The Forward-looking Equity Risk Premium of the Ross Recovery Theorem

Transcription

1 The Forward-looking Equity Risk Premium of the Ross Recovery Theorem 19 April 2017 MSc Mathematical Finance University of Oxford

2 Abstract In this paper, we investigate Ross recovery theorem and apply it to calculate a forward-looking Equity Risk Premium for S&P 500 index. We consider a number of past implementations and propose a di erent method, which in our opinion is simpler and more appropriate. It calibrates the required state-price matrix directly to option prices and applies restrictions across available maturities using the time-homogeneity property of transition matrices. We also apply a new regularization method which results in a meaningful and stable solution. Applying this algorithm over the historical data allows us to extract Equity Risk Premium for di erent tenors and construct a term structure. We find that the short term tenors are more sensitive to market changes than the long term ones and that the curve can be upward or downward slopping depending on market conditions. The median Equity Risk Premium over the available history was calculated as 3.87% annually, consistent across all considered maturities. i

3 Acknowledgements Iamverygratefultomysupervisorforhistime,valuableinsightsandpatiencewhileI worked on this thesis. I m glad that I had a chance to work with him on such interesting topic. I m also thankful to my manager at work for his support and understanding throughout my time at Oxford. ii

4 To my parents and my lovely wife. iii

5 Contents 1 Introduction 1 2 Technical Prerequisites Notation Key Financial Concepts Equity Risk Premium (ERP) Stochastic Discount Factor (SDF) and the Pricing Kernel Breeden-Litzenberger Formula and the Risk-Neutral Density Arrow-Debreu Securities and State Prices Mathematical Tools Markov Chains Perron-Frobenius Theorem Ross Recovery Theorem Overview Model Assumptions State-Price by Tenor Matrix S State-Price Matrix P Real-World Matrix F Steps to Recovery Implementation of the Recovery Theorem Ross, Spears, Audrino, Huitema and Ludwig, Backwell, Kiriu & Hibiki, Flint & Mare, Analysis of the Theorem Evaluation of Model Assumptions Comparison with Black-Scholes Implementation Challenges Verification of Results Proposed Implementation Method Calibrating the State-Price Matrix P Additional Market Constraints Interest Rate Curve iv

6 4.2.2 Dividends Forwards Formulating the Minimisation Problem Further Matrix Regularisation Restrictions on the Matrix Interest Rates in Di erent States Assessment of the Methodology Implementation Details Financial Data Implied Volatility Dividends Interest Rates Algorithm Overview Estimating the Regularisation Parameter Robustness Tests Empirical Results Probability Distributions Real and Risk-Neutral Probabilities Pricing Kernel Equity Risk Premium Historical Time Series Analysis Term Structure Volatilities Conclusion 42 8 References 44 9 Appendix A: Key Abbreviations Appendix B: More Probability Distributions 47 v

7 1 1 Introduction For anyone studying Mathematical Finance, it is fascinating to learn that financial derivatives can be priced without any knowledge of the real probability distribution of the underlying asset. In fact, it is completely irrelevant since in a complete market any financial claim can be hedged by trading in other assets and thus, one can replicate any derivative pay-o. The cost of replication would be its market price. The famous result by Black and Scholes [5] demonstrates this concept. The flip side of this powerful theory is equally fascinating. Given a market-traded financial derivative, whose final pay-o depends only and entirely on its underlying asset, its price will not reflect the true likelihood of this asset rising or falling. Instead, the price will indicate how much it costs to hedge it (plus commission costs, of course). Luckily, the relationship between the hedging costs of derivatives and the true probability distribution of underlying assets does exist and is based on investors beliefs and their perception of risk. In order to see why these come into play, consider a contract that pays an investor $100 if a fair coin flip is heads and loses them $100 if it turns tails. Since there s equal chance of winning or losing, a fair price for such security should be $0. However, would an investor accept this contract for free? Generally it is believed that individuals have concave utility functions, meaning that a loss of $100 would be more painful in absolute terms, than the pleasure received from a $100 gain. Hence, in order to accept the terms of this contract, an investor is likely to demand extra premium to compensate for the risk of potentially losing $100. The size of this premium depends on each individual investor and represents their risk preferences. If the risk preferences are known, one can uncover a true probability distribution of a financial asset, for instance, through knowledge of a Stochastic Discount Factor (section 2.2.2). Unfortunately, the information on investor sentiment is not easily observed in the market and manifests itself only in combination with other factors, as shown in the following simple example. AstandardgeometricBrownianmotionthatdescribesthedynamicsofsomeasset,S t, with constant drift and volatility parameters µ and,isgivenby: ds t = µs t dt + S t dw t

