Mathematical Finance and Systemic Risk

Transcription

1 Mahemaical Finance and George Papanicolaou Sanford Universiy IPAM h Anniversary Meeing November 2, 2 G. Papanicolaou, IPAM h /26

2 Ouline: Imporan issues: Why is modern finance mahemaical? Wha is sysemic risk?. A brief hisorical overview of mahemaical finance 2. The role of volailiy 3. Curren research direcions in mahemaical finance 4. Mean field models of sysemic risk 5. Large deviaions for mean field models 6. Concluding remarks G. Papanicolaou, IPAM h 2/26

3 A hisorical review 9: Bachelier defends hesis on Theory of Speculaion and invens Brownian moion. Probabiliy heory eners finance : Grea depression, conracion of financial markes. Qualiaive Macroeconomic Theory dominaes : Black-Scholes-Meron heory of opions pricing. Chicago Board of Opions Exchange opens. Opions become financial insrumens wih which risk (in currency exchange) can be managed. Migraion of mahemaicians and physiciss o Invesemn Banking : Golden age of financial mahemaics. Banking, invesmen and finance become a quaniaive and daa-driven indusry. Thousands of scieniss, engineers and mahemaicians ener he field. More ha 3 op universiies around he world esablish degree programs in Financial Mahemaics and Engineering. Research publicaions on mahemaical problems in invesmen and finance increase dramaically. 28-2: Criical reorienaion of research prioriies in quaniaive finance wih emphasis on risk. Acceleraion of mahemaizaion of invesmen and finance. G. Papanicolaou, IPAM h 3/26

4 The S&P 5 Index of US Equiies (Fig: 26-) S&P 5: An index ha measures he value of he op 5 American companies by capializaion. Long erm (5 year) saisics: Growh rae 2% annualized (bu negaive in las years), Realized Volailiy 2 3%. Derivaives: Financial conracs (insrumens) ha are raded and whose value depends on anoher raded insrumen Index Opions (pus and calls): Derivaives ha are highly liquid and can be used o conrol he risk of holding equiies G. Papanicolaou, IPAM h 4/26

5 The volailiy index (VIX) (Fig: 99-2) Why is volailiy imporan? Because risk managemen and he insrumens used for i, such as opions, depend essenially on volailiy. They do no depend essenially on he gain or he loss of he value of he equiy. Volailiy is an indicaor of marke healh, like he emperaure of he human body. VIX is a special, very imporan volailiy index derived implicily from opions. Volailiy generaion and liquidiy. G. Papanicolaou, IPAM h 5/26

6 VIX and S&P 5 When markes are healhy volailiy is low and liquidiy high When volailiy is high markes can be unsable VIX and S&P 5 are srongly negaively correlaed G. Papanicolaou, IPAM h 6/26

7 Some mahemaical research rends in finance Modeling flucuaions of prices (sared wih Bachelier; complexiy increases) Pricing financial conracs (commodiies, fuures, insurance, ec) Pricing opions (Black-Scholes heory) Managing invesmen porfolios Pricing bonds (credi insrumens) Pricing credi defaul swaps (insurance agains defaul and loss of value of bonds) G. Papanicolaou, IPAM h 7/26

8 Wha is sysemic risk and how o model i Consider an evolving sysem wih a large number of iner-conneced componens, each of which can be in a normal sae or in a failed sae. We wan o sudy he probabiliy of overall failure of he sysem, ha is, is sysemic risk. There are hree effecs ha we wan o model and ha conribue o he behavior of sysemic risk: The inrinsic sabiliy of each componen The srengh of exernal random perurbaions o he sysem The degree of iner-connecedness or cooperaion beween componens G. Papanicolaou, IPAM h 8/26

9 Possible applicaions Engineering sysems wih a large number of ineraced pars. Componens can fail bu he sysem fails only when a large number of componens fail simulaneously. Power disribuion sysems. Individual componens of he sysem are calibraed o wihsand flucuaions in demand by sharing loads. Bu sharing also increases he probabiliy of an overall failure. Banking sysems. Banks cooperae and by spreading he risk of credi shocks beween hem can operae wih less resricive individual risk policies (capial reserves). However, his increases he risk ha hey may all fail, ha is, he sysemic risk. G. Papanicolaou, IPAM h 9/26

