).
Probability
0.30.20.10.00
20
FIGURE 11.5B
40
60
80
Probability of Deal Break (INTC-LEVL).
184
RISK ARBITRAGE PAIRS
RISK MANAGEMENTOne of the important tasks in risk measurement is the estimation of VAR, anacronym for value at risk.
VAR MeasurementLet us now detail what we mean by VAR measurement starting with the outcome of the measurement process. The outcome is a statement as follows:“Under usual market conditions, there is a 2.5-percent chance of losing Xdollars in one day given the current list of positions.” The value X is the valueat risk and is based on the current (day’s close) mark to market. The value ofX is typically based on the statistics of the daily market movement in the pasttwo to three years—that is, the distribution of price movements—and, moreimportantly, on the correlation between the price movements of various assets. The exercise is therefore to determine what the likely value of X is.Also note that there is nothing magical about use of the 2.5-percentvalue in the VAR statement. It is an artifact left from the cultural conditioning of dealing with Gaussian distributions and represents two standarddeviations in the Gaussian case. That value could therefore be 1 percent orany number that one fancies.Another key aspect of the preceding statement is the use of the phrase,“usual market conditions.” This is because the measurement approach thatworks under usual market conditions is not sound under extreme marketconditions. This is because the correlations that may be considered as relatively stable under usual market conditions become erratic under extrememarket conditions causing the VAR approach to break down.Applying the same reasoning based on the correlation between assetprices, it is easy to deduce that commonly applicable methods to measure theVAR on a portfolio may not be used in the case of risk arbitrage. The announcement of the deal causes a qualitative shift in the way the two companies are viewed in the marketplace. The historical correlation between themis no longer relevant. The market dynamic now also reflects the possibilityof merger between the two companies and there is an increased level of correlation between the returns of the bidder and target stocks. Therefore, anyVAR methodology that relies on long-term historical correlation is renderedinappropriate.Furthermore, use of the historical correlation could potentially inflatethe VAR numbers, increasing risk capital requirements, resulting in an unrealistic increased cost of doing business. This leads us to look for reasonable ways to evaluate the value at risk involved in risk arbitrage trades.The risk arbitrage trade at hand can be viewed in two ways. On eventof deal failure, it is equivalent to holding two separate securities, and tradi-
The Market Implied Merger Probability
185
tional VAR methodologies will apply. However, when the deal is on, the riskis mostly due to the spread volatility, and the value at risk is best obtainedby measuring the spread volatility. Thus, based on the view we choose toadopt, we now have two VAR values. We now reconcile between the twoviews and arrive at a final VAR number by weighing each scenario. Fortunately for us, the consensus probabilities of deal success and failure provideus with a ready mechanism by which to weigh the two views.Putting things more precisely, each of the two views may be associatedwith a return distribution describing the probability of return of the trade inthe next time period. The overall probability distribution is then a weightedsum of the two distributions, the weights corresponding to the probabilityof deal success and deal failure, respectively. Distributions that are constructed as a weighted sum of other distributions are called mixed densitydistributions.To better understand the model, let us consider the process of drawinga sample from a mixture distribution. This is actually a two-step process. Inthe first step, we randomly choose the distribution from the available set ofdistributions. The choice is guided by the probability weights assigned toeach distribution in the set. In the next step, we draw a random sample fromthe chosen distribution.Applying the mixture model to our situation, the probability of an outcome x dollars or lower is given by the distribution function shown in Equation 11.9F(x) = π success Φ spread (x) + π failureΦ stockPortfolio (x)
(11.9)
where Φspread is the distribution of spread returns, and ΦstockPortfolio is the distribution of the returns assuming that there is no deal. The failure and success probabilities serve as weights. Thus, given a value of x dollars, we canfind the probability that the returns will be less than or equal to x using theabove formula.Note that VAR value is that value of x that results in the probabilityvalue of 2.5 percent in Equation 11.9. Additionally, the distribution functionlike every other distribution function is a nondecreasing function. As a consequence, the VAR value may be deduced by applying a standard binarysearch method. Thus, knowledge of the consensus probability estimates ofdeal success and failure may be used to calculate the VAR in risk arbitragetrades.
Event Risk ManagementOftentimes in risk arbitrage, the outcome of merger success or deal breakmay hinge on a single event like the shareholders’ vote or a court decision.
