BAYESIAN ESTIMATION OF PROBABILITIES OF TARGET RATES USING FED FUNDS FUTURES OPTIONS

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1 BAYESIAN ESTIMATION OF PROBABILITIES OF TARGET RATES USING FED FUNDS FUTURES OPTIONS MARK FISHER Preliinary and Incoplete Abstract. This paper adopts a Bayesian approach to the proble of extracting the probabilities of the FOMC s target Fed Funds rate fro option prices, extending the fraework provided by Carlson et al. (2005) in a nuber of directions. Likelihood-related novelties include (i) the allowance for slippage between the target rate and the onthaverage rate and (ii) accounting for the nonnegativity of options prices by truncating the easureent error. In addition, we generalize the likelihood for target rates to include the likelihood for target-rate paths. Prior-related novelties include (i) enforcing nonnegativity constraints on the probabilities and (ii) the infored ignorance prior that allows for the inclusion of a large nuber of possible target rates (which is especially iportant for joint/path estiation). 1. Introduction As Carlson et al. (2005) point out, the proble of extracting the probabilities of the FOMC s target Fed Funds rate fro option prices is greatly siplified if one assues the target rate can take only a sall nuber of values. They use Classical least-squares regression techniques to extract the target-rate probabilities. The paper in hand adopts a Bayesian approach to inference instead, and extends the fraework provided by Carlson et al. (2005) in a nuber of directions. 1 The ain advantage to the Bayesian approach is that one gets to specify a prior distribution. The prior allows us to deal with iportant features of the target-rate probabilities in a sensible way. First, the target-rate probabilities should satisfy an equality constraint (they should su to one) and a nuber of inequality constraints (they should each be nonnegative). Although both Classical and Bayesian approaches handle the equality constraint easily, the Classical approach has difficulty handling the inequality constraints while the Bayesian approach handles the quite naturally via the prior. 2 Second, the Bayesian approach akes it easy to allow for a large nuber of target rates (or target-rate paths) in the odel. There is a siple prior for the unobserved probabilities that encapsulates what ight be called partially infored ignorance. In effect, this Date: March 21, 11:09a. I thank Jerry Dwyer, Mark Jensen, and Dan Wagonner for helpful discussions. The opinions expressed herein are those of the author and do not represent those of the Federal Reserve Bank of Atlanta or the Federal Reserve Syste. 1 The reader is referred to Carlson et al. (2005) for oitted institutional details. 2 For an early exaple of the Bayesian approach to handling inequality constraints, see Geweke (1986). 1

2 2 MARK FISHER prior asserts that only a few of the target-rate probabilities are nonnegligible without specifying which those are. Forally, this prior is a syetric Dirichlet distribution with a concentration paraeter that encourages parsiony. In addition, the concentration paraeter itself is assued to be unknown and therefore has its own prior distribution. (A closely-related prior is used in ixture odels to allow for a potentially large nuber of ixture coponents while at the sae tie encouraging parsiony. One should keep in ind, however, the current exercise is one of functional-for fitting, rather than density estiation per se.) This prior allows one to include a large nuber of target rates (greater even than the nuber of observations) and still obtain a stable and well-estiated posterior distribution. This feature has the potential to allow robust estiation of paths of interest rate targets, where the nuber of paths (and the nuber of associated joint probabilities) is quite large. Third, the Bayesian approach provides a coherent treatent of the possibility of unscheduled FOMC eetings. The ain disadvantage to the Bayesian approach is that the coputational burden is significantly higher than for the Classical approach. (Regarding this point, however, one should recognize there are no free lunches, even in statistics.) Novelties. Likelihood-related novelties include (i) the allowance for slippage between the target rate and the onth-average rate and (ii) accounting for the nonnegativity of options prices by truncating the easureent error. In addition, we generalize the likelihood for target rates to include the likelihood for target-rate paths. Prior-related novelties include (i) enforcing nonnegativity constraints on the probabilities and (ii) the infored ignorance prior that allows for the inclusion of a large nuber of possible target rates (which is especially iportant for joint estiation). Caveats. The options are Aerican, not European. The risk-neutral easures for different horizons (different onths) are not utually copatible and consequently the proposed joint estiation is not copletely coherent. Siilarly, futures prices are used in place of forward prices. Liquidity is not taken into account in weighing the quotes. (This could enter via the uncertainty of the easureent error.) Outline of paper. Section 2 presents on overview of the odel, including the likelihood, the prior, and the posterior sapler. Section 3 introduces onth-average target rates, deals with the joint distribution of ultiple target rates (target rate paths), and outlines of how to extend the odel to include unscheduled eetings. Section 4 describes soe issues related to putting the available data into a for suitable for estiation. Section 5 discusses stochastic-process dynaics of the target-rate probabilities. Appendix A includes soe additional aterial. The epirical section of the paper is currently absent. Although a few proof of concept tests have been run, a systeatic treatent of a substantial data set is only just beginning.

