{ In general, we are presented with a quadratic function of a random vector X

Transcription

1 Quadratc VAR odel Mchael Carter à Prelnares Introducton Suppose we wsh to quantfy the value-at-rsk of a Japanese etals tradng fr that has exposure to forward and opton postons n platnu. Soe of the postons are denonated n USDs. We dentfy three key rsk factors æ spot prce of platnu n yen (X ) æ a representatve pled volatlty of platnu (X ) æ spot JPY/USD exchange rate (X 3 ) and assue that X = HX, X, X 3 L s norally dstrbuted NH, SL wth =(53.50, 0.670,07.80) and y S = k z { We value the portfolo usng applcable forward and opton prcng forulas, and quadratcally approxate ths as Y = a + D T X + X T G X Our obectve s to estate certan quantles of Y. In general, we are presented wth a quadratc functon of a rando vector X Y = a + D T X + X T G X where X ~ N H, S L. Ths ght arse as above fro a delta-gaa approxaton to a VAR easure of a portfolo contanng dervatves, or soe alternatve approxaton. Then, as we wll show, Y can be expressed as a lnear cobnaton of ndependent ch-squared and noral rando varables. Based on ths representaton, we can calculate the oents and the dstrbuton functon of Y. For exposton, let us start wth a nuercally spler specfc exaple. Suppose that X s ultvarate noral wth ean and varance µ = 8,, 0<; Σ = 0 y 0 z ; k 5 {

2 VARQuadratc.nb and that Y s gven by wth Y = a + D T X + X T G X a = ; = 88, 3, <; Γ = y z ; k { In other words, Y s deterned by the followng quadratc functon of correlated noral rando varables. Y = + 8 X + 3 X + 3 X + X X + 6 X - X 3-6 X X 3 - X X X 3 Defne new rando varables as follows Z = X -, Z = X - + HX - L - HX 3-3 L, Z 3 = - HX - L + HX 3-3 L The Z are ndependent standard noral rando varables. Exercse: Verfy that the Z, Z, Z 3 are ndependent standard noral rando varables. The rando vector Z = HZ, Z, Z 3 L s defned by the transforaton where Z = A HX - L 0 0 y A = - z k 0 - { Z has ean 0 and varance-convarance atrx A S A T = I. Solvng for X gves X = A - Z + Substtutng n the equaton defnng Y and splfyng, we have Y = + 8 X + 3 X + 3 X + X X + 6 X - X 3-6 X X 3 - X X X 3 = Z + Z + 3 Z + 6 Z 3 By "copletng the square" for Z, we can express ths as Y = Z + 3 HZ + 4 Z L + 6 Z 3 = Z + 3 HZ + L Z 3 = Z + 3 HZ + L + 6 Z 3

3 VARQuadratc.nb 3 We have expressed Y as a lnear cobnaton of 3 ndependent rando varables: Z ~ c H, 0L, Z ~ c H, 4L, Z 3 ~ NH0, L. In atheatcal ters, we have acheved two thngs. æ We have expressed Y n ters of ndependent standard noral rando varables Z (Cholesky decoposton). æ We have dagonalzed the quadratc for wth respect to these varables so that there are no crossters Z Z (Prncpal axs theore). The transforaton A s the coposton of these two steps. We now exane the condtons requred for ths n general. Suppose that Y s a quadratc functon of rando vector X Y = a + D T X + X T G X where X ~ N H, S L. Wthout loss of generalty, we can assue that = 0, snce any ean effect can be ncorporated nto a new constant ter a. Further, we can assue that S s postve defnte. There exsts a lower trangular atrx H such that S =HH T and X = H Z è where Z è s standard noral. Ths s known as the Cholesky decoposton. Substtutng where Y = a + D T H Z è + HH Z è L T G H Z è = a +D T Z è + Z è T GZ è D = H T D and G = H T G H Snce G s syetrc, so s G and there exsts an orthogonal atrx P such that P T GP = L or G = P L P T where L s a dagonal atrx contanng the egenvalues of G (Spectral theore). Therefore where Y = a +D T Z è + Z è T GZ è = a +D T PP T Z è + Z è T P L P T Z è = a + B T Z + Z T L Z B =P T D = P T H T D and Z = P T Z è Snce P s orthogonal, Z = HZ, Z,,Z L s also a vector of ndependent standard noral rando varables. Therefore, Y can be expressed as a lnear cobnaton of noral and ch-squared rando varables Y = a + B T Z + Z T L Z = a + Hb Z + l Z L = We suarze n the followng theore. Theore Suppose that Y s a quadratc functon of rando vector X