8 1 Introduction 2 Under this log-normal assumption, the parameters µ and fully describe the probability distribution of S t. Assuming a complete market, the risk of S t can be hedged away, which gives rise to a risk-neutral probability measure, equivalent to the real-world one. Under the risk-neutral measure, the drift component of financial assets is adjusted by a product of its volatility and a term representing an amount of return required per unit risk, known as market price of risk (MPR), : ds t =(µ )S t dt + S t dŵt (1) where market price of risk is: = µ r Equation (1) of course simplifies to a standard risk-neutral equivalent for S t,wherethedrift term µ is replaced with a risk-free rate, r. Since derivatives are priced using the dynamics in (1), their prices reflect terms in the equation and can be used to estimate them. This will tell us the value of and the (µ ) term. Unfortunately however, this is not enough to calculate the exact probability distribution of S t, since the drift term µ remains unknown. It features only in the combination with the market price of risk, which is also unknown. Hence more information is needed to separate these two variables and calculate the real-world probability distribution. At this stage, one may be tempted to disregard financial derivatives and estimate µ from the historical evolution of S t. However, in addition to the classical disclaimer that past data is not indicative of the future, doing this will not yield an accurate estimate for µ, evenwhen applied to an asset with very long history, such as the S&P 500 index. In a simple exercise, Monoyios [16] showed that one needs about 1537 years of observations in order to estimate the drift parameter within a 95% confidence interval, assuming a lognormal model. The error of mean estimation in fat tail processes in also highlighted by Taleb [19] in his famous book Black Swan. Hence, one needs to resort to other methods of estimating these variables. Recently, Ross [17] has developed a theorem which under certain assumptions allows to calculate the relationship between the real-world and risk-neutral probabilities and hence calculate the drift parameters. Given the impossibility of this under the classic Black-Scholes framework, as shown above, the paper has attracted much interest and many researchers investigated the potential of this theorem. In order to achieve recovery, Ross [17] puts many restrictions on the dynamics of the underlying asset and some research has been done in order to address this. As the theorem assumes a discrete environment, Tran and Xia [21] looked into the e ects of the state-space discretisation on the recovered probabilities. Carr and Yu [8] have instead looked into extending the recovery theorem to a continuous setting using a concept of a numeraire portfolio. In Carr and Yu [9] they also show how the theorem is unsuccessful within an unbounded

9 1 Introduction 3 Black-Scholes model, but achieves recovery when applied to a recurring CIR model. Ross restriction on the bounds has been also investigated by Dubynskiy and Goldstein [11], who criticise the theorem and show that bounding the state space is the reason why the recovery is possible. Instead of looking into the theoretical foundations of the recovery theorem, this paper has focused on its implementation and application within the financial markets. Our goal is to apply the theorem to S&P 500 index and investigate forward-looking dynamics of Equity Risk Premium for di erent tenors and over the available historical data. In order to achieve this, we propose a di erent implementation approach of Ross recovery theorem and a new regularisation method. It allows us to calibrate the required parameters directly to market data without having to apply any transformations. It also results in meaningful probability matrices across maturities, which then permits to extract Equity Risk Premium for di erent time horizons. We find that the risk premium curve is dynamic and changes depending on the market conditions, with the short term section being more sensitive than the long term. There have been many successful attempts at implementing the theorem (Spears [18], Audrino, Huitema and Ludwig [1], Backwell [2], Kiriu and Hibiki [15], Flint and Mare [13]) and we will start this report by first reviewing and analysing these methods. We will then discuss the details of our implementation approach and investigate its accuracy and stability. Lastly, we will apply the algorithm to real market data and analyse the resulting probability distributions, risk premium time-series and volatilities.

10 4 2 Technical Prerequisites In this section we ll outline the notation and some financial and mathematical theory used in the rest of the paper. 2.1 Notation P =[p i,j ] specifiesamatrixwithentriesp i,j,whereirefers to a row index and j denotes a column. P[:,j] denotesavector,whichisaj th column of the matrix P. P[i, :] denotes a vector, which is an i th row of the matrix P. P[k : m, :] selects a matrix consisting of rows in P from k to m, inclusive. P Truereal-worldprobabilitymeasure. Q Classicrisk-neutralprobabilitymeasure. R Marketimpliedreal-worldprobabilitymeasure. CanbeequaltoP if markets are correct. 2.2 Key Financial Concepts Equity Risk Premium (ERP) Equity Risk Premium is defined as the extra return above the risk free rate that an investor can make by investing in a stock market. Generally, it is believed that due to higher variability of returns, and hence higher risk, investing in equities should yield an additional return to what can be achieved through, for example, government bonds. An equivalent risk premium concept exists in other markets as well, however, in this paper we will be focusing at the extra return within the equities asset class. Due to its importance, ERP has been the focus of research for many years and there have been numerous approaches at estimating it calculating from historical data, conducting surveys or calculating the premium as implied by prices of financial instruments. An excellent

