CENTRAL BANK OF THE REPUBLIC OF TURKEY WORKING PAPER NO: 10/03 Recoveing Risk-Neutal Densities fom Exchange Rate Options: Evidence in Tukey Mach 2010 Halil İbahim AYDIN Ahmet DEĞERLİ Pına ÖZLÜ
Cental Bank of the Republic of Tukey 2010 Addess: Cental Bank of the Republic of Tukey Head Office Reseach and Monetay Policy Depatment İstiklal Caddesi No: 10 Ulus 06100 Ankaa Tukey Phone: +90 312 507 54 02 Facsimile: +90 312 507 57 33 The views expessed in this woking pape ae those of the autho(s) and do not necessaily epesent the official views of the Cental Bank of the Republic of Tukey. The Woking Pape Seies ae extenally efeeed. The efeeeing pocess is managed by the Reseach and Monetay Policy Depatment.
Recoveing Risk-Neutal Densities fom Exchange Rate Options: Evidence in Tukey Halil İbahim Aydın γ Ahmet Değeli γ Pına Özlü γ Abstact This pape uses ove-the-counte cuency options data to investigate maket expectations on Tukish Lia-U.S. Dolla exchange ate. We extact option implied density functions to examine the evolution of maket sentiment ove the possible values of futue exhange ates. Uncetainty is well measued by option-implied pobabilities. Estimated densities fo selected days point out an incease in uncetainty in foeign exchange maket duing financial tubulence peiods. We make infeences about the effectiveness of policy measues and see how the maket peception changed thoughout the cisis. We uncove the effectiveness of policy measues by obseving shinking densities and confidence bands. Keywods: Options Risk neutal density Maket expectations. JEL Codes: G13 G19 F31 This pape has benefited fom Refet Gükaynak s comments. We ae gateful to Selim Elekdağ Özgü Özel and Güsu Keleş fo helpful comments on a pevious daft. The views expessed in this pape do not necessaily epesent those of the Cental Bank of the Republic of Tukey. All emaining eos ae ous. γ Reseach and Monetay Policy Depatment Cental Bank of the Republic of Tukey; Head Office Istiklal Caddesi: No.10 Ulus; Ankaa; TURKEY 06100. Coesponding autho: Halil İbahim Aydın e-mail: halil.aydin@tcmb.gov.t; phone: (312) 3103646; fax: (312) 3240998. 1
I. Intoduction Options makets ae vey impotant fo academics maket paticipants and policy makes. They povide a ich souce of infomation about investos expectations and maket uncetainty about the futue couse of the financial asset pices. Options contacts give the ight to buy o sell an asset in the futue at a pice (stike) set now. Options have a value since thee is a chance that the options can be execised that is the undelying asset pice will be moe/less than a paticula stike pice. Hence when we look at options with diffeent stike pices the pices at which such contacts tade now give us infomation about the maket s view of the chances that the pice of the undelying asset will be above/below these stike pices. The common way of dealing with the infomation embedded in option pices is to extact implied isk-neutal pobability density function (RND) a commonly used indicato to gauge expectations about futue pice of a financial asset unde isk neutal assumption of the pefeences of the investos. The impact of monetay policy decisions on maket expectations the pobability that inteest ate may pevail inside a specific ange o the iskiness of a cuency position can be evealed though the infomation content of RNDs. Recently an inceasing numbe of studies have investigated the shape of the isk-neutal pobability distibution of futue asset pices using options on inteest ates equity pices and exchange ates in line with the deepening of the deivative makets. Many studies on diffeent assets indicate the usefulness of RNDs. 1 The evidence is in favo of the intepetation that the estimates of RNDs do not help foecast the futue but eveal the maket sentiment that is useful fo the policy stance of the cental banks. Most cental banks tack the changes in maket sentiment via the estimates of RNDs. 2 Estimates of RND ae commonly studied in the context of fundamental economic questions elated to an impotant event o a possible change in egime o announcements of economic news that could affect expectations. Fo example Baha (1997) compaes RNDs aound announcements of inflation epots to test the change in maket sentiment using options on thee-month steling inteest ate. He povides a vey infomative discussion about RNDs in assessing monetay conditions and 1 See Coutant et al. (2001) Jondeau and Rockinge (2000) Melick and Thomas (1997). 2 See Csavas (2008) Bha and Chiaella (2000) Sydal (1999) Nakamua and Shiatsuka (1999). 2
