: IP ( min T ( σ W (i η ( = IP min W η T σ ( η = 2Φ σ, (6 T where W is a sandard Brownian moion. Therefore, using classical equivalen for he Gaussian cumulaive disribuion funcion, we obain ( ( lim σ log IP min W (i η = η2 T 2σ 2 T. (7 In oher words, for a large number of banks, he probabiliy ha he ensemble average reaches he defaul barrier is of order exp( η 2 /(2σ 2 T. Recalling (4, we idenify { min T ( σ as a sysemic even. Observe ha his even does no depend on α >, in oher words, increasing sabiliy by increasing α (ha is increasing he rae 9 Y (i η }

10 of borrowing and lending does no preven a sysemic even where a large number of banks defaul. In fac, once in his even, increasing α creaes even more defauls by flocking o defaul. This is illusraed in he Figure 6, where α = and he probabiliy of sysemic risk is roughly 3% (obained using formula (6. One could objec ha wih his definiion of a sysemic even, in fac, only one bank could defaul (far below he barrier and all he ohers be above he defaul barrier since only he average couns. Bu, his ype of even is easily seen o be of probabiliy of smaller order. Wha we ry o capure here, is he fac ha for large α, he Y (i s are close o each oher and once in he defaul even hey will all be a (or near he defaul level. 5 Conclusion We proposed a simple oy model of coupled diffusions o represen lending and borrowing beween banks. We show ha, as expeced, his aciviy sabilizes he sysem in he sense ha i decreases he number of defauls. Indeed, and naively, banks in difficuly can be saved by borrowing from ohers. In fac, he model illusraes he fac ha sabiliy increases as he rae of borrowing and lending increases. I shows also ha his coupling hrough he drifs is very differen from correlaion hrough he driving Brownian moions or volailiies as i is he case in he srucural approach for credi risk (see for insance [2]. This can be seen by comparing loss disribuions as we did in Secion 2. In he laer case, he loss disribuion is shaped as a binomial while in he former case, i is bimodal wih a large mass on he lef on small numbers of defauls and a small mass on he righ on very large numbers of defauls. This las observaion is explained hrough he mean-field limi of he sysem (for large number of banks combined wih a large deviaion argumen. The model is rich enough o exhibi his propery and simple enough o be racable. In paricular, he mean-field limi is easy o derive. The diffusions mean-rever o he average of he ensemble, and his average converges, as he number of banks becomes large, o a level away from he defaul level. Tha explains he sabilizaion of he sysem. However, here is a small probabiliy, compued explicily in our model, ha he average of he ensemble reaches he defaul level. Combined wih he flocking behavior ( everybody follows everybody, his leads o a sysemic even where almos all defaul, in paricular when he rae of borrowing and lending is large.

11 To summarize, our simple model shows ha lending and borrowing improves sabiliy bu also conribues o sysemic risk. We have quanified his behavior and idenified he crucial role played by he rae of borrowing and lending. References [] L. Alili, P. Paie, and J.L. Pedersen. Represenaions of he firs hiing ime densiy of an ornsein-uhlenbeck process. Sochasic Models, 2(4:967 98, 25. [2] J.-P. Fouque, B.C. Wignall, and Zhou X. Firs passage model under sochasic volailiy. Journal of Compuaional Finance, (3:43 78, spring 28. [3] I. Karazas and S. Shreve. Brownian Moion and Sochasic Calculus Second Ediion. Springer, 2. [4] A.S. Szniman. Topics in propagaion of chaos. Ecole d Eé de Probabiliés de Sain-Flour XIXX989, pages 65 25, 99.