FX-IR Hybrids Modeling

Transcription

1 FX-IR Hybr Moeln Yauum Oajma Mubh UFJ Secure Dervave Reearch Dep. Reearch & Developmen Dvon Senor Manaer Oaka Unvery Workhop December 5

2 h preenaon repreen he vew o he auhor an oe no necearly relec he opnon o he Mubh UFJ Secure. Oaka Unvery Workhop 5//

3 Oulne Example o FX-IR hybr prouc Bac prcn moel ~ F lo-normal F Hull-Whe Prcn loc o PRDC an FX-ARN. FX mle moel Aympoc expanon ormula or x opon mple volaly 3 Oaka Unvery Workhop 5//

4 Inroucon Recenly orex-lnke rucure prouc uch a power-revere-ual-currency noe PRDC an orex-are-reempon noe FX-ARN are popular amon Japanee nveor. hee prouc have lon maure -3 year are ypcal I neceary o moel po x rae ynamc an omec an oren nere rae ynamc or prcn an hen hee prouc appropraely. 4 Oaka Unvery Workhop 5//

5 Example ~ PRDC Prncpal Amoun: Selemen Dae: December 5 Maury Dae: December 35 Coupon: x opon Year : 5.% Year : Max5.% x SpoFX /..%.% Year 3: Max5% x SpoFX /..%.% 5 Oaka Unvery Workhop 5//

6 Example ~ FX-ARN Coupon mlar o PRDC. Year : 8.% Year ~ 3: MaxSpoFX..%.% Early Reempon Conon: On each repecve Coupon Paymen Dae I he um o coupon pa beore reache he [are coupon] he noe ermnae beore maury % n JPY. are Coupon: 6.% per Prncpal Amoun Prncpal Reempon: 5 $ x Reempon x rae Reempon x rae: 5 bune ay pror o maury 6 Oaka Unvery Workhop 5//

7 Example ~ Varaon here are many varaon. Coupon - ep up / own cap bnary nowball... ec Callably - call rer call & rer an her combnaon Reempon - orwar ype pu opon ype ec 7 Oaka Unvery Workhop 5//

8 5// Oaka Unvery Workhop 8 Bac FX-IR Hybr moel Probably pace rk neural meaure. Spo x rae ~ F lo-normal moel Domec an oren nere rae ~ F Hull-Whe moel Here 3-m Brownan moon an correlaon marx Q Ω F / W r r S S W r a r W r a r θ θ W W W Q

9 Calbraon We nee o calbrae o - x opon volale o maury - nere rae wapon volale { L N Snce we ue lo-normal x moel we canno calbrae o x opon volaly mle. We can calculae normal wapon value by Hull- Whe moel analycally. here a amou meho; C : J.Hull Opon Fuure & Oher Dervave } 9 Oaka Unvery Workhop 5//

10 5// Oaka Unvery Workhop Calbraon Calculae x call opon o rke an maury. Uner rk neural meaure; I beer o calculae uner orwar meaure; Le enoe orwar x rae a me o maury. We can calculae by Uner orwar meaure orwar x rae exponenal marnale o we can apply Black-Schole ormula. K K C ] [ K S B E K C Q ] [ K S E P K C F P P S F

11 5// Oaka Unvery Workhop Uner Forwar meaure Uner orwar meaure po x rae an zero coupon bon ae ollown SDE; Here W r P P W r P P W r r S S a : exp : } { : ϕ ψ ϕ ψ ψ ϕ

12 5// Oaka Unvery Workhop Imple volaly o x opon Forwar x rae ae nex SDE; Imple volaly hereore we can calbrae o marke olvn quarac equaon a maure nucvely. W W W F F / c b c b F } { N L

13 5// Oaka Unvery Workhop 3 Markov unconal I convenen o ene hree Gauan proce; We can expre po x rae an coun bon a he unconal o ; W W W z W y W x : : : ψ ϕ ψ ϕ ϕ ϕ exp exp exp K K K y x z S S y P P x P P ψ ψ ψ ψ ψ ψ z y x

14 Prcn o PRDC We conruc 3D lace on each call / rer ae No cree o me recon. L N We evaluae every PRDC cahlow xe beween an on 3D-lace a. P x y z : E[cahlow x y z ] Snce our hybr moel analycal we can calculae hee cononal expecaon by BS-lke ormula. We have o ake care o mn; call / rer ae o no necearly equal xn ae. 4 Oaka Unvery Workhop 5//

