Correlation Smile Structures in Equity and FX Volatility Markets

Transcription

1 Correlaton Smle Structures n Equty and FX Volatlty Markets Nasr Afaf My current afflaton s wth Commerzbank AG, where I am n charge of foregn exchange dervatves. Ths artcle was frst wrtten n late July 2004, when I took 3 months out of the market and was not afflated wth any nsttuton. Abstract Reconclng and explanng observed volatlty surfaces of equty ndces from observed volatlty surfaces of ts consttuents s an mportant ssue for both relatve value tradng, and the prcng and hedgng of equty optons books. The same ssues are to be found n the case of cross rates n foregn exchange markets. Ths note develops the motvaton behnd movng to a correlaton structure type approach and dscusses ts ratonale n the case of both equty and foregn exchange markets. A method for estmatng the correlaton structure s then descrbed n the local volatlty framework. It s shown how the presence of lqudly quoted cross vanlla optons n the foregn exchange market leads to unque correlaton structures, and how ths s not the case for equty optons markets. It s further conjectured that a specfc choce of ft, whereby par wse correlatons are dependent on the whole state of the system rather than just the two underlyng stocks, may n fact have a bass n actual equty market dynamcs. Keywords volatlty, correlaton, smles, mult-factor dervatves, equty, foregn exchange. Background Volatlty quotes n terms of full moneyness by term matrces are normally avalable for sngle market factors these may be ndvdual FX rates, say USD/JPY, or ndvdual stocks, say ING Groep, as quoted on the EUROSTOXX 50 ndex. In equty markets as well, volatlty matrces are avalable for stock ndces the S&P 500, EUROSTOXX50, and Nkke are just some of the examples that leap to mnd. Equty and FX markets dffer n a varety of ways. One dfference s the typcal defnton of moneyness. FX markets prefer usng a normalsed verson of moneyness of a vanlla opton often the delta of the underlyng opton. Equty markets n contrast have had a few years of debate about the merts of dfferent verson of moneyness stcky strke and stcky delta are two of the more well known examples. It appears that stcky strke s stll commonly used especally on the short end of the curve, and many market partcpants have varous arguments n ts favour not least the smplcty of defnton. One smplcty s that prcng s consstent wth Black-Scholes senstvtes a trader s P/L at the end of day s consstent wth the reported greeks (or at least ought to be!). Ths s not the case wth stcky delta, for example, where under sgnfcant skews/smles the Black-Scholes senstvtes of a sngle volatlty may be sgnfcantly far removed from the actual senstvtes. Many a trader and nsttuton has come to gref on ths pont and t s far to say not all have adapted, even at the tme of wrtng. An opton premum may be faked by nputtng one volatlty number nto the Black-Scholes prcng formulas but the senstvtes cannot be obtaned n ths manner. All these dfferences notwthstandng, there are a number of smlartes across FX and equty markets, albet wth some dfferences here as well. One s the presence of mult-factor dervatves nstruments where correlaton becomes an mportant nput. Cross FX rates, whle traded as a sngle factor, are really a mult-factor, albet smple, nstrument for traders who book ther P/L n a currency dfferent from ether of the currences that consttute the cross rate. The obvous example s the EUR/JPY cross for a trader who does hs accountng n USD. Because of the lqudty of the EUR/JPY volatlty market, the trader need not estmate the volatltes from USD/JPY and EUR/USD volatltes and an assocated correlaton (structure) between these two rates. Instead, the mpled EUR/JPY volatlty surface contans wthn t mplct nformaton about ths correlaton (structure). Most often, and gven the way both FX vanlla and exotcs markets