3 BAYESIAN ESTIMATION OF PROBABILITIES 3 2. The big picture The goal is to estiate the risk-neutral probability distribution for the target fed funds rate fro put and call options data. Let R denote the target rate as set by the Federal Open Market Coittee (FOMC). 3 We assue the target rate is restricted to a finite set of known values R r = {r 1,..., r K }. Let the risk-neutral probabilities for the values be given by p(r = r j β) = β j, (2.1) where β = (β 1,..., β K ), β j 0 and K β j = 1. We can express the (risk-neutral) density for R as a ixture of point asses: p(r β) := β j δ(r r j ), (2.2) where δ( ) is the Dirac delta function. 4 We assue β is unknown. It is the focus of our investigation. We odel the onth-average fed funds rate S as the su of the target rate R and the slippage between S and R: S = R u. (2.3) Given (2.2) and (2.3), we can express the risk-neutral density for S as a related ixture of point asses: p(s β, u) := β j δ(s r j u). (2.4) We assue u is unknown as well. Consider a European option written on the onth-average rate S with a strike price of K. The payoff to the option at expiration is φ(k, γ, S) := ( γ (S K) ), (2.5) where x := ax(x, 0) and { γ = 1 for a call 1 for a put. (2.6) Let B denote the value of a risk-free zero-coupon bond that atures on the expiration date. The value V of the option is the present value of the expected payout coputed using the 3 See Section 3 for institutional details. 4 The ain features of δ( ) are (i) δ(x x0 ) = 0 if x x 0, (ii) δ(x x 0) dx = 1, and (iii) f(x) δ(x x0) dx = f(x0).

4 4 MARK FISHER risk-neutral probabilities: V = B = B = B φ(k, γ, S) p(s β, u) ds β j φ(k, γ, S) δ(s r j u) ds β j φ(k, γ, r j u). (2.7) Equation (2.7) suggests the task ahead: Given soe options data, fit a functional for using the basis functions φ(,, ). Assue the data are coposed of n observations, {K i, γ i, V i } n i=1, and suppose they are generated according to y i = β j φ(k i, γ i, r j u) ε i (2.8) where y i := B 1 V i is the deflated option value and ε j is a easureent error. Let X denote the n K atrix where X ij = φ(k i, γ i, r j u) = ( γ i (r j u K i ) ). (2.9) (Note that X depends on the unknown u. We will write X(u) when it is convenient to ake this dependence explicit.) We can express (2.8) as y i = X i β ε i, (2.10) where X i denotes the the i-th row of X. Stacking the n observations, we have y = Xβ ε, (2.11) where y = (y 1,..., y n ) and ε = (ε 1,..., ε n ). It is convenient to suppose ε N(0 n, σ 2 I n ). 5 We assue σ 2 is unknown. Let θ = (β, u, σ 2 ) denote the vector of unknown paraeters. The likelihood for θ follows fro (2.11) and the distribution for the easureent error: ( ) S(β, u) p(y θ) = N(y X(u)β, σ 2 I n ) = (2 π σ 2 ) n/2 exp 2 σ 2, (2.12) where 6 S(β, u) := (y X(u) β) (y X(u) β). (2.13) Reark. The novelty in (2.12) relative to Carlson et al. (2005) is to allow for u 0. 5 One drawback of this assuption is that the sapling distribution for the data p(y θ) [see (2.12)] iplies we ay observe negative option prices. This can be reedied by truncating the distribution for y. We incorporate this below. 6 A denotes the transpose of A.

5 BAYESIAN ESTIMATION OF PROBABILITIES 5 Predictive distributions. At this point it is convenient to ake a few rearks about predictive distributions, both prior and posterior, for R and S. Let p(β) and p(β y) denote the arginal prior and posterior distributions for β and let E[β j ] = β j p(β) dβ and E[β j y] = β j p(β y) dβ denote the prior and posterior expectations of β j. The prior and posterior predictive distributions for R are both ixtures of point asses where the ixture weights are the applicable expectations of β: p(r) = p(r y) = p(r β) p(β) dβ = E[β j ] δ(r s j ) (2.14) p(r β) p(β y) dβ = E[β j y] δ(r s j ). (2.15) Note that E[β y] is the only feature of the posterior distribution that atters if the goal of the inferential exercise is to copute the posterior predictive distribution for R. 7 Also note that as we approach the end of the onth, the posterior distribution for β should approach a point ass at β = I j for soe j, where I j is the j-th row the the identity atrix, so that E[β y] I j. It is instructive to exaine the predictive distributions for S as well. The prior predictive distribution for S inherits the for of the prior distribution for the slippage paraeter u. We assue prior independence between β and u: p(β, u) = p(β) p(u). The prior predictive distribution for S has the for of a ixture: where p(s) = = = p(s β, u) p(β) p(u) du dβ ( β j p(β) E[β j ] p(s j), ) δ(s s j u) p(u) dv dβ (2.16) p(s j) := p(u) u=ssj. (2.17) Therefore, if p(u) is continuous, then p(s) will be a ixture of continuous distributions with different locations. 7 In a decision-aking setting where one can either act now wait for new inforation before acting, the posterior uncertainty regarding β ay play a role.