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The arbitrageur may find the reward adequate but the risk too high. In suchsituations, the arbitrageur can protect himself against a widening spreadthrough the use of options.Typically, the target stock experiences higher volatility relative to thebidder stock. The common strategy under these circumstances is to purchase a put on the target stock, thus protecting against a steep drop in theprice of the target. While the price paid to enter into the hedge eats away atsome of the profit in the trades, the risk due to the put purchase is a lotlower.It may also be that the deal poses a high degree of risk for some time.Once the event resolves itself favorably, the risk of deal break diminishessubstantially. The timing, strike, and expiration of the put used will therefore depend on the appetite for risk and the degree of risk aversion of the arbitrageur.It must therefore be noted that while at the surface the risk arbitrageprocess seems cut and dried, there is still a substantial amount of decisionmaking and judgment calls that are left to the discretion of the arbitrageur.
SUMMARYThe risk neutral probability of merger is the probability implied by theobserved spread between the bidder and target firms.It is similar to the implied volatility parameter for options.The evolution of the risk neutral probability of merger can be related tothe spread dynamics.It is useful in the design of more appropriate value at risk measures.
FURTHER READING MATERIALRisk-Neutral ProbabilityJarrow, R. and S. Turnbull Derivative Securities, (Cincinnati, Ohio: 2nd Edition.South Western Publishing, 2000).
The Market Implied Merger Probability
187
APPENDIXNotationp0A
Price of security A on the eve of merger announcement
p0BpTABpTApTB
Price of security B on the eve of merger announcement
psuccess
Probability of successful merger
pfailure
Probability of deal break
g
Ratio
Price of combined entity at time TPrice of security A in case of deal break at time TPrice of security B in case of deal break at time T
We now construct Table 11.2 similar to Table 11.1.
TABLE 11.2InvestmentScenario
InitialPrice
Risk-Free
e–rT
Buy Stock ABuy Stock B
p0Aγ p0B
Payoff (Price at Time T)Merger Successful
Merger Failed
1
pTABpTAB + cash
1
pTAγ pTB
The Arrow-Debreu equations that need to be satisfied written in matrixform is as follows:e − rT 11 π′pTABpTA   success  =  p0A ′  pB  p AB + cash γ pB   π failureT T 0 Solving for the values of π successand normalizing so that they addand π failure′′up to 1, we have
π failure = [e rT (p0A − γ p0B ) + cash] / [(pTA − γ pTB ) + cash]
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Applying the fact thatS0 = p0A − γ p0BST = pTA − γ pTBWe have
π failure = (e rT S0 + cash) / (ST + cash).
CHAPTER
12
Spread Inversion
INTRODUCTIONIn Chapter 11 we discussed that the risk neutral probabilities of merger canbe evaluated from the value of the spread as observed in the marketplace.The reasoning was that market forces that affect the value of the spread takeinto account the risk of deal completion. It is probably true that in mostcases the deal completion risk is the predominant factor affecting the spreaddynamics. However, the spread is also subject to idiosyncratic movement onthe part of any one of the two stocks in question. This movement can be dueto a variety of reasons that have nothing to do with the deal completion riskand hence contributes to what we shall term noise in the evaluation of thespread-implied probabilities. This fact is rather evident from the raw plot ofthe spreads.Some of the reasons for this behavior could be due to the bid-ask spreadof the individual stocks and the order of the buys and sells as they occur oneach individual stock. It could also be due to the market maker, looking toadjust inventory. The market maker may move the price sufficiently higheror lower albeit on a temporary basis to produce the desired supply or demand in the stock and thereby adjust his inventory levels. Another reasonfor the spread movement could be attributed to short covering on the targetstock especially right after the merger announcement. Therefore, when attempting to evaluate the deal break probabilities, we need to bear in mindthat the observed spread is not a consequence of the deal risk alone.However, if the deal risk is the predominant driver of the spread dynamics, the other effects may be treated as noise and filtered using classicalfiltering methods. In this chapter, we propose the Kalman filter for the task.The primary reason for proposing the Kalman filter is that it is robust andeasy to use. Unlike Weiner filtering methods, which are valid only for stationary stochastic signals, the Kalman filter may be applied to a wide variety of situations without restriction. This property conveniently precludes usfrom coming to any conclusions on the stationarity of the spread and makesit a suitable candidate method.189
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A comprehensive introduction to the Kalman filter is provided in Chapter 4. However, for sake of continuity we will summarize the method herein a few sentences. The idea of Kalman filtering revolves around the notionof state of the system. The process involves a sequence of state predictionsfollowed by observations. The predictions are then reconciled with the observations to obtain the best estimate of the state. Applying the idea to oursituation, let the state correspond to the logarithm of the spread. Thisprocess then translates as follows. First, we make a prediction of the nextvalue for the log-spread and follow this with an observation after the elapsedtime. The predicted log-spread and the observed log-spread are then reconciled to form the best estimate of the spread at that time instance. Theprocess is then repeated for the next time step.Based on the discussion here, it is now obvious that Kalman filter design for our problem involves two main steps. First, we need to design theprediction method that we plan to use, to provide us with an a prioriestimate of the state. Next, we need to model the observation process. Theoutcome from the observation modeling process is a means of observingthe state along with a method to estimate the error variance in the observation. Once we have the prediction and observation, the Kalman filteringprocess provides us with the means to reconcile between them to come upwith an estimate of the state; that is, the logarithm of the spread. Thisvalue can in turn be used to estimate the spread implied probability of dealbreak. Let us now discuss the modeling process for the prediction and observation equations.