11 2 Technical Prerequisites 5 summary and analysis on the breadth of di erent methodologies can be found in a popular paper by Damodaran [10], which has been very recently updated. In this paper, we will be looking at estimating a forward-looking risk premium as implied in option prices, which is possible with Ross recovery theorem. We will make use of its timehomogeneity assumption and apply the theorem over multiple option expiries. This allows us to calculate Equity Risk Premium term structure implied by the market and investigate how investor sentiment changes with the investment horizon. Whether or not such term structure exists is debatable and there is research to both sides of the argument. For example, Binsberger, Brandt and Koijen [4] use dividend strips in order to extract the risk premium and investigate whether the tenor e ects the level of returns. They found that the short term dividend strips have a higher risk premium, suggesting that the risk premium curve is downward sloping. On the other hand, Boguth, Carlson, Fisher and Simutin [6] have shown that there is little evidence to suggest that returns of dividend claims depend on the time horizon. Using a di erent methodology and extracting implied Equity Risk Premium from the CDS market, Berg [3] has found the premium term structure to be flat before the financial crisis and downward slopping during the crisis. In section 6, we will look at the risk premium curves obtained from applying Ross recovery theorem and investigate whether Ross model supports any of the above results Stochastic Discount Factor (SDF) and the Pricing Kernel Stochastic Discount Factor (SDF) represents the idea of risk-neutralisation and describes the link between the risk-neutral and real-world probability distributions. Also known as the pricing kernel, it can be used to price derivatives in the real-world measure. Given an option V (S t,t)withapay-o functionf(s t ), it s value can be written as where V (S t, 0) = E P [ (S T )f(s T )] (2) (S T ) denotes the pricing kernel. Similarly, under the Black-Scholes framework, using risk-neutral pricing, the value of the same option is given by: Writing the expectation in an integral form yields: V (S t, 0) = e rt E Q [f(s T )] (3) Z 1 V (S t, 0) = e rt f(s T )Q(S T )ds T 0 Z 1 = e rt f(s T ) Q(S T ) 0 P (S T ) P (S T )ds T apple = E P e rt Q(S T ) P (S T ) f(s T )

12 2 Technical Prerequisites 6 In the above equation P (S T ) denotes the real-world probability distribution of S T and Q(S T ) denotes the risk-neutral. Comparing this equation with (2), it is easy to see that given by: (S T )is (S T )=e rt Q(S T ) P (S T ) This is simply a discounted ratio of two probability measures. It neutralises investor risk preferences and allows pricing of derivatives without full knowledge of the real-world probability distribution. As we will see in section 3.1.5, SDF is exactly what Ross recovery theorem allows one to find and this knowledge permits the calculation of a real probability distribution Breeden-Litzenberger Formula and the Risk-Neutral Density In order to use the knowledge of the Stochastic Discount Factor and solve for real-world probability density, we also need to know the risk-neutral measure. One of the most common ways to calculate it is using the Breeden-Litzenberger [7] formula, which was developed back in Even though, we will not be applying it within our methodology, we briefly outline the formula below, as it is used in many implementations of the Ross theorem considered in this paper. The idea of the formula is very simple. Assuming a call option, C t,withapay-o function f(s T )=max(s T which in the integral form becomes: K, 0), we can rewrite the equation (3) as: C t = e rt E Q [max(s T K, 0)] Z 1 C t = e rt (S T K)Q(S T )ds T K As before, Q(S T )denotestherisk-neutralprobabilitydistribution,whichwecansolveforby di erentiating twice with respect to strike, = e rt Z 1 K Q(S T )ds 2 C 2 = e rt Q(K) Re-arranging, gives the risk-neutral probability distribution in continuous time: Q(K) =e C 2 (4) One of the strengths of Breeden-Litzenberger result is that it s completely model-independent and only requires knowledge of option prices. However, since it calculates a second derivative of the price surface, it is highly sensitive to any small inaccuracies and special care must be taken when calculating the risk-neutral measure. For more information about applying Breeden-Litzenberger in practice, please refer to Figlewski [12].

13 2 Technical Prerequisites Arrow-Debreu Securities and State Prices Arrow-Debreu security is a derivative contract, with an expiry time T, which pays one unit of a given numeraire if a certain state of the world occurs and expires worthless otherwise. Even though, these contracts represent a theoretical construct, they have found many uses in derivative pricing. If we assume a money market account as the numeraire, then Arrow- Debreu security price, or a state price, is equivalent to a risk-neutral probability of a given state occurring. Let s = {S 1,S 2,S 3,...,S n } be a vector representing a discrete and finite number of states that a financial asset S i can take at time T. Then an equivalent vector of Arrow-Debreu security prices at time zero can be defined as p = {p 1,p 2,p 3,...,p n }.Eachofthesesecurities will pay $1 if S i transitions to a given state at expiry. An investor who purchases all of them is thus guaranteed to receive a pay-o of $1 at time T and as such has invested in a risk-free portfolio. Consequently, such portfolio must yield a risk-free rate and it s price at time zero must be: using, nx i=1 p i = 1 1+r Using this relationship, it is possible to convert state prices into risk-neutral probabilities q i = p i P n j=1 p j (i =1,...,n) which consequently allows to use state prices to value financial derivatives in a discrete setting. 2.3 Mathematical Tools Below we give a brief overview of the main concepts that Ross [17] makes use of in development of the recovery theorem Markov Chains At a basic level, Markov chain is a probabilistic construct that aims to capture conditional probabilities of various events happening. As before, let s = {S 1,S 2,S 3,...,S n } be a vector representing a discrete and finite number of states that a random variable can take. If at some time t, a random variable X is in one of these states, then it can transition to a di erent state at the next time step t +1. The probability q i,j of moving to the next state j depends only on the current state i and is independent of the past states and the current time: q i,j = P (X t+1 = j X t = i) =P (X t+k+1 = j X t+k = i)