6 6 MARK FISHER The posterior predictive distribution for S is ore coplicated owing to potential dependence between β and u in the posterior distribution: p(s y) = p(s β, u) p(β, u y) du dβ where = = If p(s j, β, y) p(s j, y), then ( β j p(β y) β j p(β y) p(s j, β, y) dβ ) δ(s s j u) p(u β, y) du dβ (2.18) p(s j, β, y) = p(u β, y) u=ssj. (2.19) p(s y) k E[β j y] p(s j, y). (2.20) As we approach the end of the onth, the posterior predictive distribution should collapse to a point ass on the actual onth-average funds rate. The prior. It reains to adopt a prior distribution for the unknown paraeters θ = (β, u, σ 2 ). We assue prior independence: p(θ) = p(β) p(u) p(σ 2 ). For σ 2 we adopt the Jeffreys prior: p(σ 2 ) 1/σ 2. Here we provide soe introductory rearks regarding the prior for u. Typically we will adopt a syetric prior for u, centered on zero. They ay be ties, however, when we will center it elsewhere (the beginning of the financial crisis, for exaple, when the onthaverage rate dropped significantly with no change in the stated target rate). The for of the prior for u should allow the scale to be chosen endogenously. One could for exaple adopts a Gaussian prior for u, perhaps a ixture to allow for differing regies. Now we focus on the prior for β. A benefit of the Bayesian approach is the ability to incorporate iportant considerations via the prior for β. In particular, the prior for β should ebody two features. First the prior should ensure the constraint β K1, where K1 is the (K 1)-diensional siplex. Second the prior should be capable of expressing the idea that while any target rates are possible, only a few should have nontrivial probability. The Dirichlet distribution ebodies both of these features. Let p(β α, ξ) = Γ(α) K Γ(α ξ j) K β α ξ j 1 j, (2.21) where α > 0 and ξ K1. Note E[β α, ξ] = ξ. We will refer to α as the concentration paraeter. If ξ j = 1/K and α = K, then the prior is flat: p(β α, ξ) = (K 1)!. As K get large relative to n, the flat prior for β tends to doinate the likelihood (2.12) and the posterior can becoe quite flat itself. Setting α < K encourages parsiony, pushing the ass of the prior toward the vertices of the siplex, thereby iplicitly suggesting that

7 BAYESIAN ESTIMATION OF PROBABILITIES 7 only a few of the coponents are nonnegligible. (This prior could be described as partially infored ignorance. ) On the other hand, if ξ is well-infored (because, for exaple, it is based on closelyrelated data) then α > K ay be suitable. Below we discuss incorporating uncertainty about α via a prior. The posterior sapler. The ain cost of the Bayesian approach is drawing fro the posterior distribution when an analytical solution is absent (as it is here). The posterior distribution for the paraeters can be expressed as p(θ y) p(y θ) p(β α, ξ) p(u)/σ 2, (2.22) where θ = (β, u, σ 2 ). We take a Gibbs sapler approach, applying Metropolis Hastings where necessary within. Drawing fro the posterior for σ 2 (conditional on β and u) is straightforward. Note σ 2 y, β, u Inv-χ 2 (ν, s 2 ), where ν = n and s 2 = S(β, u)/n. 8 The posterior distribution for the slippage paraeter u conditional on (β, σ 2 ) can be expressed as ( ) S(β, u) p(u y, β, σ 2 ) exp 2 σ 2 p(u). (2.23) We can ake draws fro p(u y, β, σ 2 ) using a Metropolis schee. We now turn to drawing fro the posterior for β (conditional on σ 2 and u). First we show how to draw fro the posterior assuing the prior for β is flat; then we show how to draw fro the posterior with the ore general prior. A technical detail. Since K β j = 1, the conditional distribution β j β j is degenerate: the value of β j is fixed by β j. By reoving one of the coponents, say β k, we can then cycle through the reaining K 1 coponents via a one-at-a-tie Gibbs sapler. However, the (average) agnitude of β k will affect the efficiency of the sapler. Let β j βj k denote conditioning on β j \ {β k }. If β k is close to zero, we will find ourselves back in the previous trap with little or no wiggle roo for β j βj k. Therefore, for each sweep of the Gibbs sapler we will reove the largest coponent fro the previous sweep and saple over the reaining coponents. Back to the ain thread. If the prior for β were flat, then the posterior for β (conditional on σ 2 ) would be a ultivariate noral distribution truncated to the siplex. This truncated noral distribution can be hard to draw fro when only a sall fraction of the ass of the unrestricted distribution is contained in the siplex. Moreover, with K large relative to n, X X will be singular and hence not invertible. Nevertheless, a one-at-a-tie Gibbs sapler works well. The posterior distribution for β i βi k is a univariate truncated noral distribution: where p(β i y, β k i, σ 2, u) = N [0,b k i ] (β i i, s 2 i ), (2.24) b k i = β i β k (2.25) 8 If the prior for σ 2 were different, we could use this as a proposal for a Metropolis step.

8 8 MARK FISHER and ( i, s 2 i ) can be readily calculated. In particular, note S(β, u) = c i0 c i1 β i c i2 βi 2 for soe coefficients (c i0, c i1, c i2 ) that are functions of (βi k, X(u), y). Consequently, i = c i2 /(2 c i3 ) and s 2 i = σ 2 /c i3. Draws fro the univariate truncated noral can be ade using the inverse CDF ethod (for exaple). We can accoodate the general Dirichlet prior for β by incorporating a Metropolis Hastings step. We use the truncated noral distribution in (2.24) to draw the proposal β i. Because the proposal is a draw fro the conditional posterior given a flat prior, the Hastings ratio equals the inverse of the likelihood ratio, q(β, β ) q(β, β) = p(y β, u, σ2 ) p(y β, u, σ 2 ). (2.26) Consequently, the acceptance condition depends only on the prior ratio: p(β ( ) α, ξ) β α ξi 1 ( p(β α, ξ) = i β ) α ξk 1 k u, (2.27) β i β k where u U(0, 1) and where β k = bk i β i. Note that if the prior for β were flat, then (of course) the proposal would always be accepted. Uncertainty about α. The hyperparaeter α plays an iportant role in the prior for β. Instead of specifying a fixed value for α, we can adopt a prior for α and incorporate an additional Metropolis step to saple α. The likelihood for α is given by (2.21). Let the prior for α be given by 9 p(α ζ, τ) = α 1 τ 1 ζ 1 τ ( ) τ α 1 τ ζ 1 2, (2.28) τ where ζ, τ > 0. The ean of the distribution does not exist. We can see how to interpret the paraeters via the quantile function where α = Q(x) for x (0, 1) and ( ) x τ Q(x) = ζ. (2.29) 1 x Thus ζ is the edian and τ deterines the spread. For our purpose, it is natural to choose ζ = K. 10 For aking draws fro the posterior, it is convenient to change variables: let z = log(α). Given (2.28), the prior for z is given by p(z ζ, τ) = ζ 1/τ e z/τ (ζ 1/τ e z/τ ) 2 τ, (2.30) 9 This is a special case of the Singh Maddala distribution. 10 The essence of this prior is captured by the following change of variables: Let η := log(α/k) and let ζ = K; then p(η τ) = e η/τ (1 e η/τ ) 2 τ, which is syetric around zero with a standard deviation of (π/ 3) τ.