functon at the tme of wrtng, ths s not a concern to most market partcpants. Whle ths s generally true, for anyone nvolved n prcng baskets on USD/JPY and EUR/USD, and perhaps wth a precse or pedantc bent of mnd, questons of consstency and value do arse what correlaton should one use? Relatve value correlaton traders and mult-factor exotcs traders would be askng the same queston n a dfferent context namely queston of consstency and value across the three quoted mpled volatlty surfaces. To provde a concrete and stll topcal example, the concern mght be the relatve value of a EUR/JPY 10 delta butterfly trade aganst ts 25 delta The usual dsclamers apply. The vews expressed are those of the author and not Commerzbank AG. The author s solely responsble for any naccuraces, omssons or errors. Comments and opnons to nasr.afaf@commerzbank.com 72 Wlmott magazne

2 TECHNICAL ARTICLE 2 counterpart gven nformaton about the whole EUR/USD and USD/JPY volatlty surfaces. A quck and fast answer would be along the lnes of supply and demand, and market effcency etc. Whle not obvously ncorrect, such an answer would fal to address the real underlyng ssue. A thoughtful relatve value trader would (should) realse that ths queston concerns ssues of unrealsed correlaton what correlaton does he expect (antcpate) gven prevalng market and antcpated market condtons? At the tme of wrtng the story of the US current account defct s wrt large everywhere f and when the bg deprecaton n USD comes, what would t mean for volatlty surfaces n a relatve sense across EUR/USD, USD/JPY and EUR/JPY respectvely? Would t be sensble to expect the same correlaton for large moves n the underlyng markets that one would expect for smaller moves? A thoughtful relatve value trader would, f he s lucky n the resources made avalable to hm, be able to nfer correlaton nformaton from the three volatlty surfaces, and see f hs expectatons are prced n. If not, he could put on some trades n antcpaton expectng the markets to converge favourably to hs postons as events unfold. Whle n the above dscusson I have hghlghted some ssues that are of concern to a relatve value correlaton trader, the same concerns would apply to a thoughtful exotcs trader as well one who s concerned wth puttng on approprate vega hedges for the mult-factor exotcs n hs books. The ssues reman the same, though the end objectve s slghtly dfferent. The foregong dscusson on FX markets leads us to nto equty volatlty (correlaton) markets. Here, the dfferences les prmarly n lack of lqudty to hedge the correlaton exposure (recent growth of explct correlaton exotcs, lke correlaton swaps, not wthstandng), and n the number of underlyng factors at hand. In the FX example above, the trader was concerned only wth EUR/JPY, whch s a smple product of EUR/USD and USD/JPY, just two underlyng factors. Secondly, whle an exotc wrtten on EUR/USD and USD/JPY has a correlaton exposure, ths can n prncple be hedged away by tradng the cross vega n the lqud mpled EUR/JPY optons market. By and large, the equty trader does not have such a luxury. The equty markets trader, unlke hs FX counterpart, does not have access to complete nformaton prmarly resultng from the large number of possble combnatons that can be readly created. EURSTOXX50 volatlty surfaces for ndvdual consttuents may be farly complete and readly avalable as s the case for optons on the ndex tself (the whole basket). Yet volatlty surfaces for par wse combnatons are rarely fully avalable whch s where the mssng nformaton les. What an equty trader observes, on the other hand, are the full set of volatlty matrces for the underlyng stocks, and a volatlty matrx for the ndex (basket). A thoughtful equty trader, lke hs FX counterpart, would be keen to reconcle the two. How does one move from ndvdual volatlty matrces to the ndex volatlty surface? It would surely be nce f smple estmates