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% PRESENTATION INFORMATION
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\newcommand*{\mytitle}{Filtering bond and credit default swap markets} % Title
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% PRESENTATION SLIDES
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\section{Filtering credit markets}
\subsection{A visual introduction}
\clearpage
\includegraphics[width=\textwidth]{typical_bond.png}
\clearpage
\includegraphics[width=\textwidth]{dark_bond.png}
\includegraphics[width=\textwidth]{good_bond.png}
\clearpage
\includegraphics[width=\textwidth]{newly_issued_bond.png}
\includegraphics[width=\textwidth]{intermittent_bond.png}
\clearpage
\includegraphics[width=\textwidth]{floating_bond.png}
\clearpage
%------------------------------------------------
\subsection{A unified state space for bond and CDS markets }
Hazard, funding curves can be represented as hidden processes. From this basic state, bond and credit default swaps can be computed using the usual present value computations. Additional state, representing the width of the market at various maturities, can be used to provide a rough estimate of where trading will occur. The actual observed trades $y_{t_i}$ are considered noisy measurements of a non-linear function of state $x_{t_i}$.
\begin{equation}
y_{t_i} = h\left( x_{t_i} \right) + \epsilon_i
\end{equation}
where $h$ incorporates credit default swap or bond pricing formulas, depending on the observation in question.
\subsubsection{Kalman filtering}
Linearize by differentiation, for now. Then consider a sequence of observations times \(t_1,...,t_k\) at which our latent vector process \(x\) is observed indirectly, via an observation equation
\begin{equation} y_{t_i} = H_i x_{t_i} + \epsilon_i
\end{equation}
We assume \(\epsilon_i\) is mean zero multivariate gaussian with covariance \(R_i\). For brevity we refer to \(y_{t_i}\) as \(y_i\), \(x_{t_i}\) as \(x_i\) and so forth. We assume the evolution of \(x\) in between the times specified can be written
\begin{equation}
x_{i+1} = A_i x_i + u_i
\end{equation} where \(u_i\) are also gaussian. In this linear gaussian system the recursive estimation of \(x_t\) is achieved by the well known Kalman filter.
\myblue{I think we knew that already!}
\clearpage
\subsection{Issues}
Stare at bond price time series for a while and you'll quickly discover that every assumption is violated
\begin{enumerate}
\item Brokered trades
\item Jumps
\item Lags
\item Off-market quotes
\item Funding peculiarities
\end{enumerate}
\clearpage
\includegraphics[width=\textwidth]{paired_bonds.png}
\includegraphics[width=\textwidth]{jumpy_bonds.png}
\clearpage
\includegraphics[width=\textwidth]{takeover_bonds.png}
\clearpage
\includegraphics[width=\textwidth]{slowdealer_bonds.png}
\clearpage
\includegraphics[width=\textwidth]{quoted_bonds.png}
\clearpage
\includegraphics[width=\textwidth]{quoted_badly_bonds.png}
\clearpage
\includegraphics[width=\textwidth]{quoted_very_badly_bonds.png}
\clearpage
\includegraphics[width=\textwidth]{cds_driven_bonds.png}
\clearpage
\subsection{Explaining and justifying pricing}
The contemporaneous impact of an observation \(y_{k+1}\) is proportional to the Kalman gain.
\myblue{(I think we knew that already too)}. This creates the pricing narrative.
But most observations are not contemporaneous ...
\subsubsection{Historical importance of observations}
\includegraphics[width=\textwidth]{price_inspector.png}
The derivatives of the Kalman filter estimate with respect to a {\bfseries past} observation \(y_i\) is not something we see too often.
\begin{enumerate}
\item Re-represent the Kalman estimate in the form of a weighted least squares problem (c.f. Duncan Horn representation).
\item Compute sensitivities of the solution of the weighted least squares problem. Sometimes a little adjoint trick helps here.
