The LogNormal Mixure Variance Model & Is Applicaions in Derivaives Pricing and Risk Managemen Summary The LNVM model is a mixure of lognormal models and he model densiy is a linear combinaion of he underlying densiies, for insance, log-normal densiies. The resuling densiy of his mixure is no longer log-normal and he model can hereby beer fi skew and smile observed in he marke. The model is becoming increasingly widely used for ineres rae/commodiy hybrids. In his firs par of my review of he model, I examine he mahemaical framework of he model in order o gain an undersanding of is key feaures and characerisics. In a laer second sage of he review we will exend our analysis of he mahemaical framework and assess he implemenaion of he model and is applicaion o price and hedge more complex derivaive srucures. Model Descripion Le s assume ha he forward rae F follows a log-normal process under he forward measure Q as follows 1 : df( ) V (,F )F( )dw As i s well known ha in he Black-Scholes framework he model does no show any skew or smile. One way o overcome his is o combine several same disribuions, for insance, log-normal disribuions, bu wih differen means and variances. Le s define N underlying processes as below: dg ( ) ( )G ( )d ( )G ( )dw, G ( 0) F( 0),i 0, 1,...,N i i i i i i Le i be he probabiliy of he forward process F following he ih underlying dynamics and i 1. Le p(,x) be he densiy for he forward rae process and N i N i 1 p (,x) be he densiy funcion for he ih underlying process so ha p(,x ) ip i(,x ). In order o keep he mean of he forward rae unchanged, he following i 1 condiion should be saisfied: N i( s )ds 0 ie i 1 1 We can verify ha he mean is acually kep unchanged: N N 0 0 0 N i( s )ds i( s )ds i i i i i i i 0 i 0 0 0 i 1 i 1 i 1 Fp(,F )df G p (,G )dg G ( )e F( ) e F( ) As o he form of he volailiy erm in F, he process mus solve Kolmogorov forward equaion (Fokker Planck equaion). We have: 2 p(,x) 1 2 2 2 2 x V (,x)x p(,x) 1 Alhough we use forward rae as an example, his equally applies o oher asse classes, such as FX, equiy, ec. Jonahan Kinlay The Lognormal Mixure Model Page 1
The underlying processes G i also mus solve Kolmogorov forward equaion: 2 1 2 2 i i i 2 i i p (,x) ( )xp (,x) (,x)x p (,x) x 2 x By he lineariy of he derivaive operaor, we have: N N 2 1 2 2 i i i 2 i 2 x i1 i1 and p (, x ) V (, x )x p (, x ) N N N x 2 x 1 2 2 i p i(, x ) i i( ) xp i(, x ) i 2 i (, x )x p i(, x ) i 1 i 1 i 1 Afer some simple algebra, we finally obain: 2 2 N N 2 2 i i i 2i i i x i1 i1 N 2 x ip i(, x ) i 1 V (, x ) x (, x )p (, x ) ( ) yp (,y )dy V, x n i 2 i i 1 i n i 1 i i e e 1 2 i 1 2 i Log x S Log x S 1 2 i 1 2 i 2 2 Where i 0 2 i u u The above formula is used o calculae he local volailiy for F based on he underlying processes of G i. Jonahan Kinlay The Lognormal Mixure Model Page 2
Fig 1. Local Volailiy Surface for Lognormal Mixure Model wih S 0 =100; ={0.5,0.1,0.2};={0.2,0.3,0.5};r=0.035;q=0.035 Brigo and Mercurio show ha under sandard assumpions of coninuiy in he i he SDE has a unique soluion who marginal densiy is given by he mixure of lognormals. p, x n i 1 i 1 x 2 i 1 2 i Ln x S 1 2 i ^2 Jonahan Kinlay The Lognormal Mixure Model Page 3
Fig 2. Marginal Densiy for Lognormal Mixure Model wih S 0 =100; ={0.5,0.1,0.2};={0.2,0.3,0.5};r=0.035;q=0.035 To calculae he undiscouned value of a call opion wih srike K, we have: N N Q Q Q call E F K E i G i K i E G i K i1 i1 We can see ha he opion value is acually he weighed average of opion values from he underlying processes, given explicily as: P r n i 1 i S Ln S k i 1 2 i 2 k Ln S k i 1 2 i 2 Where ф is he cumulaive Normal disribuion funcion, ω is -1 for a pu and +1 or a call and i 0 2 i u u Jonahan Kinlay The Lognormal Mixure Model Page 4