of correlaton would help create a volatlty matrx for the ndex n lne wth the observed one. Ths would perhaps be askng too much and n fact, from anecdotal evdence, appears to be so. Consstency between volatlty surfaces of ndvdual stocks and the volatlty surface of the overall ndex s not easly acheved by ascrbng a sngle correlaton number for each par. One would expect rsk prema to be an mportant factor n the determnaton of the ndex volatlty surface. Ths n partcular mmedately leads us nto the dea of a correlaton structure. Most market partcpants would generally agree that trendng markets n equtes generally exhbt hgher correlaton (at least on the downsde) and there are a number of easly concevable reasons why ths mght be so. Non trendng markets tend to get de-correlated even though sgnfcant postve correlaton s stll the norm. Let s assume for the moment that t s so what would one then expect for the relatve observed skew between ndvdual stocks and the basket? A perceptve trader would say that the above scenaro for correlaton suggests that skew for larger moves (far out of the money optons) on the ndex would be sgnfcantly above observed skews on the underlyng consttuents (for far out of the money optons). For optons not very far from the money, ths effect would be less pronounced. The overall pont s that ntroducng a reasonable correlaton structure n terms of observed dynamcs of equty markets, and market partcpant behavour, would ntroduce a relatve spread between skews(smles) on the ndex and skews (smles) n the underlyngs. The correlaton structure that leads to such an ndex volatlty surface, and conversely provdes the explanaton for observed mpled data, would not just be an abstract mathematcal construct and ft wth lttle or no fnancal justfcaton. Qute the contrary n fact. In the hands of an astute correlaton trader, such a structure would shed lght on the underlyng markets, and dentfy opportuntes both for relatve value tradng, and for hedgng correlaton exposure. Wth the above motvaton n mnd, I propose an ansatz for calculatng mpled or consstent correlaton structures. I use the term mpled when a unque correlaton structure can be unveled, n the sense of complete nformaton and lqudty, as s the case n FX markets, and consstent for ncomplete nformaton, as s the case n equty markets. It should be clear that a consstent correlaton structure so obtaned may not be unque. Correlaton Structure n the Local Volatlty Framework I hghlght the method n the local volatlty 1 framework. It s useful to keep n mnd the schematc chart depcted n Fgure 1: I shall take the above graph to mean that we can move from the volatlty surface to mpled dstrbutons, or from the volatlty surface to local volatlty n fact, n all possble drectons. Namely, that knowledge of any one of the three crcles above s enough for us to recreate the other two. 2 Assume that the ndex Y s a functon of N underlyng factors: Y Y(x 1,..., x N ) (1) and that the SDE s for the sngle factor optons (underlyngs) are gven as follows: dx = a (x, t)dt + σ (x, t)dw ; 1 N (2) Note that I have used a slghtly dfferent defnton of local volatlty above from the one conventonally used. I now assume the presence of an ndex, called Y, whch s a functon of the above x. The SDE for the ndex ^ Wlmott magazne 73

3 s then gven by Ito s lemma: dy = Y a (x, t)dt + x,j 2 Y σ σ j ρ j dt + Y x σ dw (3) However, we also observe the volatlty surface for the ndex Y drectly. Assume that ts SDE s gven by: dy = a(y, t)dt + σ y (Y, t)dw y (4) Equatons 2 and 3 provde two dfferent SDE s for the ndex Y. Takng the nstantaneous varance of equatons 2 and 3 and equatng them gves: σ 2 Y (x,..., x N ; t) =,j Y Y σ (x, t)σ j (x j, t)ρ,j (5) Equaton 5 shows that correlaton ρ.j contrbute to the local volatlty of Y, and hence may be regarded as an nstantaneous and tme and spot dependent correlaton. Let us now term t the