EXTRACTING INFORMATION FROM OPTIONS PREMIA: THE CASE OF THE RETURN OF THE ITALIAN LIRA TO THE ERM OF THE EMS* Bernardino Adão** Nuno Cassola**

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1 EXRACING INFORMAION FROM OPIONS PREMIA: HE CASE OF HE REURN OF HE IALIAN LIRA O HE ERM OF HE EMS* Bernardno Adão** Nuno Cassola** Jorge Barros Luís**. INRODUCION he prces of fnancal dervatves (forward contracts, futures and optons) reflect, at each moment, the expectatons of economc agents regardng the future path of the prces of the underlyng assets. On the other hand, the prce of the underlyng assets (e.g., reasury blls, reasury bonds, stocks and commodtes) reflect market expectatons of the future path of ther economc determnants. Whle forward and futures contracts provde nformaton on the expected value of the prces of the underlyng assets, optons prema allow the estmaton of the rsk-neutral probablty densty functon (PDF) of the prces of the underlyng assets. In ths context, the prces of fnancal dervatves contan potentally useful nformaton for monetary authortes, namely n buldng ndcators for monetary condtons, n assessng the mpact of monetary polcy actons, and to help detect anomales n the functonng of the fnancal markets. hese ssues have been studed by several authors and central banks (see for nstance Abken (995), Bahra (996), Deutsche Bundesbank (995) and Söderlnd and Svenson (996)). Naturally, ths nformaton s also relevant from the vewpont of * he opnons of ths paper represent the vews of the authors, they are not necessarly those of the Banco de Portugal. he authors benefted from the generosty of LIFFE concernng the statstcal data, and thank Vtor Gaspar and José Ferrera Machado for ther comments. he authors are lable for remanng errors and omssons. hs research also benefted from the vst of Charles homas to the Banco de Portugal. ** Economc Research Department. portfolo and rsk management by the prvate sector, and by fnancal nsttutons n partcular. he comparson between the PDF of the prce of a fnancal asset, estmated at dfferent nstants for a gven maturty, provdes a measure for the path of market expectatons, and for ts dsperson. For nstance, Campa, Chang and Reder (997) analyse the reacton of foregn exchange markets to the re-entrance of the Italan lra n the Exchange Rate Mechansm of the European Monetary System (ERM-EMS) on 5 November 996, and fnd that the mplct exchange rate volatlty of the Italan lra decreased wth ts re-entrance n the ERM-EMS. hs fndng s consstent wth the dea that the change n the exchange rate regme amed at stablsng the exchange rate. hs study analyses the same epsode, but usng the prema of futures optons of the 3-month nterest rates of the Italan euro-lra (). We use daly observatons of the settlement prces for the call-optons and put-optons traded n the London Internatonal Fnancal Futures Exchange (LIFFE). he remander of the paper s structured as follows: secton presents the most essental aspects of the estmaton methodology; all techncal detals are left for Appendx. he thrd secton apples the methodology and the last one concludes. () he analyss of Campa, Chang and Reder (997) s exclusvely based on the prema of over-the-counter currency optons, wth no reference to nterest rates. Banco de Portugal / Economc bulletn / December