cedibility of the monetay policy. He agues that the cental bank cedibility measued as the diffeence between the makets peceived distibution of the futue ate of inflation and taget inflation can be assessed via RNDs. Since thee is no diect instument available to compute RND fo inflation he suggests using the RND of long-tem inteest ates to measue uncetainty ove futue inflation. Campa et al. (1999) compute RNDs on coss ates within the Exchange Rate Mechanism of the Euopean Monetay System to assess the cedibility of commitments to exchange ate taget zones and detemine whethe the bandwidths ae consistent with the maket expectations. The RND could also be utilized in testing the ationality and measuing isk attitude of investos which ae highly impotant fo policy makes. This study aims to extact the isk-neutal densities embedded in ove-thecounte (OTC) exchange ate options in Tukey. To the best of ou knowledge this is the fist attempt to ecove RND fo TL/USD exchange ates. While the numbe of studies on RNDs fo developed economies is lage thee ae only few studies fo emeging makets due to unavailability of the options data. Gowth in the deivative activities in Tukey motivates us to undestand elevant infomation concening maket expectations contained in the pices of options. The liteatue on RND estimation coves a wide ange of techniques including paametic non-paametic and stuctual models. 3 While some methods may not be convenient due to the lack of data fo options the othes may yield distoted density functions inconsistent with empiical statistics o pobability theoy. Fo instance models with volatility smile cuve fitting as in Shimko (1993) may lead to negative pobabilities. To cicumvent these shotcomings we apply Malz (1997) method which ensues positive density. Moeove the Malz method is analytically tactable since it equies only thee obsevations to uncove the undelying density. We investigated the infomation content of RNDs when financial makets wee subject to a geate uncetainty duing the peiod 2006 and 2008. We make infeences about the effectiveness of monetay policy measues and see how the maket peception changed thoughout the cisis. Fo both peiods we obseved that the estimated densities signaled moe uncetainty duing the cisis times and the monetay policy measues wee successful in educing the uncetainty. 3 Studies such as Baha (1997) Campa et al. (1998) Jondeau and Rockinge (2000) Bliss and Panigitzoglu (2004) povide infomative discussions about vaious RND methods. 3
The plan of the emainde of this pape is as follows. We intoduce ou data in section 2. In section 3 we biefly pesent the method employed. Section 4 descibes the case studies and section 5 concludes. II. Data Deivatives ae fowad maket instuments and thei volume has exceeded the volume of spot maket instuments in the global economy. 4 Thee ae two diffeent types of deivatives makets distinguished by whethe they ae taded in an oganized maket o not. Exchange-taded deivatives ae taded in specialized and oganized makets. The othe type is the OTC maket whee tading takes place diectly between paties without an oganized maket o intemediay. The volume of OTC deivatives maket is geneally highe than that of exchange-taded deivatives maket. In Tukey thee ae both oganized and unoganized deivatives makets. Oganized deivatives maket Tukish Deivatives Exchange (TURKDEX) is located in İzmi. The poducts taded in TURKDEX ae futue contacts on stock index inteest ates cuencies and commodities. Thee ae no options contacts taded in TURKDEX. Howeve thee ae options contacts on exchange ates taded in OTC maket. It is woth mentioning a few wods on options quotations befoe documenting the details of the data and the methodology we adopt. The options ae taded in both exchange taded (oganized) and OTC (unoganized) makets. But quotations in two makets diffe fom each othe. In exchange taded makets the quotations ae given in tems of stike levels and option pices. Wheeas in OTC makets the quotations ae given in tems of deltas 5 and implied volatilities. Using the Black-Scholes fomula it is easy to eveal the stike pice fom delta and option pice fom implied volatility. The OTC quotation facilitates the scaling of the asset pice movements in makets whee financial assets display volatile pice pattens. Fo example foeign exchange ates ae quickly changing in shot peiods of time. It is vey had to detemine constant stike levels fo such assets. A pactical solution adapted in the maket is to quote in tems of deltas instead of stike pices. Ultimately the implied volatility takes the ole of option pice and delta takes the ole of stike pice. 4 See BIS statistics fo global OTC deivatives. 5 Delta is a paamete showing how close the asset pice to the stike pice (i.e. moneyness of the option). 4
Thee ae thee option combinations which ae heavily employed in cuency options makets. These ae at-the-money staddle (atm) isk evesal () and stangle (st). 6 While at-the-money staddle options have 50 delta isk evesals and stangles can have diffeent delta levels. Since 25 delta is the most liquid one we choose 25 delta isk evesal and stangle in line with the Malz (1996) appoach we adopt. These option types in cuency options makets enable investos to get infomation about the behavio of moments of expected values of the undelying exchange ate. Fo instance the staddle indicates the expected volatility of the exchange ate. Similaly isk evesal and stangle give infomation about the skewness and kutosis of the distibution of futue exchange ate espectively. The dataset consists of OTC foeign cuency options quoted in tems of implied volatility and delta. We used options on Tukish Lia against US Dolla. Matuity is taken as one month. Risk