4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 Implications: • Implied trinomialtrees • Practical calibrationof • Lecture 8:LocalVolatility Models:Implications The valuesofexoticopti The deltasofstandardoptions. Models: Implications Page 1 of19 Page Implications Models: Volatility localvolatilitymodels ons: barriers,lookbacks,etc. smile-lecture8.fm 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 where are bywriting We thenestimate isasmoothfunctionwithsec where 8.1 PracticalCalibrationofLocalVol Models h orsodn oens soasto the correspondingmoneyness represent thediscreteimplie corresponding impliedvolatil isthemoneyness,i.e.st ration, where Let is anexample. cal relationbetweensmoothdi tion. Onecanagaindeterminetheresultan small contiguousandoverlappingregions variations inmarketimpliedvolatilities The methodillustratedhere book. other papersonthistopic, hard toobtain.Wilmott’s bookhasonepa realistic formoftheparametrization,pa tives oftheimplieds.Onedifficulty wi One canthencalculatethelocalvolatil mize thedistancebetweencomputed a form fortheimpliedvolatili The moststraightforwardwaytodothis create asmooth implie implied volatilitysurfacethat cal useoflocalvolatilitymodels.To problem, andfindingmethodstosolveitar these discretely specified values.Earlier strikes andexpirations, In practicewearegivenimpliedvolat {} x i w mx , y in )ε () , i n Volatility Models: Implications Page 2 of19 Page Implications Models: Volatility i = x i 1 n egtfntosta u o1 n ah peaksaround weight functionsthatsum to1,andeach mx () d volatilitysurface. mx someofthemmentioned () y must calibrate asmoothlo ties, andthencomputetheparametersthatmini- will besemiparametric.Th i isatleast twicedifferen ity. The aimistofi = fferentiable impliedsand x = i mx i ∑ = n )ε () 1 use Dupire’s equation,weneedasmooth w i lte overarangeofdiscrete ilities nd observedstandardoptionsprices. th thismethodishowtodeterminea ities bytakingtheappropriatederiva- in rticularly onthewing byaveragingthedatainaseriesof + , ieso o ahoto,ad isthe rike/spot foreachoption,and is towritedownasmoothparametric we mentioned thatth iehge egtt oaiiis give higherweighttovolatilities () usingaparametricsmoothingfunc- rametrization.There areavarietyof t localvolatilitiesfromthetheoreti- n eiaie.ad istheerror. ond derivatives.and x i e criticallyimportanttothepracti- Σ y d volatilitydataforagivenexpi- () i K nd asmoothedregression i , T in Chapter4ofFengler’s i tiable. We musttherefore cal volatilityfunctionto e ideaistosmooththe their derivatives.Here w s wherepricesare in , is anill-posed smile-lecture8.fm i y Eq.8.2 Eq.8.1 i y i 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 Fengler’s Chapter4providesmuchmoreinformationonthismethod. soasto makes theestimatorfunctionfoll regression andthesmoothedestimator discuss howtochoosethe 8.4,lead weights aregivenbyEquation and between theobservedvolatilitiesandth One canalsoshowthatminimizingthe true regressionfunctionas One canshowthatthis andstillobtainsmoothing. a smaller tion thereisinasmallregionaround ,thegreaterdensityof ,allsmoothi volatilities at theobservedmoneynessvalues.The smoothing, As a functionwhichintegratesto1.Small hr isafunctionthatpeaksar .Oneexampleisthe determined by where As anexample,wecanchoose . closestto getsacontribut est contributioncomingfromthose in thefunction thatcorr closer tothemoneyness K h () h u n Volatility Models: Implications Page 3 of19 Page Implications Models: Volatility h h mx → () 0 w Nadaraya-Watson K in x , h i () K () u h x h → () u = = 0 ow everywiggleinthedata. h -- 1 - i ------ng vanishesandthef ∑ ------= n esponds toaparticul K x 1 2 1 o rmal ,withthegreat- ion fromall i h oens ,andsoonecanchoose the h Gaussian withstandarddeviation K π () n while avoidingtheoversmoothingthat information,andsothemoreinforma- greater thenumberof h oteslto qain8.3.Fengler s tothesolutionEquation e estimated volatilities,wherethe weighted squaresofthedifferences e ound zerowithadegreeofpeaking h xx () → – producesgreaterlocalizationofthe etmtrfr converges tothe estimatorfor ------– xx u minimize thebiasbetweentrue 2 – ∞ ⁄ i 2 h withtheirproductkeptfinite. i 2 - x {} x i , x y i unction isdefinedonly mx ar . Any argument n () i y = i observedimplied 1 smile-lecture8.fm Eq.8.4 Eq.8.3 h , x

4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 8.2 Trinomial Trees withConstantVolatility owr rc ,where 1.Both trinomial andbinomialtrees mial case,theexpectedvalueofstoc in thecontinuumlimit.First,fora must besatisfiedinorderforthetree There aretwoconditions –onthemeanan five unknownparameters: value stock pricecanmovetooneof The stockpriceatthebegi Figure 8.1belowillustra We wanttomodeltherisk-neutral process methods. more efficient. Wilmott hasathorough methods ofsolvingpartialdifferential Binomial andtrinomialtreesaremerely rial onthis. Strickland, andthebookbyEspen Haug. below aretakenfromthere.Otherre describes theconstructiona 22; aversionofthisisonmywebsi nomial Trees oftheVolatility Smile ence ontrinomialtreesisthepa partial differential equationfortheopti Both trinomialandbinomialtreesaresi steps, sothatonecanavoidar the descriptionofimpliedstochastic ment, analogoustobinomialtrees Trinomial treesprovideanotherdiscrete prices nient in each isallowed to grow without limit. Nevertheless, one kind of tree may sometimes be more 1 – p than anotherwhenyouare working indiscrete – S S u q u, ; withprobability tothemiddlenode,value S m Volatility Models: Implications Page 4 of19 Page Implications Models: Volatility and ----- dS S FSe - = S = d . rdt ()Δ r + – tes asingletimestep δ σ q nning ofthetimestepis the twoprobabilities approach the same cont dZ to the nd calibrationoftrinomial t bitrage violations threenodes:withprobability o . or per byDerman,KaniandChriss, 1 down risk-neutral . Theiradvantageisagreaterflexibilityin , JournalofDerivati S te, andtheappendixofthatpaperhas ferences arethebookbyClewlowand m to representgeometricBrownianmotion δ dS equations, someofwhichmaybemuch . Attheendof ons valuationmodel. is thedividendyield.Therefore: time, beforeyo and moregeneraldiscussionofthese ln mple discretemethodsofsolvingthe node,value specialinstancesofmoregeneral representationofstockpricemove- k attheendofperiodmustbeits process forthestockpriceindiscrete Rebonato’s bookalsohassomemate- d thevarianceofprocess–that = inuous timetheory as the number of periods inatrinomialtree. trinomialtree,asinthebino- ⎝⎠ ⎛⎞ more easily. p r and – u reachthe continuous limit. S ----- σ 2 . Duringthis timestepthe S 2 - d q, trees. Someofthenotes ; andwithprobability ves, 3(4)(1996)pp7- dt and thethreenode time step,thereare + An initial refer- σ p dZ tothe smile-lecture8.fm Implied Tri- up node, conve- 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 ipe,w hoe sothat th simpler, wechoose match themeanandvari metric. Because of thesymmetry Figure 8.3illustratesatrinomialtreefo Because trinomialtreesaremoregenera duce atrinomial tree. Figure 8.2belowillustrate tral probabilitywithnoflexibility and