2 . MAIN CONCEPS AND ESIMAION MEHODOLOGY As a rule, fnancal nsttutons prce European optons wth the Black-Scholes formula (3). he man assumpton underlyng to the Black-Scholes model s that the rates of return have normal dstrbuton (hence the prces of the underlyng assets are lognormally dstrbuted) and are ndependent and dentcally dstrbuted. he problem s that the assumpton of log-normalty of the prces of the underlyng assets s frequently n contradcton wth realty. For nstance, ths model consders that the mpled volatlty s constant for all strke prces and for all expraton dates, whch regularly does not apply. In fact, for the same expraton date and underlyng asset, the volatlty s a convex functon of the strke prce, assumng hgher values for the optons wth strke prces more dstant to the expected future prce of the underlyng asset. he relaton between mpled volatlty and the strke prce s currently defned as volatlty smle, beng usually consdered as a sgnal of rejecton of the log-normalty assumpton. Consder the valuaton of an European call-opton wth expraton date. Let S be the prce of the underlyng asset n moment, and X the strke prce of the European opton. he premum at t of a callopton wth term to maturty equallng τ= t, C( X, τ), s the expected return of the opton, dscountng that return f the premum s pad upfront at date t. hs corresponds to: [ ] ( ) rt, τ τ CX (, τ) = e max S X, qt S ds () where r t, τ s the relevant nterest rate at the moment t (for maturty τ) and q t (S ) s the (rsk-neutral) PDF of the asset prce S, condtonal on the current asset prce S t. In prncple, the PDF can be assessed drectly through a dscrete approxmaton from the opton prema for the dfferent strke prces observed, snce theoretcally the rsk-neutral PDF corresponds to the second dervatve of the call-opton prce functon (4) : r CX ( ) qt( S) = e t, τ, ττ. () X However, the results obtaned through ths method are n general unsatsfactory (5). Many alternatve technques for estmatng the PDF from the opton prema have been suggested (6). One of the most currently used conssts of estmatng the parameters of a combnaton of two log-normal dstrbutons: ( ) = θ ( α β ) + ( θ) ( α β ) q S L, ; S L, ; S (3) t where L( α, β ; ) and L( α β ) S, ; S are lognormal dstrbutons, α and α are the means of the the respectve normal dstrbutons, β and β are the standard devatons of the respectve normal dstrbutons and θ s the weght of the frst dstrbuton (7). 3. HE REURN OF HE IALIAN LIRA O HE ERM-EMS he mpact of the return of the Italan lra to the ERM-EMS on the expected exchange rate was analysed n Campa, Chang and Reder (997). Here we analyse the mpact of ths epsode on the expected value of short-term nterest rates. We use a daly sample of settlement prces of futures optons exprng on 8 December 996, for the varous strke prces, throughout the whole perod these were traded (from 9 March up to 6 December 996). Estmatng the parameters of equaton (3) yelds the PDF that characterse the daly path of the expected 3-month nterest rate of the Italan lra for 8 December 996. Snce the expraton date s fxed, one may observe that the varance of the dstrbuton naturally decreases wth the approxmaton of the expraton date. For nstance, f the prce of the underlyng asset follows a lognormal dstrbuton, when t ncreases a fracton of tme (e.g.) one day, wth tme expressed n years) (3) he Black-Scholes prcng formula was developed n Black and Scholes (973), for prema of optons on non-dvdend-payng stocks. hereafter, ths formula was adapted to optons on fnancal assets wth dfferent features. (4) See Appendx for a dervaton of the result. (5) See for nstance Soderlnd and Svenson (996). (6) See for nstance Adão et al. (997) and Bahra (997) on the methods of estmatng PDF from opton prema. (7) he estmates for the parameters are obtaned solvng the optmsaton problem presented n Appendx. 34 Banco de Portugal / Economc bulletn / December 997

3 the standard devaton of ln S t decreases by σ( τ τ / 365). herefore, f the comparson s made between qute close days, ths correcton s rrelevant. However, for comparsons between days reasonably far away, any conclusons on the expectatons' dsperson n dfferent moments requre the correcton of the standard-devaton of ln S by the tme effect. hs same correcton prncple was used for the estmated dstrbutons. Chart dsplays the PDF adjusted for the tme effect estmated for three dstnct dates: 9 March, 9 November and 5 November, respectvely, the date of the begnnng of trade on optons exprng on 8 December 996, one week before and the day the Italan lra returned to the ERM-EMS. he means of the dstrbutons equalled 8.8, 7. and 7.6 per cent, respectvely. It should be noted that the mean of the dstrbuton dd not change sgnfcantly wth the return of the lra to the ERM- EMS. Nevertheless, the probablty assocated to hgher nterest rates decreases.e., the skewness of the dstrbuton decreases. A sgnfcant reducton n the dstrbuton's mean occurred after wth the legslatve electon whch took place on Aprl 996, as shown n chart. Indeed, on 7 Aprl the dstrbuton mean amounted to 8.6 per cent, whle on Aprl the mean stood at 7.5 per cent. Contrary to what occurred wth the return of the Italan lra to the ERM-EMS, no sgnfcant changes n the skewness of the PDF were recorded. 4. CONCLUSIONS Chart PROBABILIY DENSIY FUNCIONS OF HE HREE-MONH IALIAN LIRA INERES RAE Densty,7,6,5,4,3,, Interest rate (%) Chart PROBABILIY DENSIY FUNCIONS OF HE HREE-MONH IALIAN LIRA INERES RAE Densty,7,6,5,4,3,, Interest rate (%) Fnancal dervatves provde relevant nformaton of the expectatons of economc agents regardng the futures path of prces of the underlyng assets. Whle the prces of futures contracts delver ndcatons on the expected values, the prces of optons allow for a more complete charactersaton of the varous future values. akng the 3-month Italan lra nterest rate from futures optons prema traded at LIFFE, we estmated the rsk-neutra PDF of the 3-month Italan lra nterest rate. We found that the return of the Italan lra to the ERM-EMS dd not change sgnfcantly the expected short-term nterest rates, despte reducng the skewness of the probablty dstrbuton. On the contrary, the results of the Italan legslatve electons led to a sgnfcant shft of the dstrbuton, thereafter charactersed by a lower expected short-term nterest rate. he electoral results resulted n a lessenng of poltcal uncertanty n Italy, and may have consoldated the expectatons regardng the return of the Italan lra to the ERM-EMS. herefore, when the return of the lra to the ERM took place, ths uncertanty had already been dscounted by the markets. Banco de Portugal / Economc bulletn / December