fee inteest ates used in the option pices ae detemined fom USD and TL LIBOR makets at the coesponding matuities. The dataset coves the daily obsevations between 2006 and 2009. We used the settlement pices of options which ae detemined at the close of tading day. 7 It is impotant to take into account the liquidity of USD/TL options maket. Unfotunately the depth and volume of the ove-the-counte maket in Tukey is low. The liquidity in the options makets may be even lowe duing the cisis peiod. Theefoe the RND functions may not tuly eflect all changes in maket sentiment in shot peiods. Thus fo event study puposes it may be bette to look at changes in the RND function at somewhat lowe fequencies. We choose the time intevals at least as one week. The dates efe to the time peiods befoe the financial tubulence duing the tubulence and afte the policy measues taken by Cental Bank of the Republic of Tukey (CBRT). We focus on two case studies to undestand how the maket sentiment changed duing financial tumoil expeienced in the second half of 2000 s May-June 2006 and Septembe-Octobe 2008. Fo the fist peiod we gaph RNDs fo the days of May 8 May 15 June 13 and July 21 2006. The second case study deals with the global financial cisis which was intensified by the bankuptcy of Lehman Bothes in Septembe 2008. To investigate the effects of the cisis on the financial maket we 6 See Appendix fo details. 7 See Melick and Thomas (1998) Södelind and Svensson (1997). 5
examine the days of Septembe 8 Septembe 16 Octobe 22 and Octobe 31 2008. We make infeences about the effectiveness of policy measues and see how the maket peception changed thoughout the cisis. III. Methodology The density extacted fom the taded options is called the isk neutal density (i.e. RND) which povides the set of pobabilities that investos would attach to futue asset pices in a wold in which investos ae isk-neutal. The intuition behind RND is closely linked to Aow-Debeu pices of contingent claims. Since deivatives ae contacts on futue events the pices of deivatives today eflect the possibility of the ealization of those events. The picing of contingent claims in finance liteatue is based on the isk neutal valuation pinciple. The concept of isk neutal valuation dates back to ealy effots of Samuelson (1965) on option picing. Samuelson (1965) deives the andom value of the option at execise. His fomulation included two paametes the expected ate of etun on the stock pice and the discount ate of the options. Retun paamete is the expected etun fom the stock pice if investos hold it until the matuity of the option. Discounting paamete is the oppotunity cost of investing in option. Since the option pice is embedded with uncetainty the discounting paamete should include the isk within options as well as altenative iskless investments. Both paametes of the models depend upon the unique isk chaacteistics of the undelying stock and the option. Howeve in an options contact both the deivative and its undelying asset ae subject to the same souce of pice changes o the same souces of isk. Thus uncetainty suounding the teminal stock pice affects isk attitude of the investos towads both assets. Decisions on holding the stock o the option ae both affected by the same isk. Accodingly option pice does not depend on the isk pefeence of the investos as long as the discounting and etun paametes eflect the same degee of isk avesion. This famewok fist advocated by Cox and Ross (1976) implies the ielevance of isk pefeences of investos to the option pice and captues the essence of isk neutal picing. The mechanics of isk neutal valuation is as follows. One can calculate the option pice by consideing isk neutal investo s behavio. In this setup the expected etun on undelying asset and discounting ate substituted with isk-fee inteest ate. 6
A isk neutal investo can hedge the shot/long position (whateve he pefes) by taking the opposite in the othe asset. Shot position in call option fo instance can be hedged by long position in the undelying asset. By this appoach the option pice can be calculated whateve the isk pefeence the investos have; because isk avesion is ielevant to the option pice. It is impotant to note that the state pices of contingent claims in an Aow- Debeu economy depend on the likelihood of the states and on the attitude of the agents towads those states. In ou appoach investos ae assumed to be isk neutal. Thus the effects of isk-avesion ae embedded in isk neutal pobabilities. That is likelihood of financial asset pices embedded in the options pices eflects the investos expectations as well as isk avesions about the unknown futue. This peception seves as the main pilla of the liteatue on RNDs and the basis fo ou analysis. The geneal conclusion of isk neutal valuation is that pice of options can be deduced fom the expectation of option payoff. Mathematically speaking the expectation of asset payoff is an integal with espect to teminal asset pice. Accodingly the pice of a Euopean style call option C (K) at time t with matuity τ and stike pice K can be simplified as an integal unde isk neutal pobability measue τ C( K ) = e ( S K ) p( S ) ds K T T T (1) whee S T denotes the asset pice at matuity. p (S T ) is the pobability associated with pice S T at matuity based on the teminal asset pice. Almost all option picing models in mathematical finance liteatue deal with this integal epesentation. The models ae based on vaious pobability distibution assumptions fo the teminal asset pice. The seminal wok of Black and Scholes (1973) is founded upon the assumption that the distibution of the pice S T is lognomal i.e.; the asset pice at matuity follows a geometic Bownian motion with constant volatility. This assumption yields a closed fom fomula fo the option pice. Howeve empiical facts show that financial asset pices do not admit geometic Bownian motions. Statistical popeties of asset etuns exhibit significant dispesion fom Black-Scholes assumptions. The etuns ae not nomally distibuted; leptokutic kenels have bette 7
fit pefomance than nomal densities. Moeove the volatility is not constant fo the options with diffeent stike pices. Volatilities acoss stikes show vaiations pesenting volatility smile pattens. Volatilities fo options with diffeent time-tomatuities ae also diffeent fom each othe leading to a tem stuctue of volatilities. The vaiations of volatilities fo diffeent stike pices and matuities endeed the bith of the concept of implied volatility. Implied volatility is the volatility implied by the maket pice of the option. When plugged into the Black-Scholes fomula implied volatility yields a theoetical option pice that is equal to the maket pice of the option. Implied volatility is used to emedy the shotcomings of constant volatility assumption of Black-Scholes model. It shows the chaacteistic featues of options with diffeent stikes. Fo this eason it is used as the actual pice of option. Maket pactitiones use implied volatility in Black-Scholes model as a tool of quotation. Especially in OTC makets a tade pesents his bid-ask pices fo the options in tems of its volatility. Howeve this does not mean that Black-Scholes model is the pope model fo option picing o ageed upon model of the maket. Rathe it is the model with which the pactitiones use to detemine the option pice fom volatility quotes. Due to the limitations of Black-Scholes model the liteatue on options picing focused on flexible distibutions fo financial asset etuns. A numbe of diffeent stochastic pocesses ae examined fo the etuns of the undelying asset. Results indicate that a numbe of stochastic pocesses can be matched fo asset etuns but the estimation of model paametes gets complicated as the numbe of paametes inceases. Thus assigning a stochastic pocess fo the undelying asset is the geneal methodology of option picing but its use in RND estimation is limited. Due to the lage numbe of paametes involved in estimation we opt to use a non-paametic method fo RND estimation. Most of the RND estimation techniques ae based on the wok of Beeden and Litzenbege (1978) who showed that RND could be deived fom the second deivative of call pices with espect to stike pices. 2 C = e 2 K τ p( K) One can infe the RND by numeical diffeentiation of call options with espect to stikes when thee ae sufficient obsevations. Howeve such diect estimation of (2) 8
RND is not geneally possible due to the lack of available option pices. Also these kinds of models lead to unstable density estimates 8. Thus diect numeical evaluation is egaded as an inaccuate way of density extaction and eseach focus shifted to indiect methods. A numbe of indiect density estimation methods ae poposed to cicumvent the shotcomings of diect density estimations. One of these attempts is documented in Malz (1997). The pocedue is a non-paametic estimation of RND. It is based on the idea of fitting a cuve fo volatility smile on implied volatility and delta space. Malz suggests estimating RND of exchange ates by using the infomation embedded in staddle stangle and isk evesal options. Since volatilities fo diffeent stike levels vay the model needs to build volatility as a function of stike pice (delta). To this aim Malz (1997) poposes a second ode Taylo appoximation to delta-implied volatility cuve aound the point δ 0 =0.5 as follows ˆ t ( δ ) atm + b ( δ δ ) + b st ( δ δ ) 2 0 = 0 t 1 t 0 2 t b (3) 0 whee atm t t and st t ae volatilities of at-the-money isk evesal and stangle (o buttefly) options. The intuition behind this appoach is such that the atm and st show the level skew and cuvatue of volatility smiles espectively. This infomation is captued by Taylo appoximation. Plugging implied volatilities of options with deltas 0.50 0.25 and 0.75 one can deduce the paametes as b 0 =1 b 1 =-2 and b 2 =16. The cuve fitting step needs caution since both sides of the equation includes implied volatility. Right hand side is the implied volatility and left hand side includes the tem delta which is itself a function of volatility. This equies us to find a unique solution fo implied volatility using numeical pocedues. Once the implied volatility function is estimated option pices ae found by plugging implied volatility into Black-Scholes fomula. Then indiect estimation of the isk neutal density is caied out via Beeden and Litzenbege (1978) fomula given in equation 2. 8 See Jondeau Poon and Rockinge (2007). 9
