Volatility Interpolation
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1 Volaly Inerpolaon Prelmnary Verson March 00 Jesper Andreasen and Bran Huge Danse Mares, Copenhagen Elecronc copy avalable a: hp://ssrn.com/absrac=69497
2 Inro Local volaly models such as Dupre (994), Andersen and Andreasen (999), JPMorgan (999), and Andreasen and Huge (00) deally requre a full connuum n expry and sre of arbrage conssen European opon prces as npu. In pracce, we, of course, only observe a dscree se of opon prces. In hs paper, we show how a varan of he fully mplc fne dfference mehod can be appled o effcenly nerpolae and exrapolae a dscree se of opon quoes o an arbrage conssen full connuous surface n expry and sre. In a numercal example we show how he model can be fed o all quoed prces n he SX5E opon mare ( expres, each wh roughly 0 sres) n approxmaely 0.05s of CPU me. Dscree Expres Gven a me grd of expres 0 0 and a se of volaly funcons { ( )} we consruc European opon prces for all he dscree expres, by 0,, recursvely solvng he forward sysem: [ ( ) ] c(, ) c(, ), c(0, ) ( s(0) ), 0,, () where. If we dscrese he sre space j 0 j, j 0,,, n and replace he dfferenal operaor by he dfference operaor, we ge he followng fne dfference scheme f ( ) ( f ( ) f ( ) f ( )) [ ( ) ] c(, ) c(, ), c(0, ) ( s(0) ), 0,, () The sysem () can be solved by recursvely solvng rdagonal marx sysems. One can hus vew he sysem () s a one-sep per expry mplc (fne-dfference) dscresaon of he Dupre (994) forward equaon c 0 (, ) c (3) For a se of dscree opon quoes forward expry by expry o fnd pecewse consan funcons j, j j { cˆ (, )}, he sysem () can be boosrapped j ( ) a, b b (4) ha mnmze he prcng error n (). I.e. we solve he opmsaon problems nf ( (, ) ˆ(, )) () j c j c j, (5) Elecronc copy avalable a: hp://ssrn.com/absrac=69497
3 sequenally for,,. The pon here s ha for each eraon n (5) only one rdagonal marx sysem () needs o be solved. Fllng he Gaps The sysem () ranslaes he local volaly funcons no arbrage conssen prces for a dscree se of expres bu does no drecly specfy he opon prces beween he expres. We fll he gaps by leng he opon prces beween wo expres be consruced accordng o [ ( ) ( ) ] c(, ) c(, ), ], [ (6) Noe ha we do no sep from expry o expry ha le beween he orgnal expres; for all expres ], [, we sep from. Absence of Arbrage and Sably Carr (008) shows ha he opon prces generaed by () are conssen wh he underlyng beng a local varance gamma process. From hs or from sragh calculaon we have ha (6) can be wren as c e g u du /( ) (, ) u (, ), 0 (7) where gu (, ) s he soluon o g u g(0, ) c(, ) 0 ( ), u 0 g (8) In Appendx B we use hs o show ha he opon prces generaed by () and (6) are conssen wh absence of arbrage,.e. ha c (, ) 0, c (, ) 0 for all (, ). For he dscree space case we noe ha wh he addonal (absorbng) boundary condons c (, 0) c (, n) 0, () can be wren as Ac( ) c( ) (9) where A s he r-dagonal marx 3
4 A z 0 z z z j j z z z ( ) z z z n n n 0 (0) The r-dagonal marx A s dagonally domnan wh posve dagonal and negave off-dagonals. Nabben (999) shows ha for hs ype of marx A 0 () Ths mples ha he dscree sysem () s sable. As we also have A, (,,)', () we can furher conclude ha he dscree sysem () s arbrage-free. Ths also holds f he spacng s non-equdsan. If he problem s formulaed n logarhmc space hen he dscree sysem () becomes x ln, as would ofen be he case, f ( x) ( f ( x x x) f ( x x)) f ( x) ( f ( x x x) f ( x) f ( x x)) x [ ( x) ( xx x)] c(, x) c(, x), c(0, x) ( s(0) e ), 0,, x xx I follows ha he sysem s sable f x ln( j / j), no a consran ha wll be breached n any praccal applcaon. As shown n Appendx B, () and (6) can be slghly generalsed by nroducng a deermnsc me-change T: () (3) [ ( T( ) ) ( ) ] c(, ) c(, ), ], ] (4) where T( ) and T'( ) 0. In hs case, he local volaly funcon (3) conssen wh he model s gven by 4