THE PREDICTION EQUATIONThe prediction equation is essentially an equation specifying the state transition for the Kalman filter. Since in our case the state corresponds to thelogarithm of the spread, the state equation in our case involves coming upwith a prediction scheme for the value of the spread at the next time step.We of course have at our disposal all the past values of the spread right upto the current time step. Let us therefore discuss how we can achieve that.In order to motivate our choice for the state equation, we recall the difference equation from the previous chapter.
(
t
)
(
t
)
(
)
( )
( ) (12.1)
21log π failure− log π failure= − r t 2 − t1 + log St2 − log St1
The left-hand side of the equation is equivalent to the rate of informationflow. The right-hand side is the return in excess of the risk-free rate. The
Spread Inversion
191
implication is that under static interest rate conditions and a constant rateof information arrival at the market, the rate of decay of the spread wouldbe a constant. However, the various milestones leading to the merger likethe filing of documents, the proxy vote, and so forth, all happen at discretetimes. So, how realistic is the assumption of a constant rate of informationflow? It turns out that it is not terribly unrealistic. When the merger announcement is made and events are set into motion, it creates an expectation in the marketplace as to the timing of the various milestones leading tothe merger. If a day passes without any news contrary to expectation, thenit strengthens the expectations. In that sense, the mere passage of time maybe construed as providing information leading to the strengthening of expectations—one of the very rare cases in the market where no news is considered good news.A constant rate of information flow, according to the model, carrieswith it the implication that the logarithm of the spread approaches zero in alinear fashion. In fact, this would be a very reasonable assumption, if thedeal were to proceed without any hiccups. However, given the vicissitudesof the deal process, such an assumption might prove to be untenable. Nevertheless, it may still be reasonable to assume that the logarithm of thespread is piecewise linear; that is, the rate of spread reduction is about thesame as observed in the recent past. Therefore, we model the spread at thenext time instant to be the current spread decremented by the instantaneousrate of reduction of the spread. The state equation is given asXt = Xt–1 + ∆t–1 + et
(12.2)
where ∆t–1 is the instantaneous rate of change of the spread. Note that Equation 12.2 can also be viewed as the first two terms of a Taylor expansion ofthe spread about Xt–1.This, therefore, brings us to the requirement of specifying a scheme toevaluate the instantaneous rate of change of the spread, ∆t–1. A naive methodwould be to use the correction rate of the past time step given by∆ t −1 = Xˆ t −1| t −1 − Xˆ t − 2| t − 2
(12.3)
A little more sophisticated method would be to take the minimum varianceweighted average of the past two correction rates given by
(
) (
)(
∆ t −1 = rt Xˆ t −1| t −1 − Xˆ t − 2| t − 2 + 1 − rt Xˆ t − 2| t − 2 − Xˆ t − 3| t − 3
) (12.4)
where rt is the weight. Alternately, one could obtain an estimate of the correction rate by taking the mean correction rate over the past d time steps;that is,
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192
∆ t −1 =∆ t −1 =
(Xˆ(
ˆ
t −1| t −1 − X t − 2| t − 2
)+, … , +(Xˆ
Xˆ t −1| t −1 − Xˆ t −1 − d | t −1 − d
)
d
t − d |t − d
− Xˆ t −1 − d | t −1 − d
) (12.5)
d
Thus, the process of evaluating the instantaneous rate of reduction of thespread could vary depending on d, the number of time steps in the past thatwe use in our estimation. Consequently, we could have different state equations corresponding to different values of d, the lag parameter, and for eachsuch state equation, a version of the Kalman filter could be implemented.But which one of the state equations is most suited for our purpose? Letus defer this question for now and address it a little later in the chapter.Another point that is noteworthy in the modeling of the predictionequation is the fact that the actual equation used is different for each timestep. The exact values for the coefficients in the prediction equation are estimated at each time step based on the instantaneous rate of spread reduction. This is a little different from typical Kalman filter implementations thathave a fixed prediction equation with fixed coefficients. We are now readyto move on to the observation equation.
