iplying the above by X̂t −2 and evaluating the expected value, we have
(
)
(

)
(
cov Xˆ t −1| t −1 , Xˆ t − 2| t − 2 = g t −1 2 var Xˆ t − 2|t − 2 − cov Xˆ t − 2|t − 2 , Xˆ t − 3|t − 3
)
Kalman Filter Design: Lag 2In order to enhance readability, we use X̂t and Xˆ t | t interchangeably to denote the posteriori state estimate. The state equation of the Kalman filter isobtained as a first order approximation of the Taylor expansion about thecurrent point. To do that, an estimate of the derivative is required. The previous design used the first difference of the previous step as an estimate ofthe derivative. In this design, the derivative is estimated with two samplepoints. The mean and variance of the first sample are as follows:E(sample1) = Xˆ t −1 − Xˆ t − 2
(
)
(
)
(
var(sample1) = var Xˆ t −1 + var Xˆ t − 2 − 2 cov Xˆ t −1 , Xˆ t − 2
)
The mean and variance of the second sample are as follows:E(sample 2) = Xˆ t − 2 + Xˆ t − 3
(
)
(
)
(
var(sample 2) = var Xˆ t − 2 + var Xˆ t − 3 − 2 cov Xˆ t − 2 , Xˆ t − 3
)
The covariance between the two samples are as follows:
)()(cov(samples) = cov( Xˆ , Xˆ ) − var( Xˆ ) − cov( Xˆ , Xˆ ) ++ cov( Xˆ , Xˆ )
cov(samples) = cov Xˆ t −1 − Xˆ t − 2 , Xˆ t − 2 − Xˆ t − 3t −1
t −2
t −2
t −2
t −1
t −3
t −3
The minimum variance linear combination of the two samples is given byrt (Xˆ t −1 − Xˆ t − 2 ) + (1 − rt )(Xˆ t − 2 − Xˆ t − 3 )where rt is given asrt =
var(sample 2) − cov(samples)var(sample 1) + var(sample 2) − 2 cov(samples)
RISK ARBITRAGE PAIRS
202
Now the state equation is given as
(
) (
)(
Xˆ t | t −1 = Xˆ t −1 + rt Xˆ t −1 − Xˆ t − 2 + 1 − rt Xˆ t − 2 − Xˆ t − 3
(
)
(
)
(
)
)
Xˆ t | t −1 = 1 + rt Xˆ t −1 + 1 − 2rt Xˆ t − 2 − 1 − rt Xˆ t − 3
(
) (
) var(Xˆ ) + (1 − 2r ) var(Xˆ ) + (1 − r ) var(Xˆ )+ 2(1 + r )(1 − 2r ) cov( Xˆ , Xˆ )− 2(1 + r )(1 − r ) cov( Xˆ , Xˆ )− 2(1 − 2r )(1 − r ) cov( X̂X , Xˆ )
var Xˆ t | t −1 = 1 + rt
2
2
t −1
t
t −1
t
t
t −3
t −1
t
t −3
t
t −2
t −1
t
t
2
t −2
t
t −3
The observation equation is given asYt = Xˆ t + ηtThe variance of Yt is calculated as described in the discussion of the observation equation. We now definegt =
( )
var Yt
( )
(
var Yt + var Xˆ t | t −1
)
gt = 1 – Kt , where Kt is the Kalman gain as described in the standard predictorcorrector framework. The Kalman equations providing the minimum variance linear estimate are as follows:
(
)
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 now proceed to obtain the recursive relation for cov Xˆ t −1 , Xˆ t − 2 . TheKalman equation for subscript t – 1 is as follows:
(
)
Xt | t −1 = g t −1 Xˆ t −1| t − 2 + 1 − g t Yt −1Substituting for Xˆ t −1| t − 2, we have
Spread Inversion
203
[(
)
(
)
Xˆ t −1| t −1 = g t −1 1 + rt −1 Xˆ t − 2 + 1 − 2rt −1 Xˆ t − 3 −
(
)
] (
)
− 1 − rt −1 Xˆ t − 4 + 1 − g t −1 Yt −1Multiplying the above by X̂t −2 and evaluating the expected value, we have
(
)
[(
) (
) (− (1 − r ) cov( Xˆ , Xˆ )]
) (
)
cov Xˆ t −1 , Xˆ t − 2 = g t −1 1 + rt −1 var Xˆ t − 2 + 1 − 2rt −1 cov Xˆ t − 2 , Xˆ t − 3 −t −1
t −2
t −4
Multiplying by X̂t −3 and evaluating the expected value, we have
(
)
[(
) (
) (
) (
)
cov Xˆ t −1 , Xˆ t − 3 = g t −1 1 + rt −1 cov Xˆ t − 2 , Xˆ t − 3 + 1 − 2rt −1 var Xˆ t − 3 −
(
) (
− 1 − rt −1 cov Xˆ t − 3 , Xˆ t − 4
)]
Kalman Filter Design: Lag d ( d >= 3)To enhance readability, we use Xˆ t | t and X̂t interchangeably to denote theposteriori state estimate. The slope is estimated as the mean of the last dslope samples. This is given asXˆ t −1 − Xˆ t −1 − ddThe state equation is thereforeXˆ t | t −1 = Xˆ t −1 +
(Xˆ
ˆ
t −1 − X t −1 − d
)
d11Xˆ t | t −1 = 1 +  Xˆ t −1 − Xˆ t −1 − ddd
The state variance is given by2
11ˆˆ1 +  var Xt −1 + 2 var Xt −1 − ddd
(
)
(
)
RISK ARBITRAGE PAIRS
204
