Jens Carsten Jackwert Marco Menner University of Konstanz jens.jackwert@ni-konstanz.e ttp://www.wiwi.ni-konstanz.e/jackwert/
2 Motivation State prices q pricing kernel m pysical probabilities f Normally only one qantity can be obtaine from te oter two Ross 205: A teorem, tat allows to etermine te pricing kernel an te pysical probabilities from state prices To apply te teorem, frter an severe assmptions are reqire Or goal: Use S&P 500 options ata, recover pysical probabilities wit te Ross recovery teorem, an compare tem to te tre pysical probabilities
3 Two State, One Perio Example Example: We ave two states state an state 2 an transition state prices q ij of moving from state i to state j in one time step q 0.9 q 2 q 0. 05 2 e.g.: 0.02 q 22 0.95
4 Ross Recovery Assmptions: Aing Sfficient Strctre Pysical transition probabilities can be linke to te transition state prices by te pricing kernel m: q ij f ij ij q ij m ij f ij Assmption : Transition state prices 0 for all i an j q ij Assmption 2: Transition state prices are time inepenent q ij Assmption 3: Te pricing kernel is transition inepenent: m ij j i f ij j q ij i
5 Soltion Trns Ot to be an Eigenvale Problem Pysical transition probabilities of moving from one specific state to any oter state ave to sm p to : Tis reslts in an eigenvale problem: f 2 f f 22 f 2 0.9 2 0.05 2 0.95 2 2 0.02 2 2 22 2 2 q q q q 2 2 0.95 0.02 0.05 0.9 f ij
6 Soltion of te Two State, One Perio Example For te largest absolte eigenvale Perron Frobenis teorem: 0.608 2 0.794 0.965 Tis allows to solve for te pricing kernel an te pysical transition probabilities: m m 2 m m 2 22 0.965.26 0.739 0.965 an f f 2 f f 2 22 0.932 0.06 0.068 0.984 Ross assmptions ensre tat te pricing kernel is always positive an tat tis positive soltion is niqe
7 First Observations an Problems Te pricing kernel is proportional to te i in tis example, tat is ecreasing in states an negatively correlate to te row sms: low interest rate ig kernel ts, one can pick te interest rates an etermine te kernel econometrically, te interest rates are poorly ientifie conitional pricing kernels are lower for low states vice versa Te iscont factor tens to be close to te igest row sm econometrically, te largest interest rate is poorly ientifie Assmption of time inepenent transition state prices is not realistic an row sms nee to iffer else obtains risk netrality
8 Literatre: Extensions Martin an Ross 203 sow tat can be obtaine via te long en of te yiel crve an te eigenvector in te recovery teorem can be inferre from te time series beavior of te long bon repeate applications of Q/ price te bon an ave row sms wic coincie wit te eigenvector an te inverse pricing kernel we are not yet writing tat empirical paper! Carr an Y 202 exten te teorem to continos time by assming nerlying stocastic processes. Teir approac works for bone iffsion processes Walen 204 investigates problems in an extension of te teorem to continos time if te iffsion process is nbone. He fins tat recovery is still possible for many of tese processes
9 Literatre: Analysis Arino, Hitema an Lwig 204 apply a neral network approac to get te state transition matrix. Tey investigate mean, variance, an iger moments of te recovere istribtion an evelop a traing strategy base on tat. No ceck tat actal retrns come from tat istribtion Tran an Xia 204 sow in a simlation tat te reslt from Ross recovery epens igly on te imension of te state transition matrix e.g. combine tre states: row sms will cange, ts te pricing kernel Borovicka, Hansen an Sceinkman 204 sow tat te assmptions in te Ross recovery teorem leas to a misspecifie stocastic iscont factor
0 Te Proof of te Ping is in te Eating Martin an Ross 203 alreay int at te ifficlty of fining goo state transition matrices. Tran an Xia 204 arge mc te same We want to take Ross 205 at face vale: fit a state transition matrix to ata on S&P 500 options se Ross recovery to fin te pysical transition probabilities se te Berkowitz 200 to test if ftre realize retrns o inee come from te recovere pysical transition probabilities to o so, we work ot te cmlative probability of te retrns ner te recovere istribtions is ii niform ner te Nll transform to a normal istribtion ner te Nll an test for normality mean zero, nit variance, no serial correlation Ross recovery oes NOT work in te ata
Data From OptionMetrics S&P 500 Eropean pt an call options wit matrities p to one year S&P 500 inex vales Risk free rate Daily ata from Janary 996 Jly 203 Filter ata: remove arbitrage violations an in te money options Solve for te implie ivien yiels by pt call parity
2 Fin Implie Volatility From Options Prices Transform observe option prices into implie volatilities Inter /extrapolate implie volatilities: smoot an bi ask compliant
3 Interpolate Implie Volatilities Transforme Back into Call Prices
4 From Option Prices to Spot State Prices For All Different Matrities Get state prices qk for every call price C wit strike K Breeen an Litzenberger 978: Note, tat tis reslts in spot state prices, not transition state prices 2 2 2 ) ( ) ( 2 ) ( ) ( ) ( K C K C K C K C K K q
5 Observe Implie Volatilities: 3-08-203
6 Observe an Interpolate Implie Volatilities: 3-08-203
7 Observe an Interpolate Call Prices: 3-08-203
8 (Spot) State Price Srface: 3-08-203
9 Transition State Price Srface: 3-08-203 Transition states prices base on te spot state price srface Minimize te istance of spot state prices at matrity t an spot prices at matrity t mltiplie by te transition state prices: q q ( t ) ( t) j i i Reqire tat transition state prices are always positive Transition state prices are not nimoal an ts o not make mc sense: econometrically, te transition state prices are not well ientifie bt move aron qite freely q ij
