CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS

Transcription

1 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS GRAEME WEST Abstract. Recently the SABR model has been developed to manage the opton smle whch s observed n dervatves markets. Typcally calbraton of such models s straghtforward as there s adequate data avalable for robust extracton of the parameters requred as nputs to the model. We consder calbraton of the model n stuatons where nput data s very sparse. Although ths wll requre some creatve decson makng, the algorthms developed here are remarkably robust and can be used confdently for mark to market and hedgng of opton portfolos. 1. Why another skew model? Vanlla OTC European optons or European futures optons are prced and often hedged usng respectvely the Black-Scholes or Black model. In these models there s a one-to-one relaton between the prce of the opton and the volatlty parameter σ, and opton prces are often quoted by statng the mpled volatlty σ mp, the unque value of the volatlty whch yelds the opton prce when used n the formula. In the classcal Black-Scholes-Merton world, volatlty s a constant. But n realty, optons wth dfferent strkes requre dfferent volatltes to match ther market prces. Ths s the market skew or smle. Typcally, although not always, the word skew s reserved for the slope of the volatlty/strke functon, and smle for ts curvature. Handlng these market skews and smles correctly s crtcal for hedgng. One would lke to have a coherent estmate of volatlty rsk, across all the dfferent strkes and maturtes of the postons n the book. The development of local volatlty models n Dupre 1994), Dupre 1997), Derman & Kan 1994), Derman, Kan & Chrss 1996) and Derman & Kan 1998) was a major advance n handlng smles and skews. Another crucal thread of development s the stochastc volatlty approach, for whch the reader s referred to Hull & Whte 1987), Heston 1993), Lews 2000), Fouque, Papancolaou & Srcar 2000), Lpton 2003), and fnally Hagan, Kumar, Lesnewsk & Woodward 2002), whch s the model we wll consder here. Date: November 28, Key words and phrases. SABR model, equty dervatves, volatlty skew calbraton, llqud markets. Thanks to Adam Myers of ABSA Bank, Leon Sanderson of Nedcor Bank, Arthur Phllps of RskWorX, and Patrck Hagan of Bloomberg s. 1

2 2 GRAEME WEST Local volatlty models are self-consstent, arbtrage-free, and can be calbrated to precsely match observed market smles and skews. Currently these models are the most popular way of managng smle and skew rsk. Possbly they are often preferred to the stochastc volatlty models for computatonal reasons: the local volatlty models are tree models; to prce wth stochastc volatlty models typcally means Monte Carlo. However, t has recently been observed Hagan et al. 2002) that the dynamc behavour of smles and skews predcted by local volatlty models s exactly opposte the behavour observed n the marketplace: local volatlty models predct that the skew moves n the opposte drecton to the market level, n realty, t moves n the same drecton. Ths leads to extremely poor hedgng results wthn these models, and the hedges are often worse than the nave Black model hedges, because these nave hedges are n fact consstent wth the smle movng n the same drecton as the market. To resolve ths problem, n Hagan et al. 2002) the SABR model s derved. The model allows the market prce and the market rsks, ncludng vanna and volga rsks, to be obtaned mmedately from Black s formula. It also provdes good, and sometmes spectacular, fts to the mpled volatlty curves observed n the marketplace. More mportantly, the SABR model captures the correct dynamcs of the smle, and thus yelds stable hedges. 2. The model Stochastc volatlty models are n general charactersed by the use of two drvng correlated Brownan motons, one whch determnes the ncrements to the underlyng process and the other determnes the ncrements to the volatlty process. For example, the model of Hull & Whte 1987) can be summarsed as follows: 1) 2) 3) df = φf dt + σf dw 1 dσ 2 = µσ 2 dt + ξσ 2 dw 2 dw 1 dw 2 = ρdt where φ, µ and ξ are tme and state dependent functons, and dw 1 and dw 2 are correlated Brownan motons. Smlarly, the model of Heston 1993) proceeds wth the par of drvng equatons 4) 5) df = µf dt + σf dw 1 dσ = βσdt + δdw 2 where ths tme µ, β and δ are constants.

