Estimating Forward Looking Distribution with the Ross Recovery Theorem

Transcription

1 Estimating Forward Looking Distribution with the Ross Recovery Theorem Takuya Kiriu Norio Hibiki July 28, 215 Implied distribution is a forward looking probability distribution of the underlying asset derived from option prices. There are a lot of studies estimating implied distribution in the risk neutral probability framework. However, a risk neutral probability generally differs from a real world probability, which represents actual investors view about asset return. Recently, Ross [215] has showed remarkable theorem, named Recovery Theorem. It enables us to estimate the real world distribution from option prices under a particular assumption about representative investor s risk preferences. However, it is not easy to derive the appropriate estimators because it is necessary to solve an ill-posed problem in estimation process. We propose a new method to derive the more accurate solution by configuring the regularization term considering prior information and compare it with the Tikhonov method used by Audrino et al. [215]. We conduct the numerical analyses using simulated data, and find the following three points. 1. To stabilize the solution by introducing regularization term is an effective method in terms of estimating a real world distribution with the Recovery Theorem. 2. Proposed method can estimate a real world distribution more accurately than the Tikhonov method. 3. We can offer the appropriate solutions even if the number of maturities is less than that of state. Graduate School of Science and Technology, Keio University, Hiyoshi, Kohoku, Yokohama, , Japan. Corresponding author. kiriutakuya@gmail.com Faculty of Science and Technology, Keio University, Hiyoshi, Kohoku, Yokohama, , Japan. hibiki@ae.keio.ac.jp 1

2 1. Introduction The payoff of option is determined by the future price of underlying asset and therefore the option prices contain the forward looking information. Implied distribution is a forward looking distribution of the underlying asset derived from option prices, which is useful for decision making in financial market such as development of investment strategy and monetary policy. It is possible to derive a risk neutral distribution from option prices in a complete market, and there are a lot of studies on the distribution. For example, Breeden and Litzenberger [1978], Melick and Thomas [1997], Bliss and Panigirtzoglou [22] and Ludwig [215] study the methods to estimate the risk neutral implied distribution from the option prices. Bliss and Panigirtzoglou [24] and Alonso et al. [25] examine the predictability of the risk neutral implied distribution. Zdorovenin and Pézier [211] and Kiriu and Hibiki [214] analyze the investment performance of asset allocation model with a risk neutral implied distribution. However, the risk neutral probability is generally different from the real world probability, and the real world distribution expresses actual investor s view. Recently, Ross [215] has showed remarkable theorem, named Recovery Theorem. It enables us to estimate a real world distribution from option prices under a particular assumption about representative investor s risk preferences. There are two types of studies related to the Recovery Theorem. The first is the theoretical extension into the continuous time case (See Carr and Yu [212], Dubynskiy and Goldstein [213], Walden [214], Park [215] and Qin and Linetsky [215]) and the fixed income market (Martin and Ross [213] ). The second is the development of the practical methodology to estimate real world distribution from option prices. Spears [213] indicates that estimators derived by the simple and instructive method of Ross [215] are intuitively inaccurate, and compares the estimators under various constraints. Audrino et al. [215] point out that it is necessary to solve an ill-posed problem in estimation process, and propose to apply the Tikhonov method, which is a standard regularization method for ill-posed problems. In addition, they estimate a real world distribution from 13 years of S&P5 option data and investigate the effectiveness of simple investment strategy based on moments of the distribution. To the best of our knowledge, this is the only research that uses time series data. Backwell [215] denotes time-homogeneity of state prices, which is hypothesized when estimating real world distribution, cannot be realized in a real market. The estimation method is also proposed to reduce the bias. Our paper is included in the second type and discusses about the method to estimate a real world distribution accurately with the Recovery Theorem. The previous studies propose the methods to estimate the real world distribution, whereas they do not investigate the estimation accuracy. It is important to examine the problem because it is ill-posed in estimation process. Hence, we test the effectiveness of Tikhonov method used by Audrino et al. [215], in the numerical analysis with simulated data. We propose a new method to derive the more accurate solution by configuring the regularization term reconsidering prior information, and compare it with the Tikhonov method. Moreover, we discuss regularization parameter selection to get the accurate real world distribution. We find the following three points through the numerical analysis, 2