14 2 Technical Prerequisites 8 Such process is called time-homogeneous, as the same probability applies over a given timeframe regardless of the initial time. Probabilities of moving to each possible state from any other other state are called transition probabilities and defined in a transition matrix. This matrix is n by n, whererows represent the current state and columns denote a state that X can transition to. Since a transition matrix specifies probabilities for the entire state space, it is a non-negative matrix and it must hold that: nx q i,j =1 8i j=1 As we will see, Ross recovery theorem assumes a Markov chain process for a financial asset, which will require estimation of an appropriate transition matrix Perron-Frobenius Theorem In the following, we provide a partial statement of the Perron-Frobenius theorem, which is required to apply the Ross recovery framework. Given a square matrix A, avectorx and ascalar,wecanwriteasimpleeigenvalueproblem: Ax = x The Perron Frobenius theorem states, that if A has strictly positive elements, then there exists an eigenvector whose elements are also strictly positive and which is associated with the largest, in absolute terms, eigenvalue. In anticipation of applying this theorem to a Markov transition matrix, we note that transition matrices may have zero value entries, and hence would be non-negative, rather than strictly positive. However, Perron-Frobenius theorem is also applicable to non-negative matrices if they are also irreducible.

15 9 3 Ross Recovery Theorem 3.1 Overview Under certain assumptions, Ross Recovery theorem allows to extract real-world transition probabilities, as implied by market prices. In this paper, we will refer to this probability as the real-world measure, denoted by R, even though, the market can be wrong and it may di er from a true real-world probability, P Model Assumptions In order to achieve recovery, Ross [17] makes a number of assumptions, which allow him to apply the Perron-Frobenius theorem and obtain a matrix with real-world probabilities. Below we list some of these assumptions. Ross model assumes a discrete world in which a finite number of states exist. A state variable, which we ll denote by X, isdrivenbyamarkovchainprocesswithaprobability matrix F =[f i,j ]. Each element, f i,j represents a probability of moving from state i to state j. ThismatrixisunknownandrecoveredintheprocessofapplyingRoss theorem. A driving Markov chain process implies a number of other assumptions. A state space is bounded, hence there is a lower and an upper limit for the value of X. Moreover,X is path independent and its past states have no influence on the characteristics of the current state. The transitions are assumed to be time-homogeneous, meaning that the same transition matrix applies over a given time interval, regardless of when that time interval starts. In other words, transition from time t to time t +1isindependentoft. In order to recover the real-world transition matrix F, Ross assumes that Arrow-Debreu prices across all possible states are known. They are estimated from prices of other financial derivatives traded in the market and contained in a state-price matrix P =[p i,j ]. This also implies knowledge of the interest rates in each state. For the recovery theorem to work properly, Ross implicitly imposes a restriction for interest rates to be di erent in each state. In order words, the sum of rows in matrix P must be di erent. In case when the sums are equal, Ross Recovery returns a risk-neutral matrix, as an estimate for the real-world one. Lastly, market participants are assumed to be maximising their expected utilities and there exists a representative agent whose view reflects an aggregate view of all agents. Their

16 3 Ross Recovery Theorem 10 utilities are assumed to be state-independent State-Price by Tenor Matrix S Let an m by n matrix S denote the current Arrow-Debreu prices for a finite number of states n and a number of time horizons m: S =[s i,j ] (i =1,...,m; j =1,...,n) Each row, therefore, denotes a state space vector for a given tenor. This matrix represents the first step in Ross Recovery implementation. There are a number of ways to calculate it the most straightforward is via the Breeden- Litzenberger formula from section Using their formula, one can obtain a risk-neutral distribution from option prices, which is then converted into interval probabilities and discretized into n states. Further multiplying each probability element by a discount factor yields the S matrix. In section 3.2, we ll look into other estimation approaches of the matrix S State-Price Matrix P State-price matrix P is key to Ross recovery and is the main focus of this paper and others that have implemented the theorem. Matrix P is a square, n by n matrix that contains Arrow-Debreu prices, p i,j,maturingatsometimet for all the states: P T =[p i,j ] (i, j =1,...,n) Astatepricep i,j pays out if we transition from state i to state j at time T.Therowsof P represent the state we re currently in and the columns denote the state we will transition to. Each state is denoted by a value of the state variable X and represents X being in the interval between the mid points with neighbouring states. Since the state space is bounded, first and last columns represent states where X is below or above a certain value, rather than being within an interval. Time horizon of the matrix only refers to a time period and, due to time-homogeneity assumption, is independent of when that period starts. The current state row of the P matrix equals to S[i, :], where i represents the row with the same maturity as that of P. As with S, thesumofrowsisequaltoadiscountfactorandeachelementinp can be converted to an equivalent Q-measure probability using: p i,j q i,j = P n k=1 p (i, j =1,...,n) (5) i,k In the above, q i,j represents a risk-neutral probability of going into state j, conditionalon being in state i. This property allows to use P in derivatives pricing and is applied during the calibration stage.

