Estimating Forward Looking Distribution with the Ross Recovery Theorem

Transcription

1 roceedigs of the Asia acific Idustrial Egieerig & Maagemet Systems Coferece 5 Estimatig Forward Lookig Distributio with the Ross Recovery Theorem Takuya Kiriu Graduate School of Sciece ad Techology Keio Uiversity, Yokohama, Japa Tel: (+8) , kiriutakuya@gmail.com Norio Hibiki Faculty of Sciece ad Techology Keio Uiversity, Yokohama, Japa Tel: (+8) , hibiki@ae.keio.ac.jp Abstract. The payoff of optio is determied by the future price of uderlyig asset ad therefore the optio prices cotai the forward lookig iformatio. Implied distributio is a forward lookig distributio of the uderlyig asset derived from optio prices. There are a lot of studies estimatig implied distributio i the risk eutral probability framework. However, a risk eutral probability geerally differs from a real world probability, which represets actual ivestors view about asset retur. Recetly, Ross (5) has showed remarkable theorem, amed Recovery Theorem. It eables us to estimate the real world probability distributio from optio prices uder a particular assumptio about represetative ivestor's risk prefereces. However, it is ot easy to derive the appropriate estimators because it is ecessary to solve a ill-posed problem i estimatio process. This paper discusses about the method to estimate a real world distributio accurately with the Recovery Theorem. The previous studies propose the methods to estimate the real world distributio, whereas they do ot ivestigate o the estimatio accuracy. Hece, we test the effectiveess of the Tikhoov method used by Audrio et al. (5) i the umerical aalysis with hypothetical data. We propose a ew method to derive the more accurate solutio by cofigurig the regularizatio term cosiderig prior iformatio ad compare it with the Tikhoov method. Moreover, we discuss regularizatio parameter selectio to get the accurate real world distributio. We fid the followig three poits through the umerical aalysis. () To stabilize the solutio by itroducig regularizatio term is a effective method i terms of estimatig a real world distributio with the Recovery Theorem. () roposed method ca estimate a real world distributio more accurately tha the Tikhoov method. (3) We ca offer the appropriate solutios eve if the umber of maturities is less tha that of states. Keywords: fiace, implied distributio, Recovery Theorem, regularizatio, prior iformatio. INTRODUCTION The payoff of optio is determied by the future price of uderlyig asset ad therefore the optio prices cotai the forward lookig iformatio. Implied distributio is a forward lookig distributio of the uderlyig asset derived from optio prices, which is useful for decisio makig i fiacial market such as developmet of ivestmet strategy ad moetary policy. It is possible to derive a risk eutral distributio from optio prices i a complete market, ad there are a lot of studies o the distributio. However, the risk eutral probability is geerally differet from the real world probability, ad the real world distributio expresses actual ivestor's view. Recetly, Ross (5) has showed remarkable theorem, amed "Recovery Theorem". It eables us to estimate a real world distributio from optio prices uder a particular assumptio about represetative ivestor's risk prefereces. There are two types of studies related to the Recovery Theorem. The first is the theoretical extesio ito the cotiuous time case (See Carr ad Yu (), Dubyskiy ad Goldstei (3), Walde (4), ark (5) ad Qi

2 ad Lietsky (5) ad the fixed icome market (Marti ad Ross (3)). The secod is the developmet of the practical methodology to estimate real world distributio from optio prices. Spears (3) idicates that estimators derived by the simple ad istructive method of Ross (5) are ituitively iaccurate, ad compares the estimators uder various costraits. Audrio et al. (5) poit out that it is ecessary to solve a ill-posed problem i estimatio process, ad propose to apply Tikhoov method, which is a stadard regularizatio method for illposed problems. I additio, they estimate a real world distributio from 3 years of S&5 optio data ad ivestigate the effectiveess of simple ivestmet strategy based o momets of the distributio. To the best of our kowledge, this is the oly research that uses time series data. Backwell (5) deotes time-homogeeity of state prices, which is hypothesized whe estimatig real world distributio, caot be realized i a real market. The estimatio method is also proposed to reduce the bias. Our paper is icluded i the secod type ad discusses about the method to estimate a real world distributio accurately with the Recovery Theorem. The previous studies propose the methods to estimate the real world distributio, whereas they do ot ivestigate the estimatio accuracy. It is importat to examie the problem because it is ill-posed i estimatio process. Hece, we test the effectiveess of the Tikhoov method, i the umerical aalysis with hypothetical data. We propose a ew method to derive the more accurate solutio by cofigurig the regularizatio term recosiderig prior iformatio, ad compare it with the Tikhoov method. Moreover, we discuss regularizatio parameter selectio to get the accurate real world distributio. We fid the followig three poits through the umerical aalysis. () To stabilize the solutio by itroducig regularizatio term is a effective method i terms of estimatig a real world distributio with the Recovery Theorem. () roposed method ca estimate a real world distributio more accurately tha Tikhoov method. (3) We ca offer the appropriate solutios eve if the umber of maturities is less tha that of states.. RECOVERY THEOREM I this sectio, we summarize the Recovery Theorem. We assume a arbitrage free ad complete market i discrete time with fiite state oe period model. Market states θ i (i =,, ) are defied by r i, which are uderlyig stock idex returs from time. p i,j is a trasitio state price matrix. p i,j is a state price from θ i to θ j. We similarly defie a risk eutral trasitio probability matrix Q q i,j ad a real world trasitio probability matrix F f i,j. We also describe the otatio Q as "risk eutral distributio" ad F as "real world distributio" depedig o the cotext. is assumed to be irreducible, ad therefore Q ad F are also irreducible. I this sectio, we suppose that is kow because it ca be estimated from optio prices. 3 Q is easily derived from, sice q i,j is expressed as follows, q i,j = p i,j k= p i,k (i, j =,, ). () O the other had, it is difficult to derive F because the state price is simultaeously a fuctio of both a real world probability ad market risk prefereces. However, Ross (5) showed F ca be derived from uder the assumptio that there is a represetative ivestor with Time Additive Itertemporal Expected Utility Theory prefereces over cosumptio (TAIEUT ivestor). A utility fuctio of the TAIEUT ivestor is give by U(c i ) + δ j= f i,j U c j (i =,, ), () where c i is the cosumptio at θ i, U(c) is a utility for the cosumptio ad δ (> ) is the discout factor of the utility. We assume that U(c) holds osatiatio coditio U (c) > but do ot restrict its parametric form. The the relatioship betwee f i,j ad p i,j is expressed as f i,j = U (c i ) δ U c j p i,j (i, j =,, ). (3) The ratio of p i,j to f i,j is called pricig kerel, ad it is expressed as φ i,j p i,j = δ U c j (i, f i,j U j =,, ). (4) (c i ) It is depedet o ivestor's risk prefereces. Sice is o-egative ad irreducible, the erro- Frobeius Theorem asserts that has a uique strictly positive eigevector v associated with the maximum eigevalue λ. The Recovery Theorem says that δ = λ ad U (c i ) = v i (i =,, ) hold, where v i deotes the ith elemet of v. The state price p i,j shows the price of the security at θ i which pays oe dollar if the ext state becomes θ j ad othig otherwise. Irreducibility is defied as existig k N which satisfies ( k ) i,j > for all i, j. This assumptio is very likely to be held. 3 This is explaied i Sectio 3 i detail.