9 BAYESIAN ESTIMATION OF PROBABILITIES 9 where p(z ζ, τ) is syetric around its the ean of log(ζ) with a standard deviation of (π/ 3) τ. The Metropolis step proceeds as follows: Given soe average step-size s, ake rando-walk proposals of z N(z, s 2 ) and accept z if p ( β α, ξ ) p(z ζ, τ) p ( β α, ξ ) u, (2.31) p(z ζ, τ) where u U(0, 1) and where α = e z and α = e z. A typical setting for the hyperparaeters is be ξ = (1/K,..., 1/K), ζ = K, and τ = 2. However, in going fro one observation day to the next, it ay ake sense to use ξ t = E[β t1 y t1 ]. (Liited experience indicates that the posterior distribution for α will be quite different for these two priors.) The nonnegativity of options prices. A proble with the likelihood (2.12) is that it ay iply Pr[y i < 0 β, u, σ 2 ] 0. We believe one would never observe such out-of-bounds prices. (Such an observation would be discarded as a data istake and not interpreted as the result of easureent error as we have construed it.) One way to fix this iplication is to truncate the sapling distribution for the data at zero; this has the effect of ibuing the easureent error with an upward bias, where the bias gets larger as the option price gets closer to zero. The likelihood (for a single observation) that does not take into account the positivity of option prices can be expressed as p(y i θ) = N(y i µ i, σ 2 ) where µ i := X i (u)β. By coparison, consider the likelihood given a truncated Gaussian distribution: p(y i θ) := 1 [0, ) (y i ) H(µ i, σ 2 ) N(y i µ i, σ 2 ), (2.32) where 11 { ( )} µ 1 H(µ, σ 2 ) := 1 Φ. (2.33) σ Note 0 p(y i θ) dy i = 1 and Ê[y i θ] = 0 y i p(y i θ) dy i = µ i σ 2 H(µ i, σ 2 ) N(0 µ i, σ 2 ) > µ i. (2.34) The inequality in (2.34) can be interpreted as an upward bias in the easureent error. This upward bias is largest at µ i = 0 and declines onotonically to zero as µ i. Let us now consider the likelihood of the entire data set: n p(y θ) := p(y i θ) = 1 [0, ) n(y) C(θ) p(y θ), (2.35) i=1 where p(y θ) = n i=1 p(y i θ) is given in (2.12) and ( ) 1 C(θ) := p(y θ) dy = [0, ) n n H ( µ i, σ 2). (2.36) Note, for fixed σ, H(µ, σ 2 ) declines onotonically fro H(0, σ 2 ) = 2 to li µ H(µ, σ 2 ) = 1. Therefore 1 C(θ) 2 n. 11 Φ( ) is the standard noral cuulative distribution function (CDF). i=1

10 10 MARK FISHER Let M [0, ) n denote the odel where y [0, ) n. Assuing the data are all nonnegative, the likelihood of this odel is p(y M [0, ) n) = C(θ) p(y θ) p(θ) dθ. (2.37) Thus, the Bayes factor in favor of this odel (relative to the unrestricted odel) is C(θ) p(y θ) p(θ) dθ BF = = E p(θ y) [C(θ)], (2.38) p(y θ) p(θ) dθ where the expectation is taken with respect to the posterior distribution of θ fro the unrestricted odel. Given draws of {θ (r) } R r=1 fro the posterior of the unrestricted odel, it follows fro the right-hand side of (2.38) that we can copute the Bayes factor as R C ( θ (r)) = BF. (2.39) li R R1 r=1 In addition, the weights z (r) C ( θ (r)), where R r=1 z(r) = 1, can be used to resaple the draws. If the weights do not vary very uch, then the two odels will predict siilar distributions for the paraeters even if the Bayes factor is large. The upshot is that proceed by first estiating without the option-price nonnegativity restriction and then to check afterwards to see how the restriction changes the inferences. (Liited experience suggests the restriction does not change the inferences noticeably.) 3. Month-average target rates, target-rate paths, and unscheduled eetings The target fed funds rate is set by the Federal Open Market Coittee (FOMC) at its scheduled eetings. There are eight scheduled eetings per year, so there are four onths each year without a (scheduled) eeting. (Occasionally it is changed at unscheduled eetings. We will deal with this possibility in the Appendix.) It is set in increents of 25 basis points (0.25 percent). 12 [Currently, however, the target rate is a band: 0 to.25%.] Consider a given eeting. We adopt the setup outlined in the previous section for the target rate R where there are K possibilities, r = (r 1,..., r K ), and the risk-neutral probability is p(r = r j β) = β j. We say that r is contiguous if r j = r % (j 1) for j = 1,..., K. 13 We will often assue the odel is contiguous. Now consider a given onth. The fed funds futures contract depends on the onthly average of fed funds rates. (For exaple, a Friday rate counts three days in the average, for Friday, Saturday, and Sunday. If the funds arket were closed the following Monday due to a holiday, then the preceding Friday rate would count 4 days, assuing all four days were in the sae onth.) Up to this point, we have iplicitly assued a single target applies for a given onth. In general, however, the situation is ore coplicated. More often than not, there is a scheduled eeting within a given onth. Prior to the eeting the previously established 12 Cite soe history % =