3 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 3 As another example, the models of Fouque et al. 2000) are varatons on the followng ntal set-up: 6) 7) df = µf dt + σf dw 1 dy = αm y)dt + βdw 2 where ths tme α, m and β are constants, and for example y = ln σ. Here, the process for y s a mean revertng Ornsten-Uhlenbeck process. The model we consder here s known as the stochastc αβρ model, or SABR model. Here 8) 9) 10) df = αf β dw 1 dα = vα dw 2 dw 1 dw 2 = ρ dt where the factors F and α are stochastc, and the parameters β, ρ and v are not. α s a volatlty-lke parameter: not equal to the volatlty, but there wll be a functonal relatonshp between ths parameter and the at the money volatlty, as we shall see n due course. The constant v s to be thought of as the volatlty of volatlty, a model feature whch acknowledges that volatlty obeys well known clusterng n tme. The parameter β [0, 1] determnes the relatonshp between futures spot and at the money volatlty: β 1 ndcates that the user beleves that f the market were to move up or down n an orderly fashon, the at the money volatlty level would not be sgnfcantly affected. β << 1 ndcates that f the market were to move then at the money volatlty would move n the opposte drecton. The closer to 0 the more pronounced would be ths phenomenon. Furthermore, the closer β s to 1 respectvely, 0) the more lognormal- respectvely, normal-) lke s the stochastc model. 3. The opton prcng formula A desrable feature of any local or stochastc volatlty model s that the model wll reproduce the prces of the vanlla nstruments that were used as nputs to the calbraton of the model. Materal falure to do so wll make the model not arbtrage free and render t almost useless. A sgnfcant feature of the SABR model s that the prces of vanlla nstruments can be recovered from the model n closed form up to the accuracy of a seres expanson). Ths s dealt wth n detal n Hagan et al. 2002, Appendx B). Essentally t s shown there that the prce of a vanlla opton under the SABR model s gven by the approprate Black formula, provded the correct mpled volatlty s used. For gven

4 4 GRAEME WEST α, β, ρ, v and τ, ths volatlty s gven by: α ) σx, F ) = 1 β) 2 α ρβvα F X) 1 β ρ2 F X) 1 β)/2 ) ) 24 v 2 τ ] F X) 1 β)/2 [ β)2 24 ln 2 F X + 1 β) ln4 F X z χz) 12) 13) z = v α F X)1 β)/2 ln F X ) 1 2ρz + z2 + z ρ χz) = ln 1 ρ Although the formula appears fearsome, t s closed form, so practcally nstantaneous. Ths formula of course can be vewed as a functonal form for the volatlty skew, and so, when ths volatlty skew s observable, we have some sort of error mnmsaton problem, whch, subject to the caveats rased n Hagan et al. 2002), s qute elementary. The thess of ths artcle s the same calbraton problem n the absence of an observable skew, n whch case, we need a model to nfer the parametrc form of the skew gven a hstory of traded data. Note as n Hagan et al. 2002) that f F = X then the z and χz) terms are removed from the equaton, as z then χz) = 1 n the sense of a lmt1, and so α ) σf, F ) = 1 β) 2 α ρβvα F 2 2β 4 F 1 β F 1 β ) ) + 2 3ρ2 24 v 2 τ 4. The market we consder for ths analyss We consder the equty futures market traded at the South Afrcan Futures Exchange. For detals of the operaton of ths market the reader s referred to SAFEX 2004), West 2004, Chapter 10). Ths market s charactersed by an llqudty that s gross compared to other markets. We wll focus on the TOP40 the ndex of the bggest shares, as determned by free float market captalsaton and lqudty) futures optons contracts. year. Contracts exst for expry n March, June, September and December of each Amongst these, the followng March contract s the most lqud, along wth the nearest contract. Nevertheless, the March contract only becomes lqud n anythng lke a meanngful manner about two years before expry. Packages of optons on the March 2004 contract traded a total of perhaps 800 tmes, for whch there were about double that number of dfferent strkes. By a package, we mean not only a sngle trade, but a collar, butterfly, condor, etc.) The full set of strke and volatlty hstory s not publshed. Nevertheless, we sourced, va one of the largest brokers, a sgnfcant porton of the hstory possbly 70% or more), whch we have taken as a representatve sample for the purposes of buldng our model. Despte ths, there s a sgnfcant dervatves market, chefly comprsng over the counter structures sold by the merchant banks to asset and other wealth managers. The banks need to hedge ther exposures, and they 1 Some care needs to be taken wth machne precson ssues here. One can have that z 0 and χz) = 0 to double precson. Ths needs to be trapped, and the lmt result nvoked, agan puttng z χz) = 1.