varianceconditionsdeterminedthe more flexibilityinbuildingthetree.In Because therearetwoconstraints onfive become negligible inthecontinuum limit. tral trinomialtreeshave idly asweapproachthecontinuumlimit prices andtransitionprob Second, ifthestockpricevol hr denotestermsofhigherorderthan where that probability. Thethree transitionprobabilities sumtoone. move tooneofthree possiblefuture tree stepofatrinomial time can Inasingle stockprice 8.1. the FIGURE p () S u – O F () Δ 2 t ++ 2 Volatility Models: Implications Page 5 of19 Page Implications Models: Volatility qS () d p – mr S F u = S ++ 2 ifrn ihrodrtrsi qain8.6.Theyall different higherordertermsinEquation neo over time ance of abilities mustproducethe ⎝⎠ ⎛⎞ qS s twomethodsofcombiningbinomialtreestopro- () – d 1 o atility duringthistime ----- σ – 2 2 - () p , wehavetosolve onlyfor ln 1 inavoidingarbitrageviolations. Δ – SS – t q ⁄ p () – contrast, inthebinom h that’s chosentobemore sym- r the 0 S 1-p-q p location ofthenodesandrisk-neu- q l therearemorewaystobuildthem. . Different discretizationsofrisk-neu- values,eachwithitsrespective q m parameters inthetree,onehasmuch – S m ln F e centralnodealwayscoincides S = o o o 2 = F Δ S S S Δ t whichvanishmorerap- Eq.8.5 appropriate variance,so eidi ,thenthenode period is u S d m t. To make the tree even 2 σ 2 Δ tO ε + ial case, themean

σ and smile-lecture8.fm () Δ q t in orderto 2 Eq.8.6 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 the stock price here.the stockprice probability. Thethree transition probabilities sumtoone. We draw the log o move tooneofthree possible futu can tree of atrinomial price step time thestock In asingle FIGURE 8.3. (a) Combining two steps of S p = p Sd = S q = trinomial trees withspacing FIGURE 8.2.Two equivalentmeth binomial tree withspacing binomial tree witha spacing of m u a Cox-Ross-Rubinsteina binomialtree = = ln(S/S ⎝⎠ ⎜⎟ ⎛⎞ ⎝⎠ ⎜⎟ ⎛⎞ e ------e ------

e Se σΔ Se σΔ e S r S Δ σΔ σ – t Volatility Models: Implications Page 6 of19 Page Implications Models: Volatility σ t ⁄ 0 t 2 ⁄ 2 ⁄ ) q p 2 t 2 Δ 2 – ⁄ – Δ 2 t – e t – e – e – σΔ – e σΔ σΔ r Δ 0 S S S t t d m u t ⁄ ⁄ t 2 2 ⁄ ⁄ 2 - 2 -

2 2 o (1-q)/2 Δτ/2 . Δ Δτ t q . (a)Combiningtwostepsofa CRR (1-q)/2 Δτ/2. ods forbuilding constant volatility re eachwithitsrespective values, (b) CombiningtwostepsofaJR o o o m+ m- m ε ε (b) Combining two stepsof (b) Combining p 14 1/4 = p S S S = 1/4 q m u d = = a Jarrow-Rudd binomialtree a = Se Se Se

S ()Δ ()Δ()Δ r r r – – – p

q σ

σ σ 2 2 2 smile-lecture8.fm ⁄ ⁄ ⁄ 2 2 2 t t t –2 + S S σ S m u d σ Δ 2 Δ t t f 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 Figure 8.4illustratesarisk-neutraltr higher ordertermsbecomenegligible. This isaccurateonlytoO( stock become It’s oftenconvenienttochooseq=2/3. or symmetric. To getthevarianceofreturnsrightwemusthave The expectedvalueofthe andwe symmetric about the center. with theexpectedvalueof Volatility Models: Implications Page 7 of19 Page Implications Models: Volatility log term isthen exactly Δ ln DMe Me t), butinthelimitasspacinggoestozero, UMe εσ SS = ()ε = = = ⁄ 1 – 0 ⎝⎠ ⎛⎞ q r inomial treewithconstantvolatility. –3 – 1 ------σ σ Δ ----- σ 2 – 2 2 Then themultiplicativefactorsfor 3 t ≈ q Δ - Δ Δ t t t σ also choosetheprobabilitiestobe 2 Δ t m , sincetheprobabilitiesare smile-lecture8.fm Eq.8.8 Eq.8.7 FIGURE 8.4. Example of a risk-neutral ofarisk-neutral 8.4. Example FIGURE 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718 call strike 0.9034 0.9034 pv ofstock 0.9034 0.9034 0.9034 0.9034 