11 BAYESIAN ESTIMATION OF PROBABILITIES 11 target rate applies, while after eeting a new (possible different) applies. Assuing the Open Market Desk keeps the daily funds rate at the appropriate target rate throughout the onth, then the onth-average funds rate will be R = (1 ω) R 1 ω R 2, where ω is the fraction of the days in the onth for which R 2 is the target rate. If there were no eetings between the observation date and the eeting in the onth in question, then R 1 can be identified with the current rate (assuing the absence of unscheduled eetings). More generally, both R 1 and R 2 will be unknown on the observation date and consequently the joint distribution of the two target rates coe into play. In addition, the daily effective funds rate does not equal to the target rate on a daily basis. More iportantly for our purposes, the onth-average funds rate S does not equal the onth-average target rate R. We account for this difference by letting S = Ru, where v is the onth-average slippage. Notation. Let τ = (τ 1,..., τ w ) denote a sequence of eeting dates and let R τi denote the target rate established at tie τ i. Let T 0 denote the first day of onth. Let τ denote the date of the last eeting prior to the beginning of onth and let τ denote the date of the following eeting: τ < T 0 τ. Let ω denote the fraction of onth fro τ to the beginning of the onth 1: { T 0 ω = ax 1 τ } T1 0 T 0, 0 (3.1) In particular, if τ = T 0 then ω = 1 and if τ T 0 1 then ω = 0. The onth-average target rate for onth is R = (1 ω ) R τ ω R τ. (3.2) Let t denote the quote date (also known as the observation date). If t < τ i then R τi is unknown while if t τ i then R τi is known. Note τ1 is the date of the last eeting prior to the beginning of onth 1. If onth has a eeting then τ1 occurs in onth ; otherwise τ1 occurs in an earlier onth. If t τ 1, then R is known. If t < τ1, then R is unknown. If t < τ and there is a eeting during onth, the R involves two unknowns: R τ and R τ. A nuber of special cases. Now we consider a nuber of special cases of (3.2). In each case we show how the likelihood of an observation can be expressed as y i = X i β ε i, thereby allowing us to adopt both the likelihood and the prior described in Section 2. Case 1. First consider a onth that involves a single target rate. This occurs if either ω = 0 (a onth with no eeting) or ω = 1 (a onth with a eeting on the first day of the onth). In either case, the onth-average target rate involves a single target rate: { R R = τ ω = 0 R τ ω = 1. (3.3) Let us refer to this rate as R. Let R r = {r 1,..., r K }, p(r = r j β) = β j, S = R u, and X ij = φ(k i, γ i, r j u). We have established y i = X i β ε i.

12 12 MARK FISHER Case 2. Next suppose ω (0, 1) but that R τ is known. (Thus the onth contains a single unknown target rate.) Let us refer to R τ and R τ as R 0 and R (respectively) so that R = (1 ω ) R 0 ω R. Let R r = {r 1,..., r K }, p(r = r j β) = β j, S = R u, and X ij = φ(k i, γ i, r j u) where r j = (1 ω ) R 0 ω r j. Again, we have established y i = X i β ε i (in which r j has taken the place of r j ). In passing, note that the saller ω is, the less inforation there will be regarding R. Case 3. Now consider (3.2) ore generally. Let us refer to R τ and R τ as R 1 and R 2, respectively, so that R = (1 ω ) R 1 ω R 2. Assue (R 1, R 2 ) r 1 r 2 where r 1 = (r 1 1,..., r1 K 1 ) and r 2 = (r 2 1,..., r2 K 2 ). Let b denote a K 1 K 2 atrix of joint probabilities such that where b kl 0 and K 1 row-ajor order) k=1 p(r 1 = rk 1, R 2 = rl 2 b) = b kl (3.4) K2 l=1 b kl = 1. Let β denote a vectorized version of b, where (using β j = b kl for j = (k 1) K 2 l. (3.5) Thus β K 1 where K = K 1 K 2. Assue S = R u and let x ikl = φ(k i, γ i, r kl u) where Then r kl = (1 ω ) r 1 k ω r 2 l. (3.6) y i = K 1 k=1 K 2 l=1 x ikl b kl ε i. (3.7) Let X i denote the vectorized version of x i. Then (3.7) can be expressed as y i = X i β ε i. Stacking the observations produces y = Xβ ε. The atrix X has n rows/observations and K coluns/paths and the vector β has K eleents/joint probabilities. There are a nuber of factors that ilitate against the ability to identify the probabilities in this case: the (potentially) large nuber of unknown probabilities relative to the nuber of observations, the (potentially) close spacings of the states s kl (relative to the agnitude of the slippage and the easureent error), the possibility that ω is near either zero or one. One possible way to overcoe these probles is to use inforation fro additional contract onths. We now turn to that approach. Two contract onths at once. Suppose there is no eeting in onth (ω = 0) and there is a eeting in onth 1. Let t < τ so that both R = R τ and R 1 = (1 ω 1 ) R τ ω 1 R 1 τ are unknown. The onth-average rates share a coon 1 target rate: R τ = R 1 τ. We refer to R τ and R τ as R 1 and R 2, respectively, and we adopt the previous setup (including r 1, r 2, b, and its vectorization β). In addition, we refer to the onths and 1 as onths 1 and 2, respectively. Thus, R 1 = R 1 and R 2 = (1 α 2 ) R 1 α 2 R 2.