local correlaton structure: Case 1 Cross FX ρ j ρ j (x 1,..., x N ; t) (6) In the precedng dscusson we dscussed the relaton of EUR/JPY volatlty surfaces to those of EUR/USD and USD/JPY. Regardng EUR/JPY as a functon of just the two factors, EUR/USD and USD/JPY, t s clear that one can back out a unque correlaton structure from equaton 5. In terms of a local volatlty descrpton, ths correlaton structure provdes consstency across all three mpled volatlty surfaces. It may be used for relatve value tradng or for the prcng of other exotcs. From the above example, we can see that FX s a slghtly easer case, as volatlty surfaces on the major crosses are readly avalable, so mpled local correlaton structures may be nferred for all exchange-rate pars. Case 2 Equty Indces Here equaton 6 ndcates that we have great freedom n choosng the local correlatons,ρ j (x 1,..., x N ; t), to match the observed left hand sde the mpled local volatlty of the ndex as derved from the ndex volatlty surface. For N factors, we have N(N 1)/2 correlaton to play around wth. Whle the flexblty s welcome, t s clear that some fts acheved may have lttle meanng from an economc pont of vew. One way around ths problem s to try a proportonal fttng technque across all correlatons. By ths I mean that f fttng a flat correlaton structure does not yeld the local volatlty of the ndex, all correlaton should be perturbed n the same drecton. In a world of just 3 underlyng stocks, for example, f the chosen correlaton (say the mpled ATM correlatons) do not yeld the correct local volatlty of the ndex (say are below), then all 3 correlatons may be proportonately moved up tll equalty s acheved. Ths s equvalent to wrtng equaton 5 n the followng manner: σ 2 Y (x,..., x N ; t) = ( Y + 2 =j x ) 2 σ 2 Y Y σ (x, t)σ j (x j, t)ρ j (ATM)α(x 1,.., x N ; t) Here the ρ j (ATM) are defned to be mpled correlatons usng ATM volatltes whch are known and the choce s reduced to fndng the functon α(x 1,.., x N ; t), whch s the only unknown n equaton 7. Dscusson One mportant pont to note about equatons 6 and equatons 7 s that the par wse correlatons, ρ j (x 1,..., x N ; t), may be state-dependent. In other words, nothng precludes ρ j from dependng on all the (x 1,..., x N ; t) respectvely rather than just (x, x j ; t). Whle there s no reason why ths should be the case (ndeed equaton 5 could concevably be ftted wth the constrant that the ndvdual ρ j are functons of only (x, x j ; t)), t s clear that we can make the correlatons dependent on the nformaton for the entre state at a gven tme n other words, the (x 1,..., x N ; t). The method suggested n equaton 7 to acheve a ft n fact acheves ths explctly. All correlatons move up or down by the same multplcatve factor α(x 1,.., x N ; t). In fact we have the equaton: ρ j (x 1,..., x N ; t) = ρ j (ATM)α(x 1,.., x N ; t) (8) Ths smply acheves the followng: as the market moves from state to state, all nstantaneous correlatons move up and down proportonately as determned by α(x 1,.., x N ; t). Instantaneous par wse correlatons become dependent on the entre state of the market, (x 1,..., x N ; t), not just on the sub-state (x, x j ; t). Ths s perhaps not an unwelcome effect of the fttng method chosen above. Anecdotal evdence suggests that par wse correlatons tend to move together. In other words, as the market moves from one state to another, par wse correlatons tend to move n unson at least n some average manner. The followng two graphs from the Global Ttans Index are llustratve. The tme seres of correlaton above s on a data set of 3500 days so roughly 10 years. Each data pont was constructed from 90 day perods chosen to be non overlappng here. A quck glance suggests that par-wse correlatons do tend to move together. In fact, t s nterestng to look at the above graph n lght of correlaton of the par wse correlaton tme seres. Table 1 s nterestng n that the correlaton of correlaton s roughly around 50% on average. Note that snce IBM s the base stock above, one would really expect zero correlaton across the grey row and column that s ndeed the case, but roundng errors n Excel gve non-zero numbers. Note as well that the average correlaton of correlaton s pretty (7) 74 Wlmott magazne