The opion price leads o a wide variey of shapes in he implied volailiy srucure, such as he smile illusraed in figure 3 below. As demonsraed in figure 4, smiles produced by he LNVM model end o become increasingly pronounced as ime o opion expiraion decreases, in line wih observed behavior in he grea majoriy of derivaives markes. Noe, however, ha smiles are symmeric and in order o produce volailiy skews, we need o exend he model o allow he underlying sochasic process o be shifed. 0.255 0.250 0.245 0.240 0.235 90 100 110 120 Fig 3. Implied Volailiy Curve for Lognormal Mixure Model wih S 0 =100; ={0.5,0.1,0.2};={0.2,0.3,0.5};T=1;r=0.035;q=0.035 Jonahan Kinlay The Lognormal Mixure Model Page 5
Fig X. Implied Volailiy Curve for Lognormal Mixure Model wih S 0 =100; ={0.5,0.1,0.2};={0.2,0.3,0.5};T=1;r=0.035;q=0.035 Model Generalizaion A more general model can be creaed by shifing he original process wih an affine ransformaion of he form: A A 0 S Where α is a real consan. By Io s lemma, he asse price process evolves according o: da A d, A A 0 A A 0 dw If k A 0 > 0, we can wrie he opion price as follows: P r n i 1 i 1 A 0 Ln 1 A 0 k A 0 i 1 2 i2 k Ln 1 A 0 k A 0 i 1 2 i2 For α = 0 he process A naurally coincides wih he process for S, while preserving he drif. Jonahan Kinlay The Lognormal Mixure Model Page 6
The parameer α affecs he shape of he implied volailiy surface in wo ways. Firsly, changing α produces almos a parallel shif of he surface downwards (upwards) as α is increased (reduced) in value. Secondly, i moves he locaion of he volailiy minimum o he lef, (lower srike), when α > 0, and o he righ, (higher srike) when α < 0. Examples of hese effecs are shown he figures 5 and 6 following. IVol 0.30 0.25 0.20 0.15 90 100 110 120 Srike Fig 5: Implied Volailiy Skews for Lognormal Mixure Model wih S 0 =100; ={0.5,0.1,0.2};={0.2,0.3,0.5};T=0.5.2;r=0.035;q=0.035 and α = -0.5 (blue), α = -0 (burgundy) and α = 0.5 (yellow) Jonahan Kinlay The Lognormal Mixure Model Page 7
Fig 6. Implied Volailiy Surface for Lognormal Mixure Model wih S 0 =100; ={0.35,0.1};={0.6,0.4};T=0.25;r=0.045;q=0 Model Applicaion and Calibraion If he asse is a sock or index paying a coninuous dividend yield q, we assume ha ineres raes are consan and equal o r for all mauriies. Then every forward measure coincides wih he risk-neural measure having B() = e r as numeraire and he sock price dynamics are as described for he process A, wih μ = r - q. Discree dividends can be handled by seing q = 0 and reducing A 0 by he presen value of all fuure dividends. Skews found in equiy markes can be fied saisfacorily wih large negaive values of α, alhough he qualiy of he fi is likely o deeriorae for near-erm opions where he skews become more pronounced. For FX asses we again assume ha a consan domesic risk-free rae applies, ogeher wih a consan foreign riskfree rae q. Since FX implied volailiy curves end o be smiles raher han skews, he simpler un-shifed version of he model will usually suffice. For forward Libor raes we again se r = q, since he forward rae F(, S, T) is a maringale under he forward measure Q T. The α parameer again can be expeced o play an imporan role in he calibraion of he ofen pronounced skews in ineres rae markes. Calibraion of he LNVM model is rendered slow and somewha problemaic by he large number of degrees of freedom. Brigo and Mercurio recommend using a global search algorihm wih a few model parameers and hen refining he search wih a local algorihm around he las soluion found. The se of consrains i T j 1 i T j T j mus be inroduced in order o avoid imaginary values of he T j 1 σ i s and here is anoher consrain on he parameer α, which mus saisfy k A0 > 0 Brigo and Mercurio advise ha i usually sufficien o limi he number of mixed disribuions o 3 or less, and counsel agains assuming consancy of volailiy over ime, since he fi will worsen considerably. By way of simple illusraion, we fi a mixure of wo lognormal densiies, calibraed o 2-year Libor caple volailiies wih he underlying Libor rae reseing a 1.5 years. The underlying forward rae is 5.32%, he srikes and 4%, 