3 We ca calculate F from with the Recovery Theorem as follows. We solve the eigevalue problem of ad derive the maximum eigevalue λ ad the correspodig eigevector v. The, we ca calculate the elemets of the matrix F as v j f i,j = p λ v i,j (i, j =,, ). (5) i I additio, Ross (5) also proves that the real world distributio becomes equal to the risk eutral distributio, or F = Q, whe the sum of the row elemets of is the same for each row, ad it is a special case of the Recovery Theorem. 3. IMLEMENTATION We assume that is kow i Sectio, however it is ecessary to estimate from market optio prices i practice. We represets the estimatio procedure as referred to Spears (3) i Figure. This sectio discusses Step ad Step because the Recovery Theorem is simply applied i Step 3. Moreover, we poit out the problem which occurs i Step ad propose a ew method. Market Data optio prices Step Scurret state price matrix Figure : rocess of recovery 3. Step : From Optio rices to S A m curret state price matrix is defied as S s j,τ, where s j,τ is a curret state price for τ (=,, m) periods trasitio from curret state θ i to θ j. For simplicity, we assume the umber of states is odd ad θ i is the ceter state (i = ( + )/). We estimate S from optio prices i Step. A method proposed by Breede ad Litzeberger (978) is ofte used to estimate S ad it is used to calculate the state price more accurately i a lot of literatures. It is ot difficult to estimate S, ad therefore we focus o Steps ad 3 i the aalysis. 3. Step : From S to Step (Ill-posed problem) trasitio state price matrix Step 3 (Recovery Theorem) F real world trasitio probability matrix I Step, we estimate the matrix from the m matrix S assumig that trasitios of the states follow time-homogeeous Markov chai. We assume it is satisfied that m, except the aalysis i Sectio 4.5. Deote the first colum vector of S by s, ad the i -th row vector of by p i. The j-th elemet of both vectors are p i,j accordig to the defiitio. Namely, s = p i. (6) Because represets the state trasitio i oe period, we have the followig relatioship amog s τ, s τ+ ad. s τ+ = s τ (τ =,, m ) (7) Deote the (m ) matrix trasposed from S except the last colum by A, ad the (m ) matrix trasposed from S except the first colum by B. Equatio (7) ca be expressed as follows. AA = B (8) should be estimated by miimizig the differeces i both sides of Equatio (8) uder the o-arbitrage coditios ad Equatio (6). The mathematical formulatio is mi AA B (9) subject to s = p i () p i,j (i, j =,, ). () Audrio et al. (5) idicate that the average coditio umber of matrix A estimated from S& 5 optio data from to is a very large value of.7 8, ad therefore the problem is ill-posed. The ill-posed problem has a set of cadidates of optimal solutios whose objective fuctio values are almost the same due to low idepedecy of data. Cosequetly, it has the bad characteristics that the solutio is highly sesitive to a small oise. The, Audrio et al. (5) propose to use the Tikhoov method, which is a stadard regularizatio method, i order to solve the ill-posed problem. The regularizatio method is formulated by addig the regularizatio term to the objective fuctio to stabilize the solutio agaist a small chage of the iput parameter. The regularizatio term gives the prior iformatio about the expected characteristics of solutio. Specifically, the objective fuctio is reformulated as follows, mi AA B + ζ. () The secod term is a regularizatio term ad deotes the Euclidea orm. ζ is called a regularizatio parameter ad cotrols the trade-off betwee fittig ad stability. Equatio () ca be trasformed usig a uit matrix I ad a ull matrix O. mi A ζi B O (3) I the Tikhoov method, the problem is solved with the prior iformatio that the small p i,j is preferable. However, it is iadequate to estimate usig the iformatio because p i,j should be larger for the higher trasitio probability. I additio, the matrix is ot

4 irreducible i the special case of ζ. This meas that the Recovery Theorem caot be always applied for ay ζ. Therefore, it is difficult to iterpret the relatioship betwee ζ ad the real world distributio F. We cofigure the regularizatio term for the two preferable prior iformatio to estimate as follows.. s is equal to p i (Equatio (6)). It is theoretically derived as above-metioed.. p i,j is similar to p i+k,j+k (i, j =,, ; k Z, i + k, j + k ). This meas the state prices with the equal differece of trasitio betwee states are similar to each other. It is ot the theoretically-derived coditio, but it is empirically expected. We propose a ew method so that we ca cofigure the regularizatio