5 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 5 do a sgnfcant amount of ths n the exchange market. Furthermore, as s usual, the relevant models of the skew, whch wll be appled equally to over the counter products as well as exchange traded products, wll be parametersed va exchange traded nformaton. Thus t s necessary to have a robust model of the dervatve skew for mark to market and hedgng of postons. Untl Aprl 2001 SAFEX calculated margn requrements on a flat volatlty. At that tme a skew was ntroduced nto the mark to market and margnng of exchange postons, although most players were aware of the skew and had and have) models of the skew snce sgnfcantly before that tme. 2 The constructon of the skew was ntally supposed to be va an aucton system, but has become merely a monthly poll, of moneyedness and correspondng addton or subtracton of volatlty bass ponts, from the quoted at the money volatlty. So here we note explctly that bds and offers for the usual set of at and away from the money strkes smply do not exst n ths market. Beng merely a poll, the dervatves desks do not have to put ther money where ther mouth s concernng ther contrbuton to the poll, and so, although not grossly naccurate, t s common knowledge that the exchange quoted skew cannot be used for tradng, and by preference should not be used for mark to market although many rsk/back offce/audt functons, n order to satsfy the requrement of suffcent ndependence from the front offce, mght do so). It s the scenaro descrbed here that led to the requrement from some major players n the South Afrcan market for an accurate and objectve skew constructon methodology, whch the model descrbed here s amed at provdng. One pont that needs to be noted here s that futures optons are Amercan and fully margned, that s, the buyer of optons does not pay an outrght premum for the opton, but s subject to margn flow, beng the dfference n the mark to market values on a daly bass. It can be shown see West 2004, Chapter 10)) that the approprate opton prcng formula n ths settng s 15) 16) 17) 18) V C = F Nd 1 ) XNd 2 ) V P = XN d 2 ) F N d 1 ) d 1,2 = ln F X ± 1 2 σ2 τ σ τ τ = T t It can be shown that t s sub-optmal to exercse ether calls or puts early, and so one should not be surprsed that the opton prcng formula are Black lke even though the opton s Amercan. Furthermore, the fact that the optons are fully margned has the attractve consequence that the rsk free rate does not appear n the prcng formulae. Ths s ndeed fortunate as the South Afrcan yeld curve tself s subject to a paucty 2 Although sgnfcantly, not all. Real Afrca Durolnk, a smaller bank, but major player n the equty dervatves market, faled wthn days of the ntroducton of the skew, as they were completely unprepared for the dramatc mpact the new methodology would have on ther margn requrements. See West 2004, 13.4).