17 3 Ross Recovery Theorem Real-World Matrix F As discussed earlier, F represents a real-world transition matrix governing the evolution of the state variable X. If X is assumed to represent the price of a financial asset, such as an equity index, then F can be obtained using the recovery theorem, as we shall see next. Like P, matrixf is n by n, where rows represent the current state and columns, the next: F T =[f i,j ] (i, j =1,...,n) Steps to Recovery A short overview of the theorem, based on Kiriu and Hibiki [15], is given below. For more details, we refer an interested reader to the original paper by Ross [17] or an excellent summary and evaluation by Carr and Yu [8]. Assume that a state-price matrix P T is known. In order separate investor risk preferences from the real-world distribution, Ross [17] assumes an existence of a representative agent, whose utility function is: nx u(c i )+ f i,j u(c j ) (i =1,...,n) (6) j=1 In the above equation, is the discount factor of the utility and u(c i )istheutilityof consumption in state i. Equation(6)canalsoberewrittenas: u(c i )+ E R [u(c j )] (i =1,...,n) where the expectation is taken under the real-world measure R and represents the expectation of future utility. An investor is looking to maximize their utility by controlling consumption at di erent states, subject to the initial wealth constraint:! nx max u(c i )+ f i,j u(c j ) (i =1,...,n) c i,c j subject to: c i + j=1 nx p i,j c j = w j=1 (i =1,...,n) Setting up a Lagrangian gives:! nx nx L = u(c i )+ f i,j u(c j )+ w c i p i,j c j j=1 If we partially di erentiate with respect to each c i and c j,weget: j=1 u 0 (c i ) =0 (i =1,...,n) (i =1,...,n)

18 3 Ross Recovery Theorem 12 and f i,j u 0 (c j ) p i,j =0 (i, j =1,...,n) Solving each equation for and equating them, we get a relationship between p i,j and f i,j : p i,j = u0 (c j ) f i,j u 0 (c i ) (i, j =1,...,n) (7) The equation (7) is the value of a Stochastic Discount Factor, mentioned in section The equation states that an n by n matrix with the pricing kernel values on the right-hand side is parametrized by a constant and a vector with n elements, which gives a total of (n +1)unknownsthatweneedtoestimate. Rememberingthatprobabilitiesforeachstate sum to one, this gives us n equations one short needed to solve the system. However, since matrix P is non-negative and irreducible, Ross [17] applies the Perron- Frobenius theorem, which states that there exists a unique and positive eigenvector associated with the largest eigenvalue. Letting v denote the eigenvector and denote the corresponding eigenvalue, we have: = u 0 (c i )= 1 v i (i =1,...,n) Substituting the above equations into (7), we can solve for real-world transition probabilities: f i,j = 1 v j v i p i,j (i, j =1,...,n) (8) 3.2 Implementation of the Recovery Theorem The di culty in implementing and applying Ross Recovery theorem comes from estimating an accurate state-price matrix P. Once it s known, then it s trivial to obtain the transition matrix F via Perron-Frobenius theorem using the procedure discussed above. Past papers that have attempted to implement the theorem mostly focused on di erent approaches to compute a valid P matrix. Overall, the implementation process for Ross Recovery consists of the stages outlined in Figure 1, as shown in Spears [18] and Kiriu and Hibiki [15]. Going through these stages involves solving two ill-posed problems. In the first step, we need to obtain Arrow-Debreu security prices for the current state. A standard approach is via a risk-neutral probability distribution that can be calculated by applying a Breeden- Litzenberger formula (4). However, di erentiating option prices twice yields a highly sensitive function that will amplify any inaccuracies in the option price surface. Furthermore, options are traded at a finite number of strikes and will not always be of required granularity to

19 3 Ross Recovery Theorem 13 Q Risk-Neutral Transi,on Matrix Market Data (Op,on prices) S State-price by Tenor Matrix P State-price Transi,on Matrix F Real-World Transi,on Matrix Step 1 (Ill-posed problem) Step 2 (Ill-posed problem) Figure 1: The path to recovery. Step 3 (Ross Recovery) populate the state prices. Hence in order to obtain a clean probability measure, option prices need to be smoothed, interpolated and potentially extrapolated. The non-triviality of this task has been highlighted by a nubmer of papers. Figlewski [12] discusses the issues and develops a method of obtaining risk-neutral densities which involves filtering, smoothing and fitting tails of the distributions separately. In an attempt to implement Ross Recovery theorem, Audrino, Huitema and Ludwig [1] resort to neural networks to estimate robust volatility surfaces. Flint and Mare [13] addressed this step by fitting market data to a deterministic SVI volatility model that gives a smooth surface from which to calculate option prices and densities. Once risk-neutral probabilities have been calculated and a state-price by tenor matrix S is populated, we need to estimate a state-price transition matrix P. Thisisanotherillposed problem, as we only have state prices for the current state of the world. Often the number of equations that can be generated is less than the number of matrix entries and hence the problem is underdetermined. It is necessary to impose additional constraints on the system to solve for a unique state-price matrix. We ll now look at some of the approaches taken at implementing the steps in Figure Ross, 2015 The original paper by Ross [17] proposes to apply Breeden-Litzenberger to calculate state prices for each tenor in S and then use the time-homogeneity assumption to solve for entries in P via the law of total probability. Given a state vector S[i, :] and assuming the time interval between states in S equals to