IV. Empiical Evidence One of the impotant goals of the CBRT is to ensue financial stability in domestic makets. Afte the adopt of fee-float exchange ate egime the intevention of the Bank to FX maket diminished consideably. The beginning of fee float peiod coincided with the independence of CBRT and an ageement with IMF to intevene only in limited amounts in the foeign exchange maket. The Bank was committed to intevene to smooth out exteme movements in exchange ates. Nevetheless in cetain peiods when financial makets wee hit by extenal shocks the Bank actively intevened to stabilize the makets. Tukish economy expeienced two financial lage maket tubulences duing the second half of 2000 s one in May-June 2006 and the othe in Septembe-Octobe 2008. The Bank took seveal pecautionay measues to stabilize the financial maket volatility. Duing the May-June 2006 peiod the fist measue taken was to suspend the daily foeign exchange puchase auctions that the Bank had been caying out stating fom May 16 2006. Aftewads the Monetay Policy Committee (MPC) held an inteim meeting on June 7 when policy ates wee hiked by 175 basis points. Though the fluctuations in financial makets eased to some extent the volatility intensified in the last week of June putting pessue on exchange ates and medium and long-tem inteest ates. The MPC held anothe inteim meeting on June 25 and not only inceased policy ates by a futhe 225 basis points but also took a numbe of measues egading TL and foeign exchange liquidity duing the meeting. To incease the flexibility of monetay policy and gadually educe the excess liquidity the Bank also initiated Tukish Lia Deposit Buying Auctions with standad matuities of one and two weeks as of June 26. In the meeting held in July 2006 the MPC aised inteest ates by a futhe 25 basis points in ode to alleviate the seconday effects of the exchange ate inceases and impove inflation expectations. Consequently the ovenight boowing ate was inceased by 425 basis points in total fom June 7 to July 20. The policy measues taken by the Bank povided stability in the financial makets in a shot peiod of time. Poblems in global cedit makets in autumn 2008 have aised concens on the global financial system and advesely affected the global liquidity flows. Cental banks acted pomptly and took actions in a coodinated manne. The cisis had its 10
advese effects on Tukish foeign exchange makets in Octobe. The tubulence intensifies on Octobe 22. In line with the effots in the global economy the Bank attempted to maintain healthy functioning of the domestic maket though pe-cautionay measues. Ongoing foeign exchange buying auctions wee suspended and selling auctions wee intoduced to inject foeign exchange liquidity into the maket. The Foeign Exchange Deposit Maket within the Bank was e-opened. This decision led to foeign exchange deposit tansactions among banks in tems of both US dolla and Euo and pevented the decline in the flow of foeign exchange liquidity. The matuity of the FX deposit boowed has been extended and the lending ate was educed. Simila to the tumoil in 2006 the policy measues taken by the Bank wee successful in achieving stability in the maket by the end of Octobe 2008. It is woth noting the diffeences between the implied volatility and RND gaphs befoe we depict the figues. While the figues of implied volatility povides infomation about the futue dispesion of the asset pice fom one obseved option pice the implied isk-neutal density on the othe hand povides infomation about the entie distibution of the ultimate asset pice fom seveal option pices. Also the RND could yield infomation about the highe moments of the teminal asset pice as long as these moments ae consistent acoss diffeent estimation models. Much of the infomation contained in RND functions can be captued though a ange of summay statistics shown in Table 1. The mean is the aveage value of all possible futue outcomes. The standad deviation is a measue of dispesion of the implied RND and commonly used to measue maket uncetainty. Skewness chaacteizes the distibution of pobability eithe side of the mean and povides a measue of asymmety fo the distibution. Fo example a positively skewed distibution is one fo which thee is less pobability attached to outcomes highe than the mean than to outcomes below the mean. Kutosis is a measue of how peaked a distibution is o the likelihood of exteme outcomes. The geate this likelihood the fatte the tails of the distibution. 11