6 6 GRAEME WEST of data compared to many markets, and hence may requre some art n constructon, whch wll typcally be propretary. 5. The nterpretaton of parameters 5.1. The β parameter. As n Hagan et al. 2002), 14) shows that 19) ln σf, F ) = ln α 1 β) ln F + and so the value of β s estmated from a log-log plot of σf, F ) and F. Some emprcal analyss suggests that the value of β depends on tme regmes: whether the contract s far, mddle, or near to expry. We use these terms nformally. By far we mean about two years to expry, near s perhaps sx months or less, mddle s n-between.) The contract does not trade meanngfully untl at least two years to expry; although the underlyng futures may trade, there wll be lttle or no opton actvty. See Fgure 1 where we see that the qualty of a ftted regresson lne as n 19) would very much depend on a data selecton crteron. Ths naturally suggests a tme weghted regresson, very much as for Exponentally Weghted Movng Average volatlty calculatons. 3 In Fgure 2 we show the resultant evoluton of the β value and the evoluton of the correlaton coeffcent, whch s also calculated usng tme weghtng. Fgure 1. A log-log plot for the March 04 contract 3 The only dfference here s that we do not make the assumpton of zero means, whch we do when usng returns to calculate volatltes. The mplementaton s elementary.

7 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 7 Fgure 2. The evoluton of the estmate of β for the March 04 contract One feature we can note s that the value of β deterorates towards 0 as the contract draws to expry. Ths was a feature found to be common to all expres. An analyss of the March 05 contract wll be ncluded later, n a further analyss, where we wll see that there s an argument - as n Hagan et al. 2002) - to smply choose a value of β, and stck wth t for the entre lfe of the contract The α parameter. Ths parameter s calbrated to the level of at-the-money volatlty. There s perhaps a common percepton amongst some market partcpants that t s the at-the-money volatlty. However, what one rather does s retool the SABR model to have the at-the-money volatlty as an nput and that the correct value of α be calculated nternally. Thus, as the at-the-money volatlty has a term structure and changes frequently, so too does the value of α, albet nvsbly. To obtan α from σ atm, we nvert 14). Dong so, we easly see that α s a root of the cubc 1 β) 2 τ 20) 24F 2 2β α3 + ρβvτ 4F 1 β α ) 3ρ2 v 2 τ α σ atm F 1 β = 0 24 where we are assumng that we have already solved for ρ and v. For typcal parameter nputs, ths cubc has only one real root, but t s perfectly possble for t to have three real roots, n whch case we seek the smallest postve root 4. One wants a rapd algorthm to fnd α to double precson, as then when n code fndng the skew volatlty for an opton whch s n fact at-the-money, one recovers the at-the-money volatlty exactly. 4 When there are three real roots, they are of the order of -1000, 1 and So we take the root of order 1.

8 8 GRAEME WEST We use the Tartagla method as publshed by Cardano n the 16th century!) to fnd the desred real root. For ths, we use the mplementaton and code n Press, Teukolsky, Vetterlng & Flannery 1992, 5.6). See Wessten ) for a synopss of the hstory of these root fndng methods. Havng now reformulated the opton skew wth σ atm and not α as nput n other words, α s not a constant), the skew volatlty s n fact nvarant under X F, n other words σx, F ) = σx/f, 1). Thus we can perform calbraton on relatve strkes rather than absolute strkes. Ths s very convenent, as the trader should thnk n terms of relatve strkes. It should be remarked that n an llqud market such as the one we are consderng, even the at-the-money volatlty can be a tenuous nput. Ths s because of the mark-to-market mechansm employed by the exchange. Even f new optons trade, f they are not near the money, the mark-to-market at-the-money volatlty wll not be altered. Therefore, the sophstcated model user may, on a day by day bass, wsh to modfy the at-the-money volatlty nput to ths model, n order to attempt to nfer - from the away from the money optons traded - at what level at-the-money optons would have traded f they had ndeed done so. Of course, even ths s subject to error, not only because t s outrght speculatve, but because the volatltes that were ndeed dealt could be parts of packages, as we wll see later. 6. Calbraton to exstng market data The calbraton procedure s as follows: we ft β usng the log-log plot. Accordng to Hagan et al. 2002), t may be approprate to ft ths parameter n advance, and never change t. We wll return to ths pont later. The values of ρ and v need to be ftted. As already dscussed, the value of F and the value of σ atm are nputs, and gven these and the values of ρ and v, α s no longer a requred nput parameter. It s possble to smply specfy a dscrete skew nput by the dealer) and fnd the SABR model whch best fts t. But we can be more ambtous, and ask ourselves to fnd the SABR model whch best fts gven traded data, ndependently of any dealer nput as to the skew. Thus, we wll not a pror have a dscrete skew to whch we calbrate the SABR model; rather, we seek that SABR model whch provdes a best ft to the traded data. As already dscussed, we fx n advance the value of β. Then, for any nput par ρ, v), we determne an error expresson err ρ,v, whch s a measure of the dstance from optmally re-mapped) traded volatltes to the skew mpled by these parameters. The trades that have been observed n the market may be weghted for age, for example, by usng an exponental decay factor: the further n the past the trade s, the less contrbuton t makes to the optmsaton. Then, we seek the mnmum of these error expressons err ρ,v amongst all pars ρ, v), for whch we use the Nelder-Mead smplex search. See Press et al. 1992, 10,4). Also see Spradln 2003) whch we have used as