THE OBSERVATION EQUATIONThe observation at a given time instant is the logarithm of the spread. Associated with an observation is a measure of the error. The magnitude of thiserror is quantified in terms of the error variance. The observation equationis written asYt = Xt + ht
(12.6)
where Yt is the logarithm of the observed spread at time t and ht is the observation error with zero mean and variance σ η2t. We now therefore need toestimate the variance of our observation.To estimate the variance of the observation, we draw on the notion ofrealized volatility. Realized volatility is essentially an empirical volatilitymeasure that sums the squared tick-by-tick returns over a given period. Theconstruction of the realized volatility measure is based on some theoreticalresults of integrated volatility defined in the context of continuous stochastic processes. A detailed discussion on realized volatility measures and itsscaling properties can be found in the reference material.Bear in mind that if we use the realized volatility as a measure of thevariance in the observation, then we would need access to tick data. The
Spread Inversion
193
spread can then be evaluated on a tick-by-tick basis and the sum of squaredreturns on the spread could be used as a measure of observation variance. Inthe absence of tick data we use the following approximation to estimate therealized volatility measure. Initially, the four spreads corresponding to theopen high low and close prices for the day of the two stocks are computed.Note that the times in which the two stocks registered their highs could bedifferent; nevertheless, we treat them as if the highs were registered simultaneously. The sum of squared returns on the spread is calculated on the twopossible spread paths; namely, open-high-low-close and open-low-highclose. The smaller of the two values is then chosen to be the variance of theobservation. The question the reader might ask at this point is, “Is this really the variance of the error in the observation?” It is probably not. However, the variances thus calculated are not used to price any instrument. Itshould suffice that the measured variances are proportional to the actualerror variances and that the relative order of the errors associated with theobservations is preserved. This ensures that the weighting of the observations in the evaluation of the state estimate in done in a consistent mannerresulting in state estimates that seem to be satisfactory.To get an intuitive feel for the volatility measure, consider the following.As the spread gets closer to zero, the bid-ask spread of the individual stocksmeasured as a percentage of the spread between the two stocks becomeshigher. It then gets increasingly harder to tell whether the change in spreadis real or due to microstructure or bid-ask effects. Thus, as the spread reduces, we should expect an increase in its variance.
APPLYING THE KALMAN FILTERSummarizing the discussions so far, in order to apply the Kalman filteringapproach to smooth the spread, we went through the exercise of modelingthe prediction equation and the observation equation. The Kalman state inour situation corresponds to the logarithm of the true spread, and the observation corresponds to the logarithm spread as observed in the market.The state equation as applicable in our situation is given asXt = Xt–1 + ∆t–1 + et
(12.7)
where Xt is the state at time t, ∆t–1 is the time derivative of Xt–1, and et is thestate noise. The time derivative ∆t–1 may be estimated by measuring the average change in the spread over different lag values. The observation equation is given byYt = Xt + ht
(12.8)
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where Yt is the observation and ht is the observation noise with zero meanand variance of σ η2t.With this information we are now ready to apply the Kalman-filteringideas in a recipe-like fashion (as discussed in Chapter 4). We list the recipebelow once again for sake of continuity. The a priori estimate of the state attime t given all observations up to time t – 1 is denoted as Xˆ t | t −1, and theposteriori estimate of the state at time t given all the observations up to timet is given as Xˆ t | t . The various steps are then as follows:
(
)
1. Evaluate Xˆ t | t −1 and var Xˆ t | t −1 using the state equation.2. Find the observation Yt and var Yt by observing the system.
( )
3. Evaluate Kt, also known as the Kalman gain, which will be used to obtain the linear minimum error variance estimate.4. Evaluate Xˆ t | t given by Xˆ t | t −1 + Kt Yt − Xˆ t | t −15. Finally, evaluate var Xˆ t | t
( )
(
)
These steps are repeated for the next time step. The exact formulas to use forthe various steps are derived in simplified form in the appendix.
