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Interest Rate Models

Interest Rate Models - Interest Rate Models Copyright...

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Interest Rate Models Copyright © 1996-2006 Investment Analytics
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 2 Interest Rate Models Model types, characteristics Model Taxonomy One-Factor models Vasicek Ho & Lee Hull & White Black-Derman-Toy Model Two Factor Models Fong & Vasicek Longstaff & Scwartz Hull & White Heath-Jarrow-Morton
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 3 Interest Rate Models Used to: Value derivatives, esp. non-standard Compute hedge ratios Assess portfolio risk Provides a consistent framework for valuation, hedging & risk-management
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 4 Types of Model Extensions of Black-Scholes Widely used for caps/floors (Black’s model) Models of the short term interest rate Easy to implement Many varieties, e.g. BDT, HW Models of entire yield curve Most difficult Usually simplified to two factors (e.g. HJM)
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 5 Caps, Floors & Collars Very popular instruments Great demand for caps due to increased interest rate volatility Market very liquid Used to calculate market’s view of interest rate volatility
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 6 Caps, Floors & Collars Caps: Limits Upside Risk / Gain Series of interest rate call options Caps interest rate, or equity index return Floor: Limits Downside Risk / Gain Series of interest rate put options Collar Combines Cap & Floor Fixes interest rate or equity index within a band
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 7 Interest Rate Caps Contract terms Cap strike rate, R x (7%) Term (3 years) Reset frequency (quarterly) Reference rate (LIBOR) Principal ($1MM) Payment from seller to buyer: 0.25 x $1MM x Max(LIBOR - R x , 0) In arrears, usually starts after 3 months Each piece is called a “caplet”
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 8 Collar Combines Floor(s) and Cap(s) Limits upside potential and downside risk Sale of call(s) & purchase of put(s) Premium from calls offsets cost of puts Zero Cost Collar: Special case where Put Premium = Call Premium Net cost is zero Typically used to lock in gains after market rally
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 9 Collared FRN Coupon(%) LIBOR
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 10 Black’s Model Simple extension of Black-Scholes Originally developed for commodity futures Used to value caps and floors Let F = forward price, X = strike price Value of call option: t d d t t X F d d XN d FN e C rt = + = = σ σ σ 1 2 2 1 2 1 ) 2 / ( ) / ln( )] ( ) ( [
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Copyright © 1996-2006 Investment Analytics Interest Rate Models Slide: 11 Application to Caps Example: 1-year cap NP = notional principal R j = reference rate at reset period j R x = strike rate Then, get NP x Max{R j - R x ,0} in arrears But this is an option on R j , not F j Use F j as an estimator of R j and apply Black’s model to F j Previously was a forward price, now a forward rate
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