LIBOR Market Models with Stochastic Basis. Swissquote Conference on Interest Rate and Credit Risk 28 October 2010, EPFL.

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1 LIBOR Market Models with Stochastic Basis Swissquote Conference on Interest Rate and Credit Risk 28 October 2010, EPFL Fabio Mercurio, Discussant: Paul Schneider 28 October, 2010 Paul Schneider 1/11

2 II Extension of LMM model accounting for spread-widening between discount rates and LIBOR rates Related papers Schoenbucher (1999) Mercurio (2009) Mercurio (2010) Formulae for IRS, caps, swaptions and guideline modeling different tenors simultaneously Very tractable application using SABR and shifted Lognormal models Fit to data is extraordinarily good Paul Schneider 2/11

3 II The framework models OIS forward rates Fk x (t) and spreads (t) as independent processes S x k (t) := Lx k (t) F x k But F and S are dependent by construction Also conditional variance of the two is likely related mx12m FRA - 6mx12m EONIA /01/06 01/01/07 01/01/08 01/01/09 01/01/10 01/01/11 s Paul Schneider 3/11

4 Linear dependence regression F = θ 0 + θ 1 S + ε Coefficient Est. Prob. θ θ II Variance dependence regression F 2 = η 0 + η 1 S 2 + ε Coefficient Est. Prob. η η Paul Schneider 4/11

5 Model Suppose that F and L are both driven by the same variance factor II d lnf x k (t) = 1 2 V k(t)dt + V k (t)dz F k (t) d lns x k (t) = 1 2 V k(t)dt + V k (t)dz S k (t) dv k (t) = (b + βv k (t))dt + αv k (t)dz V k (t). They are instantaneously uncorrelated. For fixed time T > 0 correlation proportional to V Note that conditional on IV k (t, T) := T t V k (s)ds log F x k (T x k 1 ) IV k(t, T x k 1 ) and log Sx k (T x k 1 ) IV k(t, T x k 1 ) are independently normally distributed Paul Schneider 5/11

6 Caplet Prices with Model Condition on IV k (t, T x k 1 ) instead of Sx k (T x k 1 ) Cplt(t, K; Tk 1 x, T k x ) = τx k P D(t, Tk x [ [ ) E T k x D E T k x [L ] x D k (Tk 1 x ) K] + Ft IV k (t, Tk 1 x ) F t ] II Inner expectation is an integration against Lognormal convolution f(z) := E T x k D [ [L x k (T x k 1 ) K] + Ft IV k (t, T x k 1 ) = z ] Denote with g(z) the conditional density of IV k (t, T x k 1 ) V k(t). We need to solve 0 f(z)g(z)dz Paul Schneider 6/11

7 Closed-form II Approximate the conditional density of IV k (T) V k (t) using Filipović, Mayerhofer, and Schneider (2010) likelihood expansions Generate affine Markov process through embedding V k (t) (V k (t), t 0 V k(s)ds) =: IV k (t)) dv k (t) = (b + βv k (t))dt + αv k (t)dz V k (t) div k (t) = V k (t)dt This process is polynomial and polynomial moments can be computed in closed-form Approximate the marginal distribution of IV k (T) V k (t) through polynomial expansion in a weighted L 2 space Expansion performs very accurately. Paul Schneider 7/11

8 of Correlated Spread Model II The Picture shows the percentage deviation of the conditional expectation obtained from true density (Fourier inversion) and Filipović, Mayerhofer, and Schneider (2010) expansion log 0 f(z)g(z)dz log 0 f(z)g (12) (z)dz 0-1e-05-2e-05-3e-05-4e-05-5e-05-6e K Paul Schneider 8/11

9 II Praise Easy to use LMM adapted to current economic environment Formulae for IRS, caps, swaptions Guideline for modeling different tenors simultaneously Future Topics Where does the spread come from? How can we make the model more realistic while maintaining tractability? Paul Schneider 9/11

10 II Filipović, D., Mayerhofer, E., and Schneider, P. (2010), Density Approximations for Multivariate Affine Jump-Diffusion Processes. working paper Mercurio, F. (2009), Interest Rates and The Credit Crunch: New Formulas and Market Models. working paper Mercurio, F. (2010), Modern Libor Market Models: Using Different Curves for Projecting Rates and for Discounting. International Journal of Theoretical and Applied Finance, 13(1): Schoenbucher, F. (2000), A Libor Market Model with Default Risk. working paper Paul Schneider 10/11

11 Model Revisited II Consider d lnfk x (t) = 1 2 V k(t)dt + V k (t) (ρdz k V (t) + ) 1 ρ 2 dzk F (t) d lnsk x (t) = 1 2 V k(t)dt + V k (t) (ηdz k V (t) + ) 1 η 2 dzk S (t) dv k (t) = (b + βv k (t))dt + αv k (t)dzk V (t). Since T t Vk (s)dz V k (s) = b(t t) + βiv k(t) + V k (t) V k (T) α, lnf(t) lnf(t) V (T), IV (T) N ( 12 T ) IV (T) + ρ V (s)dz V (s), (1 ρ 2 )IV (T) By approximating V k (T), IV k (T) V k (t) we could also induce instantaneous correlation t Paul Schneider 11/11

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