4 4 VARQuadratc.nb Y = a + D T X + X T G X where X ~ N H, S L, and S s postve defnte. Then there exsts a lnear transforaton A such that X = A - Z + and Y = a + B T Z + Z T L Z where L s a dagonal atrx and Z,Z,,Z are ndependent standard noral rando varables. Consequently Y can be wrtten as Y = a + Hb Z + l Z L = Ths theore shows that Y s a lnear cobnaton of ndependent noral and ch-square rando varables or non-central ch-square rando varables (see copleentary lecture note Quadratc functons of noral rando varables). Consequently, the characterstc functon of Y can be readly deterned, fro whch the exact dstrbuton functon can be calculated. Alternatvely, a nuber of coputatonally easer approxatons for the quantles are avalable, ncludng æ Cornsh-Fsher expanson æ saddlepont approxaton Both of these approxatons are based upon the cuulant generatng functon. The dstrbuton of Y Our frst goal s to deterne the oent generatng functon, fro whch we can derve both the cuulant generatng functon and the characterstc functon. Fro the latter, we can obtan the dstrbuton functon by Fourer nverson. à Moent generatng functon By drect ntegraton, we can readly deterne that the oent generatng functon of a sngle ter of the for b Z + l Z s where M HtL = EA b Z +l Z E = - Hb z +l z L t fhzl z = ÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!!!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - t l b t ÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅ l t fhzl = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!! p - z ÅÅÅÅÅ s the standard noral densty functon. Therefore, the oent generatng functon of Y = a + = Hb Z + l Z L s

5 VARQuadratc.nb 5 M Y HtL = E@ ty D = a t M HtL M HtL M n HtL = n a t ÅÅÅÅ Â ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!!!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k - t l = b t ÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅ l t y z = { a t Exp n ÅÅÅÅÅ k = n b t y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - l t z  { = y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!!!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ z k - t l { fro whch the oents can be edately derved by dfferentaton. (Holton 003: 49 gves a coplcated recursve forula for the oents of Y.) In the prevous exaple, a =5, B =H0,, 6L and L =H4, 3, 0L. The oents of Y are EV@YD EV@Y D 374 EV@Y 3 D 338 EV@Y 4 D EV@Y 5 D The varance of Y s therefore E@Y D - E@YD = = 30. à The cuulant generatng functon Cuulants are analogous to oents. The frst cuulant s the sae as the frst oent (the expected value); the second and thrd cuulants are respectvely the second (varance) and thrd central oents; but the hgher cuulants are nether oents nor central oents, but rather ore coplcated polynoal functons of the oents. The cuulant generatng functon s gven by the log of the oent generatng functon, that s KHtL = log M HtL = at - ÅÅÅÅÅ n = logh - l tl + ÅÅÅÅÅ n = b t ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - l t fro whch the cuulants can be derved by dfferentaton. In the prevous exaple, wth a =5, B =H0,, 6L and L =H4, 3, 0L, the cuulants of Y are κ κ 30 κ κ κ