3 1. To stabilize the solution by introducing regularization term is an effective method in terms of estimating a real world distribution with the Recovery Theorem. 2. Proposed method can estimate a real world distribution more accurately than the Tikhonov method. 3. We can offer the appropriate solutions even if the number of maturities is less than that of states. This paper proceeds as follows. Section 2 summarizes the Recovery Theorem of Ross [215]. Section 3 shows the procedure for estimating a real world distribution from option prices with the Recovery Theorem and proposes the new method. In Section 4, we show the results of the numerical analysis. The final section concludes and describes future works. 2. Ross Recovery Theorem In this section, we summarize the Recovery Theorem. The deviations and details are shown in appendix A. We assume an arbitrage free and complete market in discrete time with finite state one period model. Market states θ i (i = 1,..., n) are defined by r i, which are underlying stock index returns from time. P := (p i,j ) is a n n transition state price matrix. p i,j is a state price from θ i to θ j. 1 We similarly define a n n risk neutral transition probability matrix Q := (q i,j ) and a n n real world transition probability matrix F := (f i,j ). We also describe the notation Q as risk neutral distribution and F as real world distribution depending on the context. P is assumed to be irreducible 2, and therefore Q and F are also irreducible. In this section, we suppose that P is known because it can be estimated from option prices 3. Q is easily derived from P, since q i,j is expressed as follows, p i,j q i,j = n k=1 p (i, j = 1,..., n). (1) i,k On the other hand, it is difficult to derive F because the state price is simultaneously a function of both a real world probability and market risk preferences. However, Ross [215] showed F can be derived from P under the assumption that there is a representative investor with Time Additive Intertemporal Expected Utility Theory preferences over consumption (TAIEUT investor). A utility function of the TAIEUT investor is given by n U(c i ) + δ f i,j U(c j ) (i = 1,..., n), (2) j=1 1 The state price p i,j shows the price of the security at θ i which pays one dollar if the next state becomes θ j and nothing otherwise. 2 Irreducibility is defined as existing k N which satisfies (P k ) i,j > for all i, j. This assumption is very likely to be held. 3 This is explained in Section 3 in detail. 3

4 where c i is the consumption at θ i, U(c) is a utility for the consumption and δ(> ) is the discount factor of the utility. We assume that U(c) holds nonsatiation condition U (c) > but do not restrict its parametric form. Then the relationship between f i,j and p i,j is expressed as f i,j = 1 δ U (c i ) U (c j ) p i,j (i, j = 1,..., n). (3) The ratio of p i,j to f i,j is called pricing kernel, and it is expressed as ϕ i,j := p i,j = δ U (c j ) f i,j U (c i ) (i, j = 1,..., n). (4) It is dependent on investor s risk preferences. Since P is non-negative and irreducible, the Perron-Frobenius Theorem asserts that P has a unique strictly positive eigenvector v associated with the maximum eigenvalue λ. The Recovery Theorem says that δ = λ and U (c i ) = vi 1 (i = 1,..., n) hold, where v i denotes the i-th element of v. We can calculate F from P with the Recovery Theorem as follows. We solve the eigenvalue problem of P and derive the maximum eigenvalue λ and the corresponding eigenvector v. Then, we can calculate the elements of the matrix F as f i,j = 1 v j p i,j (i, j = 1,..., n). (5) λ v i In addition, Ross [215] also proves that the real world distribution becomes equal to the risk neutral distribution, or F = Q, when the sum of the row elements of P is the same for each row, and it is a special case of the Recovery Theorem. 3. Implementation of Recovery Theorem We assume that P is known in Section 2, however it is necessary to estimate P from market option prices in practice. We represent the estimation procedure as referred to Spears [213] in Figure 1. This section discusses Step 1 and Step 2 because the Recovery Theorem is simply applied in Step 3. Moreover, we point out the problem which occurs in Step 2 and propose a new method Step 1: from option prices to S A n m current state price matrix is defined as S := (s j,τ ), where s j,τ is a current state price for τ (= 1,..., m) periods transition from current state θ i to θ j. For simplicity, we assume the number of states is odd and θ i is the center state (i = (n + 1)/2). We estimate S from option prices in Step 1. A method proposed by Breeden and Litzenberger [1978] is often used to estimate S (See appendix B) and it is used to calculate the state price more accurately in a lot of literatures (See Bliss and Panigirtzoglou [22](natural cubic spline method), Melick and Thomas [1997](mixed log normal method), and Ludwig [215](neural network method)). It is not difficult to estimate S, and therefore we focus on Steps 2 and 3 in the analysis. 4

5 Figure 1: Process of Recovery 3.2. Step 2: from S to P In Step 2, we estimate the n n matrix P from the n m matrix S assuming that transitions of the states follow time-homogeneous Markov chain. We assume it is satisfied that n m, except the analysis in Section 4.6. Denote the first column vector of S by s 1, and the i -th row vector of P by p i. The j-th element of both vectors are p i,j according to the definition. Namely, s 1 = p i. (6) Because P represents the state transition in one period, we have the following relationship among s τ, s τ+1 and P. s τ+1 = s τ P (τ = 1,..., m 1) (7) Denote the (m 1) n matrix transposed from S except the last column by A, and the (m 1) n matrix transposed from S except the first column by B. Equation (7) can be expressed as follows. AP = B (8) P should be estimated by minimizing the differences in both sides of Equation (8) under the no-arbitrage conditions p i,j (i, j = 1,..., n) and Equation (6). The mathematical formulation is min P AP B 2 2 (9) subject to s 1 = p i (1) p i,j (i, j = 1,..., n). (11) Audrino et al. [215] indicate that the average condition number of matrix A estimated from S&P 5 option data from 2 to 212 is a very large value of , and therefore the problem is ill-posed. The ill-posed problem has a set of candidates of optimal solutions whose objective function values are almost the same due to low independency of data. Consequently, it has the bad characteristics that the solution is highly sensitive to a small noise. Then, Audrino et al. [215] propose to use the 5

