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Title

A MULTI-CURVE LIBOR MARKET MODEL WITH UNCERTAINTIES DESCRIBED BY RANDOM FIELDS

Creator

Xu, Shengqiang

Date

2012-12-19, 2012-12

Description

The LIBOR (London Interbank Offered Rate) market model has been widely used as an industry standard model for interest rates modeling and...
Show moreThe LIBOR (London Interbank Offered Rate) market model has been widely used as an industry standard model for interest rates modeling and interest rate derivatives pricing. In this thesis, a multi-curve LIBOR market model, with uncertainty described by random fields, is proposed and investigated. This new model is thus called a multi-curve random fields LIBOR market model (MRFLMM). First, the LIBOR market model is reviewed and the closed-form formulas for pricing caplets and swaptions are provided. It is extended to the case when the uncertainty terms are modeled as random fields and consequently the closed-form formulas for pricing caplets and swaptions are derived. This is a new model called the random fields LIBOR market model (RFLMM). Second, local volatility models and stochastic volatility models are combined with the RFLMM to explain the volatility skews or smiles observed in market. Closedform volatility formulas are derived via the lognormal mixture model in local volatility case, while the approximation scheme for the stochastic volatility case is obtained by a stochastic Taylor expansion method. Moreover, the above work is further extended to a multi-curve framework, where the curves for generating future forward rates and the curve for discounting cash flows are modeled distinctly but jointly. This multi-curve methodology is recently introduced lately by some pioneers to explain the inconsistency of interest rates after the 2008 credit crunch. Both LIBOR market model and RFLMM mentioned above can be categorized as models in singe-curve framework. Third, analogous to the single-curve framework, the multi-curve random fields LIBOR market model is derived and caplets and swaptions are priced with closedform formulas that can be reduced to exactly the Black’s formulas. This model is called a multi-curve random fields LIBOR market model (MRFLMM). Meanwhile, xii local volatility and stochastic volatility models are also combined with the multi-curve LIBOR market model to explain the volatility skews and smiles in the market. Fourth, the calibration of the above models is considered. Taking two-curve setting as an example, four different models, single-curve LIBOR market model, single-curve RFLMM, two-curve LIBOR market model and two-curve RFLMM are compared. The calibration is based on the spot market data on one trading day. The four models are calibrated to European cap volatility surface and swaption volatilities, given the specified parameterized form of correlation and instantaneous volatility. The calibration results show that the random fields models capture the volatility smiles better than non-random fields models and has less pricing error. Moreover, multi-curve models perform better than single-curve models, especially during/after credit crunch. Finally, the estimation of these four models, including pricing and hedging performance, is considered. The estimation uses time series of forward rates in market. Given a time series of term structure, the parameters of the four models are estimated using unscented Kalman filter (UKF). The results show that the random fields models have better estimation results than non-random fields models, with more accurate in-sample and out-sample pricing and better hedging performance. The multi-curve models also over-perform the single-curve models. In addition, it is shown theoretically and empirically that the random fields models have advantages that it is unnecessary to determine the number of factors in advance and not needed to re-calibrate. The multi-curve random fields LIBOR market model has the advantages of both multi-curve framework and random fields setting.
PH.D in Applied Mathematics, December 2012
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