4.25%, 4.50%, 4.75%, 5.00%, 5.25%, 5.50%, 5.75%, 6.00%, 6.25% and 6.50%, wih associaed mid-volailiies of 15.22%, 15.14%, 15.10%, 15.08%, 15.09%, 15.12%, 15.17%, 15.28%, 15.40%, 15.52%and 15.69%. Wih N=2, v i = η i (1.5), i = 1, 2 and λ 2 = 1- λ 1, we look for admissible values of λ 1, v 1, v 2, and α minimizing he squared percenage differences beween model and mid-marke prices. The calibraed model parameers are found o be λ 1 = 0.2859, λ 2 = 0.7140, v 1 = 13.02%, v 2 = 19.85%, and α = 0.1538. The resuling implied volailiies are ploed in figure X, where hey are compared wih marke mid-volailiies. Jonahan Kinlay The Lognormal Mixure Model Page 8
Volailiy 15.7 15.6 15.5 15.4 15.3 15.2 4.5 5.0 5.5 6.0 6.5 Srike Fig. 7: Plos of calibraed volailiies and marke mid-volailiies for a wo-lognormal mixure model Model Risk Characerisics In he following analysis we focus on comparing he risk characerisics of call opions wih varying srikes and mauriies of beween 1 monh and 5 years, under he sandard Black-Scholes model and a mixure of wo shifed lognormal disribuions wih S 0 =100; ={0.35,0.1}; ={0.6,0.4}; r=5%; q=0% and shif parameer α in he range {-0.5, 0.5}. We consider he form and properies of various risk sensiiviies, being firs and second derivaives of he LNVM pricing funcion wih respec o he underlying, S, and ime o mauriy,, for his double-lognormal mixure densiy. Derivaions of he Greeks are given in Appendix 1. SMILE CHARACTERISTICS LNVM is a sicky-dela model: he implied volailiy smile moves wih he underlying. This means ha if spo changes, he implied volailiy of an opion wih a given moneyness doesn' change. So if spo moves from $100 o $110, LNVM would predic ha he implied volailiy of he $110 srike opion would be whaever he $100 srike opion's implied volailiy was before he move (as hese are boh ATM a he ime). We illusrae his characerisic of he LNVM model in figure 8 below. Sicky-dela models, such as LNVM, assume ha ATM volailiy is he raional esimae of he fuure cos of replicaing fuure opions issued now and ha, on average, over he long run, ATM volailiy should be independen of he level of he underlying. One implicaion of his is opion delas under LNVM will exceed Black-Scholes delas, as he analysis in he nex secion of he repor confirms. Jonahan Kinlay The Lognormal Mixure Model Page 9
IVol 0.5 IVol 0 0.32 0.24 0.30 0.22 0.28 0.20 90 100 110 120 Srike 90 100 110 120 Srike 0.5 IVol 0.20 0.18 0.16 0.14 0.12 0.10 80 90 100 110 120 Srike Fig. 8: LNVM Implied volailiy curves for S 0 = 90 (blue), S 0 = 100 (burgundy), S 0 = 110 (yellow) Jonahan Kinlay The Lognormal Mixure Model Page 10
SMILE DYNAMICS The inroducion of he shif parameer α in he exended LNVM model enables asymmeric smiles o be calibraed, as required in equiy and ineres rae markes. A furher aracive feaure of he LNVM model (boh symmeric and exended) is ha smile dynamics parallel he characerisic behavior of many markes, in which he smile becomes increasingly pronounced nearer o expiraion. Boh feaures are illusraed in he example in figure 9. However, alhough he smile dynamics in LNVM are broadly in line wih marke behavior, he model is likely o perform less well for near-erm mauriies, where smiles are ofen much more pronounced and where smile dynamics are ofen more successfully capured by some form of sochasic jump model. This concern applies more o equiy raher han FX markes, in which U shaped smiles are roughly similar for all mauriies. Fig 9. Implied Volailiy Surface for Lognormal Mixure Model wih S 0 =100; ={0.5,0.1,0.2};={0.2,0.3,0.5};T<0.2;r=0.035;q=0.035; α =-0.5; DELTA The chars in figure 10 compare he behavior of he opion dela as he underlying asse changes in value for he Black Scholes model and a lognormal mixure model wih shif parameer α = 0. As foreshadowed in he discussion of smile dynamics, for he LNVM model opion delas are significanly higher han Black-Scholes for all bu he shores opion mauriies. In addiion, he value of dela as a funcion of opion mauriy varies wih he shif parameer α, as he char in figure 11 makes clear: For he LNVM model wih non-negaive α, he dela-decay curves (i.e. he rae of change of opion dela over ime, approximae ha seen in he Black- Scholes model (i.e. he curves are approximaely parallel) For he LNVM model wih negaive α, he rae of dela-bleed exceeds ha derived from he Black-Scholes model. Jonahan Kinlay The Lognormal Mixure Model Page 11