6 Tikhonov method, which is a standard regularization method, in order to solve the illposed problem. The regularization method is formulated by adding the regularization term to the objective function to stabilize the solution against a small change of the input parameter. The regularization term gives the prior information about the expected characteristics of solution. Specifically, the objective function is reformulated as follows, min P AP B ζ P 2 2. (12) The second term is a regularization term and 2 denotes the Euclidean norm. ζ is called a regularization parameter and controls the trade-off between fitting and stability. Equation (12) can be transformed using a n n unit matrix I and a null matrix O. min P [ ] A ζi P [ ] B 2 O 2 In the Tikhonov method, the problem is solved with the prior information that the small p i,j is preferable. However, it is inadequate to estimate P using the information because p i,j should be larger for the higher transition probability. In addition, the matrix P is not irreducible in the special case of ζ. This means that the Recovery Theorem cannot be always applied for any ζ. Therefore, it is difficult to interpret the relationship between ζ and the real world distribution F. We configure the regularization term for the two preferable prior information to estimate P as follows. (13) Info 1. Info 2. s 1 is equal to p i (Equation (6)). It is theoretically derived as above-mentioned. p i,j is similar to p i+k,j+k (i, j = 1,..., n; k Z, 1 i + k n, 1 j + k n). This means the state price with the equal difference of transition between states are similar to each other. It is not the theoretically-derived condition, but it is empirically expected. We propose a new method so that we can configure the regularization term for the prior information mentioned above. Specifically, we rewrite Equation (9) into min P min P AP B ζ P P 2 2 (14) [ ] [ ] A ζi B 2 P ζ (15) P 2 6

7 where, p 1,1 p 1,2 p 1,i 1 p 1,i p 1,i +1 p 1,n 1 p 1,n p i 1,1 p i 1,2 p i 1,i 1 p i 1,i p i 1,i +1 p i 1,n 1 p i 1,n P = p i,1 p i,2 p i,i 1 p i,i p i,i +1 p i,n 1 p i,n p i +1,1 p i +1,2 p i +1,i 1 p i +1,i p i +1,i +1 p i +1,n 1 p i +1,n p n,1 p n,2 p n,i 1 p n,i p n,i +1 p n,n 1 p n,n (16) i k=1 s k,1 s i +1,1 s n 1,1 s n, k=1 = s k,1 s 3,1 s i,1 s i +1,1 s i +2,1 s n,1 s 1,1 s 2,1 s i 1,1 s i,1 s i +1,1 s n 1,1 s n,1 s 1,1 s i 2,1 s i 1,1 s i,1 s n n 2,1 k=n 1 s. k, s 1,1 s 2,1 s n i 1,1 k=i s k,1 (17) Our method 4 stabilizes the elements of the estimated matrix P by getting closer to P, which expresses the prior information. However, the values are accumulated in the first and last columns of the matrix, and we set zero to the other elements. We have the same sensitivity of P to small change of input value for the Tikhonov method and our method because the first term of Equation (13) is the same matrix as that of Equation (15). Our method can clarify the effects of the regularization term on the real world distribution F. The sum of the elements for every row of a matrix P is identical, and it is n k=1 s k,1. In the case of ζ, we obtain P = P, and the real world probability coincides with risk neutral probability. (See Section 2 and appendix A) Therefore, as ζ gets larger, it is expected that the estimated matrix F gets closer to Q. Our method can derive a risk neutral distribution as forward looking distribution even in the framework of the Recovery Theorem. 4. Numerical Analysis We examine the effectiveness of the method in Step 2 in estimating the matrix F accurately. We conduct the analysis using the hypothetical real world distribution in place of the true real world distribution, because it is impossible to know a true real world distribution and it is difficult to verify the accuracy of estimation using real data. 4 Mathematical formulation like Equation (14) is called generalized Tikhonov regularization. The proposed method is a special case where P is set as Equation (17), and we set P = O in the ordinary Tikhonov method. 7

8 Φ Φ Figure 2: Framework of analysis Figure 2 represents framework of analysis. Firstly, we provide the two simulated matrices; true real world transition probability matrix F T rue and true pricing kernel matrix Φ T rue. Then, we calculate the true state price transition matrix P T rue and true state price by maturity matrix S T rue in backward order. These matrices are unobservable from the analyst who attempts to estimate the real world distribution, but we suppose that the analyst can observe S Est which is the matrix generated by adding white noise to S T rue. In this way, we eliminate the impact of estimation method of Step 1. Then, the analyst estimates P Est from S Est (Step 2), and derive F Est and Φ Est applying the Recovery Theorem to P Est (Step3). The estimation accuracy is evaluated by comparing the estimated value F Est (or Φ Est ) with true value F T rue (or Φ T rue ). This section is organized as follows. Section 4.1 shows the setting. Section 4.2 explains the method to generate the simulated data. Section 4.3 shows the result for the base case and discusses about it. In Section 4.4, we investigate the effect of prior information. Section 4.5 verifies the robustness of result showed in Section 4.3 through the analyses of the various simulated data. Section 4.6 analyzes the case where the number of maturities of option is insufficient relative to the number of states. Section 4.7 discusses a selection method of regularization parameter Setting We explain the definition of states, the procedure of generating the simulated data in place of the analyst s observed data, and the evaluation criteria of estimation accuracy. Definition of states Market state is defined by return from time. We provide 31 returns placed by 2% symmetrically from the return of %, which is equally divided from 3% to 3% and i = 16. The analyst s observed data We apply the number of maturities of option traded in the market to the number of period m when estimating S from real data. However, we can apply any value 8