stock d otnos01010101010.1 1.1247 1.0080 1.1247 0.2000 0.1000 0.1 1.0080 1.0101 1.1247 0.2000 0.1000 1.0080 1.0101 1.1247 0.1 0.2000 0.1000 1.0080 1.0101 1.1247 0.2000 0.1000 1.0080 1.0101 0.1 1.1247 0.2000 0.1000 1.0080 1.0101 0.2000 0.1000 0.1 1.0101 u 0.1 m sig dt f r continuous Risk-neutral trinomialtreewithconstantvolatility 100.0000 0.001083 0.19102.4290 101.6129 100.8032 100.0000 103.2518 102.4290 101.6129 100.8032 100.0000 Copyright 3.2518 4.6552 5.6828 Emanuel 6.5036 7.1968 Derman 2008 1.721336 114.2872 113.3766 112.4732 115.2052 114.2872 113.3766 112.4732 597 559 522 15.2052 15.2822 15.5599 15.9075 91.8012 91.0697 90.3441 92.5386 91.8012 91.0697 90.3441 .91112 .360.0000 0.5366 1.1223 1.6931 2.01127.5182 126.5021 128.5425 127.5182 126.5021 842 853 28.5425 28.5132 28.4823 82.2761 81.6206 82.9370 82.2761 81.6206 .85000 0.0000 0.0000 0.0885 trinomial tree with constant volatility 142.2810 143.4238 142.2810 Volatility Models: Implications Page 8 of19 Page Implications Models: Volatility 326 43.4238 43.2760 73.7394 74.3316 73.7394 .000.0000 0.0000 160.0279 60.0279 66.6192 0.0000 C=[1/6(up)+2/3(middle)+1/6(dn)]/f m*exp(-sig*sqrt( m*exp(sig*sqrt( exp(r-sig^2/2)dt withprob2/3 3 3 dt)) withprob1/6 dt)) withprob1/6 smile-lecture8.fm 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 8.3 Trinomial Trees withLocalVolatility pin rcsi h ii s .We can firstchoosethegridandthen lie onan initial grid formed simply determine theprobabilities. options pricesinthelimit as ferent probabilities Below aretwoexamplesoftreesbuilt with rect forwardandvariance. determine We cantherefore chooseagridofstockprices Given thenodes , with theforward.Then To makelifeeasier, wechoose In thefigureatright, easily avoidingnegativeprobabilities. are chosenindependently, therebymore probabilities afterthestockpricenodes atilities canbedonebyadjustingthe will show thattheca trage principle.Fortrinomialtrees,we probabilities orviolat volatility sometimesleadtonegative ibrating abinomialtree trees, wediscoveredth In dealingwithbinomiallocalvolatility p () S u p ’s and – Volatility Models: Implications Page 9 of19 Page Implications Models: Volatility F S 2 p ++ u q p ()

pS and ’s thatliestrictlybetween () S libration tolocalvol- and 1 u ions oftheno-arbi- u tfrfnt cal- forfinite at – – pS q p to avariablelocal ++ S p F conditions tosatisfyare d q () = = , wecansolveforpandq: u – 1 onthetree,butwillneve equations abovesimplifiesto 2 + In theexamplebelowwe choosestockpricesthat q () ------() – + Δ FS S qS p t () u qS S – S → () d – – m m S S q F Δ = d ≡ d 2 2 0 – by usingaCRRstockprice generators. σ σ t – S F ------() () F () 2 2 S S m F pq Δ Δ u u 2 so themiddlenodealwayscoincides + t t 2 – – qS ≈ S S different gridsandthatleadtodif- qS d d S F () d S - - 2 0 and1stillmatchthecor- inthefuturethatallowsusto σ = d 2 – Δ F F o t rtheless producethesame 2 ≈ Δ S t 2 σ 2 1-p-q p Δ q t smile-lecture8.fm σ o o o Eq.8.10 Eq.8.11 (S,t) Eq.8.9 S S S u d m 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 t0.0100 1.0000 a+b 0 local sig dt f r continuous 1.0000 0.9790 1.0214 M D U vol generator For generationofinitiallattice Risk-neutral trinomialtreewithconstantvolatility strike q_dn p_up stock statespace ecos n ,Notethatvolatility and We choose tility ofprices. Here below isarisk-neutraltr local) volatilityusedtogene atility, butjustsomeconstant(conveni σ σ 102.0000 g 0.001000 0.001000 100.0000 100.0000 100.0000 100.0000 100.0000 () S (the .13012 .130.1123 0.1123 0.1123 0.1123 .09019 .09019 .78077 .780.7778 0.7778 0.7778 0.7778 0.1099 0.1099 0.1099 0.1099 .95017 .65005 0.0000 0.0158 0.0645 0.1273 0.1955 = generator S^2*sig^2*dt/((F-S_d)(S_u -S_d)) S^2*sig^2*dt/((S_u -F)(S_uS_d)) 0.1 0.401214 0.40102.1440 102.1440 102.1440 102.1440 ( /0 a S/100 - 791 791 791 97.9010 97.9010 97.9010 97.9010 U .71000 0.0701 0.1656 0.0701 0.1656 0.0701 0.1656 .66008 .66081 .630.8613 0.6723 0.8613 0.6723 0.8613 0.6723 0.0686 0.1621 0.0686 0.1621 0.0686 0.1621 .04001 .000.0000 0.1440 0.0000 0.4752 0.0011 0.7004 0.0054 0.8723 0.1500 oaiiyMdl:Ipiain Page10of19 Implications Models: Volatility + = () S UD=M^2 toclosetree exp ⁄ volatility)usedtogeneratethegridisnottruelocalvol- 100 0.391433 104.3339 104.3339 104.3339 586 586 95.8461 95.8461 95.8461 .340.0384 0.0384 0.2307 0.2307 .36007 .210.9241 0.9241 0.5434 0.5434 0.0376 0.0376 0.2259 0.2259 .00000 0.0000 0.0000 2.3339 0.0000 2.3339 2.4103 () σ – g 1 2 0.78106.5708 106.5708 builtonalatticegeneratedwith15%CRRvola- Δ 384 93.8343 93.8343 rate thelatticeofprices. 0.0165 0.3083 .120.9673 0.3898 0.0162 0.3019 .00b1.0000 b 0.1000 .000.0000 0.0000 4.5708 4.5708 t inomial tree withlocal volatility U 108.8557 D 91.8647 0.0000 6.8557 ent, fake,approximatelyrepresentative == == = x .50.02 0.15 exp σ exp 1-p-q () with 15%volatility Jarrow-Rudd generatedlattice S () () σ – discounted callvalueforstrike102 = σ 2 g 0.1 Δ t 2 Δ + t () S x 1.0214 exp ⁄ 100 () smile-lecture8.fm – 1 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 Below isanotherrisk-neutr one needsmuchsmallertimesteps. an example. Ofcourse,foraccuratec with localvolatility=0.15-0.1(S/1001) For generationofinitiallattice t1.0000 1.0500 a+b local sig 0. dt f r continuous 1.0500 0.7913 1.3932 M D U vol Risk-neutral trinomialtree p_up stock statespace strike q_dn call option 102.0000 0.001500 1.501572 121.5505 115.7624 110.2500 105.0000 100.0000 g 056 030 001 975 19.5505 19.7653 20.0519 20.3209 20.5266 enerato .45015 .240.1157 0.1254 0.1350 0.1445 .09011 .95007 .46073 .800.7971 0.7800 0.7632 0.7466 0.0872 0.0945 0.1018 0.1089 r S^2*sig^2*dt/((S_u -F)(S_uS_d)) S^2*sig^2*dt/((F-S_d)(S_u -S_d)) 3.211620 5.08161.2850 153.6048 146.2903 139.3241 oaiiyMdl:Ipiain Page11of19 Implications Models: Volatility ( /0 a S/100 - 912 308 723 91.6051 87.2430 83.0886 79.1320 148 388 641 59.2850 56.4619 53.8486 51.4482 0.2000 .85018 0.1701 0.0597 0.1789 0.0691 0.1875 0.0787 .85408 .830.0000 2.3873 4.0986 5.3875 0.7017 0.8953 0.6862 0.8789 0.6712 0.8620 0.1282 0.0450 0.1348 0.0521 0.1413 0.0593 04879 UD=M^2 toclosetree 9.112387 214.0085 203.8176 194.1121 0.901664 112.0085 106.6748 101.5950 268 579 69.0371 65.7497 62.6188 .250.2180 0.2255 0.0137 0.0201 .84000 0.0000 0.0000 0.3864 0.6177 0.6046 0.9760 0.9648 0.1643 0.1699 0.0103 0.0151 al trinomialtreebuilton 7.49283.9671 270.4449 7.00181.9671 173.3020 951 52.0290 49.5515 .50b 0.1500 0.2581 0.0027 .000.0000 0.0000 0.5475 0.9953 0.1945 0.0020 onvergence tothecontinuoustimelimit, 7.99104.999983 376.7949 274.7949 39.2111 0.0000 adjust probabilities and timestepsofoneyear, justas σ 1-p-q () S -0.1000 with 20%volatility Jarrow-Rudd generatedlattice = is 20.5266 discounted callvalueforstrike102 20% vol-generatinglattice .50.1 0.15 – () S smile-lecture8.fm ⁄ 100 – 1 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 about thesame,andwillbeidenticalas tice: stock prices arediffer Here isone more risk-neutral trinomia For generationofinitiallattice t1.0000 1.0500 a+b local sig 0. dt f r continuous 1.0500 0.8737 1.2619 M D U vol Risk-neutral trinomialtree p_up stock