9 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 9 a gude for mplementng ths algorthm n two dmensons. The Nelder-Mead algorthm s a non-analytc search method that s very robust. Note that the error expresson s essentally non-dfferentable because t mplctly nvolves the root of a cubc n other words, the dfferentaton nvolved would be horrendous). Thus for the second procedure we use a non-dfferentable approach, for the frst, all analytc procedures are avalable. In ths applcaton both optmsaton procedures are extremely rapd. The Nelder-Mead method needs to have error traps bult n: that 1 ρ 1 and v > 0. The method mght stray out of these bounds n the ntal stages of expanson, for example). The frst condton s acheved by collarng ρ wthn [ 1, 1] whle the second condton s ensured by boundng v below by 0.01, whch suffces. For any nput par ρ, v), the mappng procedure for hstorcal trades wll be as follows Sngle trades. A sngle trade for Q-many optons made at date t 1 trades at σ tr := σ tr F t 1 ), X, t 1 ). If the trade had been done on the skew, t would have been done at a volatlty of 21) σ mod := σf t 1 ), X, t 1, σ atm t 1 ), ρ, v) The contrbuton to the error term err ρ,v modulo weghtng for age) wll be deemed to be 22) f = Q V σ mod) [ ] 2 σtr σ mod where the symbol V refers to the vega of the opton wth that strke. 5 Ths s a sensble modellng method as the mportance of the volatlty parameter ft s proportonal to the vega of the opton. Naturally, then, greater weght s gven to near the money optons. Ths s most sutable n markets where optons, when dealt, are typcally near the money Trade sets. An ssue whch often arses, s that certan strateges e.g. bull or bear spreads, butterfles, condors) trade for a par, trple or quadruple of volatltes whch may appear off market. In realty t s the prce of the strategy that s tradng and so a relevant set of volatltes may be found whch s closer to the market than may at frst appear. To acheve ths, we frst determne the prce of the strategy, mplemented at tme t. Ths s gven by 23) 24) 25) P = d 1,2 = n Q η [F t)nη d 1) X Nη d 2)] =1 τ = T t F t) ln X ± 1 2 σt, X ) 2 τ σt, X ) τ 5 Of course, some expermentaton wth the choce of the weght determned by the quantum s necessary. One could choose Q 2 for example, or ndeed any postve weght. There wll be no requrement of any smoothness of the weght n what follows.

