# Research seminar by Leonid Timoshouk (UBS Bank): A new non-linear method for construction of yield curves

Tomorrow, September 8 at 4.30 pm International College of Economics and Finance held Research seminar.

Venue: Pokrovski Bulvar, 11, Room Zh-822

Speaker: Leonid Timoshouk (UBS Bank)

Theme: «A new non-linear method for construction of yield curves»

**Absract:**

**The classical bootstrapping process for yield curves works as follows:**

- • a set of standard yield curve tenors is constructed, e.g. from overnight to 20y or so;
- • one instrument per tenor (usually the most liquid instrument) is identified, along with its current market price;
- • the bootstrapping process itself is run, i.e. the forward rate values are constructed iteratively from the lowest tenor to the highest one; for each tenor, the corresponding forward rate value is obtained by solving the pricing equation for that instrument against the market price, with all values but the last one already known from the previous iteration steps;
- • to obtain the forward rates (and hence the discount factors) for time instants in between the standard tenors, interpolation is used.

**However, in spite of being classical, the bootstrapping method has multiple disadvantages:**

- • It mixes up instrument of different credit risk standing (different collateralisation types). E.g., fwd rates for tenors under 1y are typically derived from OIS (over-night interest swap) rates, whereas for tenors over 1y, IRS (interest rate swap) rates are used, based e.g. on 3m LIBOR rates. Before the crisis of 2007-2010, the OIS - 3mLIBOR spread was small and absolutely constant (about 10bp), but this is not the case anymore. The problem is not really solved by building a backbone LIBOR-based forwardcurve and then applying a negative spread ("cost of funding") to get a better-collateralised curvebecause this method is quite unsystematic.
- • It does not allow us to make use of all available market information, because it is restricted to one instrument per tenor. E.g., for tenors under 1y, we may have a number of different instruments of about the same liquidity level -- OIS rates, forward rate agreements (FRAs), deposit rates, government bond futures, to name a few. But only one instrument per tenor could be used, all other market data are disregarded.
- • The forward rates in between the tenor points depend on the interpolation method used. There have been lengthy discussions among academics and practitioners as to which interpolation method would be "the best", but they are of course inconclusive: there are simply no data in between the nodes constructed.

**In response to these shortcomings, a completely new method for building forward curves has recently been developed. The main features of this method can be summarised as follows:**

- • A hierarchy of forward curves is constructed instead of just one curve. Each curve in the hierarchy corresponds to each own collateralisation type. For eample, the root of the hierarchy typically corresponds to zero credit risk and is used for discounting of all cash flows in the calibration instruments. Other collateralisation types include LIBOR etc. However, the trader can arrange the hierarchy any way they like and/or add extra collateralisation types (e.g., for non-USD ccys, ccy equivalents of USD curves can be added and calibrated against cross-ccy swaps). There is no need for a separate "cost of funding" anymore.
- • Each forward curve to be constructed is represented as a sequence of linear segments; the time intervals of these segments can be as short as necessary, they do not need to correspond to standard tenors. As a result, once the curves are constructed, they can be used directly, without any interpolation. The linear coefficients of all curves are the N unknowns to be found. (Technical details are omitted).
- • "Spikes" in LIBOR rates at year turns are automatically corrected.
- • Any number (M) of calibration instruments can be put in, providing the equations to be solved. It is possible to have N > M (the usual case), N = M, or N < M. Virtually all kinds of instruments can be used, and they may have overlapping tenors without any restrictions. For JPY it is even possible to use both LIBOR- and TIBOR-derived prices at the same time. The spec for each calibration instrument contains explicit references to the collateralisation types and the index (e.g. LIBOR/TIBOR) in depends upon.
- • As follows from the above setup, the forward curve parameters are to be derived via a constrained optimisation procedure. The constraints are market prices, and the cost function of this procedure is user-configurable: it allows the trader to put in their educated guess of the next Central Bank rate moves, the requirements for curves continuity or (otherwise) jumps, etc.
- • It is even possible to also use illiquid market instruments. Their prices are not to be matched directly -- the degree of illiquidity serves as a measure of tolerance in matching the market prices (higher illiquidity --> higher tolerance).

**Numerical implementation of the above method includes:**

- • A highly accurate numerical method for solving the constrained optimisation problem, with a large radius of convergence and a built-in error detection (i.e., the solution procedure either hits a detectable singular point or converges to the true solution).
- • Parallelisation of the solution method.