statespace strike q_dn call option 102.0000 0.001500 1.501572 121.5505 115.7624 110.2500 105.0000 100.0000 g 029 994 956 885 19.5505 18.8357 19.5063 19.9541 20.2896 enerato .26037 .830.2632 0.2853 0.3071 0.3286 .75025 .34029 .99047 .740.5178 0.4774 0.4374 0.3979 0.2190 0.2374 0.2555 0.2735 r S^2*sig^2*dt/((S_u -F)(S_uS_d)) S^2*sig^2*dt/((F-S_d)(S_u -S_d)) 2.941252 3.21146.0834 139.1271 132.5020 126.1924 ( /0 a S/100 - oaiiyMdl:Ipiain Page12of19 Implications Models: Volatility 736 174 631 101.1376 96.3216 91.7349 87.3665 842 006 194 44.0834 41.9842 40.0265 38.4824 0.1300 .83035 0.3450 0.1796 0.3659 0.2017 0.3863 0.2239 .41719 .430.0000 5.3443 7.1391 8.6451 0.3680 0.6710 0.3297 0.6306 0.2922 0.5898 0.2870 0.1494 0.3044 0.1678 0.3215 0.1863 04879 UD=M^2 toclosetree 5.431727 175.5679 167.2075 159.2453 639 015 84.1528 80.1456 76.3291 678 004 73.5679 70.0647 66.7283 .460.4214 0.4406 0.1001 0.1203 .68000 0.0000 0.0000 1.8658 0.2280 0.1929 0.8166 0.7796 0.3506 0.3666 0.0833 0.1001 ent, probabilitiesaredifferen 0.55211.0032 200.9555 0.16109.0032 103.8126 666 70.0204 66.6861 .50b 0.1500 0.4908 0.0351 .000.0000 0.0000 0.1008 0.9356 0.4084 0.0292 5.96104.999983 253.5906 151.5906 58.2614 l treebuiltona13%vol-generatinglat- adjust probabilities 0.0000 Δ σ () t S → 1-p-q with 13%volatility Jarrow-Rudd generatedlattice 0 = -0.1000 is 20.2896 discounted callvalueforstrik .50.1 0.15 – t, butoptionspricesare () S ⁄ 100 smile-lecture8.fm – 1 e 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 late theno-arbitrage principle. dom inbuildingtrinomialtrees,onecan apart togiveprobabilitiesthatliebetw local volatilitiesgeneratedbytheform generating volatilityistoo lattice thatdoesn’t work.But,because Finally, hereisatrinomialtreebuilton For generationofinitiallattice t1.0000 1.0500 a+b local sig 0. dt f r continuous 1.1269 U 1.0500 0.9783 M D call option strike q_dn vol Risk-neutral trinomialtree p_up stock statespace oaiiyMdl:Ipiain Page13of19 Implications Models: Volatility -152.2057 102.0000 0.001500 1.501572 121.5505 115.7624 110.2500 105.0000 100.0000 g enerato .11193 .331.6915 1.8333 1.9736 2.1121 .69188 .011.5760 1.7081 1.8389 1.9679 r S^2*sig^2*dt/((F-S_d)(S_u -S_d)) S^2*sig^2*dt/((S_u -F)(S_uS_d)) 1.941838 2.44130.4566 124.2444 118.3281 112.6934 ( -34.8299 /0 a S/100 - 451 581 711 28.4566 27.1016 25.8110 24.5819 371 772 869 19.5505 18.6196 17.7329 43.7111 781 0.241789 113.2525 107.8595 102.7234 97.8318 0.0500 .76206 1.8965 1.4845 2.0361 1.6275 2.1736 1.7697 .22187 1.7671 1.8971 2.0252 .49156 1.3831 1.5164 1.6489 04879 small toproperlymatchor UD=M^2 toclosetree 2.901337 140.0153 133.3479 126.9980 446 076 11.2525 10.7166 24.4765 441 625 38.0153 36.2050 34.4810 570 0.91105.5209 100.4961 95.7106 .721.143.5209 10.7124 2.6732 .362.0981 2.2346 1.2773 1.4202 .801.9549 2.0820 .221.1901 1.3232 of thegreaternumberdegreesfree- ula, andsothenodesarenotfarenough een 0and1.Thisisan a5%volatility-gener 4.14150.2743 143.1184 always findalatticethatdoesn’t vio- 595 48.2743 45.9755 365 98.3171 93.6354 .760.0000 7.1706 2.2951 1.0723 2.1384 .50b 0.1500 0.9991 6.80104.999983 161.2850 59.2850 91.6051 VIOLATIONS ARBITRAGE adjust probabilities: 0.0000 σ () represent themuch larger S 1-p-q Jarrow-Rudd generate with 5%volatility = 309 282 -2.5 -2.8125 -3.0799 -0.1000 NONSENSE is -152.2057 discounted callval .50.1 0.15 illustration of a ating lattice This smile-lecture8.fm – 318 -2.9 -3.1987 248 -2.1 -2.4186 ( S ⁄ 100 -3.3 -1.7 d 41 33 43 16 43 u – 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 times violatedtheno- contrast, withbinomialtrees ties tomatchthestochasticprocess, tices ofstockpricesthatdon’t violatear Thus, wehavemoreflexib Rule ofThumb2: strikes andmaturitiesdecrease Comment: Inequitymarketswithnegati about twiceasrapidlyimplie Rule ofThumb1:The2: slope at-the-moneyapproximation: 8.4.1 Fourrulesofthumbfo 8.4 DeltasandExotics in strikelevel. market level isabout the sameasthech and strike. Thechangeinimpliedvolatility level index oaiiyMdl:Ipiain Page14of19 Implications Models: Volatility