MODEL SELECTIONNote that based on the preceding scheme, we have multiple models for theprediction equation. The differentiating factor amongst them is d, the lagfactor used in the estimation of the instantaneous error correction rates. Foreach of the prediction equations, we can implement a unique Kalman filterimplementation. We therefore need to choose the implementation thatis most appropriate; that is, we need to make an appropriate choice for theparameter d.The principle guiding our choice for the lag value d is the maintenanceof the delicate balance between prediction and observation. Relying overlyon the prediction would mean that we rely excessively on our model and donot give enough weight to what is observed in practice. However, relying toomuch on the observation would mean that we do not have any view whatsoever on where the spread must be and so must rely excessively on the noisyobservation. The right model choice achieves a happy medium between thetwo extremes. Let us therefore look at how we can quantify this notion of ahappy medium.The basic idea is to work with the two sources of error; namely, themeasurement/observation error and the state transition prediction error.We denote the measurement cost as the sum of all measurement errorsand the prediction cost as the sum of all the prediction errors. More precisely, if X1, X2, X3, . . . , Xn are the estimated states of the Kalman filter,
Spread Inversion
195
Y1,Y2,Y3, . . . ,Yn, the measurements, and Z1, Z2, Z3, . . . , Zn, the predicted values from the state equation, thenn
measurement cost =
∑ (Xi − Yi )
2
i =1
n
prediction cost =
∑ ( X i − Zi )
2
i =1
These cost functions may be considered the measures of effectiveness ofthe measurement and the state transition models. A high prediction cost indicates a poor prediction model. Similarly, a high observation cost indicatesa poor observation model. If the two models must be equally effective, itmakes sense to require that the values of the two cost functions be more orless the same.Additionally, we also know that the Kalman estimate for the state is aconvex combination of the measured and predicted states; that is, given ki tobe the Kalman gain at the ith time step, we haveXi = (1 – ki)Zi + kiXi
(12.9)
The discussion on model selection in the case of smoothing functionscan be viewed as a situation where we attempt to separate the signalfrom noise given a sum of both. On the one hand, we can be very conservative and treat every kink as meaningful and separate out very little as noise. On the other hand, we can throw the baby out with thebathwater and oversmooth the given signal, discarding part of the signal along with the noise. The model selection criterion above providesus with a methodology to achieve a balance between both extremes.
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This dependency on the measured and predicted states requires both themeasurement model and the prediction model to be fairly precise. Havingone of them to be very precise and the other to be erroneous leads us tooverly rely on one or the other and is likely to produce a mediocre result.Thus, in some sense there is a trade-off to be made between the observationand prediction costs, and the reduction of one of the costs at the expense ofthe other is highly undesirable.With this motivation, we define the cost function associated with aKalman filter to becost function = measurement cost + prediction cost
(12.10)
2
2
1
1
log(Spread)
log(Spread)
This cost function serves to keep the system honest. If in an attempt to reduce the cost function, we try to reduce the prediction cost, it would be allright as long as it does not increase the measurement cost and vice versa. Thebest choice for our prediction equation is therefore the one that results inthe minimum value for this cost function.To demonstrate the approach, let us apply it to a real-life situation. Figure 12.1 is a plot of the spread and the corresponding Kalman smoother forvarious lags. The bidder in this case is McKesson Inc., and the target is HBO
0–1
–1
–2
–2
–3
–310
30
50
2
2
1
1
log(Spread)
log(Spread)
0
0–1
–3
–3
FIGURE 12.1
50
50
10
30
50
–1–2
30
30
0
–2
10
10
Kalman Filter Implementations (MCK-HBOC).
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197
13
Model Cost
12
11
10
9
80
2
FIGURE 12.2
4
6
8
Model Choice ( MCK-HBOC).
and Company. The exchange ratio is 0.37. The deal was announced October 19, 1998, and completed January 13, 1999. Notice that smaller lags tendto follow the data more closely. Increasing the lag results in greater smoothing and greater deviations from the data.We also have a plot of the cost function as discussed for various lags. SeeFigure 12.2.Note that the minimum cost value occurs at lag 3. A visual examinationof the smoothed series for lag 3 against the series for other lags shows thatit is indeed a reasonable choice.Thus, a low value for the lag parameter d implies a noisy set of state estimates, making the Kalman filter very sensitive to the observations. Alternately, a high value for the lag parameter d denotes a smoother set of statesand the observations are largely ignored. The most suitable value for the lagparameter is one that minimizes the cost function.
