Lecture_14 - 14 Interest Rate Exotics We study digital...

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1 14 Interest Rate Exotics We study digital options, range notes, and index-amortizing swaps. Digital options are used because they provide similar “insurance” to caps and floors, but they are cheaper. Range notes provide a partial hedge for floating rate loans with caps and floors attached. Index-amortizing swaps provide a partial hedge for the prepayment option embedded in mortgage backed securities.
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2 A Simple Interest Rates A simple interest rate of maturity T - t , denoted R(t,T) , is defined in terms of a zero-coupon bond price P(t, T) as t T 1 ) T , t ( P 1 ) T , t ( R . (14.1) Alternatively, ) t T )( T , t ( R 1 1 ) T , t ( P + = . (14.2) Unlike the other rates in this book, R(t,T) represents a percentage , i.e. it is a number between - 1 and 1.
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3 EXAMPLE: SIMPLE INTEREST RATES. Figure 14.1.
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4 Figure 14.1: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree. .923845 .942322 .961169 .980392 1 .947497 .965127 .982699 1 .937148 .957211 .978085 1 1/2 1/2 1/2 1/2 1/2 1/2 .967826 .984222 1 .960529 .980015 1 .962414 .981169 1 .953877 .976147 1 .985301 1 .981381 1 .982456 1 .977778 1 .983134 1 .978637 1 .979870 1 .974502 1 1 1 1 1 1 1 1 1 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 P(0,4) P(0,3) P(0,2) P(0,1) P(0,0) = B(0) 1 1.02 1.02 1.037958 1.037958 1.042854 1.042854 r(0) = 1.02 1.017606 1.022406 1.016031 1.020393 1.019193 1.024436 1.054597 1.054597 1.059125 1.059125 1.062869 1.062869 1.068337 1.068337 time 0 1 2 3 4
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5 Using these zero-coupon bond prices at time 0, the corresponding simple interest rates are 020608 . 4 1 923845 . 1 4 1 ) 4 , 0 ( P 1 ) 4 , 0 ( R 020403 . 3 1 942322 . 1 3 1 ) 3 , 0 ( P 1 ) 3 , 0 ( R 020200 . 2 1 961169 . 1 2 1 ) 2 , 0 ( P 1 ) 2 , 0 ( R 020000 . 1 980392 . 1 1 ) 1 , 0 ( P 1 ) 1 , 0 ( R = = = = = = = = = = = = Note that the simple interest rate of maturity one is R( 0 , 1 ) = r( 0 ) - 1 = 0 . 02. ±
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6 B Digital Options Digital options are used in the market because they provide similar “insurance” to call and put options, but they are much cheaper. A European call digital option with expiration date T and strike price k on the simple interest rate with time to maturity T* is defined by its payoff at expiration. Denote the digital's time t value as D(t) . Then the digital's time T payoff is + > + = k *) T T , T ( R if 0 k *) T T , T ( R if 1 ) T ( D (14.4) It is called a digital because the payoff is a unit (digit).
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7 Using the risk-neutral valuation procedure, the value of the digital option at time t is ). t ( B ) T ( B ) T ( D t E ~ ) t ( D = (14.5) It is interesting to rewrite expression (14.4) using expression (14.1). *) kT 1 /( 1 *) T T , T ( P * kT 1 *) T T , T ( P / 1 k *) T T , T ( R + < + + > + > + This allows us to rewrite expression (14.4) as + + + < + = *) kT 1 /( 1 *) T T , T ( P if 0 *) kT 1 /( 1 *) T T , T ( P if 1 ) T ( D ( 1 4 . 6 ) A European digital call option on the simple interest rate is equivalent to a European digital put the zero-coupon bond with maturity date T + T* .
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8 EXAMPLE: DIGITAL CALL VALUATION.
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Lecture_14 - 14 Interest Rate Exotics We study digital...

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