4 TECHNICAL ARTICLE 2 TABLE 1: CORRELATION OF CORRELATION TIME SERIES WITH IBM AS BASE STOCK RELATING TO FIGURE 2 ABOVE CORRELATION of CORRELATION - 90 day non overlappng perods, 10 year hstory back from July 22, 2004 C LN IBM PEP PFE PG ROG RDA A C 100% 45% 7% 88% 31% 50% 47% 50% 48% 44% LN 45% 100% -17% 44% 46% 41% 46% 46% 58% 44% IBM 7% -17% 100% 9% -11% -9% -15% -11% -10% 0% 88% 44% 9% 100% 37% 56% 50% 66% 53% 50% PEP 31% 46% -11% 37% 100% 66% 62% 40% 34% 61% PFE 50% 41% -9% 56% 66% 100% 71% 60% 45% 60% PG 47% 46% -15% 50% 62% 71% 100% 41% 61% 51% ROG 50% 46% -11% 66% 40% 60% 41% 100% 51% 54% RDA 48% 58% -10% 53% 34% 45% 61% 51% 100% 48% A 44% 44% 0% 50% 61% 60% 51% 54% 48% 100% Average 50.39% 46.16% 55.46% 47.01% 56.18% 51.03% 49.58% 51.52% TABLE 2: CORRELATION OF CORRELATION TIME SERIES WITH PG AS BASE STOCK RELATING TO FIGURE 3 ABOVE CORRELATION of CORRELATION - 90 day non overlappng perods, 10 year hstory back from July 22, 2004 C LN IBM PEP PFE PG ROG RDA A C 100% 15% 26% 100% 27% 45% -24% 55% 34% 34% LN 15% 100% 36% 16% 22% 49% -2% 18% 36% -3% IBM 26% 36% 100% 27% 38% 55% -7% 9% 6% -9% 100% 16% 27% 100% 26% 46% -24% 56% 35% 34% PEP 27% 22% 38% 26% 100% 30% -2% 17% 26% 5% PFE 45% 49% 55% 46% 30% 100% -12% 21% 28% -12% PG ROG -24% 55% -2% 18% -7% 9% -24% 56% -2% 17% -12% 21% 100% -19% -19% 100% -11% 38% -6% 29% RDA 34% 36% 6% 35% 26% 28% -11% 38% 100% 7% A 34% -3% -9% 34% 5% -12% -6% 29% 7% 100% Average 41.87% 23.64% 23.44% 42.39% 23.79% 32.58% 30.31% 26.10% 10.69% close to 50%. It s n fact 51.24%, excludng self-correlatons and the greyed cells. IBM s a random choce and perhaps the numbers say more about the base stock chosen than hghlghtng the sze of the effect. It s clear that credt ratngs, sectors and other essental factors and nformaton would be expected to have an effect on the numbers obtaned. Over the same perod, t would be nstructve to use some other ^stock as a base just for the sake of comparson. I have randomly chosen Wlmott magazne 75

5 TABLE 3: CORRELATION OF CORRELATION TIME SERIES WITH HSBC AS BASE STOCK RELATING TO FIGURE 4 ABOVE CORRELATION of CORRELATION - 90 day non overlappng perods, 10 year hstory back from July 22, 2004 C LN IBM PEP PFE PG ROG RDA A C 100% -9% 20% 100% 56% 38% 59% 72% 54% 71% LN -9% 100% -11% -11% 4% -18% 17% 5% -2% -15% IBM 20% -11% 100% 21% 25% 50% 25% 27% 26% 39% 100% -11% 21% 100% 55% 39% 56% 72% 55% 71% PEP 56% 4% 25% 55% 100% 35% 64% 63% 50% 52% PFE 38% -18% 50% 39% 35% 100% 35% 53% 35% 39% PG 59% 17% 25% 56% 64% 35% 100% 53% 32% 58% ROG 72% 5% 27% 72% 63% 53% 53% 100% 61% 69% RDA 54% -2% 26% 55% 50% 35% 32% 61% 100% 53% A 71% -15% 39% 71% 52% 39% 58% 69% 53% 100% Average 58.70% 29.19% 58.46% 49.86% 40.40% 47.72% 58.77% 45.75% 56.24% Volatlty Surface Local Volatlty Impled Dstrbuton Fgure 1: The relaton between volatlty surface, mpled dstrbutons and local volatlty. 90 Day NonOverlappng Correlaton - IBM as base stock PG here. Ths tme we see greater dsperson n the correlaton tme seres chosen n Table 2: Comparng Fgure 2 wth Fgure 3, we can see that the envelope of tme seres s a lttle broader. The correlaton of correlaton tme seres wth PG as a base stock s shown below: Here t s clear that the correlaton of par-wse correlaton wth PG as base stock s lower than wth IBM as base stock. In fact the average s now 28.31%, so droppng by around 24% from the results for the correspondng perod wth IBM. For one fnal example I now choose HSBC as a base stock keepng the same perod. Table 3 depcts the numbers and Fgure 4 the graphs assocated wth ths