Table 1: Desciptive Statistics of Risk Neutal Densities May-June 2006 1 Month Mean Vaiance Skewness Kutosis Standad Lowe Uppe Deviation Bound* Bound* 8-May-06 1.31 0.02 1.60 5.23 0.12 1.27 1.38 15-May-06 1.48 0.03 1.48 5.36 0.16 1.36 1.66 13-Jun-06 1.58 0.04 1.32 5.35 0.20 1.39 1.87 21-Jul-06 1.53 0.03 1.80 6.70 0.17 1.42 1.72 Septembe-Octobe 2008 1 Month Mean Vaiance Skewness Kutosis Standad Lowe Uppe Deviation Bound* Bound* 8-Sep-08 1.22 0.01 1.27 5.28 0.09 1.14 1.32 16-Sep-08 1.26 0.01 1.31 5.30 0.11 1.16 1.40 22-Oct-08 1.67 0.07 1.26 5.10 0.27 1.37 2.14 31-Oct-08 1.52 0.04 1.38 5.96 0.20 1.28 1.88 * Lowe and uppe bounds efe to 5 and 95 pecent confidence levels. Figues 1 and 2 pesent the implied volatilities acoss execise pices duing the two tumoil peiods in Tukey. The implied volatility cuve pesents the volatility pices of the options fo diffeent execise pices. The value in the y-axis is egaded as the pice of the option in tems of volatility. The x-axis denotes the execise pices. Accodingly the pice of the option is at the lowest level when the option is at-themoney (i.e. execise pice is close to spot pice). As the execise pice goes up o down the volatility pice of the option inceases. Howeve the incease in option volatility may be highe fo high stikes than low stikes and vice vesa. This effect is known as the volatility skew o smik. This patten appeaed in stock options maket afte the 1987 cash. Volatility skew may be seen in cuency options as well. This means that the maket feas the depeciation of the local cuency and willing to pay moe fo the possible depeciation than appeciation o vice vesa. The volatility skew is moe appaent when the maket volatility gets highe and highe. Befoe May 2006 tubulence the implied volatility diffeence between in-the money and out-of-the money options was low on May 8 2006 wheeas it widens duing the tubulence on June 13 2006 depicted in Figue 1. Simila patten can be obseved duing the Septembe-Octobe 2008 tubulence shown in Figue 2. The figues on RND indicate the pobabilities attached to diffeent pice levels of the undelying asset. The x-axis denotes the futue values of spot TL/USD exchange ates. The peceived isk neutal occuence pobabilities of these exchange 12
ate levels ae pesented in the y-axis. A numbe of illustations of RND functions ae given in Figues 3-4 and changes in thei shapes befoe and afte the financial tumoil ae pesented in 2006 and 2008. Changes in the width of the confidence inteval infom us about changes in maket uncetainty about futue asset pice levels. Figue 1. Implied Volatility Cuve fo May-June 2006 0.40 0.35 0.30 8-May-06 13-Jun-06 15-May-06 21-Jul-06 Implied volatility 0.25 0.20 0.15 0.10 0.05 0.00 1.00 1.06 1.12 1.18 1.24 1.30 1.36 1.42 1.48 1.54 1.60 1.66 1.72 1.78 1.84 1.90 1.96 Exchange Rates Figue 2. Implied Volatility Cuve fo Septembe-Octobe 2008 0.60 0.50 8-Sep-08 22-Oct-08 16-Sep-08 31-Oct-08 0.40 0.30 0.20 0.10 0.00 1.00 1.06 1.12 1.18 1.24 1.30 1.36 1.42 1.48 1.54 1.60 1.66 1.72 1.78 1.84 1.90 1.96 Implied Volatility Exchange Rates Figue 3 compaes the one month implied RND fo TL/USD option on dates befoe and afte the financial tumoil in 2006. We see that as the tumoil intensifies by June 13 the confidence intevals get wide suggesting the feas of lage movements of exchange ates. The ight tail of RND function gets fatte indicating the ising uncetainty in the maket sentiment o the peceived incease in the pobability of a lage depeciation. The RND gaph on July 21 2006 shows that the width of the confidence inteval become naowe afte the Bank intevened in the maket. The 13
good news is that the decease of uncetainty about expected futue exchange ate level can be visualized fom RND figues and confidence bands. Figue 3. Implied RND fo the one-month TL/USD foeign exchange ate in May-June 2006. 0.20 0.18 0.16 0.14 8-May-06 13-Jun-06 15-May-06 21-Jul-06 Pobability 0.12 0.10 0.08 0.06 0.04 0.02 0.00 1.00 1.06 1.12 1.18 1.24 1.30 1.36 1.42 1.48 1.54 1.60 1.66 1.72 1.78 1.84 1.90 1.96 Exchange Rates A simila analysis on changes in implied RND function following the global cisis in Septembe 2008 is given in Figue 4. We gaph the RND functions befoe and afte the bankuptcy of Lehman Bothes on Septembe 15 2008. When we compae the RND functions befoe and afte the bankuptcy of Lehman Bothes we see that the confidence intevals widens as the cisis intensifies and then diminished afte the policy measues taken by the end of Octobe. Case studies on RND functions show that policy measues including the foeign exchange intevention deceases pospective uncetainty in the foeign exchange makets. Figue 4. Implied RND fo the one-month TL/USD foeign exchange ate in Septembe-Octobe 2008. 0.12 0.10 0.08 8-Sep-08 22-Oct-08 16-Sep-08 31-Oct-08 0.06 0.04 0.02 0.00 1.00 1.06 1.12 1.18 1.24 1.30 1.36 1.42 1.48 1.54 1.60 1.66 1.72 1.78 1.84 1.90 1.96 Pobability Exchange Rates It is impotant not to ead too much infomation fom gaphs of isk-neutal distibutions (BIS 1999). The RNDs do not povide the actual pobabilities of futue 14