13 BAYESIAN ESTIMATION OF PROBABILITIES 13 Assue each onth has its own slippage: S = R u, for = 1, 2. Let x ikl = φ(k i, γ i, s kl u ), where s 1 kl = r1 k (3.8) s 2 kl = (1 ω 2) r 1 k ω 2 r 2 l. (3.9) (Note the atrix s 1 is coposed of K 2 copies of the vector r 1.) Let Xi denote the vectorized version of x ikl. Then we have yi = Xi β ε i. (3.10) Let Y = [ ] [ ] [ ] y 1 X 1 ε 1 y 2, X = X 2, and E = ε 2. (3.11) We can express the joint likelihood as Y = X β E, where E N(0 n, σ 2 I n ) and where n = n 1 n 2. Consequently, we can use the posterior sapler described above. 14 This approach can be generalized to include ore than two onths where β is a vectorized version of the tensor of joint probabilities and β K 1, where K = u K u. (See Appendix A for an exaple involving three contract onths.) Reark. Recall that α is the concentration paraeter in the prior for β. Note that if α < K ax{k u } then there is no restricted odel that supports independence aong the probabilities. 15 In effect, a sufficient sall value of α for the joint probabilities rules out independence. Unscheduled eetings. In order to keep the analysis siple, consider a onth with no scheduled eeting. Let R 1 denote the applicable target rate absent the unscheduled eeting and let R 2 denote the target rate set at the unscheduled eeting (if it occurs). Note that R 2 R 1. Let the onth-average target rate be given by R = (1 ω) R 1 ω R 2. (3.12) The novelty is that ω is unknown. If ω = 0, there is no unscheduled eeting, while if 0 < ω 1 there is an unscheduled eeting. The distribution for ω is discrete because there are a finite nuber of days in any onth. (The values of ω with positive probability are further restricted by the fact that unscheduled eetings occur on business days in the sense that the announceent of a new target rate occurs on business days.) To treat the siplest case, suppose R 1 is known, in which case r j = (1 ω) R 1 ω rj 2 (where r j = R 1 if and only if ω = 0) and X ij = φ(k i, γ i, r j u). Note the X atrix depends on the unknown ω (as well as u). The (conditional) posterior distribution for ω is discrete where ( ) S(β, u, ω) p(ω β, u, σ 2, y) exp 2 σ 2 p(ω). (3.13) We can think of (3.12) as cobining two copeting odels: the no-unscheduled-eeting odel N characterized by ω = 0 and the unscheduled-eeting odel U characterized by 14 See Appendix A for an alternative forulation in ters of arginal and conditional probabilities, with an illustration assuing ω 2 = At least no siple odel that I can see.

14 14 MARK FISHER ω 0. The Bayes factor in favor of odel N relative to odel U is given by the posterior odds ratio divided by the prior odds ratio: BF N U = p(ω = 0 y) p(ω = 0) p(ω 0 y) p(ω 0). (3.14) Let the prior distribution p(ω q) be given by p(ω = 0 q) = q and p(ω 0 q) = 1q, where 1 q is distributed equally over the reaining possibilities. Suppose the hyperparaeter q has a prior distribution, p(q). Then p(ω = 0) = p(ω = 0 q) p(q) dq = E[q] and p(ω = 0 y) = p(ω q) p(q y) dq = E[q y]. For exaple, suppose the prior for q is a beta distribution paraeterized as follows: p(q) = B ( q α q ξ q, α q (1 ξ q ) ), (3.15) where ξ q = E[q] and α q is the concentration paraeter. The flat prior is given by ξ q = 1/2 and α q = 1. If in fact we choose ξ q = 1/2, then BF N U = E[q y] 1 E[q y]. (3.16) However, it probably akes ore sense to choose ξ q to atch the unconditional probability that a onth has no unscheduled eeting. If we choose α q 1, then the prior expresses the following view: On the one hand it is probably very unlikely that there will be an unscheduled eeting, but on the other it is possible that it is very likely. 4. Data Here we describe the data and what is required to put it in a for suitable for estiation. For each quote date there are quotes for puts and calls on a nuber of contracts. The strike prices are expressed in ters of an index P, where P = 100 (1 S). 16 For exaple, P = 97 refers to a onth-average funds rate of S = 1.97 =.03. Note that the payoff at expiration to a call option in ters of P translates into the payoff of a put in ters of S. To see this, let K P = 100 (1 K S ) denote the two expressions of a given strike price and note (P K P ) = ( 100 (1 S) 100 (1 K S ) ) = 100 (KS S). (4.1) Siilarly, put payoffs expressed in ters of P becoe call payoffs expressed in ters of S. Hereafter, assue the data are expressed in ters of S. Let t denote the quote date and denote the contract onth. The data for a given quote date include quotes for a total of M t contract onths, and for each contract onth are in the for of n t triples: {(K ti, γ ti, V ti )}n t i=1. We deflate the option values: y ti = V it /B t for i = 1,..., n t. The result is a data set for the quote date t of the for {(K ti, γ ti, y ti )}n t i=1 for = 1,..., M t. It reains to copute the appropriate X atrices fro the strike prices {K ti }. We proceed as follows. (We will suppress the dependence on the quote date occasionally to reduce the notational clutter.) First, for each contract onth in the data set we need to 16 Because the relation between S and P is linear, there are no issues relating to Jensen s inequality in changing back and forth between the two representations.