6 6 VARQuadratc.nb à Characterstc functon The characterstc functon of any rando varable Y s YHwL = E@  wy D Consequently, t can be obtaned by substtutng t = Âw nto the oent generatng functon. The characterstc functon of Y s YHwL = M H wl = Exp  aw- ÅÅÅÅÅ k = n n w b y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ -  w l z  { = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ!!!!!! ÅÅÅÅÅÅÅÅ -  w l à The dstrbuton functon The dstrbuton functon of Y can be obtaned by nvertng the characterstc functon (Holton 003: 59, Quadratc functons of noral rando varables) FHyL = ÅÅÅÅÅ - ÅÅÅÅÅ p IH YHwL - wy L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ w 0 w Substtutng the characterstc functon and splfyng gves where FHyL = ÅÅÅÅÅ + ÅÅÅÅÅ p A snhb + CL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ w 0 D A = - ÅÅÅÅÅÅÅÅ w = C = ÅÅÅÅÅ = b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + 4 l w, B = w y b y l -a+w ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + 4 l k w z, = { tan - H- l wl, D = w H + 4 l w L ê4 = For gven values of b, l ( =,,, ) and y, ths expresson can be nuercally ntegrated to gve a very accurate approxaton of dstrbuton functon. The cuulatve dstrbuton functon of Y for the exaple wth a =5, B =H0,, 6L and L =H4, 3, 0L s depcted below, together wth the CDF of a noral dstrbuton wth the sae ean and varance (red).

7 VARQuadratc.nb The CDF of Y Y Noral Fro the dstrbuton functon, the quantles can be coputed drectly. The Fast Fourer Transfor provdes a potental alternatve ethod of coputng the dstrbuton functon of Y. Approxatng the quantles of Y A nuber of ethods have been suggested for approxatng the quantles of Y. Two of the ost prosng are the Cornsh-Fsher expanson and saddlepont approxatons. Coparatve studes suggest that the Cornsh- Fsher expanson s coputatonally faster, whle the saddlepont approxaton s ore accurate, especally n the tals. à Cornsh-Fsher expanson The Cornsh-Fsher expanson approxates the quantles of a dstrbuton as a polynoal of ts cuulants. That s, the a quantle of Y s F Ȳ HaL º a 0 + a k 3 + a k 4 + a 3 k 3 + a 4 k 5 + a 5 k 3 k 4 + a 6 k 3 3 where the coeffcents a 0, a,, a 6 are polynoals of the quantles of the standard noral dstrbuton. a 0 = F - HaL, a = ÅÅÅÅÅ 6 IF- HaL - M, a = ÅÅÅÅÅÅÅÅ 4 IF- HaL 3-3 F - HaLM, a 3 = - ÅÅÅÅÅÅÅÅ 36 I F- HaL 3-5 F - HaLM a 4 = ÅÅÅÅÅÅÅÅÅÅÅ 0 IF- HaL 4-6 F - HaL + 3M, a 5 =- ÅÅÅÅÅÅÅÅ 4 IF- HaL 4-5 F - HaL + M, a 6 = ÅÅÅÅÅÅÅÅÅÅÅ 34 I F- HaL 4-53 F - HaL + 7M The cuulants are readly obtaned fro the cuulant generatng functon.

8 8 VARQuadratc.nb The followng graph shows the Cornsh-Fsher expanson superposed on the exact dstrbuton functon of Y. The Cornsh-Fsher expanson à Saddlepont approxaton The saddlepont approxaton to the cuulatve dstrbuton functon of Y (due to Lugannan and Rce) s gven by F Y HyL º FHrL - nhrl J ÅÅÅÅÅ () u - ÅÅÅÅÅ r N where F and n are respectvely the dstrbuton functon and densty functon of the standard noral dstrbuton, and r = è!!! è!!!!!!!!!!!!!!!! f y - KHfL!!!!!!! and u = f è!!!!!!!!!!!! K '' HfL K s the cuulant generatng functon KHtL = log M HtL = at - ÅÅÅÅÅ = logh - l tl + ÅÅÅÅÅ = b t ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - l t and the saddlepont f solves K ' HfL = y An alternatve approxaton, due to Barndorff-Nelsen, s gven by () F Y HyL º FJr - ÅÅÅÅÅ r log ÅÅÅÅÅ u N