10 A basic, bisable mean-field model The risk variable x j () of each componen j =,...,N, saisfies he SDEs dx j () = h y V ( x j () ) d + θ ( x() x j () ) d + σdw j () Here V (y) is a poenial wih wo sable saes. Wihou noise, he individual risk x j () says in hese saes, one of which denoes he normal sae and he oher he failed sae. A ypical bu no unique choice of V (y) is V (y) = 4 y4 + 2 y2. The parameer h conrols he probiliy wih which x j jumps from one sae o he oher. { } N w j are independen Brownian moions and σ is heir srengh. j= x = N N i= x i is he mean-field, which we ake (define) as he sysemic risk, and θ ( x x j ), wih θ >, is he cooperaive ineracion parameer. G. Papanicolaou, IPAM h /26

11 Why his model? The hree parameers, h, σ and θ conrol he hree effecs we wan o sudy: (i) Inrinsic sabiliy, (ii) random perurbaions, and (iii) he degree of cooperaion, respecively. Why mean field ineracion? Because i is perhaps he simples ineracion ha models cooperaive behavior. And i can be generalized o include diversiy, as explained laer, as well as oher more complex ineracions such as hierarchical ones. Connecion wih QMU (Quanificaion of Margins of Uncerainy): In a cooperaing or iner-conneced sysem, individual componens can be operaing closer o heir margin of failure, as hey can benefi from he sabiliy of he res of he sysem. This, however, reduces he overall margin of uncerainy, ha is, increases he sysemic risk. This is one of he main resuls of he analysis. G. Papanicolaou, IPAM h /26

12 Schemaic for he risk of one componen - + Saus G. Papanicolaou, IPAM h 2/26

13 The large sysem limi (N ) The empirical risk densiy X N () := N N j= δ x j ()( ) converges weakly, in probabiliy, as N o u (, ), he soluion of he nonlinear Fokker-Planck equaion : u = h y [U(y)u] + 2 σ2 2 y 2 u θ y U(y) = d dy V (y). {[ ] } yu(, dy) y u Exisence of bi-sable equilibrium saes in he limi: Given θ and h, here exiss a criical value σ c such ha u has one sable equilibrium for σ σ c, and has wo sable equilibria for σ < σ c. Simplificaion: If h is small, hen u have he bi-sable saes if and only if 3σ 2 < 2θ. Explanaion: The condiion 2θ 3σ 2 means ha he sysem ineracion dominaes he noise, and herefore componen cooperaion dominaes. In conras, wih srong noise forces, all x j s ac more as independen componens and roughly one half are in one sae and he res are in he oher sae. G. Papanicolaou, IPAM h 3/26

14 Simulaion - Impac of increasing θ (sabilizing) and σ (desabilizing) N=, h=., σ=, θ=5 dm()/dξ=.4, 3σ 2 /2θ=.3 N=, h=., σ=, θ= dm()/dξ=.24, 3σ 2 /2θ= N=, h=., σ=, θ= dm()/dξ=.894, 3σ 2 /2θ= N=, h=., σ=, θ=6 dm()/dξ=.26, 3σ 2 /2θ=.25 2 N=, h=., σ=2, θ=6 dm()/dξ=.2, 3σ 2 /2θ= 2 N=, h=., σ=4, θ=6 dm()/dξ=.9572, 3σ 2 /2θ= G. Papanicolaou, IPAM h 4/26

15 Simulaion 2 - Impac of increasing h(sabilizing) and N 2 N=, h=.2, σ=, θ= dm()/dξ=.7, 3σ 2 /2θ= N=, h=., σ=, θ= dm()/dξ=.86, 3σ 2 /2θ= N=, h=, σ=, θ= dm()/dξ=.44, 3σ 2 /2θ= N=4, h=., σ=, θ= dm()/dξ=.86, 3σ 2 /2θ= N=8, h=., σ=, θ= dm()/dξ=.86, 3σ 2 /2θ= N=2, h=., σ=, θ= dm()/dξ=.86, 3σ 2 /2θ= G. Papanicolaou, IPAM h 5/26