5 c (, ) ln c (, ) (, ) ( ) [ T '( ) ( T( ) ) ] c (, ) (5) The nroducon of he me-change gves a handle on he nerpolaon n he expry drecon. For example, a choce of a pecewse cubc funcons T () can be used o ensure ha mpled volaly s roughly lnear n expry. Algorhm In summary: a dscree se of European opon quoes s nerpolaed no a full connuous surface of arbrage conssen opon quoes by Sep : For each expry solve an opmsaon problem (6) for a pecewse consan volaly funcon wh as many levels as arge sres a he parcular expry. Each eraon nvolves one updae of () and s equvalen o one me sep n a fully mplc fne dfference solver. Sep : For expres beween he orgnal expres, he volaly funcons from sep s used n conjuncon wh (7) o generae opon prces for all sres. Noe ha sep does no nvolve any eraon. The process of he me seppng s shown n Fgure. Numercal Example In hs secon we consder fng he model o he SX5E equy opon mare. The number of expres s wh up o 5 sres per expry. The arge daa s gven n Table. We choose o f a log-normal verson of he model based on fne dfference soluon wh 00 grd pons. The local volaly funcon s se up o be pecewse lnear wh as many levels as calbraon sres per expry. The model fs o he opon prces n approxmaely 0.05s of CPU me on a sandard PC. The average number of eraons s 86 per expry. Table shows he calbraon accuracy for he arge opons. The sandard devaon of he error s 0.03% n mpled Blac volaly. Afer he model has been calbraed we use (6) o compue opon prces for all expres and sres and deduce he local volaly from he opon prces usng (3). Fgure shows he resulng local volaly surface. We noe ha he local volaly surface has no poles. So, as expeced, he model produces arbrage conssen European opon prces for all expres and sres. 5
6 References Andersen, L. and J. Andreasen (999): Jumpng Smles. RISK, November. Andreasen, J. and B. Huge (00): Expanded Smles. RISK, May. Andreasen, J. (996): Impled Modelng. Worng paper, Aarhus Unversy, Denmar. Carr, P. (008): Local Varance Gamma. Worng paper, Bloomberg, New Yor. Dupre, B. (994): Prcng wh a Smle. RISK, January, 8-0. JPMorgan (999): Prcng Exocs under he Smle. RISK, November. Nabben, R. (999): On Decay Raes of Trdagonal and Band Marces. SIAM J. Marx Anal. Appl. 0,
7 Appendx A: Tables and Fgures Table : SX5E Impled Volaly Quoes \ % 33.66% 3.9% 58.64% 3.78% 3.9% 30.08% 65.97% 30.9% 9.76% 9.75% 73.30% 8.63% 8.48% 8.48% 76.97% 3.6% 30.79% 30.0% 8.43% 80.63% 30.58% 9.36% 8.76% 7.53% 7.3% 7.% 7.% 7.% 8.09% 84.30% 8.87% 7.98% 7.50% 6.66% 86.3% 33.65% 87.96% 3.6% 9.06% 7.64% 7.7% 6.63% 6.37% 5.75% 5.55% 5.80% 5.85% 6.% 6.93% 89.79% 30.43% 7.97% 6.7% 9.63% 8.80% 6.90% 5.78% 5.57% 5.3% 5.9% 4.97% 93.46% 7.4% 5.90% 4.89% 95.9% 5.86% 4.88% 4.05% 4.07% 4.04% 4.% 4.8% 4.0% 4.48% 4.69% 5.0% 5.84% 97.% 4.66% 3.90% 3.9% 98.96% 3.58% 3.00%.53%.69%.84%.99% 3.47% 00.79%.47%.3%.84% 0.6%.59%.40%.3%.4%.73%.98%.83%.75% 3.% 3.84% 3.9% 4.86% 04.45% 0.9% 0.76% 0.69% 06.9% 0.56% 0.4% 0.5% 0.39% 0.74%.04%.3% 08.% 0.45% 9.8% 9.84% 09.95% 0.5% 9.59% 9.44% 9.6% 9.88% 0.%.5%.6%.9%.69% 3.05% 3.99%.78% 9.33% 9.9% 9.0% 3.6% 9.0% 9.4% 9.50% 0.9% 7.8% 8.85% 8.54% 8.88% 0.39% 0.58%.%.86%.3% 3.% 0.95% 8.67% 8.% 8.39% 9.90% 4.6% 8.7% 7.85% 7.93% 9.45% 0.54%.03%.64%.5% 3.94% 9.88% 0.54%.05%.90% 39.7% 9.30% 0.0% 0.54%.35% 46.60% 8.49% 9.64% 0.% Table shows mpled Blac volales for European opons on he SX5E ndex. Expres range from wo wees o a b less han 6 years and sres range from 50% o 46% of curren spo of Daa s as of March, 00. 7