asset pices. Rathe they give the values that option maket paticipants attach to hedges of diffeent possible outcomes. Thus the liteatue is much moe inteested in the intuitive notion captued by the change in the shape of RND. Also isk neutality of the RND stands fo the assumption that individuals in the maket ae isk-neutal. Moeove OTC makets compise of institutional investos who tend to be less isk avese. 9 The time seies epesentations of TL/USD and RND implied standad deviation povided in Figues 5 and 6 convey how the maket uncetainty about the expected outcome changed ove time and illustate the ise in uncetainty in foeign exchange maket duing financial tubulence peiods. Figue 5: Nominal Exchange Rate (TL/USD) 1.9 1.8 TL/USD 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 03.01.2006 03.03.2006 03.05.2006 03.07.2006 03.09.2006 03.11.2006 03.01.2007 03.03.2007 03.05.2007 03.07.2007 03.09.2007 03.11.2007 03.01.2008 03.03.2008 03.05.2008 03.07.2008 03.09.2008 03.11.2008 03.01.2009 03.03.2009 03.05.2009 03.07.2009 03.09.2009 03.11.2009 Figue 6: RND Implied Standad Deviation 0.25 0.2 RND Implied Standad Deviation 0.15 0.1 0.05 0 03.01.2006 03.03.2006 03.05.2006 03.07.2006 03.09.2006 03.11.2006 V. Conclusion 03.01.2007 03.03.2007 03.05.2007 03.07.2007 03.09.2007 03.11.2007 03.01.2008 03.03.2008 03.05.2008 03.07.2008 03.09.2008 03.11.2008 03.01.2009 03.03.2009 03.05.2009 03.07.2009 03.09.2009 03.11.2009 9 Gükaynak and Wolfes (2005) show that isk neutality assumption can not be ejected in a maket fo institutional investos. 15
V. Conclusion Option implied isk neutal densities povide infomation about maket expectations. In this pape we estimate the isk-neutal pobability density (RND) functions fo TL/USD cuency options. We apply the method poposed by Malz (1996) due to its analytical tactability and simplicity. We investigated the infomation content of RNDs when financial makets wee subject to a geate uncetainty duing the peiod 2006 and 2008. We make infeences about the effectiveness of monetay policy measues and see how the maket peception changed thoughout the cisis. Fo both peiods we obseved the widening of the confidence intevals as the financial tubulence hit the maket implying feas of lage movements in exchange ates. We uncove the effectiveness of policy measues by obseving shinking densities and confidence bands. The oveall esult is such that the Bank was successful in stabilizing the makets duing ecent financial tubulences and the infomation content of RNDs facilitates to undestand how maket paticipants view the futue and pices of isk associated with futue outcome. 16
REFERENCES Baha B. 1997. Implied Risk-Neutal Pobability Density Functions fom Option Pices: Theoy and Application. Bank of England Woking Pape No.66 1-56. Bank fo Intenational Settlements 1999. Estimating and Intepeting Pobability Density Functions. BIS Pape No. 6. Bha R. Chiaella C. 2000. Expectations of Monetay Policy in Austalia Implied by the Pobability Distibution of Inteest Rate Deivatives. Euopean Jounal of Finance Vol. 6(2) 113-115. Black F. Scholes M. 1973. The Picing of Options and Copoate Liabilities. Jounal of Political Economy Vol.81 (3) 637-654. Bliss R. Panigitzoglu N. 2004. Option-implied Risk Avesion Estimates. Jounal of Finance Vol.59 (1) 407-446. Beeden D. Litzenbege R. 1978. Pices of State-Contingent Claims Implicit in Option Pices. Jounal of Business Vol.51 (4) 621-651. Campa J. Chang K. Refalo J. 1999. An Options-Based Analysis of Emeging Maket Exchange Rate Expectations: Bazil s Real Plan 1994-1997. NBER Woking Pape No.6929. Campa J. Chang K. Reide R. 1998. Implied Exchange Rate Distibutions: Evidence fom OTC Option Makets. Jounal of Intenational Money and Finance Vol: 17(1) 117-160. Coutant S. E. Jondeau E. Rockinge M. 2001. Reading PIBOR Futues Option Smiles: The 1997 Fench Snap Election. Jounal of Banking and Finance Vol. 25(11) 1957-1987. Cox C. Ross A. 1976. The Valuation of Options fo Altenative Stochastic Pocesses. Jounal of Financial Economics Vol.3 (1) 145-166. Csávás C. 2008. Density Foecast Evaluation and the Effect of Risk-neutal Cental Moments on the Cuency Risk Pemium: Tests Based on Eu/Huf Option- Implied Densities Cental Bank of Hungay Woking Pape No.3. Gükaynak R. Wolfes J. 2005. Macoeconomic Deivatives: An Initial Analysis of Maket-Based Maco Foecasts Uncetainty and Risk. NBER Intenational Semina on Macoeconomics 2005 (2) 11-50. Jondeau E. Poon S. Rockinge M. 2007. Financial Modeling Unde Non-Gaussian Distibutions. Spinge-Velag New Yok. 17
Jondeau E. Rockinge M. 2000. Reading the Smile: The Message Conveyed by Methods Which Infe Risk Neutal Densities. Jounal of Intenational Money and Finance Vol. 19 (6) 885-915. Malz A. 1996. Using Option Pices to Estimate Realignment Pobabilities in the Euopean Monetay System: The Case of Steling-Mak. Jounal of Intenational Money and Finance Vol.15 (5) 717-748. Malz A. 1997. Estimating the Pobability Distibution of the Futue Exchange Rate fom Option Pices. Jounal of Deivatives Vol. 5 (2) 18-36. Melick W. Thomas C. 1997. Recoveing an Asset s Implied PDF fom Option Pices: An Application to Cude Oil Duing the Gulf Cisis. Jounal of Financial and Quantitative Analysis Vol. 32 (1) 91-115. Melick W. Thomas C. 1998. Confidence Intevals and Constant Matuity Seies fo Pobability Measues Extacted fom Option Pices. Confeence Pape Bank of Canada. Nakamua H. Shiatsuka S. 1999. Extacting Maket Expectations fom Option Pices: Case Studies in Japanese Option Makets. Monetay and Economic Studies Institute fo Monetay and Economic Studies Bank of Japan Woking Pape Vol. 17(1) 1-43. Samuelson P. 1965. Rational Theoy of Waant Picing Industial Management Review Vol.6 13-31 Shimko D. 1993. Bounds of Pobability. Risk Vol.6 33-47. Södelind P. Svensson L. 1997. New Techniques to Extact Maket Expectations fom Financial Instuments. Jounal of Monetay Economics Vol. 40(2) 383-429. Sydal S. 2002. A Study of Implied Risk-Neutal Density Functions in the Nowegian Option Maket. Noges Bank Woking Pape No.13. 18