16 Transiion o failure and is probabiliy For σ < σ c (or 3σ 2 < 2θ for small h), he value of he sysemic risk remains around x ±ξ b. Because of he randomness, he ransiion in (, T) (or he sysem collapse in he risk sense): x () ξ b, x (T) ξ b happens wih nonzero probabiliy. Quesion: Wha is he probabiliy of his happening? G. Papanicolaou, IPAM h 6/26

17 Large deviaion principle The asympoic probabiliies, for large N, can be compued hrough a large deviaion principle. [Dawson & Garner, 987] M (R) is he space of probabiliy measures on R, and A is a se of M (R)-valued coninuous process on [, T]. Then where I h (φ) = T 2σ 2 ( ) P(X N A) exp N inf I h (φ) φ A f: φ,( y f sup ) 2 φ L φ φ hm φ, f 2 ( φ, y f) 2 d L ψ φ = 2 σ2 2 y 2 φ θ {[ ] } yψ(, dy) y φ y M φ = [( y 3 y ) φ ]. y To compue he ransiion probabiliy, A is he se of all coninuous ransiion pahs: A = { φ : [, T] M (R),E φ() X = ξ b, E φ(t) X = ξ b }. G. Papanicolaou, IPAM h 7/26

18 Small h (inrinsic sabiliy) analysis Why consider small h? The problem is nonlinear and infinie-dimensional, and is generally inracable. If h is small hen he problem can be reduced o a finie-dimensional problem. For V (y) = 4 y4 + 2 y2, i is a four-dimension problem. Numerical simulaions show ha he probabiliy of ransiions is almos zero even for moderae h. When h =, he sysemic risk is effecively a Brownian moion: x () = σ N w(). We expec, herefore, ha for small h, a ransiion pah of empirical densiies is Gaussian wih a small perurbaion: [ A = φ = p + hq : p(,y) = (y a()) 2 ] 2πb 2 exp () 2b 2,a() = ξ () b,a(t) = ξ b. G. Papanicolaou, IPAM h 8/26

19 Small h analysis (wih J. Garnier and T-W Yang) For h small, he large deviaion problem is solvable approximaely by a Chapman-Enskog expansion, and he ransiion probabiliy is P (X N A) ( [ exp N ) σ 2 2 ( 3 σ2 + 24h T 2θ σ 2 ( σ 2 Here are some commens of his resul: 2θ ) 2 ) ]) ( 2 σ2 + O(h 2 ). 2θ A large sysem is more sable han a small sysem. In he long run (T large), a ransiion will happen. Increase of he inrinsic sabilizaion parameer h reduces sysemic risk. Mean ransiion imes are simply relaed o ransiion probabiliies in his approximaion (use for his M. Williams 82) G. Papanicolaou, IPAM h 9/26

20 Modeling of diversiy in cooperaive behavior The cooperaive behavior of componens can be differen across groups: dx j () = hκ y V (x j ())d + σdw j () + θ j ( x() x j ()) d. The componens are pariioned ino K groups. In group k, he componens have cooperaive parameer Θ k. In he limi N he empirical densiies of each groups converge o he soluion of he join Fokker-Planck equaions: u = hκ [ ] 2 U (y)u + σ2 y 2 y 2 u Θ K y ρ y k u k (,dy) y u k=. u K = hκ [ ] 2 U (y)uk + σ2 y 2 y 2 u K Θ K K y ρ y k u k (,dy) y u K k= where ρ k N is he size of group k. G. Papanicolaou, IPAM h 2/26

21 Impac of componen diversiy on he sysemic risk Why is he diversiy ineresing? The model is more realisic and more widely applicable. Diversiy significanly affecs he sysem sabiliy by reducing i. Impac from he diversiy: Analyical and numerical sudies show ha even wih he same parameers and wih {θ j } whose average equals θ he sysem sill changes significanly. G. Papanicolaou, IPAM h 2/26

