The model proposed by Vasicek in 1977 is a yield-based one-factor equilibrium model given by the dynamic

Transcription

1 h Vsick modl h modl roosd by Vsick in 977 is yild-bsd on-fcor quilibrium modl givn by h dynmic dr = b r d + dw his modl ssums h h shor r is norml nd hs so-clld "mn rvring rocss" (undr Q. If w u r = b/, h drif in inrs r will disr. So his vlu rrsns h mn vlu of h shor r. So is msur of how fs h shor r will rch h long-rm mn vlu. h modl is oulr in h cdmic communiy (minly du o is nlyic rcbiliy. Bcus h modl is no ncssrily rbirg-fr wih rsc o h cul undrlying scuriis in h mrklc, h modl is no usd much. Wih ( = -, b( = b, c( = nd d( = (s bov h quion of B is givn by: B (, B (, = B (, = his cn sily b solvd, nd h nswr is B (, = ( { } Insring his in h quion of A, w obin A (, bb + B = A (, = (, (, b ( s ( s A(, = bb( s, ds + B (, s ds { } ds { } ds = + ( s ( s ( s b = s s b ( ( ( = { B(, + } b = B (, 4 hr r good robbilisic rsons why som of h modls in our lis r sir o hndl hn ohrs. W s h h modls of Vsick, Ho-L nd Hull-Whi (xndd Vsick ll dscrib h shor r using linr SDE. Such modls r sy o solv nd h r- rocsss cn b shown o b normlly disribud.

2 For h Vsick modl w g from: h d( r = dr + rd = b r d + dv + r d = bd + dv u u r( = r( + b du+ dv u which simlifis o b ( u r ( = r( + ( + dv h clculion should hv srd ny im, hus ( s b ( s ( u r ( = rs ( + + dv h mn vlu in his modl is givn by: b E r r [ (] ( = + ( nd h vrinc by Vr r E dv du ( u ( u [ (] = u = = ( h bond rics r hn givn by Q (, = E x rudu F s I is lil ricky o clcul h bond rics wih his xrssion. o do so, w cn us h following horm: horm: For X ~ N(m, nd γ w hv u u E = x γ m+ γ γ X (for roof, s lcur nos r i.. for γ = w hv: = x + = x [ ] + [ ] X E m E X Vr X

3 If w l X = rudu W cn wri h bond rics (, s: Q (, = x E r( u du + Vr r( u du king r by r w obin b r( b b E r( u du = r( + du = + + b = ( r( b( + Ingring from o givs: ( b E r( u du = ( b r( + ( In ordr o clcul u u Vr r( u du w nd h following wo rsuls from sochsic clculus: Rsul : If W( is Brownin Moion nd g( non-rndom funcion, hn: X = g( u dw( u is Gussin Procss wih E[X(] = nd Vr[ X ] = g ( u du. Rsul : If W( is Brownin Moion nd g( nd h( non-rndom funcions dfind s: X = gudwu Y = huxudu

4 hn Y( is Gussin Procss wih E[Y(] = nd Vr[ Y ] = g ( u h( y dy du u Now, l g(u = u nd h(y = -y. Hnc w g: u u y u Vr r( u du = Vr[ Y ] = ( dy du = ( du u If w u ll oghr w find: u u ( u = ( du = ( du u ( u ( = ( + du = 3 ( + ( P(, = x r( 4 b b 3 Rmrk! his is h sm rsul obind bov whr w solvd h quions for A(, nd B(,. h so r voliliy in h Vsick modl is givn by: so = ( ( his rsul is found by h chniqu of chnging numrir, which w will discuss lr. A disdvng wih his modl is h h shor rs hv osiiv robbiliy o bcom ngiv. A simulion of h disribuion of h shor rs wih =.5 nd b =.9 % is shown in h figur blow:

5 Vsick Modl: Sdy S Probbiliy Dnsiy Funcion for Sor r 6,%,35% 9,65% Mn Sor (r -SD +SD -5% -% -5% % 5% % 5% % 5% If r( is 8% h simuld rm srucur of inrs rs is shown in h figur blow: Vsick rm Srucur of Inrs 9% 8% 7% 6% 5% 4% 3% % % % Vsick Zro R Long-rm quilibrium r r = Infinily long r im o muriy his giv discoun funcion s