9 of m for the simulated data. We analyze the case where the number of estimated variables is the same as the number of data, or m = n = 31. The case where m < n is analyzed 5 in Section 4.6. The analyst supposes to observe S Est which is the matrix generated by adding white noise to S T rue. The noise e i,j is i.i.d., and it follows normal distribution with mean and standard deviation σ. s Est i,j = s T rue i,j (1 + e i,j ) (i, j = 1,..., n) (18) The numerical results are calculated using random numbers for the specific random seed, but we derive the similar conclusions for the different seeds. Evaluation criteria of estimation accuracy We denote the i -th row vector of the matrix F by f i, which is the real world distribution at current state. We evaluate the estimation accuracy by the Kullback- Leibler divergence (KL divergence) of estimated distribution at current state f Est i from true one f T i rue. It is a measure of the difference between the two distributions and defined as D KL (f Est i f T i rue ) = n j=1 i=1 j=1 ( ) f Est fi Est i,j ln,j fi T rue. (19),j When the estimated distribution is exactly equal to the true distribution, D KL is equal to zero. We add very small values 1 2 to both f Est i and f T i rue to prevent its anti-logarithm from being zero and avoid dividing it by zero. This procedure has no impact on the result. We also have the same conclusion in the cases of evaluating entire matrix, ( ) n n f Est D KL (F Est F T rue ) = fi,j Est i,j ln fi,j T rue, (2) and using a different criteria such as Euclidean distance D Euclid (f Est i f T i rue ) = n ( f Est i,j fi T rue ) 2,j. (21) j=1 Hence, we show only the result using D KL (f Est i 4.2. Simulated data f T i rue ) hereafter. We generate simulated data used for the base case analysis in Section 4.3. The procedure is as follows. 5 We omit to show the result in the case where m > n, because it is almost identical with the case where m = n. 9

10 Pricing kernel Assuming TAIEUT investor, ϕ can be decomposed into U and δ shown in Equation (4). We assume that the investor has a CRRA utility function U(c) = c 1 γ R/(1 γ R ) with relative risk aversion γ R. Denote the (i, j) element of Φ T rue by ϕ T rue i,j ( ) 1 + γr rj = δ (i, j = 1,..., n). (22) 1 + r i γ R = 3, δ =.999 are used in the base case. In the case of CARA utility function U(c) = exp( γ A c)/γ A with absolute risk aversion γ A = 3 in Section 4.5, the elements are denoted by ϕ T rue i,j = δe γ A(r j r i ) (i, j = 1,..., n). (23) Real world probability The matrix F T rue is generated from the S&P 5 historical data. We set a reference date, and calculate returns from the reference date to the twelve dates which are set as every 3 calendar days. If it is a holiday, the day before a holiday is used. A matrix is generated by counting the number of state transitions in one period from the return sequence. Denote the return of state θ j by r j in the matrix, which is discretely described by every 2%. When a real historical return is between r j 1% and r j +1%, it is assigned to state θ j. For example, suppose that a return is 12.5%. It is between 11%(12% 1%) and 13%(12% + 1%), and therefore 12% is assigned to the return. A return more than or equal to 29% (less than or equal to 29%) are assigned to 3% ( 3%). This is repeated daily by changing the reference date from Jan 3, 195 to Jan 3, 214. Then, all the matrices are summed up. Finally, each element of summed matrix is divided by each sum of the row elements to make it probability matrix. The simulated real world transition probability matrix F T rue is shown in Table 1. The optimization problem in Step 2 is still ill-posed because the condition number of A T rue calculated backward using Φ T rue and F T rue is very large, and Base case KL divergence Figure 3 displays the KL divergence with the proposed method and Tikhonov method for different values of ζ where ζ = 1 14, ,..., 1 1.6, 1 2. The KL divergence of q T i rue from f T i rue, D KL (q T i rue f T i rue ), is shown as a RND (Risk Neutral Distribution). q T i rue is the most accurate distribution when we estimate the forward looking distribution in risk neutral probability framework. Getting a smaller KL divergence than RND is one of the important points to judge that a real world distribution estimator is good. Firstly, we discuss about the results of the case of σ = % where analyst observes S Est without white noise. Theoretically, the KL divergence where ζ = becomes because the distribution estimated without using regularization methods equals the true 1