Fig. 10: Comparison of Black-Scholes (upper) and LNVM (lower) opion Dela wih shif parameer α=0. Jonahan Kinlay The Lognormal Mixure Model Page 12
1.0 BS Dela LNVM 0.5 0.9 LNVM 0 LNVM 0.5 0.8 0.7 0.6 0.5 0.4 0 1 2 3 4 5 T Fig. 11: Comparison of Black-Scholes and LNVM ATM opion Dela wih shif parameer α = {-0.5, 0, 0.5} GAMMA The chars in figure 12 and 13 compare he gamma surfaces for call opions wih varying srikes and mauriies. The well-known characerisics of he surface in he Black-Scholes framework are ha gammas decline very quickly for OTM opions and accelerae very rapidly for ATM opions as hey near mauriy. The picure for he LNVM model is similar, bu wih higher gammas for near-he-money opions han Black-Scholes. Of course, he shape of he gamma curve is impaced direcly by he shif parameer α, increasing in kurosis wih α, as demonsraed in figure 14. For negaive values of α, he LNVM gamma curve has a higher dispersion han he Black-Scholes curve, producing larger opion gammas for srikes around {- 20%, +40%} ou-of-he-money and lower gammas compared o Black-Scholes around he ATM srikes. As α increases, he posiion is reversed and LNVM model gamma of ATM opions exceeds ha of Black-Scholes, while Black-Scholes gammas dominae for OTM opions. In figure 15, we compare gamma as a funcion of mauriy for an ATM opion under he Black-Scholes and LNVM models, wih differen values of he shif parameer α. Each of he gamma curves shows is characerisic exponenial increase as ime o mauriy decreases. For opions modeled under LNVN wih negaive shif parameer, opion gamma is uniformly lower han ha derived under he Black-Scholes framework a all mauriies. As he shif parameer is increased, he gamma curve is shifed upwards and for posiive α opion gammas from he LNVM model exceed hose from Black-Scholes a all mauriies. So, oo, does he rae of change of gamma wih respec o ime o expiraion, dγ/d, increase in absolue magniude wih he shif parameer α. Jonahan Kinlay The Lognormal Mixure Model Page 13
Fig. X: Black-Scholes opion Gamma wih shif parameer α = 0 Fig. 13: LNVM opion Gamma wih shif parameer α = 0. Jonahan Kinlay The Lognormal Mixure Model Page 14
0.08 Black Scholes LNVM 0.5 LNVM 0 0.06 LNVM 0.5 0.04 0.02 0.00 40 60 80 100 120 140 160 180 Srike Fig. 14: Comparison of Black-Scholes and LNVM Gamma for call opion wih 6m mauriy for differen values of shif parameer α. 0.20 0.15 0.10 Black Scholes LNVM 0.5 LNVM 0 LNVM 0.5 0.05 0.00 0 1 2 3 4 5 T Fig. 15: Comparison of Black-Scholes and LNVM Gamma for ATM call opion for differen values of shif parameer α. THETA Thea, he rae of opion decay, is highly non-linear around he ATM srike in he Black-Scholes framework, as shown in figure 16, and he paern of he hea surface under he LNVM model is much he same (figure 17). Jonahan Kinlay The Lognormal Mixure Model Page 15
Fig 16: Opion Thea from he Black-Scholes model Fig 17: Opion Thea from he LNVM model Jonahan Kinlay The Lognormal Mixure Model Page 16
The rae of opion decay is correlaed wih he shif parameer α, as can be seen from figure 18: he rae of decline in opion value increases as α becomes more negaive. As in he Black-Scholes model, he rae of opion decay acceleraes as opions move owards expiraion and he rae of decay is correlaed wih he shif parameer α, as illusraed in figure 19. Decay raes in he LNVM framework will exceed hose in he Black-Scholes model for sufficienly negaive values of α. Price 40 BS LNVM 0.5 LNVM 0 LNVM 0.5 30 20 10 0 1 2 3 4 5 T Fig 18: ATM opion value decay in he Black-Scholes and LNVM model frameworks Thea 5 10 BS Thea 15 LNVM 0.5 LNVM 0 LNVM 0.5 20 0 1 2 3 4 5 T Fig 19: ATM comparison of Thea in he Black-Scholes and LNVM model frameworks Jonahan Kinlay The Lognormal Mixure Model Page 17