15 BAYESIAN ESTIMATION OF PROBABILITIES 15 find α, R τ and R τ in order to for R as given in (3.2). Next copute the union of the unknown rates across entire collection of onths. Let R 0 denote the known target rate as of tie t and let {R u } Ut u=1 denote the set of unknown rates that appear in the onthaverage rates. (At this point, soe discretion on the part of the researcher coes into play.) For each unknown rate, specify the set of possible values: R u r u = {r1 u,..., ru K u }. Let r 0 = {R 0 }. It would probably ake sense to have r 0 r 1... r u. The total nuber of target-rate paths is Kt = U t u=1 K u. If all of the data were to be used in a single joint estiation, the X atrix would have n t = M t =1 n t rows/observations and Kt coluns/paths. It ay not be possible to learn uch for such a odel, given the nuber of observations and the nuber of paths. Instead a subset of the onths and a corresponding subset of the target rates ay be used. Given these subsets, let Xij = φ(k i, γ i, s j u ), where the index j = 1..., K is coputed according the vectorization of the join probability tensor for the given subset (and K now denotes the product coputed fro the subset). The data are now in a for suitable for estiation. Additional thoughts. We will want to copare futures prices (as proxies for forward prices) with E[S y] coputed fro the posterior predictive distribution [see (2.18)]. We ay want to use the futures prices to help for the prior for β. We can use put call parity to assess the agnitude of the easureent error. Questions for e. What is the expiration day if the last day of the onth falls on a nonbusiness day? Does the funds rate for a given day include trades after the FOMC announceent. What is the effective funds rate on eeting days, both with and without target rate changes. Use this inforation to decide whether to include the eeting day with the previous target rate or with the next target rate. 5. Dynaics We can extend this approach to incorporate day-to-day dynaics. The idea here is that β is a stochastic process that evolves through tie. Since β is a vector of probabilities, it ust be a artingale, where 17 E t1 [β t ] = β t1. Let p(β t β t1 ) denote the pdf for β t given the paraeter β t1 and let Y t1 := (y 1,..., y t1 ). Then p(β t y t, Y t1 ) p(y t β t ) p(β t Y t1 ) = p(y t β t ) p(β t β t1 ) p(β t1 Y t1 ). (5.1) There are a nuber of ways to copute the updating in (5.1). The particle filter is discussed below (as an illustrative exaple). The central idea here is that the probabilities on a given day (for a fixed set of target rates) will be closely related to the probabilities on the preceding and succeeding days. One consequence is that ore efficient estiation could use the inforation in the data fro the neighboring days (y t1, y t1 ). Another consequence is that if we wish to infer whether the probabilities have changed fro one day to the next, we ust copute their joint distribution p(β t1, β t ). 17 Warning: The notation in this section ay conflict with that in other sections.

16 16 MARK FISHER Particle filter. The updating in (5.1) can be coputed using a particle filter. In particular, the draws fro p(β t1 Y t1 ) constitute a particle swar. There are a nuber of related ways to ipleent a particle filter. The ost straightforward is the sapling iportance resapling (SIR) approach. Following this approach, ake a rando draw fro p(β t β (i) t1 ) for each β(i) t1 in the swar. Denote these draws {β (i) t }. These draws represent draws fro p(β t Y t1 ). At this point, iportance sapling coes in to play. Copute w (i) t := p(y t β (i) t ) and resaple {β (i) t } using these weights. A proble one encounters is when the bulk of p(y t β t ) is located in the tail of p(β t Y t1 ) then the weights {w (i) t } ay be doinated by a few large values with the result that the swar provides a poor representation of the posterior. An alternative approach is the independent particle filter (IPF). Make draws {β (j) t } fro the likelihood p(y t β t ) assuing it is proper. 18 We need to copute the weights p(β (j) t Y t1 ) = p(β (j) t β t1 ) p(β t1 Y t1 ). The factor p(β t1 Y t1 ) is encoded in the posterior draws fro the previous day. 19 For each draw fro the second day, choose L draws fro the first day, where 1 L and is the nuber of draws fro the first day. Then copute z (i,j) t = p(β (j) t β (i) t1 ). Let z (j) t = L i=1 z(i,j) t and use these weights to resaple {β (j) t }. (By sapling fro p(β t1 Y t1 ), we incorporate its weight iplicitly.) We can tweak the IPF by drawing fro p(y t β t ) p(β t ) where p(β t ) is the axiu entropy distribution with ean = 1 M M i=1 β(i) t1. (We could use the Dirichlet distribution as well.) This will tend to reduce the variation of the weights coputed as p(β t β t1 )/p(β t ). In order to obtain the joint distribution p(β t1, β t Y t1, y t ), we can apply the so-called soothing step. We do this as follows. Make a draw fro {β (j) t } and conditional on that draw, ake a draw fro {β (i) t1 } i=1 using the weights {z(i,j) t } i=1 (fixing the j drawn). The pairs thus generated are fro the joint distribution conditional on (Y t1, y t ). The arginal distribution for β t1 encoded in these draws reflects the data fro both days. The coputation of {z (i,j) t } can be quite expensive. One can reduce the expense by discretizing by binning and coputing p(β (j) t β (i) t1 ) at the idpoints of the bins and using the bin counts to adjust. Let z (i,j) t := p(β (j) t β (i) t1 ) n(i,j) t where p(β (j) t β (i) t1 ) is coputed at the idpoint of bin (i, j) and n (i,j) t denotes the corresponding bin count. Then z (j) t := i z(i,j) t. Appendix A. Addition aterial Marginal and conditional target-rate probabilities. Given the joint probabilities (3.4), we can copute arginal probabilities p(r 1 = r 1 k b) = K 2 l=1 b kl = β 1 k and p(r 2 = r 2 l b) = 18 The draws of βt are ade independent of β t1. For ore on the IPF, see Lin et al. (2005). 19 One way to proceed is to take a draw fro p(βt β (i) t1 ) for each i fro the first day and then approxiate the density.