11 Table 1: Simulated real world transition probability matrix F T rue [a] General view [b] Enlarged view Figure 3: Base case: KL divergence of estimated distribution from true distribution 11

12 PricingKernel ζ= ζ=1^-1 ζ=1^-6 ζ=1^-2 ζ=1^2 Proposed Tikhonov legend Estimated True Return [a] σ = % PricingKernel ζ= ζ=1^-1 ζ=1^-6 ζ=1^-2 ζ=1^2 Proposed Tikhonov legend Estimated True Return [b] σ = 1% Figure 4: Base case: Pricing kernel distribution. However, the estimated KL divergence is.273 due to the calculation error. This shows that how it is difficult to get an accurate estimator of ill-posed problem. Using the proposed method, the minimum KL divergence is where ζ = and the estimation accuracy is improved drastically. In addition, the proposed method stabilize the estimators by achieving them closer to RND as ζ gets larger. Using the Tikhonov method, the KL divergence is also improved to.134 where ζ = However, the KL divergence with the Tikhonov method is always lower than that with the proposed method in any ζ. We check the cases with noise (σ = 1%, 2%, 5%). For small ζ, the estimation accuracy significantly deteriorates, compared with no noise case (σ = %), because the problem is still ill-posed. The KL divergence decreases in both regularization methods, as ζ is greater to some extent. The result indicates that it is effective to introduce the regularization term in order to stabilize the solution, and the estimation accuracy in the proposed method is better than the Tikhonov method for any ζ. We find the low bias estimator is derived by the proposed method because the regularization term is configured more appropriately using the prior information Pricing kernel Figure 4 shows the estimated pricing kernel ϕ Est i and true pricing kernel ϕ T rue i with [a] the proposed method and [b] Tikhonov method for ζ =, 1 1, 1 6, 1 2, 1 2. As shown in Equation (4), the pricing kernel is defined as the ratio of the state price to the 12

13 [a] Mean [c] Skewness [b] Standard Deviation [d] Excess kurtosis Figure 5: Base case analysis: Moments real world probability. When the real world probability coincides with the risk neutral probability, each element of ϕ i is equal to δ, regardless of the return. Figure 4a shows the pricing kernel for the case of σ = %. When we do not use the regularization method (ζ = ), the shape of pricing kernel is largely distorted, and it stems from the calculation error. Using the proposed method, the estmated pricing kernel is very close to the true pricing kernel at ζ = 1 1 and the shape gets smoother and flatter as ζ is getting larger. The graph becomes almost flat for ζ = 1 2, because this method stabilizes the solutions by bringing the real world distribution to risk neutral distribution. On the other hand, the Tikhonov method gives more complicated relationship between the pricing kernel and return for larger ζ. It is difficult to comprehend how it affects the estimates. In addition, the Tikhonov method tends to provide less accurate estimates of ϕ i than the proposed method. Figure 4b shows the result of σ = 1%. Although the main features are identical to the case of σ = %, the pricing kernel estimated with each method is largely distorted when ζ is less than or equal 1 6 because the larger the noise added to the input data is, the larger ζ needed to stabilize the solution is. 13

14 Moments Figure 5 displays the statistics of mean, standard deviation, skewness and excess kurtosis of f i. We focus on the result of the proposed method because it shows the high accuracy in the previous analysis. In the case of σ = 1%, the estimates for mean, standard deviation and skewness are relatively good around ζ = It almost matches the value of ζ that gives the low KL divergence, shown in Figure 3. On the other hand, the excess kurtosis tends to be overestimated. This indicates that it is difficult to get good estimates of higher moments even using the proposed method. This is similar to the cases of σ = 2% and 5%. Nevertheless, it is preferable for the proposed method to get a better result regarding the mean because the existence of risk premium is the primary difference between the real world probability and risk neutral probability Effect of prior information We examine the effect of prior information by decomposing the information considered in Section 3.2. The regularization term of the proposed method is configured based on both the theoretical information (Info 1) and empirically expected information (Info 2). Because a state price is a product of a discount factor and a risk neutral probability, Info 2 can be decomposed into the following two information. Info 2a. The risk neutral probability with the equal difference of transition between states are similar to each other (q i,j q i+k,j+k (i, j = 1,..., n; k Z, 1 i + k n, 1 j + k n)). Info 2b. The discount factors (the row sums of matrix P ) are almost equal regardless of the state ( n k=1 p i,k n k=1 p i,k (i, i = 1,..., n)). This information corresponds to the condition that the real world distribution is close to the risk neutral distribution. If the both Info 2a and Info 2b are satisfied, then Info 2 holds. The optimization problem to estimate P under Info 1 and 2a is formulated by introducing variables x i (i = 1,..., i 1, i + 1,..., n) to the regularization term. The formulation is, min P,x AP B ζ P P 2 2 (24) where P = x 1 i k=1 s k,1 x 1 s i+1,1 x 1 s n 1,1 x 1 s n, x 2 i 1 k=1 s k,1 x i 1s 3,1 x i 1s i,1 x i 1s i+1,1 x i 1s i+2,1 x i 1s n,1 s 1,1 s 2,1 s i 1,1 s i,1 s i+1,1 s n 1,1 s n,1 x i+1s 1,1 x i+1s i 2,1 x i+1s i 1,1 x i+1s i,1 x i+1s n 2,1 x n i+1 k=n 1 s k, x n s 1,1 x n s 2,1 x n s i 1,1 x n n k=i s k,1. (25) 14