8 Table : SX5E Calbraon Accuracy \ % 0.00% 0.00% 58.64% 0.00% -0.0% 0.08% 65.97% 0.00% 0.0% -0.3% 73.30% 0.00% -0.0% 0.05% 76.97% -0.0% -0.0% 0.00% 0.00% 80.63% -0.0% -0.0% 0.00% 0.0% 0.00% 0.00% 0.0% 0.06% 0.00% 84.30% 0.00% 0.00% 0.00% -0.0% 86.3% 0.0% 87.96% -0.07% -0.05% 0.0% 0.0% 0.0% -0.0% 0.0% 0.00% 0.00% -0.0% -0.0% 0.00% 89.79% 0.0% 0.0% 0.00% 9.63% 0.0% 0.0% 0.00% 0.0% 0.0% 0.00% -0.0% 93.46% -0.0% -0.0% 0.00% 95.9% 0.00% 0.00% 0.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.0% -0.0% 0.00% 97.% 0.0% 0.0% -0.0% 98.96% -0.0% -0.0% 0.00% 0.00% 0.00% 0.00% 0.00% 00.79% 0.0% 0.00% 0.00% 0.6% 0.0% -0.0% 0.00% 0.00% 0.00% -0.0% -0.0% 0.00% 0.00% -0.03% 0.00% 0.00% 04.45% 0.0% 0.00% 0.0% 06.9% -0.06% -0.0% 0.00% 0.0% 0.00% 0.03% 0.0% 08.% 0.00% 0.00% -0.0% 09.95% -0.0% -0.09% 0.00% -0.0% 0.00% 0.0% -0.0% 0.00% -0.0% 0.0% -0.0% 0.00%.78% -0.0% 0.03% -0.04% 3.6% 0.03% 0.00% -0.0% 0.00% 7.8% -0.03% 0.00% 0.0% 0.00% 0.00% 0.0% -0.0% 0.00% 0.00% 0.95% 0.0% 0.00% -0.0% 0.00% 4.6% 0.00% 0.0% 0.07% 0.0% -0.03% 0.0% -0.0% 0.00% 3.94% 0.00% -0.05% 0.0% 0.00% 39.7% 0.00% 0.0% -0.0% -0.0% 46.60% 0.0% -0.0% 0.00% Table shows he dfference beween he model and he arge n mpled Blac volales for European opons on he SX5E ndex. Daa s as of March, 00. 8
9 Fgure : Model Tmelne Fgure : Local Volaly derved from Model Opon Prces 00.00% 80.00% 60.00% 40.00% 0.00% 0.00% Fgure shows he local volaly surface n he model afer has been fed o he SX5E mare. Daa s as of March, 00. 9
10 Appendx B: Techncal Resuls Proposon : Absence of Arbrage The surface of opon prces consruced by he recursve schemes () and (6) s conssen wh absence of arbrage,.e. c (, ) 0 c (, ) 0 (B) for all (, ). Proof of Proposon Consder opon prces generaed by he forward equaon g 0 ( ) g (B) gven whch s solved forward n me gven he nal boundary condon g(0, ). As also noed n Andreasen (996), (B) can also be seen as he bacward equaon for g(, ) E [ g(0, (0)) ( ) ] (B3) where follows he process d( ) ( ( )) dz( ) (B4) and Z s a Brownan moon runnng bacwards n me. The mappng g(0, ) g(, ) gven by (B) hus defnes a posve lnear funconal n he sense ha g(0, ) 0 g(, ) 0 (B5) Furher, dfferenang (B) wce wh respec o yelds he forward equaon for p g : p 0 [ ( ) p] p(0, ) g (0, ) g (0, l) ( l) dl (B6) Equaon (B3) s equvalen o he Foer-Planc equaon for he process dx( ) ( x( )) dw ( ) (B7) 0
11 where W s a sandard Brownan moon. From hs we conclude ha (B) preserves convexy: g (0, ) 0 g (, ) 0 (B8) Le Tu ( ) be a srcly ncreasng funcon. Defne he (Laplace) ransform of he opon prces by / T ( u) h( u, ) e g(, ) d 0 Tu ( ) (B9) Mulplyng (B) by / T ( u) e and negrang n yelds T u h u g [ ( ) ( ) ] (, ) (0, ) (B0) From (B5) and (B8) we conclude ha (B0) defnes a posve lnear funconal ha preserves convexy. Dfferenang (B0) wh respec o u yelds [ T( u) ( ) ] h (, ) '( ) ( ) (, ) u u T u h u (B) Usng ha (B0) s a posve lnear funconal ha preserves convexy we have ha f g (0, ) s convex hen for all ( u, ). h (, ) 0 u u (B) We conclude ha he opon prces consruced by () and (6) are conssen wh absence of arbrage.
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