term for the prior iformatio metioed above. Specifically, we rewrite Equatio (9) ito mi AA B + ζ (4) where, = = mi A B ζi ζ p, p, p,i p, p, p, p, p,i p, p, p i, p i, p i,i p i, p i, p, p, p,i p, p, p, p, p,i p, p, i k= sk, s i +, s, i k= sk, s i, s, s, s, s i, s, s, (5) s, s i, k=i +s k, s, s i, k=i s k, Our method stabilizes the elemets of the estimated matrix by gettig closer to, which expresses the prior iformatio. However, the values are accumulated i the first ad last colums of the matrix, ad we set zero to the other elemets. We have the same sesitivity of to small chage of iput value for the Tikhoov method ad our method because the first term of Equatio (3) is the same matrix as that of Equatio (5). Our method ca clarify the effects of the regularizatio term o the real world distributio F. The sum of the elemets for every row of a matrix is idetical. I the case of ζ, we obtai =, ad the real world probability coicides with risk eutral probability. Therefore, as ζ gets larger, it is expected that the estimated matrix F gets closer to Q. Our (6) (7). method ca also derive a risk eutral distributio as forward lookig distributio i the framework of the Recovery Theorem. 4. NUMERICAL ANALYSIS We verify the accuracy of estimatio ad examie the effectiveess of the proposed method. However, it is difficult to kow a true real world distributio from real data. Therefore we assume real world distributio usig hypothetical data ad verify the accuracy of the estimates. Figure : Framework of aalysis Figure represets framework of the aalysis. Firstly, we give the two hypothetical matrices; hypothetical real world trasitio probability matrix F H ad pricig kerel matrix Φ H. The, we calculate the trasitio state price matrix H ad curret state price matrix S H i backward order. I reality, it is difficult to obtai S H because of the oise cotaied optio price data ad estimatio error of Step, so we geerate the matrix S N give by addig oise to S H. We assume the oise e i,j follows i.i.d., ad ormal distributio with mea ad stadard deviatio σ. s N i,j = s H i,j + e i,j (i, j =,, ) (8) I this way, we elimiate the impact of estimatio method of Step. The, we estimates N from S N (Step ) with proposed or Tikhoov method, ad derive F N ad Φ N applyig the Recovery Theorem to N (Step3). If the estimator F N is close to the origial data F H, the estimatio method is appropriate because we ca restore the origial data. The detail of estimatio accuracy criteria is metioed i Sectio Settig We explai the defiitio of states, the way of geeratig hypothetical data, ad the evaluatio criteria of estimatio accuracy. Market state is defied by retur from time. We provide 3 returs placed by % symmetrically from the retur of %, which is equally divided from 3% to 3% ad i =6.

5 %_roposed %_roposed %_roposed 5%_roposed %_Tikhoov %_Tikhoov %_roposed %_Tikhoov %_roposed %_roposed %_Tikhoov 5%_roposed We apply the umber of maturities of optio traded i the market to the umber of period m whe estimatig S from real data. However, we ca apply ay value of m for the hypothetical data. We aalyze the case where the umber of estimated variables is the same as the umber of data, or m = = 3. The case where m < is aalyzed i Sectio 4.5. We deote the i -th row vector of the matrix F by f i, which is the real world distributio at curret state. We evaluate the estimatio accuracy by the KL divergece of f i N from f i H. It is a stadard measure of the differece betwee the two distributios ad defied as D KK f i N f i H j= N f i,j l f N i,j H f i,j. (9) Whe the two distributios are exactly equal, KL divergece is equal to zero. We also have the same coclusio i the cases of evaluatig etire matrix ad usig a differet criteria such as Euclidea distace. Hece, we show oly the result usig D KK f i N f i H hereafter. 4. Hypothetical Data Regularizatio arameter ζ (log) We give the hypothetical matrix Φ H ad F H as plausible as possible used i the aalysis. The matrix Φ H is geerated by assumig TAIEUT ivestor who has a CRRA utility fuctio U(c) = c γ R/( γ R ). γ R is relative risk aversio. Assumig TAIEUT ivestor, φ ca be decomposed ito U ad δ show i Equatio (4). So, we ca deote the (i, j) elemet of Φ H by φ H i,j = δ + r γ R j (i, j =,, ). () + r i γ R = 3 ad δ =.999 are used i the base case. The matrix F H is geerated from the S& 5 historical data. We set a referece date, ad calculate returs from the referece date to the twelve dates which Regularizatio arameter ζ (log) [a] Geeral view [b] Elarged view N H Figure 3: Base case: KL divergece of f i from f i are set as every 3 caledar days. If it is a