15 When we solve the problem under Info 1 and Info 2b, we rewrite Equation (25) into x 1,1 x 1,2 x 1,i 1 x 1,i x 1,i +1 x 1,n 1 x 1,n x i 1,1 x i 1,2 x i 1,i 1 x i 1,i x i 1,i +1 x i 1,n 1 x i 1,n P = s 1,1 s 2,1 s i 1,1 s i,1 s i +1,1 s n 1,1 s n,1 x i +1,1 x i +1,2 x i +1,i 1 x i +1,i x i +1,i +1 x i +1,n 1 x i +1,n x n,1 x n,2 x n,i 1 x n,i x n,i +1 x n,n 1 x n,n (26) subject to n n x i,k = s k,1 (i = 1,..., i 1, i + 1,..., n), (27) k=1 k=1 where x i,j (i = 1,..., i 1, i + 1,..., n; j = 1,..., n) are also variables. Table 2 shows the prior information and corresponding formulations where objective function is Equation (24). In the cases of Info 1+2a and 1+2b, the sensitivity of P to small change of input value is not the same as the case of Info 1+2 (proposed method) because variables x are included in P. Therefore, we examine the minimum KL divergence where we solve optimization problem for the different values of ζ as well as Section 4.3. Table 2: Prior information and formulation Prior information Constraint (6) ζ P No information = - Info 1 = - Info 1+2a > Equation (25) Info 1+2b > Equations (26),(27) Info 1+2a+2b (=Proposed) > Equation (17) The result is shown in Table 3. As more prior information is considered to configure the regularization term, the estimation accuracy is improved. This shows each prior information expresses the characteristics of the solution well and contributes to improvement of the estimation accuracy. We find that the information that the real world distribution is close to the risk neutral distribution is especially important, because the KL divergence of Info 1+2b is much smaller than that of Info 1+2a Robustness check We check the robustness of the result in the base case by using the different simulated data from the base case. Figure 6a indicates the relationship between the regularization 15

16 Table 3: Effect of prior information: Minimum KL divergence Prior information σ = % σ = 1% σ = 2% σ = 5% No information Info Info 1+2a Info 1+2b Info 1+2a+2b (=Proposed) [a] δ :.995 [b] γ R : 1 [c] U(c) : CARA utility [d] F T rue : Nikkei225 Figure 6: Robustness check: KL divergence 16

17 [a] σ = % [b] σ = 1% Figure 7: The case where the number of maturities is less than that of states: KL divergence with the proposed method parameter ζ and KL divergence where we use.995 as the simulated δ. Figure 6b shows the result of γ R = 1. Figure 6c displays the case of the CARA utility function with γ A = 3 and Figure 6d shows the result of F T rue estimated from Nikkei225 historical data of the same period in place of S&P5. These results show that the δ and utility function are not sensitive to the KL divergence, but γ R and F T rue are sensitive to the shapes of graph. The following two features observed in the base case are preserved in any case. Firstly, the estimators using the proposed method or the Tikhonov method are more accurate than the estimators without regularization. Secondly, the estimation accuracy of proposed method is better than the Tikhonov method. The impact of simulated data change is not so big, and it is expected to get the similar results in most cases, as long as we use the appropriate simulated data. However, the further analysis is required to demonstrate the robustness The case where the number of maturities is less than that of states The analysis is done with m = 31 so far. However, the number of option maturities traded typically in the market is less than 31. For instance, the number of S&P5 option maturities traded every month regularly in Chicago Board Options Exchanges is 12, and the number of Nikkei 225 option in Osaka Exchange is 9. In addition, m of S estimated from market data will be smaller because long-term options are likely to have low liquidity. We conduct the analysis for the case where the number of maturities (data) m is less than the number of states (estimated variables) n. 6 More specifically, we estimate the real world distribution under the 31 states and three kinds of the numbers of column of S Est (m = 6, 12, 31) by the proposed method, and calculate their KL divergences. 6 Usually, n should be less than m to estimate variables under the sufficient data. Therefore, the case in this section is analyzed under the insufficient uncertainty. 17