the assumption being that X̂t −1 and X̂t −1 − d are not correlated. The observation equation is given asYt = Xˆ t + ηtThe variance of Yt is calculated as described in the discussion of the observation equation. We now definegt =
( )
var Yt
( )
(
var Yt + var Xˆ t | t −1
)
The Kalman equations providing the minimum variance linear estimate areas follows:
(
)
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
Index
AAggressive trading, 160Akaike information criterion (AIC),27–29Alza Pharmaceuticals, 198American Home Products (AHP),180Analysis step, 24, 25APT. See Arbitrage pricing theoryArbitrage pricing theory (APT),39–42, 75Arcsine law, 113ARMA model, 124–125ARMA process, 21–22Arrow-Debreu theory, 172, 173, 177,187Asset returns, 3–5Autocorrelation, 15Autoregressive process (AR), 19–21
BBand design, 119BARRA model, 38Baruch, Bernard, 141Berkshire Hathaway, 182–183Beta, 3–5, 37Bias, 25Bidder firm, 142Bidder stock, 144Bipartite graph, 157Boesky, Ivan, 141Bonds, issuing, 139Bootstrap technique, 114, 116Bounds, on position size, 163–165Box, George, 14Box-Jenkins approach, 15Broker, 151
Brownian motion, 113, 163–164Buffett, Warren, 182Burmeister, Ibbotson, Roll, and Ross(BIRR) model, 38
CCantor, 79Capital Asset Pricing Model (CAPM),3–5, 37asset returns, 3–5beta, 3–5market portfolio, 3security market line (SML), 3–4Capital structure, 139CAPM. See Capital Asset PricingModelCash amount, in a deal, 142, 144Chinesing, 154Chi-square merit function, 110–111Close-close method, 111Cohn, Harry, 141Cointegration, 75–79application of, 80coefficient, 118common trends model, 87–90error correction, 76–77Collars, 145Common drift, 99Common factor risk, 44–46Common factor spread, 98–99Common trends cointegration model,87–90Common trends model, 77and APT, 90–93Compaq Computers Company (CPQ),154–155Complexity theory, 79
205
INDEX
206
Conditional mean, 19Conditional variance, 19Conservative trading, 160Control theory, 52Corporate events, 139Correlation, 13Correlogram, 15Cost function, 133Covariance, 12Covariance matrix, 41–44
DDeal, 141process, 141–142spread, 145–147Debt, issuing, 139Definitive agreement, 141Deviations from ideal conditions,97–99Differencing, 24–25, 76Dinic’s algorithm, 168–170blocking flow, 168flow augmenting paths, 168residual graph, 168steps in algorithm, 168, 170Discretization effect, 129Distance measure, 93–94formula, 94interpreting, 94–97calculating angle between twovectors, 95–96geometric interpretation, 96–97Dollar neutral portfolio, 6Dual, of a linear program, 156, 158
EEigen portfolios, 38Eigen value decomposition, 103Engle, Robert F., 76, 105Equilibrium price, 60–61Equilibrium value, 118Equity, issuing, 139Ergodicity, 126–127Error correction, 76–77Event risk management, 185–186Exchange offers, 140
Exchange ratio, 144–146Executionduring pricing period, 161–166verifying, 155–160
FFactor covariance matrix, 42Factor exposure matrix, 42Factor models, 37–51applications, 44–50calculating risk on portfolio,44–47calculation of portfolio beta,47–49tracking basket design, 49–50arbitrage pricing theory, 39–42covariance matrix, 42–44fundamental, 38macroeconomic, 38parameters, 42sensitivity analysis, 50statistical, 38Fibonacci numbers, 63Fibonacci retracements, 54Fixed ratio stock exchange, 144Fixed value stock exchange, 144Forecasting, 24–25Ford, 156Fulkerson, 156Fundamental factor models, 38
GGaussian distribution, 16Gaussian white noise, 16Gaussian white noise series, 120General Electric, 30Generalized auto-regressive conditionalheteroskedascity (GARCH),124, 125General Re, 182–183Geometric interpretation, 96–97Golden mean ratio, 63Goodness of fit vs. bias, 25–26Granger, C.W., 76, 105Granger representation theorem, 76Gruss, Joseph, 141
Index
H
207