holiday, the day before a holiday is used. A matrix is geerated by coutig the umber of state trasitios i oe period from the retur sequece. Deote the retur of state θ j by r j i the matrix, which is discretely described by every %. Whe a real historical retur is betwee r j % ad r j + %, it is assiged to state θ j. A retur more tha or equal to 9% (less tha or equal to 9% ) are assiged to 3% ( 3%). This is repeated daily by chagig the referece date from Ja 3, 95 to Ja 3, 4. The, all the matrices are summed up. Fially, each elemet of summed matrix is divided by each sum of the row elemets to make it probability matrix. The optimizatio problem i Step is still ill-posed because the coditio umber of A H calculated backward usig Φ H ad F H is very large, ad The umerical results are calculated usig radom oises for the specific radom seed, but we derive the similar coclusios for the differet seeds. 4.3 Base Case Figure 3 displays the KL divergece with the proposed method ad Tikhoov method for differet values of ζ where ζ = 4, 3.6,,.6,. The KL divergece of q i H from f i H, D KK q i H f i H, is show as "" (Risk Neutral Distributio). q i H is the most accurate distributio whe we estimate the forward lookig distributio i risk eutral probability framework. Gettig a smaller KL divergece tha is oe of the importat poits to judge that a real world distributio estimator is good. Firstly, we discuss about the results of the case of σ = % where S N is observed without oise. Theoretically, the KL divergece where ζ = becomes zero because the distributio estimated without usig regularizatio methods equals the origial distributio. However, the estimated KL divergece is.73 due to the

6 %_roposed %_Tikhoov %_roposed %_roposed %_Tikhoov 5%_roposed %_roposed %_roposed %_roposed 5%_roposed %_Tikhoov %_Tikhoov %_roposed %_Tikhoov %_roposed %_roposed %_Tikhoov 5%_roposed %_roposed %_Tikhoov %_roposed %_roposed %_Tikhoov 5%_roposed Regularizatio arameter ζ (log) Regularizatio arameter ζ (log) Regularizatio arameter ζ (log) Regularizatio arameter ζ (log) [a] δ =.995 [b] γ R = [c] U: CARA utility [d] F TTTT : Nikkei5 N H Figure 4: Robustess check: KL divergece of f i from f i m=6 m= m= m=6 m= m= Regularizatio arameter ζ (log) Regularizatio arameter ζ (log) [a] σ = % [b] σ = % N H Figure 5: The case where m < : KL divergece of f i from f i calculatio error. This shows that how it is difficult to get a accurate estimator of ill-posed problem. Usig the proposed method, the miimum KL divergece is.65 3 where ζ = 9. ad the estimatio accuracy is improved drastically. I additio, the proposed method stabilize the estimators by achievig them closer to as ζ gets larger. Usig the Tikhoov method, the KL divergece is also improved to.34 where ζ =.4. However, the KL divergece with proposed method is always lower tha that with the Tikhoov method i ay ζ. We check the cases with oise (σ = %, %, 5%). For small ζ, the estimatio accuracy sigificatly deteriorate, compared with o oise case ( σ = % ), because the problem is still ill-posed. The KL divergeces decrease i both regularizatio methods, as ζ is greater to some extet. The result idicates that it is effective to itroduce the regularizatio term i order to stabilize the solutio, ad the estimatio accuracy i proposed method is better tha Tikhoov method for ay ζ. We fid the low bias estimator is derived by the proposed method because the regularizatio term is cofigured more appropriately usig the prior iformatio. 4.4 Robustess Check We check the robustess of the result i the base case by usig the differet hypothetical data from the base case. Figure 4a idicates the relatioship betwee the regularizatio parameter ζ ad KL divergece where we use δ =.995. Figure 4b shows the result of γ R =. Figure 4c displays the case of the CARA utility fuctio with γ A = 3 ad Figure 4d shows the result of F H estimated from Nikkei5 historical data of the same period i place of S&5. These results show that the δ ad utility fuctio type are ot sesitive to the KL divergece, but γ R ad F H are sesitive to the shapes of graph. The followig two features observed i the base case are preserved i ay case. Firstly, the estimators usig the proposed method or the Tikhoov method are more accurate tha the estimators without regularizatio. Secodly, the estimatio accuracy of proposed method is better tha the Tikhoov method. The impact of hypothetical data chage is ot so big, ad it is expected to get the similar results i most cases, as log as we use the plausible hypothetical data. However, the further aalysis is required to demostrate the robustess.

