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2.2.4 Bond options: caps, floors and swaptions $ \diamondsuit $

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In the same manner as stock market derivatives have been introduced in section 2.1.3 for an underlying share, different types of credit market derivatives confer the holder a right to buy or sell the earnings from interest rates. The simplest is the bond option, which confers its holder the right to buy or sell an underlying discount bond with a maturity TB   that is necessarily longer than the option expiry T<TB.   Using the notation VB(r,t,TB)   for the value of the discount bond before the maturity (something that we will calculate later in chapter 5) the terminal conditions for plain vanilla call or put options are simply defined by
$\displaystyle \Lambda_\mathrm{call-bond}(r,T)=\max(V_B(r,T,T_B)-K,0)$     (2.2.4#eq.1)
$\displaystyle \Lambda_\mathrm{put-bond}(r,T)=\max(K-V_B(r,T,T_B),0)$      

Directly related is the interest rate cap (alt. floor), which can be understood as a form of insurance against underlying floating rates moving above (alt. below) a certain level. Imagine a loan for an amount A, where the a priori unknown floating rate ri=r(ti,ti + ti)   resets to LIBOR at the end of every period and leads to payments float(ti+1=Ari+1t   To guarantee that these payments do not exceed (alt. drop below) a certain level, the loan can be supplemented with a cap (alt. floor), which is itself composed of a series of caplets (alt. floorlets) having all the same cap-/floor rate K,   and paying the difference when the floating rate moves beyond:
$\displaystyle \Lambda_\mathrm{floorlet}(t_{i+1})=$   $\displaystyle \mathrm{max}(K-r_i,0)\tau_i$ (2.2.4#eq.2)
$\displaystyle \Lambda_\mathrm{caplet}(t_{i+1}) =$   $\displaystyle \mathrm{max}(r_i-K,0)\tau_i$  

No-arbitrage arguments again show that the a priori unknown future values of the rates resetting at the end of the time interval have to be equal to the projected forward rates $ r_i=F(0,t_i,t_{i+1})$ . Exercise 2.11 shows that a floorlet is closely related with a call option on a discount bond and a caplet is closely related with the corresponding put option. By holding a cap and shorting a floor with different rates Kfloor < Kcap   you get a collar, which guarantees that the interest rate remain within a pre-determined interval. Using the same rates Kfloor = Kcap   guarantees the payment of a fixed rate leading to the cap-floor parity

$\displaystyle \mathrm{cap} - \mathrm{floor} = \mathrm{swap}$ (2.2.4#eq.3)

Options can also be defined on credit derivatives: a European swap option or swaption carries the right to enter a swap (i.e. switch from variable to fixed interest rates) at a predetermined rate K':  
$\displaystyle \Lambda_\mathrm{payer-swaption}(t_{i+1}) =$   $\displaystyle \mathrm{max}(K-K^\prime,0)B$ (2.2.4#eq.4)
$\displaystyle \Lambda_\mathrm{reciever-swaption}(t_{i+1})=$   $\displaystyle \mathrm{max}(K^\prime-K,0)B$  

where B=k P(ti,ti+k) tk   Options on caps-/floors can be defined in the same manner as above and are called captions and floortions.

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