\end{enumerate}
\clearpage
\subsubsection{Kalman as least squares on the current state}
Sketch: We set up a least squares problem involving the current state \(x_k\) only. The solution to this problem is identical to the Kalman filter's current estimate. This establishes that the current estimate \(\hat{y}_k\) is a simple linear function of the current state \(x_k\), so we can compute the derivative of the current estimate with respect to all previous observations.
\clearpage
To be tidy, assume a gaussian prior on the initial state \(x_0\). To avoid annoying special cases in what follows, we clean up the notation by indexing back to $-1$ as follows:
\begin{eqnarray*}
y_{-1} & = & H_{-1} x_{-1} + \epsilon_{-1} \\
x_0 & = & A_{-1} x_{-1} + u_{-1}
\end{eqnarray*}
and here \(H_{-1}\) and \(A_{-1}\) are identity matrices, \(\epsilon{-1}\) is identically zero, \(y_{-1}\) is set equal to the mean of our prior and \(u_0\) adopts its covariance.
%------------------------------------------------
With the boundary conditions cleaned up in this fashion we can invert the dynamical equations, assuming only that \(A\)'s have left inverses \(A^{-1}\), as follows:
\begin{equation}
x_j = A^{-1}_{j}\left( x_{j+1} - u_j \right)
\end{equation}
and then re-arrange the observation equations so that the only value of \(x_i\) that appears is \(x_k\).
\clearpage
The inversion looks like:
\begin{eqnarray*}
y_k & = & H_k x_{k} + \epsilon_k \\
y_{k-1} & = & H_{k-1} x_{k-1} + \epsilon_{k-1} \\
& = & H_{k-1} \left( A^{-1}_{k-1}\left( x_{k} - u_{k-1} \right) \right) + \epsilon_{k-1} \\
& = & H_{k-1} A^{-1}_{k-1} x_{k} - H_{k-1} A^{-1}_{k-1} u_{k-1} + \epsilon_{k-1} \\
y_{k-2} & = & H_{k-2} x_{k-2} + \epsilon_{k-2} \\
& = & H_{k-2} \left( A^{-1}_{k-2}\left( x_{k-1} - u_{k-2} \right) \right) + \epsilon_{k-2} \\
& = & H_{k-2} A^{-1}_{k-2} x_{k-1} - H_{k-2} A^{-1}_{k-2} u_{k-2} + \epsilon_{k-2} \\
& = & H_{k-2} A^{-1}_{k-2} \left( A^{-1}_{k-1}\left( x_{k} - u_{k-1} \right) \right) - H_{k-2} A^{-1}_{k-2} u_{k-2} + \epsilon_{k-2} \\
& = & H_{k-2} A^{-1}_{k-2} A^{-1}_{k-1} x_{k} - H_{k-2} A^{-1}_{k-2} A^{-1}_{k-1} u_{k-1} - H_{k-2} A^{-1}_{k-2} u_{k-2} + \epsilon_{k-2} \\
& \dots &
\end{eqnarray*}
\clearpage
From which it is apparent that if we write \(Y = (y_k, y_{k-1}, y_{k-2},...,y_{-1} ) \) then
\begin{equation}
Y = G x_{k} + \eta
\end{equation}
where \(G\) is the concatenation of the coefficients of \(x_k\) given above and \(\eta\) is the gaussian random variable equal to the sum of \(u_k\)'s and \(\epsilon_k\)'s.
\myblue{ Thus we have a simple least squares problem for the contemporaneous state $x_k$} which is not dissimilar to the Duncan-Horn representation of the Kalman filter.
\clearpage
\subsubsection{Sensitivities of the least square problem}
Suppose \(x\) solves \(Qx = b(y)\). We wish to compute the derivative of \(g(x)\) w.r.t. y \myblue{ (because, to recap this will tell traders how important every historical observation is to the current price estimate whether or not the observation pertains to a bond in question}).