Relation betweensensitivityof arbitrage conditions. current Σ vol ()σ SK ility inbuildingtrinomial , , wewereforcedtoadefinitelatticewhichsome- down as themarketlevelincreases. later up d volatilityvarieswithstrike. ≈ negat Local volatilityvarieswithmarketlevel 0 provided thelatticewasreasonable.In r localvolatilitiesinthesmall – in LocalVolatility Models i β ve s () ange inimpliedvolatilityforachange bitrage, andthenadjusttheprobabili- ve skew, theimpliedvolatilityforall SK strike k + ew of agivenoptionforchangein + 2 subtree higher volatility β impliedvolatility tospot subtree lower volatility trees; wecanchoosealat- S time 0 smile-lecture8.fm 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 V moves up.SupposeS=1000. has atruehedge ratioof50%,because For example,aone-yearS&Poptionwith given bythechainruleformula Rule ofThumb3:Thecorrect exposure fore smallerthanBlack-Scholeshedgeratios. Hedge ratiosofstandardoptionsinthe spot tostrike. given strikeistheharmonicaverageof Rule 4 C BS =400dollars; Black-Scholes tree implied tree implied Black-Scholes tree C . Forshorttimestoexpiration,thei C u d oaiiyMdl:Ipiain Page15of19 Implications Models: Volatility β = -0.0002 volpointperstrikept.: ΔΔ = subtrees volatility constant BS + V volatilitymoves downasthemarket BS presence of(negative)skewarethere- the localvolatilitiesacross ln(S)from C × aB-Shedgeratioof60%probably nverse oftheimpliedvolatilityfora Δ β of anoption C C ' u ' d V BS isapproximately β ∼ 0.1 smile-lecture8.fm subtree Eq.8.12 subtree 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 of theBlack-Scholesimplie approximately at120.Thus,inalocalvol For anoptionwithstrikeat100andba local volatilitiesbetweenspotandstrike. each callhasanimpliedvol mined bytheconstantBlack-Scholesvol Theoretical8.4.2 Value ofBa intuition aboutaveragingworksreasonably. which correspondstoaBlack The valueofthedownand ity variesbetween0.1and0.0 the localvolatilitiesbetw the barrier. Inaflat-volatilityworld,th with ashortpositioninputwhosestri cate adownandout In thelectureonstatic Example 1:AnUp-and-OutCall. about them. calculate theirvalue inlocal volatility region. strikeandbarrier, whichdepends between thestrikeandbarrier, and cially sensitive tothe risk-neutral proba options, takingbarrieroptionsasanexam illustratetheeffeIn thissectionwe oaiiyMdl:Ipiain Page16of19 Implications Models: Volatility call bymeansofal hedging, weshowedthat een 100and120.Inthefigurebe out optioninthelocalvol d volatilityfortheup-and-out atility which isapproxima -Scholes impliedvolatilit 7 inthisrange,withanaverageofaabout0.085. ct oflocalvolatilitymodelsonexotic with Strike 100andBarrier110 rrier Options inLocalVol Models models andtrytogainsomeintuition ke is(logarithmically)reflectedthrough rrier at110, thereflectedstrikeis e valueofbothth hence tothe local volatilityin this ong positioninthecallitselfcombined bility ofindexremainingintheregion ontheskew:.Herewearegoingto atility. Inaskewedworld,however, atility model,theapproximatevalue ple. Barrieroptionsvaluesareespe- you canapproximatelyrepli- atility modelisabout1.1, tely theaverageof y ofabout0.09,sothis call is the average of low, the local volatil- ese calls is deter- smile-lecture8.fm 1  2 34 %      ,    0  - ,     (  / ,   !   ,,  , .   + , -"  ,, , ,  4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