22 Simulaion 3 - Impac of diversiy, change of Θ k and ρ k N=, h=., σ=, Θ=[;;] ρ=[.33;.34;.33], sdev(θ)/mean(θ)= dm()/dξ=.86, Σ i (ρ i /Θ i )(3σ 2 /2Θ i )=.85 N=, h=., σ=, Θ=[6;;4] ρ=[.33;.34;.33], sdev(θ)/mean(θ)=.3266 dm()/dξ=.92, Σ i (ρ i /Θ i )(3σ 2 /2Θ i )=.996 N=, h=., σ=, Θ=[2;;8] ρ=[.33;.34;.33], sdev(θ)/mean(θ)=.6532 dm()/dξ=.89, Σ i (ρ i /Θ i )(3σ 2 /2Θ i )= N=, h=., σ=, Θ=[5;;5] ρ=[.;.8;.], sdev(θ)/mean(θ)= dm()/dξ=.89, Σ i (ρ i /Θ i )(3σ 2 /2Θ i )=.88 N=, h=., σ=, Θ=[5;;5] ρ=[.3;.4;.3], sdev(θ)/mean(θ)= dm()/dξ=.95, Σ i (ρ i /Θ i )(3σ 2 /2Θ i )=.94 N=, h=., σ=, Θ=[5;;5] ρ=[.45;.;.45], sdev(θ)/mean(θ)= dm()/dξ=.99, Σ i (ρ i /Θ i )(3σ 2 /2Θ i )= G. Papanicolaou, IPAM h 22/26

23 Analysis in he diversiy case The criical value of he sysem flucuaion is lower: of he homogeneous case is The criical value σ homo c 2 3 /2 K ρ K k 3ρ / k Θ k= k 2Θ k= 2. k If θ = K k= ρ kθ k, hen σ homo c σ div c always. Sysem wih diversiy have larger ransiion probabiliies: θ, and σdiv c We show ha when h is zero, he average of θ j is θ, and he diversiy of θ j is small, hen he sysem has a higher ransiion probabiliy. Mahemaically, if h =, Θ k = θ( + δα k ) wih δ, and K k= ρ kα k =, hen ( P X div ) N A ( ) exp N σ 2 2 3σ2 2δ 2 ( ρ T 2θ k α 2 3σ 2 k 2θ + T ( e θs) 2 ) ds T. k is G. Papanicolaou, IPAM h 23/26

24 A Hierarchical Model of Here we consider a hierarchical model wih a cenral agen: dx = σ dw h U (x )d θ ( x N n ) x j d j= dx j = σdw j hu(x j )d θ ( x j x ) d, j =,...,N X models he cenral sable agen. I is inrinsically sable (h > ), and no subjeced o exernal flucuaions (σ = ). I can be desabilized hrough a mean field ineracion wih he oher agens. X j, j =,...,N model individual agens ha are subjeced o exernal flucuaions. They are (h > ) or are no (h = ) inrinsically sable. They are sabilized wih an ineracion wih he sable agen X. G. Papanicolaou, IPAM h 24/26

25 On-going and fuure work, analysis Compue he ransiion probabiliy of he diversiy case for small h. Show ha i is uncondiionally rue ha sysems wih diversiy are less sable han homogeneous ones. Sudy models wih diversiies everywhere: The mos general case (when he limi exiss) is ha V, σ and θ in each group can be differen, i.e. if x j is in he group k, hen he SDE is dx j () = h y V k (x j ()) d + σ k dw j () + Θ k ( x () x j ())d. Combine hierarchical and diversiy models. How general should our models be o deal wih realisic problems? G. Papanicolaou, IPAM h 25/26

26 Research direcions for sysemic risk Inerconneced financial sysems have many sources of insabiliy. Insabiliies are everywhere in he financial world. They are a consequence of he dual marke variables: volailiy and liquidiy. Using he analysis as a guide, design imporance sampling algorihms for compuing efficienly (very) small sysemic failure probabiliies Can dynamic conrol mechanisms reduce sysemic risk? Wha if he conrollers mus rely on imperfec informaion? Can ransacion fees (Tobin ax) sabilize markes in he sysemic risk sense? (I is no known if hey increase or decrease volailiy). Is saisical arbirage desabilizing? (Hedging derivaives increases volailiy bu porfolio opimizaion decreases i). G. Papanicolaou, IPAM h 26/26