VOLATILITY RISK The sandard volailiy risk sensiiviy parameer Vega is problemaic o evaluae in he conex of a model such as LNVM, which makes use of a volailiy erm srucure raher han a single volailiy parameer as in Black Scholes. Insead we focus on a DV01, i.e. he change in opion value of a 1% increase in implied volailiy. 0.6 0.5 0.4 0.3 0.2 0.1 0.5 Black Scholes LNVM Seepener LNVM Flaener LNVM Parallel 0.4 0.3 80 100 120 140 160 0 Black Scholes LNVM Seepener LNVM Flaener LNVM Parallel 0.2 0.1 0.4 0.3 0.2 80 100 120 140 160 0.5 Black Scholes LNVM Seepener LNVM Flaener LNVM Parallel 0.1 80 100 120 140 160 Fig 20: Volailiy sensiiviy (DV01) for %1 increase in ATM implied volailiy Jonahan Kinlay The Lognormal Mixure Model Page 18
In he LNVM conex here are a wide variey of ways in which he volailiy erm srucure can be perurbed and we consider boh parallel and nonparallel moves. As shown in figure 20, he impac of changes in volailiy depends no only on he level and seepness of he volailiy curve, bu also on he shape parameer α. Broadly, as migh be expeced, changes in opion value are inversely relaed o he shif parameer, largely because of is impac on he level and slope of he volailiy skew: For α = 0, he impac of a 1% increase in volailiy on ATM opion values is approximaely he same under he Black-Scholes and LNVM models, while he relaive impac on OTM opion values depends criically on wheher he change in he volailiy curve is assumed o be parallel or non-parallel. For α < 0, he impac of a 1% increase in volailiy is greaer under LNVM han Black-Scholes across all srikes (excep higher srikes in he vol-flaening scenario). For α > 0, he impac of a 1% increase in volailiy is greaer under Black-Scholes han LNVM across all srikes. The inverse relaionship beween shif parameer and volailiy sensiiviy is furher illusraed in figure 21. As in he Black-Scholes framework, in he LNVM model volailiy sensiiviy is much greaer for long-daed opions han opions close o expiraion as illusraed in figure 22. Fig 21: Impac of shif parameer α on volailiy sensiiviy (DV01) Jonahan Kinlay The Lognormal Mixure Model Page 19
Fig 22: Volailiy sensiiviy (DV01) in he LNVM model as a funcion of opion mauriy Jonahan Kinlay The Lognormal Mixure Model Page 20
Implicaions for Trading, Hedging and Risk Managemen Through judicious selecion of an appropriae value of he shif parameer α, he exended version of LNVM allows us o calibrae smiles and skews ypical of many differen markes. For example, a value of α close o zero would be appropriae for modeling he symmeric smiles found in FX markes, while a negaive value of α would be used o reproduce he negaive skews ypically seen in equiy markes. In erms of smile dynamics, however, he LNVM model has imporan characerisics ha may be difficul o reconcile wih observed marke behaviors. Firsly, i is going o be challenging for a floaing (sickydela) smile model such as LNVM o reproduce he variable smile-dynamical behavior which characerizes equiy markes. These have a persisen negaive skew and, during periods of posiive reurns, ypically behave in a way bes described by a sicky-srike model: implied volailiy ends o fall as he underlying coninues o appreciae; ATM opions are he mos liquid; and hese mos liquid opions are sold more and more cheaply as if raders never had o worry abou fuure marke declines. In his regime of marke complacency or greed, opion delas are close o Black-Scholes levels. Conversely, during periods of sress, marke behavior can ofen bes be described by a sicky-implied-ree model. In his view, he skew is aribuable o an expecaion of higher volailiy as he marke declines. ATM volailiy falls