20 3 Ross Recovery Theorem 14 the time horizon of the matrix P, thenthefollowingrelationshipholds: S[i +1, :] = S[i, :] P (9) Ross [17] then states that if the number of expiries, m, is equal to the number of states, n, then we ll have m 2 equations to solve for m 2 unknowns. In practice, this means that to obtain a granular matrix P, the price or volatility surface should be interpolated in time dimension, so that it yields more expires and hence more rows in S to match with the number states. Ross [17] also puts an additional restriction on P stating that its rows should be unimodal. The relationship in (9) can be applied across tenors and re-written as an optimisation problem that can be solved via a method of least squares: Or: where A = S[1 : m P =argmin S[1 : m 1, :] P S[2 : m, :] 2 2 P 0 P =argmin AP B 2 2 (10) P 0 1, :] and B = S[2 : m, :]. Depending on the values of n and m, thiscan be an overdetermined or underdetermined problem Spears, 2013 Spears [18] has tried to replicate Ross result using his working version of the paper and found that one needs to impose additional restrictions in order to extract a sensible P. A few of the restrictions he tested include unimodality, forcing the mode to lie on the diagonal and setting diagonal values equal to each other. For some of these, he also applied Breeden- Litzenberger formula to calculate state-price by tenor matrix S and used a least-squares and alternating least squares (ALS) algorithm to solve for P Audrino, Huitema and Ludwig, 2015 Audrino, Huitema and Ludwig [1] have applied Ross Recovery theorem to S&P 500 historical data and analysed moments of the risk-neutral and real-world probability distributions. Their approach was novel for two reasons. First, they applied neural networks to obtain agranularandsmoothvolatilitysurfaces,whichwherethenconvertedtostatepricesvia Breeden-Litzenberger. Secondly, they apply a Tikhonov regularization term to their optimisation problem in order to stabilise the solution and minimise the impact of noise: P =argmin AP B P 2 2 P 0 Such regularisation has the e ect of bringing entries of P closer to zero. If infinity, then the optimal solution will be given by a null matrix. is set to

21 3 Ross Recovery Theorem Backwell, 2015 The vital di erence between Backwell [2] methodology and others considered in this section is that he didn t resort to Breeden-Litzenberger formula in order to construct a current state matrix of Arrow-Debreu securities. Instead, he used known properties of a state-price vector, along with a restriction on smoothness, and solved for it as an optimisation problem. Specifically, we know that due to its relationship with the risk-neutral probability measure (5), a state-price vector can be used to calculate option prices and forwards, and its sum must equal to a discount factor. Taking these values as implied by a given S matrix, Backwell [2] then minimises their di erence with the market values. Matrix P is then calculated using the standard optimisation problem (10), with restrictions of positive interest rates, forward pricing across states and smoothness. The benefit of this approach is that it eliminates the need to calculate a price/volatility surface or make any decisions on its interpolation or extrapolation. Matrix P, andconsequently the state dimension of S, canbeselectedofanyspanandgranularity,asitdoesn t need available option prices in the tails. The method proposed in this paper uses the work of Backwell [2] and we will formalise these properties in more detail in section Kiriu & Hibiki, 2016 Kiriu and Hibiki [15] focus their research on Step 2 in the diagram in Figure 1 and investigate a method to calculate P from a given S. They use hypothetical data in their analysis in order to isolate this step from the impact of various methods on estimating S in the previous step. Their methodology extends the work of Audrino, Huitema and Ludwig [1]. Instead of using Tikhonov regularization to stabilise the solution, they regularise using a custom matrix P: P =argmin AP B P P 2 2 P 0 Matrix P is constructed using an appropriate state-price vector from S as the current state row, and each row above and below is set equal to current state, shifted by one to the left or right respectively. State prices that overflow the most left and right columns are added to the first or last state price in that row. Under this restriction, setting the regularisation parameter = 1 will yield a risk-neutral transition matrix as an estimate for F, sincethe sum of rows in P is equal.