6 ,,9,8,7,6,5,4,3,, - Vsick Discoun Funcion im o muriy Vsick Discoun Fcor W will in scion blow show h sm simulion for h CIR modl whr h shor r lwys is osiiv. Oion ricing In mny rm srucur modls, i is ossibl o find nlyicl soluions for Euron oions on discoun bonds. In r by Jmshidin (989, mhod for ricing oions on couon bonds is dvlod. hs oions r in fc oions on orfolio of discoun bonds. Jmshidin shows how h vluion rocdur cn b chngd so h h oion cn b clculd s orfolio of oions on discoun bonds wih rori srik rics. h mhod works for on-rmr modls, sinc ll bond rics r dcrsing funcions of h inrs r usd s h s vribl. Considr Euron oion on couon bond (or gnrl fixd csh flow rn wih srik ric X xiring imτ. h vlu of h bond ny im cn b wrin: n B(, r = c Pr (,, i= i i whr c i is h couons. h yoff from h oion muriy, τ is: mx(, B( r, τ X. h * vlu of r whn h oion is xcly -h-mony is clld r* nd dfind by: B( r, τ = X. * h oion will b xrcisd whn r( τ < r. I cn b shown h n n mx, ci P( r, τ, i X = ci mx, P( r, τ, i P( r, τ, i i= i= * { } h scond summion is h xc yoff of orfolio of oions on discoun bonds. Jmshidin hs lso shown h oions on zro-couon bonds cn b vlud using Vsick's modl. A Euron cll oion is givn by:

7 Π= L P(, S N( h K P(, N( h whr L is h fc vlu of h bond, S h bond muriy, K h oion srik nd L P(, S h = ln + K P(, ( S = ( Similrly, Euron u oion is givn by Π= K P(, N( h+ L P(, S N( h h voliliy givn will b inrrd s h voliliy of roorionl chngs in h shor r, in ordr o obin vlus in h sm unis s in h Blck-Schols modl. If h voliliy is givn in yild i hs o b convrd o ric voliliy: y dp = y P dy whr is h oion muriy nd τ h muriy of h bond. For oions on csh flows wih floing rs, n ddiionl rocdur mus b usd. I cn b shown h h rsn vlu nd inrs r snsiiviy of csh flow dnding on h imlid forwrd r cn b md idnicl o wo fixd csh flows. For swions, his rocdur is usd o firs convr ll floing csh flows nd hn ly h mhod dscribd bov. Exml: Considr Euron cll oion on zro-couon bond. im o xirion is wo yrs, h srik ric is 9, h voliliy is 3%, h mn-rvring lvl is 9%, nd h mn rvring r is.5. h fc vlu of h bond is wih im o muriy hr yrs, nd iniil risk-fr r of 8%. F =, K = 9, =, τ = 3, b =.45, =.5, r =.8, =.3..5 B (, = B(, = = B (, τ = B(,3 = = B (, τ = B(,3 = = { B(, + } A = A = B =.763 (, (, (,

8 .3 { B(, 3+ } A = A = B =.56 (, τ (,3 (,3 { } P (, = P(, = x A(,.8 B(, =.853 { } P (, τ = P(,3 = x A(,3.8 B(,3 = =.3 B(,3 = P(,3 h = ln + = P(, h cll vlu for on USD in fc vlu is c= P(,3 N( h 9 P(, N( h =.43 Wih fc vlu of h cll vlu is.43 USD ( x.43. Exml: Considr Euron cll oion on couon bond. im o xirion is four yrs, h srik ric 99.5, h voliliy is 3%, h mn-rvring lvl is.%, nd h mnrvring r is.5. h fc vlu of h bond is, nd i ys smi-nnul couon of four. im o muriy is svn yrs, nd h risk-fr r is iniilly 9%. Firs find h r r h mks h vlu of h couon bond qul o h srik ric h oion s xiry. ril nd rror givs r = 8.5%. o find h vlu of h oion, w hv o drmin h vlu of six diffrn oions:. A four-yr oion wih srik ric on 4.5-yr zro-couon bond wih fc vlu of four.. A four-yr oion wih srik ric 3.69 on fiv-yr zro-couon bond wih fc vlu of four. 3. A four-yr oion wih srik ric on 5.5-yr zro-couon bond wih fc vlu of four. 4. A four-yr oion wih srik ric on six-yr zro-couon bond wih fc vlu of four. 5. A four-yr oion wih srik ric 3.77 on 6.5-yr zro-couon ond wih fc vlu of four. 6. A four-yr oion wih srik ric on svn-yr zro-couon bond wih fc vlu of 4. h vlu of h six oions r, rscivly,.56,.493,.73,.97,.5, nd his givs ol vlu of