HBO and Company, 196–197Hedge, 10, 48, 151, 161, 162Hedge ratio, 48, 49Heuristics approach, 86Hewlett Packard Company (HWP),154–155Historic volatility, 64
Level Communications (LEVL), 146,147, 148, 183Levy, Gustave, 141Levy, P., 113Lewis, Salim, 141Linear programming, 158Linear relationship, 106–107Long–short portfolio, 6–7
I
M
Implied probabilityof deal success, 172of merger success, 172Indexing, 81Induction, 79Innovation, 22Information theory, 27, 179Intel Corporation (INTC), 146, 147,148INTC-LEVL, 183Integer linear program, 156
Macroeconomic factor models, 38Market implied merger probability,171–188implied probabilities and ArrowDebreu Theory, 178multistep model, 177–180reconciling theory and practice,180risk management 184–186event risk management, 185–186VAR measurement, 184–185single-step model, 175–177Market neutral portfolios, 5Market neutral strategy, 5–7dollar neutral portfolios, 6long–short portfolio, 6–7market neutral portfolios, 5mean-reverting behavior, 5–6zero beta portfolio, 5–6Market portfolio, 3Markowitz approach, 40Mark Twain, 127Married puts, 166Martingales, 23, 30Max-flow formulation/problem,158–159Max flow–min cut theorem, 156Maximization of profits, 118Maximum entropy methods (MEMmethods), 132Maximum likelihood criterion, 26Maximum value of Brownian motion,163McKesson Inc., 196Mean, 12Mean drift, 99, 100, 104, 119, 136
JJenkins, Gwilym, 14
KKalman, R.E., 53Kalman filter/filtering, 52–69, 189–190, 192, 194, 196, 197applications to trading, 197–199applying, 193–194design, 200–204filtering the random walk, 60–64application to S&P index, 64–67formulas, 69Kalman filter, 53, 54–57observation innovation, 56scalar Kalman filter, 57–60system state concept, 53Karp, Richard, 79Kirchoff’s law, 59
LLagged values, 23, 196–197Lazy allocation algorithm, 170Least squares criterion, 111
INDEX
208
Mean reversion, 24Mean-reverting behavior, 5–6Mergers, 140boom, 141“Mining syndrome,” 43, 50Mispricing, 74Mixture Gaussian distribution,123–124Model choice, 26–29Model error, 99Modeling stock prices, 30–32Model selection, 194–197Modern Control Theory, 53Monsanto Corp. (MTC), 180–181Morgan Stanley, 73Moving average process (MA), 17–19Multifactor approach, 107–108Multifactor models. See Factor modelsMultiple threshold design, 135–136Multistep model, 177–180
NNewbold, P., 105Newell Company, 180New Jersey Holding Company Act,140Noise, 189Nonparametric approach, 126–130Nonstationary time series, 23, 79Normal distribution, 16NP completeness, 79
OObservation equation, 192–193Ohm’s law, 59Order, specifying, 152–155Ordinary least squares, 108, 111
PPair selection in equity markets,85–103common trends cointegration model,87–90common trends model and APT,90–93distance measure, 93–94interpreting, 94–97
reconciling theory and practice,97–102deviations from ideal conditions,97–99model error, 99numerical example, 99–102stationarity of integrated specificreturns, 97Pairs trading, 8–9described, 74–75history of, 73–74risk arbitrage pairs, 8, 9spread, 8statistical arbitrage pairs, 8, 9strategy, 83–84Parity relationship, 8, 79, 123, 125,145Portfoliobeta, calculation of, 47–49calculating risk, 44–47Positive definite, 42Prediction-correction method, 55Prediction equation, 190–192Prediction step, 24, 25Preprocessing step, 24–25Pricing period, 144Probability formulas, 12–13
RRandom walk process, 22–24Random walk seriesfiltering, 60–64applied to S&P index, 64–67Ratio, 8, 9, 39, 97, 121, 153, 161Recapitalizations, 139Recursive least squares method,62Regression, 108–109Regression approach, 108–112Regularization, 130–135Relative pricing, 74Residual component, 4, 5Residual time series, 6, 108, 112,115–116Rice, S.O., 113, 114, 125Risk, 39–40common factor risk, 44–46
Index
