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期权期货及其衍生产品官方课件.pptx

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1、Chapter 31Interest Rate Derivatives: HJM and LMM,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,1,HJM Model: Notation,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,2,Notation continued,Options, Futures, and Other Derivatives, 8th

2、 Edition, Copyright John C. Hull 2012,3,Modeling Bond Prices (Equation 31.1, page 716),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,4,The process for F(t,T)Equation 31.4 and 31.5, page 717),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. H

3、ull 2012,5,Tree Evolution of Term Structure is Non-Recombining,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,6,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 cap

4、let prices,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,7,Notation,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,8,Volatility Structure,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,9,In Theo

5、ry the Ls can be Determined from Cap Prices,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,10,Example 31.1 (Page 720),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, 8t

6、h Edition, Copyright John C. Hull 2012,11,Example 31.2 (Page 720-21),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,12,The Process for Fk in a One-Factor LIBOR Market Model,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,13,Rolling

7、 Forward Risk-Neutrality (Equation 31.12, page 721),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,

8、 8th Edition, Copyright John C. Hull 2012,14,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, 8th Edit

9、ion, Copyright John C. Hull 2012,15,Monte Carlo Implementation of LMM Model (Equation 31.14, page 721),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,16,Multifactor Versions of LMM,LMM can be extended so that there are several components to the volatilityA factor a

10、nalysis can be used to determine how the volatility of Fk is split into components,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,17,Ratchet Caps, Sticky Caps, and Flexi Caps,A plain vanilla cap depends only on one forward rate. Its price is not dependent on the nu

11、mber 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, 8th Edition, Copyright John C. Hull 2012,18,Valuing European Options in the LIBOR

12、 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, 8th Edition, Copyright John C. Hull 2012,19,Calibrating the LIBOR Market Model,In theory the LMM can be exactly calibrated to cap price

13、s as described earlierIn practice we proceed as for short rate models to minimize a function of the formwhere 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,Optio

14、ns, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,20,Types of Agency Mortgage-Backed Securities (MBSs),Pass-ThroughCollateralized Mortgage Obligation (CMO)Interest Only (IO)Principal Only (PO),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 201

15、2,21,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 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, 8th Edition, Copyright John C. Hull 2012,22,

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