 #        

) !           Copyright Emanuel Derman 2008 VOLATILITYLOCAL OFSPOT ASAFUNCTION !   # ( VALUE ASA OFUP-AND-OUTCALL    .    ' %   %  ( %    ! %      oaiiyMdl:Ipiain Page17of19 Implications Models: Volatility that is1/2(0.1+\0.7)=0.85 which isabouthalfthelocalvolati The call value inthe local volatility model hasanimpliedBSvolatility  %  & %  #  %   $  !  !  !  #     !         strike     *#   "        FUNCTION OFIMPLIED VOLATILITY   barrier lity atstrikeandreflected strike,     reflected strike    smile-lecture8.fm        4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 and-out option under consideration. yield. Thearrows showthestrike(100) and barrier (130) level of the up- We assume a constant discountrateof5% riskless and a zero dividend A hypotheticalvolatilit FIGURE 8.5. 6.46. WhatBlack-Scholesvolatilitydoe skew. Theresultantvalueofthebarrier We canvaluetheUp-and-OutCall bybuild strike at100,andthebarrier130,skewasshownin get youtheexactlycorrect matched by In somecases,thelocalvolatilitiescan Example 1.AnUp-andOutCal No skew:Up-and-OutcallvalueasafunctionofBlack-Scholes Implied Volatility implied volatility 10.00 15.00 20.00 25.00 5.00 any oaiiyMdl:Ipiain Page18of19 Implications Models: Volatility 0.00 Black-Scholes implied 50.00 value. Considerthecasebelow, withaspotand strike (%ofspot) l thathasnoBlack-Schol y skew for optionsof any . 100.00 produceoptionsvaluesthatcannotbe option inthislocalvolatilitymodelis volatility. Noamountofintuitioncan s thiscallpricecorrespondto? ing animplied tree calibrated tothis 150.00 200.0 es Implied Volatility 0 Figure smile-lecture8.fm 8.5 . 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

Copyright Emanuel Derman 2008 The impliedvolatilitythatcomescloses gives thelocal-volatilit the local volatilitymodel.There is NO to a9.5%impliedvolatility. The maximumBlack-Scholesvaluein option withstrike100andbarrier130as Now, weshowedinthepreviouslect per 5strikepoints. of 2thenindicatesthattheslopeth this asfollows.Theslopeoftheskewis with strike100andareflectedcall volatility modelthantheydoinBlack-Scholes. .Theaveragelocalvolatility inthisrangeisabout9%, between inceptionandexpiration),forexam (that payoutthefinalvalueofi options, movingtheirvaluesawayfrom Local volatilitymode of thelocalvolatilities which substantiatestheapproximateclai to slope ofapproximately1vol ity thatisrelevanttovaluationrange 56 5 60 15 .Derman,Kamal, Zou: Th 1. –3% () ⁄ oaiiyMdl:Ipiain Page19of Implications Models: Volatility = ls haveanalogouseffects on y “correct”optionvalue. between spotandstrike. e LocalVolatility Surface. This valueissmallerthanthe“correct”in pt. per5strikepoints,that ndex lesstheminimumvalueofindex s betweenspotprices100and160witha ure thatyoucanthinkofanup-and-out e localvolatilitieswi ano-skewworldis6.00corresponding Black-Scholes implie strike160.Therefore,thelocalvolatil- t toitisabout10% Black-Scholesvalues.Lookbackcalls 1 volpt.per10strikepoints.Therule m theimpliedvolatilityisaverage being replicatedby ple, havehigherdeltasinalocal 1 the valuesofotherexotic isfromvaluesof15% . We canunderstand ll beabout1volpt. d volatilitywhich an ordinarycall smile-lecture8.fm