7 y_fit fit:% fit:% fit:5% reg:% reg:% reg:5% Regularizatio arameter ζ (log) y_reg Figure 6: Decompositio of the objective fuctio value 4.5 The Case where m < The aalysis is doe with m = 3 so far. However, the umber of optio maturities traded typically i the market is less tha 3. For istace, the umber of S&5 optio maturities traded every moth regularly i Chicago Board Optios Exchages is, ad the umber of Nikkei 5 optio i Osaka Exchage is 9. I additio, m of S estimated from market data will be smaller because logterm optios are likely to have low liquidity. We coduct the aalysis for the case where the umber of maturities (data) m is less tha the umber of states (estimated variables). 4 More specifically, we estimate the real world distributio uder the 3 states ad three kids of the umbers of colum of S (m = 6,, 3) by the proposed method, ad calculate their KL divergeces. Figure 5a shows the result of σ = %. Eve i the cases of m = 6 ad, the variables ca be estimated as accurately as the case of m = 3. The similar result is obtaied i the case of σ = % i Figure 5b as well. It might be cosidered that it is impossible to get the accurate estimators sice the umber of data is less tha the umber of estimated variables. However, we ca estimate the accurate estimators usig the proposed method. This is because the prior iformatio icluded i the regularizatio term offsets the isufficiet iformatio. I other words, all ecessary iformatio cocerig ivestor's risk prefereces is almost icluded i the matrix of six colums to estimate the real world distributio from the state price matrix. 4 Usually, should be less tha m to estimate variables uder the sufficiet data. Therefore, the case i this sectio is aalyzed uder the isufficiet ucertaity. Figure 7: Fuctio value of h(ζ) ad KL divergece 4.6 Selectig Regularizatio arameter We evaluate the estimatio accuracy for various regularizatio parameters ζ, usig the KL divergece. However, it is difficult to choose a appropriate value of ζ as a practical matter. The, we propose a method of how to select ζ to get the accurate estimates of the real world distributio. The objective fuctio of optimizatio problem i Step is Equatio (4), ad it cosists of two parts. The first term shows the fittig error, ad deote it by y fff, whereas the secod term except ζ shows the deviatio betwee N ad N, ad deote it by y rrr.we show them for the various ζ i Figure 6. As ζ icreases, y fff decreases ad y rrr icreases mootoically. Both y fff ad y rrr have the domai where the values greatly chage. For example, i the case of σ = %, the value of y fff greatly chages aroud ζ = 6 ad y rrr aroud ζ =. This is oe of the characteristics of the ill-posed problem. The purpose of usig regularizatio term i the ill-posed problem is to fid the optimal solutio stably amog the degeerated solutios which have almost the same fittig error, based o the prior iformatio. Therefore, the soud strategy is to select ζ i the rages where both y fff ad y rrr are relatively small. I the case of Figure 6, the appropriate value of ζ is betwee about 4 ad. I cosideratio of the fact that the rage of y fff is differet from the rage of y rrr, we propose the method of selectig ζ by miimizig the fuctio h(ζ) defied as, h(ζ): = y fff(ζ) y fff () y fff ( ) y fff () + y rrr(ζ) y rrr ( ) () y rrr () y rrr ( ) y fff (ζ) ad y rrr (ζ) are fuctios of ζ as show i Figure 6. h(ζ) is the sum of the ormalized values. y() is the value without the regularizatio term ad y( ) is the value derived uder the coditio that N = N. So, y rrr ( ) = must hold. Moreover, h() = ad