choce. The correlaton of correlaton table s gven below. In ths case the average correlaton s now back up at 50% (n fact 49.45%) Correlaton Days Back from today (July 22, 2004) C LN IBM PEP PFE PG ROG RDA A Fgure 2: 90 Day correlaton on ndvdual Global Ttans aganst IBM over last 10 years. Dscusson of Method and Suggestons for Further Research The examples chosen above were for llustratve purposes only and were randomly chosen. A full statstcal study would be n order before any level of confdence can be acheved. Nevertheless the above numbers are encouragng n that they seem to pont to a pattern that suggests that par wse correlatons tend to move together n unson dependng on the full state of the system. In partcular t does seem that par wse ρ j depend on (x 1,..., x N ; t) rather than just (x, x j ; t). In other words the full dependence may be wrtten as ρ j (x 1,..., x N ; t). It further appears that, n the local volatlty framework descrbed elsewhere n ths paper, the choce of ft suggested n equaton 8 may have some underlyng meanng n terms of dynamcs of actual markets Wlmott magazne

6 TECHNICAL ARTICLE 2 Correlaton 90 Day NonOverlappng Correlaton - PG as base stock Days Back from today (July 22, 2004) C LN IBM PEP PFE PG ROG RDA A Ths means that all par wse correlatons are the same, though state dependent clearly not a very precse assumpton, but one whch sheds some lght on the correlaton structure so obtaned. From equaton 5, we then get: σ 2 Y (Y(x 1,.., x N ); t) ( ) Y 2 σ 2 (x ; t) x ρ(x 1,.,x N ; t) = 2 (10) Y Y σ (x ; t)σ j (x j ; t) =j Equaton 10 s useful n that we see the correlaton structure obtaned s not entrely unntutve. It s smply the dfference between the nstantaneous varance on the ndex and the sum of the nstantaneous varances of ts consttuents (approprately weghted). Keepng our goal of smplest possble fts we can go a touch better by rewrtng the par-wse state dependent correlatons as: ρ j (x 1,..., x N ; t) = ρ j α(x1,.., x N ; t) Fgure 3: 90 Day correlaton on ndvdual Global Ttans aganst PG over last 10 years. Correlaton Day NonOverlappng Correlaton - HSBC as base stock Days Back from today (July 22, 2004) C LN IBM PEP PFE PG ROG RDA A Fgure 4: 90 Day correlaton on ndvdual Global Ttans aganst HSBC over last 10 years. where ρ j are some mean level of correlaton per chosen par. In ths case we obtan: σ 2 Y (Y(x 1,.., x N ); t) ( ) Y 2 σ 2 (x ; t) x α(x 1,.,x N ; t) = 2 Y Y ρj σ (x ; t)σ j (x j ; t) =j and we can then use equaton 11 to get the resultng correlaton structure. Not that n both cases above, the par-wse state dependent correlatons so obtaned wll be 100% correlated wth each other whch s clearly not the case n realty. However, t does capture some essence of underlyng markets, whle achevng a consstency of ft between the ndex volatlty surface and ts consttuent surfaces. Any nformaton n ths report s based on data obtaned from sources consdered to be relable, but no representatons or guarantees are made by the author or Commerzbank AG wth regard to the accuracy or completeness of the data. The opnons, statements and calculatons contaned heren consttute the author s opnon and work at ths date and tme, and are subject to change wthout notce. Ths report s for nformaton purposes, t s not ntended to be and should not be construed as a recommendaton, offer or solctaton to acqure, or dspose of, any partcular securtes or a recommendaton to adopt a partcular tradng strategy. Appendx Equatons 5 and 7 can be solved explctly n smple cases. Say we frst set ρ j (x 1,..., x N ; t) = ρ(x 1,.,x N ; t) for all, j (9) FOOTNOTES 1. Dupre, Bruno (1994). Prcng wth a smle, Rsk, 7 (1), Techncally, constrants have to be mposed n terms of ntegrablty, choce of stochastc dfferental equaton etc but I wll assume that they have been approprately mposed. 3. Where we had set ρ j (x 1,,x N ;t) = ρ j (ATM)α(x 1,,x N ;t) W Wlmott magazne 77