Chapter 32 HJM, LMM, and Multiple Zero Curves,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,1,HJM Model: Notation,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,2,Notation continued,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,3,Modeling Bond Prices (Equation 32.1, page 741),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,4,The process for F(t,T) Equation 32.4 and 32.5, page 742),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,5,Tree Evolution of Term Structure is Non-Recombining,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,6,Tree for the short rate r is non-Markov,The LIBOR Market Model,The LIBOR market model is a model constructed in terms of the forward rates underlying caplet prices,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,7,Notation,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,8,Volatility Structure,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,9,In Theory the Ls can be Determined from Cap Prices,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,10,Example 32.1 (Page 745),If Black volatilities for the first three caplets are 24%, 22%, and 20%, then L0=24.00% L1=19.80% L2=15.23%,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,11,Example 32.2 (Page 745-746),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,12,The Process for Fk in a One-Factor LIBOR Market Model,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,13,Rolling Forward Risk-Neutrality (Equation 32.12, page 746),It is often convenient to choose a world that is always FRN wrt a bond maturing at the next reset date. In this case, we can discount from ti+1 to ti at the di rate observed at time ti. The process for Fk is,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,14,The LIBOR Market Model and HJM,In the limit as the time between resets tends to zero, the LIBOR market model with rolling forward risk neutrality becomes the HJM model in the traditional risk-neutral world,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,15,Monte Carlo Implementation of LMM Model (Equation 32.14, page 746),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,16,Multifactor Versions of LMM,LMM can be extended so that there are several components to the volatility A factor analysis can be used to determine how the volatility of Fk is split into components,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,17,Ratchet Caps, Sticky Caps, and Flexi Caps,A plain vanilla cap depends only on one forward rate. Its price is not dependent on the number of factors. Ratchet caps, sticky caps, and flexi caps depend on the joint distribution of two or more forward rates. Their prices tend to increase with the number of factors,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,18,Valuing European Options in the LIBOR Market Model,There is an analytic approximation that can be used to value European swap options in the LIBOR market model.,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,19,Calibrating the LIBOR Market Model,In theory the LMM can be exactly calibrated to cap prices as described earlier In practice we proceed as for short rate models to minimize a function of the form where Ui is the market price of the ith calibrating instrument, Vi is the model price of the ith calibrating instrument and P is a function that penalizes big changes or curvature in a and s,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,20,Modeling LIBOR and OIS Term Structures Simultaneously (pages 753-755),Necessary when American swap options and other complex derivatives are valued using OIS discounting Need to ensure that LIBOR-OIS spread is positive First OIS zero curve is modeled (e.g., using a short-rate model or a LMM type of model) Then spreads are modeled as non-negative variable. An LMM type of model can be used for the evolution of forward spreads,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,21,Types of Agency Mortgage-Backed Securities (MBSs),Pass-Through Collateralized Mortgage Obligation (CMO) Interest Only (IO) Principal Only (PO),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,22,Option-Adjusted Spread (OAS),To calculate the OAS for an interest rate derivative we value it assuming that the initial yield curve is the Treasury curve + a spread We use an iterative procedure to calculate the spread that makes the derivatives model price = market price. This spread is the OAS.,