15 3-m Backwar We know he rbuon o Markov proce x y z. o calculae preen value we have o calculae he cononal expecaon on each ae; E [ x y z x y z x y z ] x y z P xx yy zz xyz I you calculae 3-m Gauan neral rahorwar calculaon me abou N^6 orer o very me conumn. Fa calculaon alorhm o convoluon neceary. 5 Oaka Unvery Workhop 5//

16 Prcn o FX-ARN Generally Prcn loc o FX-ARN Mone-Carlo bu beer o ue conen moel wh PRDC. We can calculae ome kn o pah-epenen opon uch a FX-ARN by backwar alorhm. Le C be he cumulave coupon beore me hen we can rear C a -m Markov proce. We conruc 4D lace on each xn ae L C C coupon x y z coupon x y z : coupon xe a x y z. N 6 Oaka Unvery Workhop 5//

17 FX volaly mle moeln I mporan o calbrae o FX volaly mle. here are ome moel ncorporae wh FX mle; - local volaly moel ochac volaly moel jump-uon unveral volaly moel. Snce we on wan o a more ochac acor we ue local volaly moel here; S / S r r S W Epecally we ue aonary mle; S S F 7 Oaka Unvery Workhop 5//

18 5// Oaka Unvery Workhop 8 Imple volaly o local vol moel o calbrae o marke we calculae mple volaly o local volaly moel. Fr aume ha nere rae are eermnc. Aympoc expanon ormula Haan K 4 x x K mpl / : : : : S P P K x

19 5// Oaka Unvery Workhop 9 Aympoc expanon We prove h ormula un aympoc expanon. Le hen. Sae nex SDE; We e he mall volaly aympoc expanon; : B S D S X K 3 3 X u W v W u W v u W W u u W u W u W 3 X F S W X X X /

20 Aympoc expanon o call opon We enoe by pay-o uncon o call opon; We calculae near AM opon; Call opon prce o rke maury x : x C K S E[ y 3 K] Un Waanabe aympoc expanon ormula we can calculae call opon prce; E[ E[ y y] k k k! 3 K] E[ K hen we can calculae mple volaly o call opon. K k K F y y 3 k ] Oaka Unvery Workhop 5//

21 5// Oaka Unvery Workhop Imple volaly o hybr local vol moel Aympoc expanon ormula uner hybr moel Noe ha r erm aylor expanon o orwar x rae volaly o more accurae o apply he volaly. L Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ 4 4 x x K Σ Σ Σ Σ : : :

22 5// Oaka Unvery Workhop -parameer aympoc expanon Snce Inere rae volaly Hull-Whe normal volaly much maller han ha o x volaly we ue - parameer aympoc expanon; hen / / W r a r W r a r W F S X X δ θ δ θ δ W W R W R X X X : exp δ δ δ δ δ <<

23 5// Oaka Unvery Workhop 3 Aympoc expanon o call opon Call opon prce We aume. Aer lon calculaon we e he aympoc expanon ormula. W y W W W h W W R h : : L L ] [ ] [ ] [ ] [ h h y E h h y E y E h h y E δ δ δ δ δ δ δ

24 Reul ~ CEV We emae he eec o ochac nere rae o x mple volaly. We ue CEV ype local volaly uncon;. % x x β 9% 8% AM Imple volaly 7% 6% 5% 4% 3% % % % Srke CEV Hybr Moel param bea.5 Maury 5Y CorXD.3 CorXF.3 CorDF.5 4 Oaka Unvery Workhop 5//

25 Reul ~ Quarac volaly In quarac volaly cae mle curve more convex. S F S c S F S c S F S 35% 3% 5% Imple volaly % 5% AM Quarac Hybr % 5% Moel param C.5 C.5 Maury 5Y % Srke 5 Oaka Unvery Workhop 5//

26 Concluon We ve a prcn loc o PRDC an FX-ARN n he ame 3-acor moel. o calbrae o FX opon volaly we ve a local volaly moel or po x rae an calculae he mple volaly a a uncon o rke by un aympoc expanon approach. Local volaly moel can creae varou ype o mle curve bu o capure he ynamc we houl ry anoher moel uch a ochac volaly moel. 6 Oaka Unvery Workhop 5//