Applying PCA: Swap Spread Price Discovery in an Illiquid Market
Team Latte 27th May 2012
During the global financial crisis there was an absence of liquidity in the USD swap market. This absence of liquidity was most pronounced during the thinner Asian trading hours. This presented challenges for market users who rely on accurate market information to construct yield curves to price a whole slew of products ranging from interest rate swaps and options to FX forwards and FX options. Trading desks use cash treasury prices and broker screens for IRS spreads to build a blended yield curve. A typical Reuters screen is presented below.
These swap spreads are quoted in basis points over ontherun treasury yields^{1}.Many brokers, including TotanIcap have a policy of adjusting their screens only when there is a bid or offer of a minimum amount shown in the market. This is to ensure the integrity of the market and transparency of the screens. In a normal functioning market an interbank market maker will "adjust" the screens by quoting an outright bid/offer spread or a spread against a more liquid tenor to adjust stale tenors.
This process broke down during the financial crisis and broker screens were not reliable especially for the less liquid tenors. The only alternative market users had was to speak to their broker and ask where he/she thought the levels are. The broker would do his/her best to provide colour on where the interests are in the market and where he/she thinks it may trade.
In 2008 an interesting paper surfaced by Leonardo M. Nogueira that offered an alternative^{2}.In that paper, the author presented a principal component based method of inferring a yield curve given an analysts' forecast of selected tenors. This method offered a solution to the problem of inferring swap spreads in an illiquid market. Below is a specific example of the adaptation of the results of that paper for our purposes.
An input into our model is a time series of daily historical swap spreads. It is important for the data to be sampled at the most liquid times in a day. In our example we use a historical database extracted from ICAP NY sampled at 4:45 PM NY time. The close is an important time with chances of a stale print for any given tenor minimized. Below is a screenshot of the data.
We calculated a time series of daily changes and it can be seen that the individual series are highly correlated. Also worth noting is the spike in volatility seen in 2008.
For our purposes we used a rolling window of 40 observations to construct a covariance matrix.
We then carried out principal component analysis and extract the eigenvalues and eigenvectors as presented below.
We can see that the first principal component explains 77% of the variation in the data. The second principal component explains an additional 11% of the variation of the data. The largest three principal components cumulatively explain 92% of the variation in the data series. Below we present a plot of the eigenvectors of the first two principal components. We can see that the first component represents a parallel move in all swap spreads while the second represents a flattening of the swap spread curve.
For our example we assume that on 25th of August 2008 NY closing levels were:
A few hours later when Asia's markets open assume that there is an active market for 5 year and 10 year swaps spreads with 41.00 and 24.75 mid. With slight variation of Nogueira's 1st Theorem we can infer the entire swap spread curve.
With
where is our implied swap spread curve, is the latest swap spread curve observed at NY close, D is a diagonal matrix of standard deviations, W is submatrix of the eigenvectors, V is a matrix that takes on 1 for each observed swaps spread tenor and 0 otherwise, and finally is the change in swap spread that we observe in the market from its NY close. In the spreadsheet that accompanies this paper we have:
is repeated here again
is simply observed change in the 5 year swap spread +.25 and the change in 10 year swap spread .50.
D is below
We only use the first two eigenvectors since we only have two observations in the market.
Applying Nogueira's 1st Theorem to our data we get the following results:
The bottom row is the implied swap spread curve based on only two observed swap spreads that are assumed to be in the market. Graphically the change looks reasonable.
We presented here a viable solution to building a USD Libor yield curve in an illiquid market. This approach was used in Asia to infer a full USD swap spread curve during the financial crisis^{3} . The implied swap spread curve was then combined with ontherun treasury curve to build a full USD Libor yield curve. This method also allowed for informed placement of bids and/or offers in the market despite poor liquidity.
^{1}For a good explanation of swap spreads and market conventions see Amir Sadr, Interest Rate Swaps and Their Derivatives, A Practitioner's Guide, Wiley Finance series, John Wiley and Sons, 2009
^{2}Nogueira, Leonardo M., Updating the Yield Curve to Analyst's Views (March 18, 2008). Available at SSRN: http://ssrn.com/abstract=1107653 or http://dx.doi.org/10.2139/ssrn.1107653
^{3} This example is a simplification of the approach used. Most important difference is the adjustment required to historical swap spreads due to new treasury issues. As a new ontherun bond is auctioned the swap spread is adjusted for the roll.
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