8 h( ) = must hold because both y fff (ζ) ad y rrr (ζ) are mootoic fuctios. We obtai the differet values of h(ζ) by solvig the optimizatio problems for the differet values of ζ, ad the we ca adopt the ζ that miimizes h(ζ). We show the fuctio values of h(ζ) ad the KL divergeces for various values of ζ i the base case i Figure 7. I the rage of small h(ζ), the KL divergece is also small i each case where σ = %, %, 5%. It idicates that ζ ca be selected well by usig h(ζ). Our selectio method could select appropriate values of ζ i most cases eve for differet hypothetical data, ad we ca fid it effectively CONCLUSION The Recovery Theorem makes it possible to estimate the real world distributio implied i optio prices. However, it is ot easy to fid accurate estimators because it is ecessary to solve the ill-posed problem i the estimatio process. This paper proposes the method to estimate the real world distributio accurately, ad the aalyzes how accurate the estimatio is by umerical aalysis usig hypothetical data. We clarify the followig three poits through the aalysis. First, the regularizatio method like Tikhoov or our method used i Step improves the estimatio accuracy. This is because the regularizatio term eables us to suppress the effect of perturbatio such as umerical error ad data oise. Secod, our method ca estimate the real world distributio more accurately tha the Tikhoov method, because our method could itroduce more adequate regularizatio term, based o the prior iformatio. Last, we fid the fact that we derive the estimators accurately by our method to some extet eve if the umber of maturities of optio is less tha the umber of states. It is sufficiet to provide the six maturities of optios i order to solve the problem with 3 states appropriately. This is likely to be less tha the umber of maturities of optio traded i the market. The result suggests the possibility of obtaiig the good estimator of the real world distributio from optio prices traded i the market. Future works are as follows, () checkig the robustess of the result uder more various coditios ad hypothetical data, ad () estimatig a forward lookig real world distributio from time-series optio data with our proposed method ad testig the predictability. REFERENCES Audrio, F., Huitema, R. ad Ludwig, M. (5) A empirical aalysis of the Ross recovery theorem. Available at SSRN: Backwell, A. (5) State prices ad implemetatio of the recovery theorem. Joural of Risk ad Fiacial Maagemet, 8, -6. Breede, D. T. ad Litzeberger, R. H. (978) rices of state-cotiget claims implicit i optio prices. Joural of busiess, Carr,. ad Yu, J. () Risk, retur, ad Ross recovery. Joural of Derivatives,, 38. Dubyskiy, S. ad Goldstei, R. S. (3) Recoverig drifts ad preferece parameters from fiacial derivatives. Available at SSRN: Marti, I. ad Ross, S. A. (3) The log bod. Workig aper, Staford Uiversity. ark, H. (5) Ross recovery with recurret ad trasiet processes. arxiv preprit arxiv:4.8v4. Qi, L. ad Lietsky, V. (5) ositive eigefuctios of Markovia pricig operators: Hase- Scheikma factorizatio ad Ross recovery. arxiv preprit arxiv:4.375v. Ross, S. (5) The recovery theorem. The Joural of Fiace, 7, Spears, T. (3) O estimatig the risk-eutral ad real-world probability measures. Doctoral dissertatio, Oxford Uiversity. Walde, J. (4) Recovery with ubouded diffusio processes. Available at SSRN: 5 We may eed to compare it with alterative methods. For istace, the followig fuctio is cosidered, h(ζ) max y fff (ζ) y fff () y fff ( ) y fff (), y rrr (ζ) y ree ( ) () y rrr () y rrr ( ) However, we could select better ζ slightly usig Equatio (), rather tha Equatio (). Comparig with other alteratives is our future research.