9 VARQuadratc.nb 9 Wthout loss of generalty, assue that l = n l and l = ax l. If l < 0, then we ust have t > ê l. Slarly, f l > 0, then we ust have t < ê l. In any case, K s defned on an nterval around the orgn. Coputng the saddlepont approxaton for a specfc y nvolves solvng equaton () for the saddlepont f, and then calculatng F Y HyL usng equaton (). The frst and second dervatves of K are K ' HtL = a + = l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - l t b + th -l tl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H - l tl = K '' HtL = = l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H - l tl + = b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H - l tl 3 SaddlePont Approxaton à Coparson The followng graph copares the errors of the two approxaton ethods, relatve to the cuulatve dstrbuton functon of Y.

10 0 VARQuadratc.nb Errors of saddlepont and Cornsh-Fsher approxatons 0.00 Cornsh-Fsher expanson Saddlepont approxaton Exaple (adapted fro Holton 003) A Japanese etals tradng fr has exposure to forward and opton postons n platnu. Soe of the postons are denonated n USDs. We dentfy three key rsk factors X æ spot prce of platnu (JPY) æ a representatve pled volatlty of platnu æ spot JPY/USD exchange rate and assue that X s norally dstrbuted NH, SL wth µ = , 0.670, 07.80< ; Σ = y k z ; { We value the portfolo usng applcable forward and opton prcng forulas, and quadratcally approxate ths as wth P = a + D T X + X T G X a = ; = , , <; Γ = y k z ; 5673 {

11 VARQuadratc.nb Hz = CholeskyDecoposton@ΣD êê TransposeL êê MatrxFor y z k { H8Λ, u< = Egensyste@Transpose@zD.Γ.zD L; 8Λ, MatrxFor@uD< , 858., 845.9<, y z = k { Transpose@uD.u êê Chop êê TableFor HA = u.inverse@zdl êê MatrxFor k Usng the substtuton y z { Z = A HX - L we transfor P nto a quadratc functon of ndependent ch-squared rando varables, naely ChopA a +.X+ X.Γ.X ê. X Inverse@AD. 8Z,Z,Z 3 < + µ êê Expand, 0 8 E Z Z Z 858. Z Z Z 3 That s P = a + Hb Z + l Z L = wth a = µ 0 9, B =H µ 0 8, µ 0 6, µ 0 6 L and L =H µ 0 6, -858., 845.9L.

12 VARQuadratc.nb A. êê Chop êê MatrxFor k y z { paraeters = 8 = 3, α = , β = , β = , β 3 = , λ = , λ = 858., λ 3 = 845.9<; The cuulants of P are Table@8κ k,d@k@td, 8t, k<d ê. t 0<, 8k, 5<D êê TableFor κ κ κ κ κ The standard devaton s è!!!!!!!!!!!!!!! K''@0D The portfolo's -day standard devaton s JPY 358 llon. Applyng the Cornsh-Fsher expanson, the 5% quantle of P s approxately JPY 8.7 bllon. CFQuantle@0.05D Based on the expected ean of JPY 9.3 bllon, the portfolo has a -day 95% VAR of approxately JPY 583 llon, whch s close to.65 standard devatons. K'@0D CFQuantle@0.05D è!!!!!!!!!!!!!!! K''@0D

13 VARQuadratc.nb 3 Suppose that we sply gnored the non-noralty, and calculated the standard devaton lnearly, n effect gnorng the quadratc ter. P = a + D T X + X T G X In ths case, we over-estate the standard devaton. è!!!!!!!!!!!!!!!!. Σ Our 95% VAR estate s then.65 è!!!!!!!!!!!!!!!!. Σ whch overestates the rsk by 6%