10 10 GRAEME WEST where T s the expry date of the optons and tme s measured n years, η = ±1 for a call/put, Q s the number of optons traded at the th strke as part of the strategy, and σt, X ) s the quoted volatlty for the th strke. We would lke to re-map ths to the dentcal strategy, wth the same prce, but booked at dfferent volatltes. These volatltes are found to be as close as possble to the volatltes on the skew curve. Thus, we would lke to mnmse 26) f σ ft 1, σ ft 2,..., σ ft ) n = n =1 Q V σ mod ) σ ft σ mod ) 2 where ft denoted ftted volatltes and mod denotes volatltes from the SABR model, subject to 27) 28) 29) P = g σ ft 1, σ ft 2,..., σ ft ) n := n Q η [F t)nη d 1) XNη d 2)] =1 d 1,2 = F t) ln X ± 1 2 σft σ ft τ τ = T t 2 τ Once done, the value of f wll contrbute to err ρ,v. Ths mnmsaton s done usng the method of Lagrange multplers. In order to mnmse f σ ft 1, σ ft 2,..., σ ft ) n subject to g σ ft 1, σ ft 2,..., σ ft ) n = P, we solve the smultaneous set of equatons 30) 31) f = λ g g σ ft 1, σ ft 2,..., σ ft ) n = P whch easly smplfes to 32) 33) Let V 2 σ ) = 2 V σ 2 2 Q V σ mod ) σ ft g σ mod ) λq V ) σ ft 1, σ ft 2,..., σ ft n be the volga of the th opton. Note that σ ft ) P = 0 = 0 1 n) 34) 35) Vσ ) = F τn d 1) V 2 σ ) = F τn d 1) d 1d 2 σ

11 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 11 For convenence, put 36) 37) L = 2 Q V σ mod ) K = L σ mod We have a system of n + 1 non-lnear equatons n σ ft 1, σ ft 2,..., σ ft n, and λ) and so we use the multdmensonal Newton-Raphson method for ths part of the problem. See, for example, Press et al. 1992, 9.6).) By the economc nature of the problem t s farly clear that the zero s unque and that pathology wll not arse n the use of the Newton-Raphson method. Let x = σ 1, σ 2,..., σ, λ) be the unknown and requred vector, where we have dropped the superscrpt ft. The teraton s σ mod 1 σ mod ) x 0 = σ mod n 0 39) 40) 41) x m+1 = x m J 1 F L 1 σ 1 K 1 λq 1 Vσ 1 ) L 2 σ 2 K 2 λq 2 Vσ 2 ) F =. L n σ n K n λq n Vσ n ) g σ 1, σ 2,..., σ n ) P L λq Wσ ) f = j n 0 f j n [J] j = Q Vσ ) f n, j = n + 1 Q j Vσ j ) f = n + 1, j n 0 f = j = n + 1 Here J s the Jacobean, the matrx of partal dervatves: [J] j = F x j. The nverse of J s found va the LU decomposton Press et al. 1992, 2.3). Convergence s very rapd. 6 Thus, for any strategy booked, equvalent volatltes can be found whch are most compatble wth the SABR model selected. The error for the nput par ρ, v) s the sum err ρ,v of the above errors, possbly weghted for age. We then seek, amongst all ρ and v, the mnmum of these expressons, usng the two-dmensonal Nelder-Mead algorthm. 6 Note that the matrx wll almost certanly not be of sze greater than 5 5.

12 12 GRAEME WEST Fgure 3. The error quanttes for ρ and v. As one can see n Fgure 3 - ths result s typcal - the choce of parameters s farly robust, wth the mnmum found at the bottom of a shallow valley. As ponted out n Hagan et al. 2002), the dea s that the parameter selecton change nfrequently perhaps only once or twce a month) whereas the nput values of F and σ atm change as frequently as they are observed. Ths s n order to ensure hedge effcency. We now consder the evoluton of parameters for the Mar 05 contract. Once agan, the features that occur are typcal. As has been mentoned, t can be argued that the use of these algorthms to fnd the best parameters should not be undertaken too frequently. We performed two analyses: n the frst nstance, the fndng of the β parameter every day, usng the exponentally weghted regresson methodology mentoned, and then fndng the consequental ρ and v. The evoluton s shown n Fgure 5. In the second nstance, we fxed by economc consderatons a value of β = 70% throughout, and agan found the consequental ρ and v. The evoluton s shown n Fgure 6.

