Recent Advances in Mathematical Finance Chicago, December 8, 2007 Yevgeny Goncharov

1. Prepayment and Mortgage Rate Modeling Recent Advances in Mathematical Finance Chicago, December 8, 2007 Yevgeny Goncharov Department of Mathematics Florida State University goncharov @ math.fsu.edu partially supported by NSF grant DMS-0703849

2. ? ? ? ? ? ? ? ? ? ? ? ? prepayment default curtailment Mortgage Securities Pooled cash flow: interest + principal payments Rule for distribution of cash flow A B C Investors

3. P0 at time “0” cash flow up to time A Mortgage Borrower Lender $ • m – mortgage rate • P(t)– outstanding principal • gt– Ft -intensity of prep. (“prepayment rate”) • c – payment rate ($/time) • t – prepayment time • Ft – “relavant” information

4. Prepayment: Empirical or Model-based or refinancing incentive market factors measure of borrower’s reaction Prepayment (Intensity) Specification Note: the incentive translates “market” to “resident” money-language

5. Intensity functiong (.) (“Low frequency”) Refinancing incentivePt (“High frequency”) Estimates the borrower’s “response” (in probabilistic terms) to certain market situations. Estimates “usefulness” of the refinancing from the borrower’s point of view. Based on the “market” information (interest, unemployment rates, home prices). Prepaymentor Intensity Modeling

6. Why to Model? sensitive to time Empirical time Modeling not sensitive to time

7. Mortgage Modeling Calibration Numerical implementation Mortgage Model

8. Mortgage Model Classification • Data, calibration • Computational Method • Interest Rate Model, • Prepayment intensity function • Additional predictors • (house prices, media effect, etc) • Refinancing incentive Not mortgage-specific. “Standard” problems Statistics 1. Mort-Rate-Based Mortgage Model = Ref. Inc. 2. Option-Based

9. The rate mt implied by the prepayment process gt: Implied Mortgage Rate Process

10. 1. The processthe 10-year Treasury yield+const. 3. Endogenous mortgage rate : Pliska/Goncharov MOATS Mortgage-Rate-Based Approaches

11. A Simple Example

13. MRB Mortgage Rate

14. Option-Based Mortgage Rate

15. T T T Citigroup: • T=360 (30 yr), Tmoats=720 (60yr) • comlpexity: (361*360/2+360*360)*N*I=194,580*N*I • Interest only? One factor only? • Historical dependence dropped, “calibrated” later… Citigroup’s MOATS(generalized) 0 Tmoats T 0 Tmoats

16. MOATS mortgage rates time to Tmoats (quarterly) interest rate

17. MOATS convergence MOATS mortgage rates time to Tmoats (quarterly)

18. MOATS convergence MOATS mortgage rates interest rate

19. MOATS convergence (interest only) MOATS mortgage rates interest rate

20. Endogenous Mort Rate Iteration • mi+1() requires estimation of L() for “every” r? • The result of L()-estimation is used at x=r only, • other values discarded? • Curse of dimensionality with growth of r-dimension?

21. r0 r0 r1 r1 r2 r2 r3 r3 Fix m then solve for r(m): Fix m then solve for r(m): the same! Refinancing region controlled Mortgage Rate “Iterations” r4 r0 r1 r2 r3 Fix r then solve for m(r): the same…

22. No need for iterations if • The conditional expectation in L() • is used on a hypersurface (level set), • i.e., “waste of one dimension” only • Number of L()-estimations is • independent of the dimension/number • of the underlying factors Computation with Level Sets r0 r1 r2 r3 r

23. Conclusion • Endogenous mortgage rate is defined • far from or implied by 10yr Treasury yield • accented nonlinear behavior • MOATS • transparent definition efficient implementation • convergence to MRB is shown • A general ‘level set’ method is proposed • flexibility of implementation: [RQ]MC or PDE • reduces/eliminates the burden of iterations • complexity of the same order as the underlying problem • efficient and simple for the computation of implied mortgage rate given any prepayment model