22 3 Ross Recovery Theorem Flint & Mare, 2016 Flint and Mare [13] apply the methodology developed by Kiriu and Hibiki [15] to the South African Top 40 index futures. As mentioned earlier, their approach at estimating S was to use an SVI volatility model to construct a valid surface and calculate state prices using Breeden-Litzenberger result. Moments of the recovered probability distributions were then plotted as a time-series, similarly to Audrino, Huitema and Ludwig [1]. 3.3 Analysis of the Theorem In this section, we ll look into some of the limitations of the theorem that might hinder its usefulness and practicality Evaluation of Model Assumptions While interesting and fascinating, the Recovery theorem is based on assumptions that might not necessarily hold in practice. For instance, assuming a Markov chain driving process raises a number of concerns related to a bounded state-space, time-homogeneity and path independence. One can argue that prices of financial instruments, such as stocks or commodities, do not have an upper bound and putting a restriction on their level would yield a di erent asset from the assumed one. Dubynskiy and Goldstein [11] argue that assuming a bounded state variable is economically implausible and show that this assumption is actually the reason why it is possible to calculate real moments of the underlying process. Assumed time-homogeneity implies that the same probability transition matrix governs the dynamics of the state variable during any fixed time interval. However, markets can have volatile and quiet periods and hence di erent transition matrices must apply. Timehomogeneity assumption may also lead to negative entries in the state-price transition matrix, as shown in Carr and Yu [8]. Moreover, there seems to be strong path-dependency in the financial markets. If the S&P 500 index is at 1,500 points, states of the world would be di erent depending if the index climbed there from 800 points or dropped from 2,000. Yet, the state corresponding to 1,500 points in P will be the same regardless of which state the index transitioned from. This is further highlighed by Taleb [20] in his comment on the theorem. Ross assumption about market participants maximising their expected utilities and an existence of a representative agent is also debatable. Carr and Yu [8] paper gives examples of studies, which imply that market players do not necessarily act in order to maximise their expected utilities.

23 3 Ross Recovery Theorem Comparison with Black-Scholes The key di erence of the Ross recovery model from the Black-Scholes framework is that it is discrete and it makes the recovery of real-world parameters possible. Due to a discrete nature of the model, in order to capture movements of financial assets with the Ross theorem, the granularity of the parameters must be suitable for a chosen time-horizon. The Black-Scholes framework, however, allows one to consider any time frame using the same parameter values. Carr and Yu [8] have successfully extended Ross theorem to a continuous state-space using recurrent stochastic processes. Unlike Ross theorem, Black-Scholes model also doesn t imply the natural distribution of the assets. In a classical risk-neutral pricing, the drift of the underlying instrument is hedged away and is eliminated from the option pricing formulas. Hence, varying the drift does not impact the price of the contract, which means that it s not possible to work backwards and recover these parameters. Ross model, on the other hand, is capable of capturing the drift dynamics and makes it possible to estimate them. Furthermore, it is interesting how Ross theorem contradicts the Black-Scholes assumption about constant interest rates. If the rates were indeed constant, then recovery theorem ceases to work and returns a risk-neutral distribution as the result. Hence, in order to get a realworld probability matrix, di erent states must have di erent interest rates, which is rather more realistic than the Black-Scholes framework Implementation Challenges Ross assumption that the state prices for all the states are known posses many di culties in implementing the recovery theorem as highlighted in section 3.2. The challenge in estimating P matrix has been the core of many research papers dedicated to Ross recovery and many di erent approaches and methods have been attempted. Since the transition matrix doesn t naturally follow form the current state prices may suggest that either we re trying to extract information which isn t necessary there or trying to enforce time-homogeneity on a process that might not have this characteristic Verification of Results In addition to being challenging to implement, Ross Recovery theorem is also challenging to verify. In order to perform a controlled experiment, we need to start with a hypothetical realworld transition martix, as has been done in Kiriu and Hibiki [15], and obtain a state-price by tenor matrix S by reversing the steps in Figure 1. Then S can be used as a starting point to test if we can recover the original real-world matrix.

24 3 Ross Recovery Theorem 18 However, this framework is testing how accurately we can estimate a P matrix from S, rather than testing how well the recovery theorem uncovers originally encoded distribution. Since we ve encoded it by reverse-applying the recovery theorem, there isn t any value in testing how well the recovery theorem decodes it back. Ideally, we d want to have some process with a known real-world probability distribution and which can be used to price options. This would then allow to test if the same realworld distribution is uncovered using only option prices. Of course taking a simple Brownian motion process with a drift µ for these purposes is not possible, as the drift won t feature in option pricing nor the volatility surface (which will also be flat). Hence, we need other means of testing the validity of the recovered probability distribution. We could test it empirically. However, this is also di cult since at the end of the forecast horizon, we only obtain a single data point rather than a realised probability distribution. So if we take a 3-month real-world implied probability distribution at time t, we won t able able to confirm it s validity at time T as we will observe only one realised path. Vincent-Humphreys and Noss [22], however, have tried estimating a real-world distribution empirically, by calculating an adjustment to the risk-neutral distribution, which would make historical real-world observations more likely under this adjusted density. However their approach, while possible, is very data intensive - just for 60 observations they needed options data from 1992 to However, even if there was some way to obtain a true real-world probability distribution to compare with and verify the output of the Ross theorem, we wouldn t be able to explain whether any di erence would be due to the model/implementation being wrong or the market being wrong, since Ross theorem gives a market implied real-world distribution R, which can be di erent from an actual real world distribution P.