APPLICATIONS TO TRADINGAlong with the smoothed version of the spread, at every time instant, theKalman filter also estimates the error standard deviation at each point. Thiscan be treated as error bands, about a mean estimate, similar to bollinger
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bands in technical analysis. Trading can be undertaken when the observedspread is on the upper and lower fringes of the band.We, however, add a note of caution. When the spread widens to theupper fringe of the band, it may be because there is some deterioration in thefundamentals of the merger and may not be just an aberration. Therefore,one needs to exercise extreme caution when putting the spread on. The matter is straightforward when timing the unwind. If the observed spread is nearthe bottom fringe of the band, then one can safely unwind the position andget back into it again at a higher spread level.Illustrated in Figure 12.3 is a plot of the spread, the Kalman smoother,and the error bands. The bidder in this case was Alza Pharmaceuticals. Thetarget was Sequus Pharmaceuticals. The exchange ratio was 0.4 share. Thedeal was announced October 5, 1998, and completed March 17, 1999.Last, it is important to bear in mind that the smoothing scheme in Figure 12.3 has been set up with daily data in mind. When looking at data onan intraday basis, there is likelihood of running into intraday effects on thespread variance at market open and market close. One can expect that thespread variance is high around those periods and lower during the middle of
1
Log (Spread)
0
–1
–2
–30
FIGURE 12.3
20
40
60
80
100
120
Kalman Smoothing with Confidence Bands (AZA-SEQU, lag 5).
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199
the day. The constant arrival of information assumption is probably not correct under such circumstances.
SUMMARYThe spread observed in the marketplace between two companies involved in a merger is likely to be distorted due to other market effectslike the bid-ask spread effects and market maker inventory adjustments.The Kalman filtering approach is a suitable smoothing technique for estimating the actual spread levels.The filtered spread could be used for the risk-neutral probability andalso assist in timing executions.
FURTHER READING MATERIALKalman FilterHarvey, A. C. Time Series Models, 2nd Edition. (Cambridge, Mass: MIT Press, 1993),pp. 82–104.Model ChoiceKalaba, R., and T. Tesfatsion. “A Multicriteria Approach to Model Specification andEstimation.” Computational Statistics and Data Analysis 21 (1996): 193–214.Realized VolatilityAnderson, T. G., and others. “The Distribution of Exchange Rate Volatility.” Symposia 99, Statistical Issues in Risk Management. Leonard N. Stern School ofBusiness, April 1999.
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APPENDIXKalman Filter Design: Lag 1The state equation of the Kalman filter is given asXˆ t | t −1 = 2Xˆ t −1| t −1 − Xˆ t − 2| t − 2
(
)
(
)
(
)
(
var Xˆ t | t −1 = 4 var Xˆ t −1| t −1 + var Xˆ t − 2| t − 2 − 4 cov Xˆ t −1| t −1 , Xˆ t − 2| t − 2
)
The observation equation is given asYt = Xt + htThe variance of Yt is calculated as described in the discussion on the observation equation. We now definegt =
( )
var Yt
( )
(
var Yt + var Xˆ t | t −1
)
where gt = 1 – Kt, Kt is the Kalman gain as described in the standard predictorcorrector framework. The posteriori estimate of the state, and its variance isgiven as
(
)
Xˆ t | t = g t Xˆ t | t −1 + 1 − g t Yt
( )
var Xˆ t | t =
(
) ( )
var Xˆ t | t −1 var Yt
(
)
( )
var Xˆ t | t −1 + var Yt
We note that the a posterori estimate is actually a convex combination of thea priori estimate and the observation. The value of gt as computed here ensures that the variance of the resulting combination is a minimum. We nowproceed to obtain a recursive relation for cov Xˆ t −1 Xˆ t − 2 . The Kalman equation for the subscript t – 1 is
(
)
(
)
Xˆ t −1| t −1 = g t −1 Xˆ t −1| t − 2 + 1 − g t −1 Yt −1Substituting for Xˆ t −1| t − 2, we have
(
) (
)
Xˆ t −1| t −1 = g t −1 2Xˆ t − 2| t − 2 − Xˆ t − 3| t − 3 + 1 − g t −1 Yt −1
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Mult