18 Figure 8: Decomposition of the objective function value with proposed method (base case) Figure 7a shows the result of σ = %. Even in the cases of m = 6 and 12, the variables can be estimated as accurately as the case of m = 31. The similar result is obtained in the case of σ = 1% in Figure 7b as well. It might be considered that it is impossible to get the accurate estimators since the number of data is less than the number of estimated variables. However, we can estimate the accurate estimators using the proposed method. This is because the prior information included in the regularization term offsets the insufficient information. In other words, necessary information to estimate the real world distribution is almost included in the state price matrix of six maturities Selecting regularization parameter We evaluate the estimation accuracy for various regularization parameters ζ, using the KL divergence. However, it is difficult to choose an appropriate value of ζ as a practical matter. Then, we propose a method of how to select ζ to get the accurate estimates of the real world distribution. The objective function of optimization problem in Step 2 is Equation (14), and it consists of two parts. The first term shows the fitting error, and denote it by y fit, whereas the second term except ζ shows the deviation between P Est and P, and denote it by y reg. We show them for the various ζ in Figure 8. As ζ increases, y fit decreases and y reg increases monotonically. Both y fit and y reg have the domain where the values greatly change. For example, in the case of σ = 1%, the value of y fit greatly changes around ζ = 1 6 and y reg around ζ = 1. This is one of the characteristics of the ill-posed problem. The purpose of using regularization term in the ill-posed problem is to find the optimal solution stably among the degenerated solutions which have almost the same fitting error, based on the prior information. Therefore, the sound strategy is to select ζ in the ranges where both y fit and y reg are relatively small. In the case of Figure 8, the appropriate value of ζ is between about 1 4 and 1 2. In consideration of the fact that 18

19 Figure 9: Function value of h(ζ) and KL divergence with the proposed method (base case) the range of y fit is different from the range of y reg, we propose the method of selecting ζ by minimizing the function h(ζ) defined as, h(ζ) := y fit(ζ) y fit () y fit ( ) y fit () + y reg(ζ) y reg ( ) y reg () y reg ( ). (28) y fit (ζ) and y reg (ζ) are functions of ζ as shown in Figure 8. h(ζ) is the sum of the normalized values. y() is the value without the regularization term and y( ) is the value derived under the condition that P Est = P. So, y reg ( ) = must hold. Moreover, h() = 1 and h( ) = 1 must hold because both y fit (ζ) and y reg (ζ) are monotonic functions. We obtain the different values of h(ζ) by solving the optimization problems for the different values of ζ, and then we can adopt the ζ that minimizes h(ζ). We show the function values of h(ζ) and the KL divergence for various values of ζ in the base case in Figure 9. In the range of small h(ζ), the KL divergence is also small in each case where σ = 1%, 2%, 5%. It indicates that ζ can be selected well by using h(ζ). Our selection method could select appropriate values of ζ in most cases even for different simulated data, and we can find it effectively. 7 7 We may need to compare it with alternative methods. For instance, the following function is considered, ( ) yfit (ζ) y fit () yreg(ζ) yreg( ) h(ζ) := max,. (29) y fit ( ) y fit () y reg() y reg( ) However, we could select better ζ slightly using Equation (28), rather than Equation (29). Comparing with other alternatives is our future research. 19

20 5. Conclusion The Recovery Theorem makes it possible to estimate the real world distribution implied in option prices. However, it is not easy to find accurate estimators because it is necessary to solve the ill-posed problem in the estimation process. This paper proposes the method to estimate the real world distribution accurately, and then analyzes how accurate the estimation is by numerical analysis using simulated data. We clarify the following three points through the analysis. First, the regularization method like Tikhonov or our method used in Step 2 improves the estimation accuracy. This is because the regularization term enables us to suppress the effect of perturbation such as numerical error and data noise. Second, our method can estimate the real world distribution more accurately than the Tikhonov method, because our method could introduce more adequate regularization term, based on the prior information. Last, we find the fact that we derive the estimators accurately by our method to some extent even if the number of maturities of option is less than the number of states. It is sufficient to provide the six maturities of options in order to solve the problem with 31 states appropriately. This is likely to be less than the number of maturities of option traded in the market. The result suggests the possibility of obtaining the good estimator of the real world distribution from option prices traded in the market. Future works are as follows, (1) checking the robustness of the result under more various conditions and simulated data, and (2) estimating a forward looking real world distribution from time-series option data with our proposed method and testing the predictability. References Francisco Alonso, Roberto Blanco, and Gonzalo Rubio Irigoyen. Testing the forecasting performance of ibex 35 option-implied risk-neutral densities. Working Paper, Banco de Espana, 25. Francesco Audrino, Robert Huitema, and Markus Ludwig. An empirical analysis of the ross recovery theorem. Working Paper, Available at SSRN , 215. Alex Backwell. State prices and implementation of the recovery theorem. Journal of Risk and Financial Management, 8(1):2 16, 215. Robert R Bliss and Nikolaos Panigirtzoglou. Testing the stability of implied probability density functions. Journal of Banking & Finance, 26(2): , 22. Robert R Bliss and Nikolaos Panigirtzoglou. Option-implied risk aversion estimates. The Journal of Finance, 59(1):47 446, 24. Douglas T Breeden and Robert H Litzenberger. Prices of state-contingent claims implicit in option prices. Journal of Business, 51: ,

