HJM, LMM, and Multiple Zero Curves

Chapter 32HJM, LMM, and Multiple Zero CurvesOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201411HJM Model: Notation

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142 P(t,T ): price at time t of a discount bond with principal of $1 maturing at TWt :vector of past and present values of interest rates and bond prices at time t that are relevant for determining bond price volatilities at that timev(t,T,Wt ): volatility of P(t,T)2Notation continued

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20143(t,T1,T2): forward rate as seen at t for the period between T1 and T2F(t,T): instantaneous forward rate as seen at t for a contract maturing at Tr(t): short-term risk-free interest rate at tdz(t): Wiener process driving term structure movements3Modeling Bond Prices (Equation 32.1, page 741)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20144

4The process for F(t,T)Equation 32.4 and 32.5, page 742)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20145

5Tree Evolution of Term Structure is Non-Recombining Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20146Tree for the short rate r is non-Markov6The LIBOR Market Model The LIBOR market model is a model constructed in terms of the forward rates underlying caplet pricesOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201477NotationOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20148

8Volatility StructureOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20149

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In Theory the Ls can be Determined from Cap PricesOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201410

10Example 32.1 (Page 745)If Black volatilities for the first threecaplets are 24%, 22%, and 20%, thenL0=24.00%L1=19.80%L2=15.23%

Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141111Example 32.2 (Page 745-746)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201412

12The Process for Fk in a One-Factor LIBOR Market ModelOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201413

13Rolling 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 201414

14The LIBOR Market Model and HJMIn 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 worldOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141515Monte Carlo Implementation of LMM Model (Equation 32.14, page 746)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 201416

16Multifactor Versions of LMMLMM can be extended so that there are several components to the volatilityA factor analysis can be used to determine how the volatility of Fk is split into componentsOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141717Ratchet Caps, Sticky Caps, and Flexi CapsA 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 factorsOptions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20141818Valuing European Options in the LIBOR Market ModelThere 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 20141919Calibrating the LIBOR Market ModelIn theory the LMM can be exactly calibrated to cap prices as described earlierIn 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 201420

20Modeling LIBOR and OIS Term Structures Simultaneously (pages 753-755)Necessary when American swap options and other complex derivatives are valued using OIS discountingNeed to ensure that LIBOR-OIS spread is positiveFirst 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 201421Types of Agency Mortgage-Backed Securities (MBSs)Pass-ThroughCollateralized Mortgage Obligation (CMO)Interest Only (IO)Principal Only (PO)Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142222Option-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 spreadWe use an iterative procedure to calculate the spread that makes the derivatives model price = market price.This spread is the OAS.Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 20142323n12345

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