20 Transition State Price Srface: 3-08-203
2 Te Recovere Pricing Kernel Moves Aron Erratically as te Row Sms Vary
22 Pysical Transition Probabilities Base on Ross Recovery Inerit Mlti-Moalities
23 Pysical (Spot) Density: 3-08-203 is Srprisingly Well-Beave: Convoltion!
24 Te Berkowitz (200) Test: Do Observations Come From a Particlar Distribtion? Do te empirical pysical spot probabilities come from te recovere pysical spot probability istribtion? Work ot cmlative probability ner te Nll of eac retrn Tose transforme vales sol be ii niformly istribte Transform again to a stanar normal istribtion ner te Nll Test te twice transforme ata for normality Also works for small sample sizes
25 Te Berkowitz Test: Transform Retrns to Cmlative Probabilities Uner te Nll
26 Te Berkowitz Test: Transform te ii Vales (Uner te Nll) to Normality
27 Te Berkowitz Test: Te Transforme Recovere Distribtion Does not Work Well 7 ay pysical probabilities in 08/203 transforme into stanar normal: recovere green, 5 years istorical re, an Nll ble istribtions
28 Te Berkowitz Test: First Reslts an Interpretation H0: Ftre realize retrns are rawn from te recovere istribtion Test if twice transforme empirical istribtion is a stanar normal as it sol be ner te Nll LR test For Ross recovery: p vale0 : Rejection For 5 year istorical istribtion: p vale0 : Rejection Ross Recovery as well as istorical ata fail to forecast ftre istribtion of retrns Room for improvement: better Q matrix, ifferent sample perios
29 Aitional Finings I Strctre of Q: i t row of Q represents state prices of receiving $ in te ftre given te crrent state is state i. Te i t row sm of Q sol eqal te iscont factor in te i t state Implications: Te mass in Q sol be centere aron te main iagonal. Te more te mass in te rows of Q is centere aron te respective state, te more te eigenvale tens to move towars te largest row sm Interpretation: Te more state prices clster along te main iagonal, te iger te eigenvale an ts, te smaller te average risk free rate
30 Aitional Finings II Simlation: Investigate te correlation of te row sm in Q an te eigenvector in qestion. Q contains niform ranom nmbers Implication: Te risk free rate nees to be state epenent an negatively correlate wit te state variable e.g. te S&P 500 inex to obtain a monotonically ecreasing pricing kernel Dimension n, were Q is n x n Mean of te correlations 5 0.998 0.0002 0 0.997 0.0006 25 0.994 0.008 50 0.988 0.0072 00 0.977 0.084 Stanar eviation of te correlations
3 Aitional Finings III Frter reslts relate to te pricing kernel: Te eigenvector inverse pricing kernel becomes more monotone, if te variance in te rows of Q is low. So, te more clstering among te main iagonal appears in Q, te less variable is te pricing kernel State epenence: If te risk free rate is not state epenent, te row sms in Q ave all to be eqal. Ten te eigenvector pricing kernel is constant an te pysical transition probabilities eqal te risk netral transition probabilities Black Scoles economy: Te state epenence of te risk free rate contraicts te Black Scoles framework. In Black Scoles, risk netral probabilities o not necessarily eqal pysical probabilities even tog te risk free rate is te same in all states
32 Sensitivity Analysis of te Pricing Kernel: Example Wit 3 States; Socks of 3% Vol Given is a volatility srface an state epenent interest rates Stress te volatility srface wit ranom, niform socks
33 Alternative Recovery Assme tat te strctre of te pricing kernel oes not epen on te crrent state: m ij m j i Ten, we obtain te eqation system: q q q 2 3 m 0 m m 2 q 2 q q 2 m 2 m 0 m q 3 q 3 q m 3 2 m m 0 f f f f 2 f 2 f 2 f 3 f 3 f 3 By setting te vales of te pricing kernel at te borers constant m m an m, we can make te system niqely solvable 2 2 m
34 Frter Ieas Use te long ate bon ynamics to fin te pricing kernel Use bon options to aitionally fit te interest rate Use options on bons an stocks to etermine teir correlation an get insigts into state epenence of te risk free rate Use interest rates implie by bons an fit Q wit te smootest pricing kernel possible Apply te teorem to te joint istribtion of S&P 500 an VIX, see Jackwert an Vilkov 204
35 Conclsion We take Ross recovery to te task an recover te pysical spot probabilities base on state transition probabilities wic price observe S&P 500 inex options Ftre realize retrns are not rawn from tis pysical spot istribtion Te Berkowitz 200 test strongly rejects sc Nll ypotesis Te searc is on for a version of Ross recovery tat works if any!
36 References Arino, F., Hitema, R. an Lwig, M. - An Empirical Analysis of te Ross Recovery Teorem, Working Paper, (204) Borovicka, J., Hansen, L., P. an Sceinkman, J. - Misspecie Recovery, Working Paper, (204) Carr, P. an Y, J. - Risk, Retrn, an Ross Recovery Te Jornal of Derivatives, 38-59, Jan (202) Jackwert, J.,C. an Vilkov, G. - Asymmetric Volatility Risk: Evience from Option Markets, Working Paper (204) Martin, I. an Ross, M. - Te Long Bon, Working Paper (203) Ross, S. - Te Recovery Teorem, Jornal of Finance fortcoming (205) Tran, N., K. an Xia, S. - Specifie Recovery, Working Paper (204) Walen, J. - Recovery wit Unbone Diffsion Processes, Working Paper (204)