21 Peter Carr and Jiming Yu. Risk, return, and ross recovery. Journal of Derivatives, 2 (1):38, 212. Sergey Dubynskiy and Robert S Goldstein. Recovering drifts and preference parameters from financial derivatives. Working Paper, Available at SSRN , 213. Takuya Kiriu and Norio Hibiki. Optimal asset allocation model with implied distributions for multiple assets. Transactions of the Operations Research Society of Japan, 57: , 214. Markus Ludwig. Robust estimation of shape-constrained state price density surfaces. The Journal of Derivatives, 22(3):56 72, 215. Ian Martin and Steve Ross. The long bond. Working Paper, Stanford University, 213. William R Melick and Charles P Thomas. Recovering an asset s implied pdf from option prices: an application to crude oil during the gulf crisis. Journal of Financial and Quantitative Analysis, 32(1):91 115, Hyungbin Park. Ross recovery with recurrent and transient processes. Working Paper, arxiv preprint arxiv: , 215. Likuan Qin and Vadim Linetsky. Positive eigenfunctions of markovian pricing operators: Hansen-scheinkman factorization and ross recovery. Working Paper, arxiv preprint arxiv: , 215. Steve Ross. The recovery theorem. The Journal of Finance, 7(2): , 215. Trent Spears. On estimating the risk-neutral and real-world probability measures. PhD thesis, Oxford University, 213. Johan Walden. Recovery with unbounded diffusion processes. Working Paper, Available at SSRN , 214. Vladimir V Zdorovenin and Jacques Pézier. Does the information content of option prices add value for asset allocation? ICMA Centre Discussion Paper No. DP211-3, 211. Appendix A. The derivation of Recovery Theorem We explain the Recovery Theorem (Theorem 1) and its special case (Theorem 2). Theorem 1 (Recovery Theorem). We assume the existence of a representative investor with TAIEUT preferences over consumption. Then, the real world distribution can be recovered uniquely with Equation (5). 21

22 Proof. The expected utility maximization problem of the TAIEUT investor can be written as max c i,c j subject to c i + U(c i ) + δ n f i,j U(c j ) (i = 1,..., n) (3) j=1 n p i,j c j = w, (31) j=1 where w is wealth of representative investor. Solving the optimization problem by the Lagrange multiplier method, we obtain δf i,j U (c j ) = U (c i )p i,j (i, j = 1,..., n). (32) Solving for f i,j, we obtain Equation (3). We define n n diagonal matrix as D := (d i,j ; d ii = U (c i ), d i,j,i j = ). Then, Equation (32) is rewritten in a matrix form, Now, since F is a probability matrix, we obtain δf D = DP F = 1 δ DP D 1. (33) F e = e (34) where e = (1, 1,..., 1) which is a n dimensional vector. Substituting Equation (33) into Equation (34), we obtain ( ) 1 DP D 1 e = e P D 1 e = δd 1 e. (35) δ Letting z = D 1 e, we obtain P z = δz. (36) Equation (36) is in the form of the eigenvalue problem for P. Although z must be strictly positive, the Perron-Frobenius Theorem asserts that P has a unique strictly positive eigenvector v associated with the maximum eigenvalue λ, so that Equation (38) is also written as δ = λ (37) z = v. (38) U (c i ) = v 1 i (i = 1,..., n). (39) Therefore, we can derive Equation (5) by substituting Equations (37) and (39) for Equation (3). 22

23 Moreover, Ross shows the Recovery Theorem in a special case where the sum of the row elements of P is the same for each row. Theorem 2. If the sum of the row elements of P is the same for each row, the real world distribution is identical to the risk neutral distribution. Proof. Because the sum of the row elements of P is the price of the security which is allowed to get one dollar regardless of next state, we have n p i,k = α i (i = 1,..., n), (4) k=1 where α i denotes a discount factor at θ i. Consequently, if the sum of the row elements of P is the same for each row, the discount factor is not dependent on the state. Denoting the state independent discount factor by α, we have That is, n p i,k = α (i = 1,..., n). (41) k=1 P e = αe. (42) So, α is the positive eigenvalue of P and e is the strictly positive eigenvector. From Equations (1), (5), (4), (43) and (44), we obtain Therefore, we derive F = Q. λ = α (43) v = e. (44) f i,j = p i,j α = q i,j (i, j = 1,..., n). (45) B. Step 1: the method of Breeden and Litzenberger [1978] In this section, we introduce the estimating method of Step 1 developed by Breeden and Litzenberger [1978]. Suppose European options are traded in the market at the call price 8 C(K j, τ) where the strike price is K j (j =,..., n + 1) and maturity is τ. We define the states as return of the underlying asset which corresponds to each strike price. C(K j, τ) can be calculated as the sum of the products of payoff and state price. C(K j, τ) = n s k,τ max(k k K j, ) (j = 1,..., n) (46) k=1 8 In the strike price range where call option is not traded, we can transform the put price into the call price with the put-call parity. 23

24 s j,τ is solved by differentiating twice for K j. s j,τ = 2 C(K j, τ) K 2 j C(K j 1, τ) 2C(K j, τ) + C(K j+1, τ) (K j K j 1 )(K j+1 K j ) (j = 1,..., n; τ = 1,..., m) (47) 24