13 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 13 Fgure 4. The SABR model for March 2005 expry, wth traded quoted) volatltes squares), and wth strateges recalbrated to a ftted skew trangles), and the ftted skew tself sold lne). Hstorcal trades have smply been shfted by the dfference n the then and current at-the-money volatlty; ths s smply for graphcal purposes. No stcky rules are assumed n ths analyss. Interestngly, n the second nstance, the parameters ρ and v only change nfrequently 7. Ths agan may be a feature that favours the choce of a sngle β whch, n the absence of extraordnary events, remans constant throughout the lfe of the contract: parameters remanng unchanged mples, as seen n Hagan et al. 2002), lower hedgng costs. References Derman, E. & Kan, I. 1994), Rdng on a smle, Rsk 72). Derman, E. & Kan, I. 1998), Stochastc mpled trees: Arbtrage prcng wth stochastc term and strke structure of volatlty, Internatonal Journal of Theory and Applcatons n Fnance 1, Derman, E., Kan, I. & Chrss, N. 1996), Impled trnomal trees of the volatlty smle, Journal of Dervatves 4summer). Dupre, B. 1994), Prcng wth a smle, Rsk 71). Dupre, B. 1997), Prcng and hedgng wth smles, n M. Dempster & S. Plska, eds, Mathematcs of Dervatve Securtes, Cambrdge Unversty Press, Cambrdge, pp Fouque, J.-P., Papancolaou, G. & Srcar, K. R. 2000), Dervatves n Fnancal Markets wth Stochastc Volatlty, Cambrdge Unversty Press. 7 Of course, to very hgh precson they are always changng. Here we mean that they are unchanged up to the farly hgh) precson that we chose n the Nelder-Mead algorthm.

14 14 GRAEME WEST Fgure 5. The SABR model for March 2005 expry, wth estmated β, and the estmated ρ and v. Fgure 6. The SABR model for March 2005 expry, wth constant β, and the estmated ρ and v. Hagan, P. S., Kumar, D., Lesnewsk, A. S. & Woodward, D. E. 2002), Managng smle rsk, WILMOTT Magazne September, * smle.pdf

15 CALIBRATION OF THE SABR MODEL IN ILLIQUID MARKETS 15 Heston, S. L. 1993), A closed form soluton for optons wth stochastc volatlty wth applcatons to bond and currency optons, Revew of Fnancal Studes 62). Hull, J. & Whte, A. 1987), The prcng of optons on assets wth stochastc volatltes, Journal of Fnance 42, Lews, A. 2000), Opton Valuaton under Stochastc Volatlty : wth Mathematca Code, Fnance Press. Lpton, A. 2003), Exotc Optons: The cuttng-edge Collecton, Rsk Books. Press, W. H., Teukolsky, S. A., Vetterlng, W. T. & Flannery, B. P. 1992), Numercal recpes n Fortran 77: the art of scentfc computng, second edn, Cambrdge Unversty Press. SAFEX 2004). * Spradln, G. 2003), The Nelder-Mead method n two dmensons. * Wessten, E. W ), Cubc formula, From MathWorld A Wolfram Web Resource. * West, G. 2004), The Mathematcs of South Afrcan Fnancal Markets and Instruments. Lecture notes, Honours n Mathematcs of Fnance, Unversty of the Wtwatersrand, Johannesburg. * Fnancal Modellng Agency, 19 Frst Ave East, Parktown North, 2193, South Afrca, and Programme n Advanced Mathematcs of Fnance, School of Computatonal & Appled Mathematcs, Unversty of the Wtwatersrand, Prvate Bag 3, Wts 2050, South Afrca E-mal address: graeme@fnmod.co.za