4 REFERENCES Abken, Peter A. (995) Usng Eurodollar Futures and Optons: Gaugng the Market s Vew of Interest Rate Movements, Federal Reserve Bank of Atlanta Economc Revew, March/ /Aprl:-3. Adão, Bernardno, Nuno Cassola and Jorge Barros Luís (997) Methods of extractng nformaton from the prema of optons on fnancal assets, Banco de Portugal, DEE, mmeo (n Portuguese). Bahra, Bhupnder (997) Impled Rsk-Neutral Probablty Densty Functons from Opton Prces: heory and Applcaton, Bank of England Workng Paper, no. 66. Bahra, Bhupnder (996) Impled Rsk-Neutral Probablty Densty Functons from Opton Prces: heory and Applcaton, Bank of England Economc Bulletn, August 996. Bates, Davd S. (995), Post- 87 Crash Fears n S&P 5 Futures Optons, NBER Workng Paper Seres, W.P Black, Fscher and Myron Scholes (973) he Prcng of Optons and Corporate Labltes, Journal of Poltcal Economy, 8 (May-June), pp Campa, José Manuel, P. H. Kevn Chang and Robert L. Reder (997), ERM bandwdths for EMU and after: evdence from foregn exchange optons, Economc Polcy, No. 4, Abrl, pp Deutsche Bundesbank (995), he nformaton content of dervatves for monetary polcy, Deutsche Bundesbank Monthly Report, November 995. Duffe, Darrell (989), Futures Markets, Prentce-Hall. Hull, John (997), Optons, Futures, and Other Dervatves (hrd Edton), Prentce-Hall. Melck, Wll and Charles homas (994), Recoverng an Asset s Impled PDF from Opton Prces: An Applcaton to Crude Ol Durng the Gulf Crses, Workng Paper, Federal Reserve Board, Washngton. Söderlnd, Paul and Lars E.O. Svensson (997), New echnques to Extract Market Expectatons from Fnancal Instruments, NBER, Workng Paper Banco de Portugal / Economc bulletn / December 997

5 APPENDIX Deducton of equaton () Dfferentatng () n order to the strke prce we obtan: CX (, τ) rt, τ τ = e qt( S) ds = X X X ( t( ) ) rt,τ τ = e q S ds (A) whch yelds, CX (, τ) τ + = [ X rt, r P S q ] (A) where P q s the probablty measure. Dfferentatng (A) n order to the strke prce yelds (). Estmaton of the parameters of equaton (3) he parameters are estmated from the mnmsaton of a dstance functon between the sample prema and the prema obtaned from the specfed functonal form. he followng optmsaton problem s solved: Mn N N [ ( ) ] ( ) [ ] α, α, β, β, θ CX, τ C + PX, τ P + = = α+ β α + β + θe + ( θ) e e rt S subject to β, β > and θ, and beng: r CX, τ = e τ θlα, β ; S + [ ( ) ( ) X (A3) (A4) ( θ) ( α, β ; )]( ) + L S S X ds r (, τ) τ θ ( α, β ; ) PX = e L S + X [ ( θ) ( α, β ; )]( ) + L S X S ds (A5) where X ( =,,N) are the strke prces and PX (, τ) s the prce of a put-opton. he frst two terms n (A3) are the sum of square of the resduals between the estmated and the sample prema of call-optons and put-optons, respectvely. he last term s an approxmatng factor of the estmated mean to the forward prce. hs method has several advantages: frstly, t allows for flexble densty functons, consequently permttng to characterse multmodal functons and functons exhbtng skewness or kurtoss; secondly, t allows to use smultaneously call-optons and put-optons prema, wthout any arthmetc transformaton. However, t s a tme-consumng procedure and the results are senstve to the ntal values. o mnmse these aspects, one may use a more comprehensve functon, for nstance, mnmsng only on call-optons prema (and/ or transformng the Put-optons prema nto call-optons prema) and elmnatng the approxmatng factor to the forward. he elmnaton of ths factor allows to use the devatons between the estmated mean and the forward prce as a measure of the degree of precson of the dstrbuton obtaned. Banco de Portugal / Economc bulletn / December