In particular, if \(y\) is the observation and \(x\) the solution of a generalized least squares problem with error co-variance \(R\) we can cast it in this form by writing:
\begin{eqnarray*}
g(x)& = & H x \\
Q & = & H^T R^{-1} H \\
b(y) & = & H^T R^{-1} y
\end{eqnarray*}
Consider now
\begin{equation}
f(x,y) = 0
\end{equation}
where
\begin{equation}
f(x,y) = Q x - b(y)
\end{equation}
We use derivatives of
\begin{equation}
\tilde{g} = g - \lambda^T f(x,y)
\end{equation}
with respect to \(y\) as a means of computing derivatives of \(g\) with respect to \(y\).
\clearpage
Note that
\begin{equation}
\frac{\partial \tilde{g}}{\partial y} = \frac{\partial g}{\partial x}\frac{\partial x}{\partial y} -
\lambda^T \left( \frac{ \partial f }{\partial x }\frac{\partial x}{\partial y} + \frac{\partial f}{\partial y} \right)
\end{equation}
and this will simplify if we choose \(\lambda\) judiciously as a solution of
\begin{equation}
\frac{\partial g}{\partial x } = \lambda^T \frac{\partial f}{\partial x}
\end{equation}
which is the adjoint equation. For then
\begin{eqnarray}
\frac{\partial \tilde{g}}{\partial y} & = & \frac{\partial g}{\partial x}\frac{\partial x}{\partial y} -
\lambda^T \left( \frac{ \partial f }{\partial x }\frac{\partial x}{\partial y} + \frac{\partial f}{\partial y} \right) \\
& = & -\lambda^T \frac{\partial f}{\partial y} \\
& = & \lambda^T \frac{\partial b}{\partial y}
\end{eqnarray}
\clearpage
Now specializing to
\begin{equation}
g(x) = H x
\end{equation}
and \(b(y)\) as above we can solve for this convenient choice of \(\lambda\) by writing
\begin{eqnarray}
H & = & \frac{\partial g}{\partial x} \\
& = & \lambda^T \frac{\partial f}{\partial x} \\
& = & \lambda^T Q \\
& = & \lambda^T H^T R^{-1} H
\end{eqnarray}
where the second equality is the adjoint equation. Thus we can compute derivatives of \(\tilde{g}\) with respect to \(y\), and thereby compute derivatives of \(g\) with respect to \(y\) which is what we set out to do.
%------------------------------------------------
\subsection{Accuracy}
At some point you have to explain the accuracy of your prices to clients. On the vendor side it is sometimes argued that customers are insensitive to pricing quality and the service is therefore sticky. This is party true but that argument presumes there will be no material change in market structure or competitive forces. In fact major buy side firms are gearing up to better quantify their \myblue{transaction costs} relative to peers. Others wish to use their inventory to generate \myblue{alpha}. And sell-side firms look for lower cost means of \myblue{making markets} and even \myblue{assessing traders}.
Unfortunately, accuracy is a subtle beast...
\subsubsection{A simple target}
The next inter-dealer trade. Issues:
\begin{enumerate}
\item Rare for many bonds
\item Noisy
\item Repeated
\item Paired
\item Serially correlated
\end{enumerate}
\clearpage
\subsubsection{A time, size and money-under-the-bridge weighted target}
Fix some moment $t$ at which a price is supplied by a vendor and consider the $J$ subsequent inter-dealer trades:
\begin{equation}
FVT(t;J) = \frac{ \sum_{j=1}^{J} p_j s_j e^{-(t_j-t)} e^{-M^-_j} }
{ \sum_{j=1}^{J} s_j e^{-(t_j-t)} e^{-M^-_j} }
\end{equation}
where \(p_j\), \(s_j\) and \(t_j\) are the price, size and time of subsequent inter-dealer trades with time measured in business days and
\begin{equation}
M^{-}_j = \frac{1}{c}\sum_{k=1}^{j-1} s_k
\end{equation}
is the cumulative trading volume up to but not including the trade in question.