19 Appendix Stike Pice(K): Stike pice o execise pice is the pice at which the owne of an option contact can execise the option Implied Volatility ( ): The volatility implied by the maket pice of the option based on an option picing model Black-Scholes (1973) model in this pape. whee S t =spot exchange ate τ =time to expiation K=execise pice =implied volatility =domestic inteest ate * =foeign inteest ate N=standad cumulative nomal distibution function Delta ) (δ : Delta is the ate of change of the option pice (C) with espect to the undelying asset pice (S). Fo instance a delta of 0.80 means that fo evey $1 the undelying asset inceases the call option will incease by $0.80. Kutosis: Kutosis measues the the peakedness of the pobability distibution. It can be calculated as the fouth cental moment of the pobability distibution nomalised by the fouth powe of its standad deviation. Skewness: Skewness measues the the asymmety of the pobability distibution. It can be calculated as the thid cental moment of the pobability distibution nomalised by the thid powe of its standad deviation. At-the-moneyness: An option is at-the-money if the execise pice is equal to the cuent pice of the undelying asset. In-the-moneyness: A call option is in-the-money if the execise pice is below the cuent pice of the undelying asset. Out-of-the-moneyness: A call option is out-of-the-money if the execise pice is above the cuent pice of the undelying asset. + + = = τ τ τ τ δ τ 2 ln ) ( ) ( 2 * * * * K S N e S K S C K S t t t t 4 4 μ = Kutosis 3 3 μ = Skewness + + + = τ τ τ τ τ τ τ 2 ln 2 ln ) ( 2 * 2 * * * K S KN e K S N S e K S C t t t t (1) (2) (4) (3)
Standad Option Combinations At-the-money-staddle: A combination of at-the-money call and put options. Since staddle is an at-the-money option its stike pice is vey close to pevailing exchange ate. The quotation in tems of call option implied volatilities is given as follows atm = (0.50 ) (5) Risk evesal: An option stategy whee the investo simultaneously puchases an out-of-the-money call option and sells an out-of-the-money put option. The quotation in tems of call option implied volatilities is given as follows = ( 0.25 ) (0.75 ) Stangle: An option stategy consisting of a simultaneous puchases of an out-of-themoney put and an out-of-the-money call option. The quotation in tems of call option implied volatilities is given as follows [ (0.75 ) + (0.25 )] (0.50 ) st = 0.5 (6) (7) Payoff Diagams of Standad Option Combination δ ( K1 ) = 0.25 δ ( K 2 ) = 0.25 δ ( K ) = 0.25 δ ( K1 ) = 0.25 δ ( K 2 ) = 0.25 20
Cental Bank of the Republic of Tukey Recent Woking Papes The complete list of Woking Pape seies can be found at Bank s website (http://www.tcmb.gov.t). Tükiye İmalat Sanayiin İthalat Yapısı (Şeef Saygılı Cengiz Cihan Cihan Yalçın Tüknu Hamsici Çalışma Tebliğ No. 10/02 Mat 2010) Dış Ticaette Küesel Eğilimle ve Tükiye Ekonomisi (Fauk Aydın Hülya Saygılı Mesut Saygılı Gökhan Yılmaz Çalışma Tebliğ No. 10/01 Mat 2010) Dayanıklı Tüketim Malı Fiyat Dinamiklei (Fethi Öğünç Çalışma Tebliğ No. 09/08 Aalık 2009) Wealth Distibution and Social Secuity Refom in an Economy with Entepeneus (Okan Een Woking Pape No. 09/07 Septembe 2009) Monetay Shocks and Cental Bank Liquidity with Cedit Maket Impefections (Piee-Richad Agéno Koay Alpe Woking Pape No. 09/06 July 2009) Paa Politikası Paasal Büyüklükle ve Küesel Mali Kiz Sonası Gelişmele (K. Azim Özdemi Çalışma Tebliğ No. 09/05 Temmuz 2009) Financial Maket Paticipation and the Developing County Business Cycle (Hüseyin Muat Özbilgin Woking Pape No. 09/04 Apil 2009) (Published in: Jounal of Development Economics (2009)doi:10.1016/j.jdeveco.2009.03.005) Design and Evaluation of Coe Inflation Measues fo Tukey (Oğuz Atuk Mustafa Utku Özmen Woking Pape No. 09/03 Mach 2009) Sticky Rents and the Stability of Housing Cycles (Edem Başçı İsmail Sağlam Woking Pape No. 09/02 Febuay 2009) Inflation Tageting and Exchange Rate Dynamics: Evidence Fom Tukey (K. Azim Özdemi Sekan Yiğit Woking Pape No. 09/01 Febuay 2009) Tükiye'de Paa Politikasının Aktaımı: Paa Politikasının Mali Piyasalaa Etkisi (Zelal Aktaş Haun Alp Refet Gükaynak Mehtap Kesiyeli Musa Oak Çalışma Tebliğ No. 08/11 Aalık 2008) On the Stability of Domestic Financial Maket Linkages in the Pesence of time-vaying Volatility (Thomas J. Flavin Ekateini Panopoulou Deen Ünalmış Woking Pape No. 08/10 Septembe 2008) İşlenmiş Gıda Fiyatlaını Belileyen Faktöle (Yusuf Sone Başkaya Tuğul Gügü Fethi Öğünç Çalışma Tebliğ No. 08/09 Ağustos 2008) Monetay Pessues and Inflation Dynamics in Tukey: Evidence fom P-Sta Model (K. Azim Özdemi Mesut Saygılı Woking Pape No. 08/08 August 2008) Tangos Sambas o Belly Dancing? O do Speads Dance to the Same Rhythm? Signaling Regime Sustainabil ity in Agentina Bazil and Tukey (Santiago Heea Fehan Salman Woking Pape No. 08/07 August 2008)