17 K1 k=1 b kl = β 2 l BAYESIAN ESTIMATION OF PROBABILITIES 17 and conditional probabilities p(r 2 = r 2 l R 1 = r 1 k, b) = p(r 1 = r 1 k, R 2 = r 2 l b) p(r 1 = r 1 k b) = b kl K2 l=1 b kl =: B kl. (A.1) Note β 2 = B β 1. Also note p(r 1 = rk 1, R 2 = rl 2 B, β1 ) = B kl βk 1. As an alternative to odeling the joint probabilities, one could odel arginal and conditional probabilities. If we choose to odel β 1 and B (instead of odeling b), the nuber of paraeters reains K 1 K 2 1, since K 1 k=1 β1 k = 1 and K 2 l=1 B kl = 1 for k = 1,..., K 1. (Each of the K 1 rows of B sus to one.) We could adopt Dirichlet priors for β 1 and for the coluns of B. This approach, however, offers no advantages in general relative to the approach of odeling the joint probabilities. In order to illustrate this approach, let us odify the exaple involving two onths by setting ω 2 = 1. In this case each onth is associated with a single target rate and consequently [ ] [ ] [ ] X 1 X1 0 β 1 X 2 β = 0 X2 β 2 (A.2) subject to β 2 = B β 1, where X ij = φ(k i, γ i, r j u ) (A.3) for rj r. Carlson et al. (2005) provide an exaple of this approach (with u = 0 in their setup). In the exaple (p. 1214), K 1 = 2 and K 2 = 3. Consequently there are K = K 1 K 2 = 6 possible paths and K 1 = 5 independent probabilities. Using a prior inforation, they set three path probabilities to zero, leaving three paths and two independent probabilities reaining. The restrictions produce the following atrix of conditional probabilities: 20 [ ] B B =, (A.4) 0 B 22 B 23 so that β 2 = B β 1 = B 11 β 1 B 22 β2 1. B 23 β2 1 (A.5) Since each row of B ust su to one, we have B 11 = 1 and B 22 B 23 = 1. In addition β1 1 β1 2 = 1. (Together these restrictions iply β2 1 β2 2 β2 3 = 1.) This leaves a total of two free paraeters. 20 The atrix of joint probabilities is [ ] b = B β 1 B11 β = 0 B 22 β2 1 B 23 β2 1, where (row-by-row) b k = B k β 1 k.

18 18 MARK FISHER An exaple involving three onths. Here we provide an exaple involving three onths. Let R = (1 ω ) R τ ω R τ (A.6a) R 1 = (1 ω 1 ) R τ ω 1 R 1 τ 1 (A.6b) R 2 = (1 ω 2 ) R τ ω 2 R 2 τ. 2 (A.6c) If ω 1 = 1, then there is no rate in coon between R and R 1. Siilarly, if ω 2 = 1, then there is no rate in coon between R 1 and R 2. Assuing ω 1 < 1, { R R τ = τ ω = 0 (A.7) 1 R τ ω > 0 and assuing ω 2 1, R τ = 2 { Rτ 1 R τ 1 Suppose ω, ω 1 (0, 1) and ω 2 = 0 so that ω 1 = 0 ω 1 > 0. R = (1 ω ) R 1 ω R 2 R 1 = (1 ω 1 ) R 2 ω 1 R 3 R 2 = R 3, (A.8) (A.9) (A.10) (A.11) where (R 1, R 2, R 3 ) represent the three distinct rates. Let r = (r 1,..., r K ) for = 1, 2, 3 and let p(r 1 = r 1 k, R 2 = r 2 l, R 3 = r 3 w B) = B klw, where B is a rank-3 tensor of joint probabilities with diensions K 1 K 2 K 3. Let β denote the vectorized version B: β j = B klw for j = (k 1) K 2 K 3 (l 1) K 3 w. (A.12) Note β K 1, where K = K 1 K 2 K 3. Assue S = R u. Let x iklw = φ(k i, γ i, s klw u ), where s 1 klw = (1 ω ) r 1 k ω r 2 l s 2 klw = (1 ω 1) r 2 l ω 1 r 3 w s 3 klw = r3 l. (A.13) (A.14) (A.15) Then we have yi = Xi β ε i (A.16) where X i is a vectorized version of the tensor x i. Again we can express the joint likelihood as Y = X β E, where E N(0 n, σ 2 I n ) and where n = n 1 n 2 n 3. References Carlson, J. B., B. R. Craig, and W. R. Mellick (2005). Recovering arket expectations of FOMC rate changes with options on federal funds futures. Journal of Futures Markets 25, Geweke, J. (1986). Exact inference in the inequality constrained constrained noral linear regression odel. Journal of Applied Econoetrics 1,

19 BAYESIAN ESTIMATION OF PROBABILITIES 19 Lin, M. T., J. L. Zhang, Q. Cheng, and R. Chen (2005). Independent particle filters. Journal of the Aerican Statistical Soceity 100, Federal Reserve Bank of Atlanta, 1000 Peachtree St. N.E., Atlanta, GA 30309