\subsubsection{Fixing an inconsistency}
The use of $M^{-}_j$ is slightly unnatural because it means the target is not invariant to splitting of future trades. We can easily fix this, however, by integrating in money instead of time.
$$
FVT'(t;J) = \frac { \int_{m=0}^{M^{+}_J} p(m) e^{-m}e^{-(t(m)-t)} dm }
{ \int_{m=0}^{M^{+}_J} e^{-m}e^{-(t(m)-t)} dm }
$$
where \(M^{+}_J := M^{-}_{J+1}\) is the total amount of money under the bridge up to and including the \(J\)'th trade, \(p(m)\) is the price when \(m\) dollars of trading has occurred, and $t(m)-t$ is the time we have progressed when $m$ dollars of trading has occurred.
\subsubsection{Multivariate filters seemingly perform worse then univariate}
A more serious issue with accuracy measures for bond pricing relates to the inefficiency in the market. One needs to be careful not to reward univariate models (treating only the bond in question) for over-fitting. A stylized simulation to prove the point following West and Papanicoloau can run as follows. Disregard funding rates and use piece-wise hazards:
\begin{equation}
\lambda(t) = \lambda_i, t_{i-1} \le t_{i}
\end{equation}
so that yields
\begin{equation}
y_i = \int_0^{T_i} \lambda(t) dt
\end{equation}
are clearly linear in the hazard rates.
\clearpage
Assume now that hazards are independent random walks
\begin{equation}
\lambda_k(t_dt) = \lambda_k(t) + N(0, dt\sigma_k^2)
\end{equation}
and for simplicity set $\sigma_1=0.1$, $\sigma_2=0.05$ and $\sigma_3=0.01$. Assume bond prices are observed at times generated by a self-exciting Hawkes process:
\begin{equation}
dc_t = (c_t-c)e^{-\kappa t} + f dN_t
\end{equation}
West and Papanicolaou studied various types of bonds based on these parameters. They then used both univariate and multivariate Kalman filters, and also a simple model where the estimate was simply the last trade observation.
\clearpage
\begin{table}
\begin{tabular}{|l|l|l|l|}
\hline
& Last Trade & Univariate K.F. & Multivariate K.F. \\ \hline
Apparent error & 0.020 & 0.0377 & 0.0383 \\ \hline
Actual error & 0.056 & 0.0512 & 0.0286 \\ \hline
\end{tabular}
\caption{Apparent error when measured against the next trade, and actual error against ground truth. In a
simulation the model that simply uses the last trade of the bond in question appears to perform better than the multivariate Kalman filter applied to the entire curve - but that is an illusion. Numbers courtesy of Nick West and George Papanicolaou.}
\end{table}
\clearpage
\subsubsection{Serial correlation distorts accuracy metrics}
A related problem occurs if we assume that trade observations are serially correlated errors with respect to some ground truth. Simple models can over-fit but appear to perform well in simple accuracy statistics.
\myblue{With all that said ...} here is a comparison between the current version of Benchmark, built by A.J. Linderman, Antoine Toussaint et al, and a prominent vendor offering real-time pricing. (We can't compare the original Benchmark pricing service as there were no other real-time services at the time).
\clearpage
\subsubsection{Vendor comparison}
\begin{table}
\begin{tabular}{|l|l|l|l|}
\hline
Time since last trade & Market leader & BMRK & Count \\ \hline
5 - 20 mins & 0.238 & 0.140 & 7,348 \\ \hline
20 mins -2 hr & 0.215 & 0.156 & 9,326\\ \hline
2hr - 4hr & 0.250 & 0.174 & 726 \\ \hline
4hr - 10 hr & 0.217 & 0.159 & 5,931\\ \hline
10hr - 2 days & 0.217 & 0.171 & 2,613 \\ \hline
2 days - 1 week & 0.234 & 0.189 & 4,738 \\ \hline
& & & 30,672 \\ \hline
\end{tabular}
\end{table}
\clearpage
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