much more quickly han he skew would appear o imply and opion delas are lower han Black-Scholes delas. During period of normalcy, we can expec he sicky-dela ype of model o describe he skew dynamics quie well and indeed, an argumen can be made ha, a leas in he long erm, he regimes of fear and greed end o average ou o produce he kind of behavior well described by floaing smile models like LNVM. However, during periods of rapid appreciaion LNVM delas may significanly over-esimae he sicky-srike delas peraining in he marke and his dela-overesimaion is likely o ge even worse in periods of marke sress. We have also noed elsewhere in his repor ha he praciioner is likely o experience difficuly in calibraing he very much seeper smiles for shor-daed mauriies, which are ofen more accuraely represened as he produc of a jump-diffusion process. Similar concerns may also hold for commodiy derivaives or hybrids, given he exended rends in meals, grains, energy and oher commodiies, and he recen sell-off and increased volailiy in hose markes. Here, oo, we migh expec o see changes in smile dynamics, from sick-srike o sicky implied-ree, creaing difficulies for he LNVM model in erms of calibraion and risk sensiiviy esimaion. As far as ineres rae markes are concerned, no only are skews ypically very sicky, bu here is he added complicaion ha hey are likely o be calibraed wih posiive shif parameer under LNVM. As our previous analysis has shown, his will inflae no only delas, bu also gammas, relaive o hose derived from Black-Scholes, o an even greaer exen han in markes where he LNVM model calibraes wih a zero or negaive shif parameer value. LNVM is likely o be more successful in modeling FX/hybrid exoics, given he floaing smile dynamics ha ypically characerize currency markes, which are fairly well described by sicky dela models, and where smiles are relaive consisen across mauriies. Jonahan Kinlay The Lognormal Mixure Model Page 21
A major concern for raders and risk managers is he complexiy of volailiy risk in he LNVM model. Volailiy sensiiviy becomes an increasingly imporan facor for longer mauriies in boh he LNVM and Black-Scholes frameworks. However, volailiy sensiiviy is greaer in he LNVM conex compared o Black-Scholes when calibraed wih negaive shif parameer, for example in equiy markes. Models calibraed for FX markes are likely o show similar levels of opion sensiiviy for LNVM and Black-Scholes, a leas for ATM opions, while LNVM models calibraed for ineres rae markes (i.e. wih posiive shif parameer) and likely o exhibi lower volailiy sensiiviy han Black-Scholes. Furhermore, he impac of volailiy changes will be fel across a much wider range of srikes, depending on wheher he perurbaions o he volailiy erm srucure affec he slope of he curve or no. Of course, some of hese concerns will fall away in he conex of hedging, where a replicaing porfolio of vanilla opions can be consruced using a consisen model framework. Here, however, he primary concern becomes parameer sabiliy, absen which he rader could easily find himself having o radically aler hedge posiions even when he marke iself is relaively sable. This is a significan concern for he LNVM model, which has several degrees of freedom. Taking he simple calibraion exercise described on page 9 as an illusraion of he problem, i can be seen from figure 23 ha he objecive funcion (shown in jus wo of is four dimensions) used o calibrae he model is quie fla. I is no difficul o envisage more complex calibraion problems, for example, where we use a mixure of hree raher han jus wo lognormal disribuions, which could easily resul in local minima being found wih very differen parameer esimaes. I can readily be demonsraed ha a model as complex as LNVM is highly suscepible o parameer insabiliy, even for he relaively simple calibraion problem presened here. Changing jus one of he observed volailiies o creae an oulier a he 4.75% caple srike produces very differen parameer esimaes: {v10.0620721,v20.179128,0.121851,10.0593295},compared wih {v10.130249,v20.198467,0.153773,10.285982}for he original daa se. The effecs are shown in he char in figure 24. In his simple case a remedy could probably be achieved wih a more sophisicaed objecive funcion ha enables ouliers o be deal wih by means of a weighing marix. All he same, i is no difficul o devise scenarios where a combinaion of very small changes in observed marke daa could produce very differen esimaes of model parameers. Jonahan Kinlay The Lognormal Mixure Model Page 22