25 19 4 Proposed Implementation Method As discussed earlier in section 3.2, implementation of Ross Recovery theorem is not a trivial task and involves solving two ill-posed problems estimation of risk-neutral probability densities and a subsequent calculation of the state-price transition matrix P. Thismakesthe theorem greatly dependent on implementation decisions and various assumptions made by practitioners, which further reduces its reliability and consistency. The method that we would like to propose extends the approach taken by Backwell [2] and uses a new regularisation scheme. His implementation does not require smoothing or interpolation of volatility surfaces and allows to specify any dimension for matrix P even when there isn t any options data in the tails, which is often the case. In addition to calculating the state prices at each maturity by calibrating them to option prices, we will do this while solving for matrix P. Thebenefitisthatiteliminatestheneed for Step 1 in Figure 1 and allows us to solve for state-price transition matrix directly from market data. Consequently the number of ill-posed problems is reduced to one. Moreover, we will further make use of known properties of state-price transition matrices and apply them to transition matrices of di erent maturities as implied by P. Since, all these restrictions are not enough to produce a unique solution, further regularization will be used. In this section we will discuss the details of this new approach. 4.1 Calibrating the State-Price Matrix P In order to solve both ill-posed problems at the same time, we exploit the time-homogeneity property of the matrix P and use it to calculate state prices for later time horizons. From section 3.2, we had the following relationship between state-price by tenor matrix S and the state-price matrix P: S[i +1, :] = S[i, :] P In this case, S contains state prices for the current state only, however, this relationship should also hold for other states of the world, whose state prices are given in P. Assuming that P T contains state prices maturing at some time T then we can use the above relationship and find P 2T : P 2T = P T P T

26 4 Proposed Implementation Method 20 Or more generally, for a positive scalar k, P kt will be given by: P kt =(P T ) k (11) If the time-homogeneity assumption holds, then by selecting an appropriate value for k, we can obtain state-price matrices for di erent time frames. Since we re looking to investigate properties of probability distributions at di erent tenors, our P matrix should be well-behaved such that it can be multiplied by itself many times over and still produce reasonable and meaningful probability distributions across all its states. By calculating P kt and we are able to ensure that certain properties apply and hence impose additional restrictions on P T. Secondly, instead of fixing the current state row of P to equal to market-implied state prices, as was done in previous implementations (section 3.2), we will solve for it along with other entries in P. Theavailabledatawillinsteadbeusedtorestrictthecurrentstaterow entries such that they generate option prices consistent with those seen in the market or within some reasonable margin of error. Let V (X t,t)denotethepriceofaeuropean-styleoptiononanassetx t, maturing at time T with a pay-o function f(x). Then, its value at time t is given by: V (X t,t)=e r(t t) E Q [f(x T ) F t ] Using the equation (5) from section 3.1.3, we can convert the P T matrix into an equivalent Q measure and apply it to discretely approximate our option pricing formula above. Hence, the price of an option in state i is given by: nx V i (X i,t)= p i,j f(x j ) (i =1,...,n) (12) j=1 In (12) p i,j represents an Arrow-Debreu security price in state i which pays out if state j occurs at time T and X j is the value of the asset in state j. Since we know option prices in the current state, we can set i accordingly and ensure that our estimate for a P T matrix gives option prices close to those seen in the market. Furthermore, using the relationship (11) above, we can use P T to calculate P kt for each available maturity and impose accurate option pricing across tenors. Alternatively, note that instead of matching to option prices, we can transform them to volatilities and match with available market implied volatilities. However, we have found that this significantly increases the computational cost of the optimisation problem. In addition to pricing options across strikes and tenors using P, wenowhavetoconvertthosepricesinto volatilities, which requires numerical inversion of the Black-Scholes formula. Since this has to be performed for each optimisation step, the overall cost to calculate the P matrix increases significantly. Nonetheless, there is no theoretical reason for not using implied volatilities to calibrate the P matrix and this approach would also work for our purposes.

27 4 Proposed Implementation Method Additional Market Constraints Since a given P matrix also defines such things as interest rates, forward prices and dividends, we can use this to further restrict our solution Interest Rate Curve When calibrating the P T matrix, we apply: nx j=1 p i,j = 1 1+r i (13) and ensure that the sum of Arrow-Debreu prices in the current state, set by i, matchesthe discount factor implied by market interest rate curve. In the above equation, r i is the deannualised interest rate corresponding to the tenor of the transition matrix. Again, this is done across maturities using P kt matrices and ensures that P T produces an interest rate curve consistent with the one observed in the market. Interest rates in other states are not known and hence are free to vary Dividends As with interest rates, only the current state dividend yield is known. In order to ensure that forwards are priced correctly under the P matrix expectation, we will assume a current dividend yield across all the states. While in real-world it is possible for a dividend yield to vary between states, our model can still capture the e ects of this variation, since a dividend yield occurs in a combination with a variable interest rate. Moreover, such assumption would still be made implicitly even if the state variable X didn t pay any dividends in the current state and a zero dividend yield was assumed for other states Forwards Since the expectation under a Q measure yields a forward price, we have: F i = E Q [X i ]= 1+r i 1+y i X i =(1+r i ) nx p i,j X i,j j=1