diagram, 41specific, 44, 46Risk arbitrage mechanics, 139–150capital structure, 139corporate events, 139deal process, 141–142deal spread, 145–147history of risk arbitrage, 140–141quantitative aspects, 149trading strategy, 147–149transaction terms, 142–145collars, 145fixed ratio stock exchange, 144fixed value stock exchange, 144stock and cash exchange, 144–145Risk arbitrage pairs, 8, 9Risk management, 184–186Risk neutral probabilities, 172Risk on portfolio, calculating, 44–47r-squared measure, 105Ross, Stephen A., 39Rubbermaid Corporation, 180
SScore/distance measure, 86–87Securities and Exchange Commission(SEC), 142Security market line (SML), 3–4Selling short, 166–167Sensitivity analysis, 50Sequus Pharmaceuticals, 198Shannon, Claude E., 179Sharpe, William T., 3Sharpe ratio calculations, 136Short sale, 151–152indicator, 154Signal-to-noise ratio (SNR), 98, 104,113, 119, 136Single-step model, 175–177Specific risk, 44, 46Specific spread, 92–93Specific variance matrix, 42Spinoffs, 139Sports Authority (TSA), 181Spread, 8, 74, 80, 118deal, 145–147dynamics, 118, 122–128
209
valve, 153volatility, 185Spread inversion, 189–199applications to trading, 197–199applying Kalman filter, 193–194model selection, 194–197observation equation, 192–193prediction equation, 190–192Spurious regression, 105Standard & Poor (S&P) index, 64–67Stationarity of integrated specificreturns, 97Stationary time series, 79Statistical arbitrage pairs, 8, 9trading, history of, 73–74Statistical factor models, 38Stock, James H., 77, 79, 104Stock and cash exchange, 144–145Stock prices, modeling, 30–32System state transition, 53
TTarget firm, 142Target stock, 144Target stock quantity, 154Tartaglia, Nunzio, 73, 74Technical analysis, 53, 54, 198Tender offer, 139–140Threshold level, 128Threshold value, 127Tikhonov-Miller regularization, 132Time-based stops, 136Time series, 14–36autocorrelation, 15correlogram, 15defined, 15defined, 14forecasting, 24–25goodness of fit vs. bias, 25–26model choice, 26–29modeling stock prices, 30–32time series models, 16–24autoregressive process (AR),19–21general ARMA process, 21–22moving average process (MA),17–19
INDEX
210
Time series (cont.)random walk process, 22–24white noise, 16–17types of, 14deterministic, 14probabilistic, 14stochastic, 14Tracking baskets, 81design, 49–50Tracking error, 49, 81Tradability, testing for, 104–117linear relationship, 106–107multifactor approach forestimating, 107–108regression approach forestimating, 108–112testing residual for tradability,112–118Trade execution, 151–167broker, 151execution during pricing period,161–166bounds on position size, 163–165short sale, 151–152short selling, 166–167specifying the order, 152–155action, 153bidder and target tickers, 153ratio and cash amount, 153short sale indicator, 154spread value, 153target stock quantity, 154verifying the execution, 155–160Trading design, 118–136ARMA model, 124–125band design for white noise,118–122hidden Markov ARMA models,125–126mixture Gaussian distribution,123–124
multiple threshold design, 135–136nonparametric approach, 126–130regularization, 130–135Tikhonov-Mille, 132Sharpe ratio calculations, 136spread dynamics, 118, 122–128time-based stops, 136Trading strategy, 82–83, 147–149
VValue at risk (VAR), 172measurement, 184–185Variance, 12Vectors, 95–96Venetor Group, 181Volatility, 5, 44, 64, 102, 108,123–124, 172historic, 64implied, 166, 172realized, 192–193Volume weighted average price(VWAP) price, 112
WWalk away right, 145Watson, Mark W., 77, 79, 104Weighted least squares, 111Weiner, Nobert, 14Weiner filtering, 14, 22, 189White noise, 16–17, 121band design for, 118–122Gaussian series, 120Wilshire Atlas model, 38Wyser-Pratt, Eugene, 141
ZZero beta portfolio, 5–6Zero crossing, 24, 112–114rate, 113–114Rice formula for, 113–114
<|end|>