Fig 23: Objecive funcion for simple calibraion problem Volailiy 15.6 15.5 15.4 15.3 15.2 15.1 4.0 4.5 5.0 5.5 6.0 6.5 Srike Fig 24: Re-calibraed volailiies and marke mid-volailiies for a wo-lognormal mixure model Jonahan Kinlay The Lognormal Mixure Model Page 23
Conclusions 1. An aracive feaure of he exended LNVM model is is abiliy o calibrae a wide variey of symmeric and asymmeric volailiy smiles and skews. 2. The model can also replicae smile dynamics o some degree, showing more pronounced smiles a shorer mauriies, as observed in marke volailiies. I is unlikely, however, ha i can successfully replicae he very pronounced smiles a near-erm mauriies, which are more successfully capured by a (sochasic) jump-diffusion model. 3. LNVM is likely o be more successful in modeling he dynamics of symmeric, sable smiles in FX markes han he unsable negaive skews in equiy markes or posiive skews in ineres rae markes. 4. LNVM is a sicky-dela model, where he smile floas wih he level of he underlying, and his resuls in delas higher han Black-Scholes. Depending on wheher he curren regime can be characerized as driven by greed or fear, many markes display dynamical smile behavior bes modeled wih a sicky srike, or sicky implied ree model, respecively. In hese circumsances LNVM is likely o produce (very) inflaed dela values during srong up-rends and even more so in periods of marke sress. 5. As far as ineres rae markes are concerned, no only are skews ypically very sicky, bu here is he added complicaion ha hey are likely o be calibraed wih posiive shif parameer under LNVM. This will inflae no only delas, bu also gammas, relaive o hose derived from Black-Scholes, o an even greaer exen han in markes where he LNVM model calibraes wih a zero or negaive shif parameer value. The rae of decay of ime value is also posiively correlaed wih he shif parameer. 6. Volailiy sensiiviy is highly dependen on he shif parameer used o calibrae he LNVM model, wih which i is inversely correlaed, and he impac of volailiy changes will be fel across a wider range of srikes if he slope of he volailiy erm srucure changes. Managing he changes in volailiy sensiiviy of an exoics porfolio modeled wih LNVM is likely o prove exremely challenging, given i high degree of dependency on key model parameers and assumpions abou how perurbaions affec he volailiy erm srucure. These difficulies will be exacerbaed for longer-mauriies and his may limi he pracical applicabiliy of he model for longer-daed securiies. 7. LNVM is a complex model wih many degrees of freedom, including he choice of n lognormal disribuions, n volailiy parameers, n-1 weighing parameers and a shif parameer. Parameer insabiliy is likely o arise because of he flaness he objecive funcion around he global minimum, resuling in parameer values associaed wih local minima. Brigo and Mercurio recommend a wo-sage search procedure, using a local opimizaion search around he minimum idenified by an iniial global search. An objecive funcion ha penalizes for he number of parameers, reduces he influence of ouliers and conrols he degree of over-fiing would probably also help sabilize parameer esimaes. References Brigo, D., and Mercurio, F, Lognormal-mixure dynamics and calibraion o marke volailiy smiles, Banca IMI inernal repor, 2001 Jonahan Kinlay The Lognormal Mixure Model Page 24