An arrear quanto constant-maturity-swap (CMS) is a swap that pays coupons in a different currency from the notional and in arrears. The underlying swap rate is computed from a forward starting CMS

Tim Xiao

Humanities Commons

Arrear Quanto CMS Model  An arrear quanto constant-maturity-swap (CMS) is a swap that pays coupons in a different currency from the notional and in arrears. The underlying swap rate is computed from a forward starting CMS.  Assumes that, under the coupon payment currency (SEK) risk-neutral probability measure, the forward swap rate process corresponding to each swap rate fixing follows Geometric Brownian motion with drift.  Each forward swap rate process is then convexity adjusted, and is furthermore expressed under the notional currency (FRF) risk neutral-probability measure by means of a quanto adjustment.  We assume that, under the SEK risk-neutral probability measure, the forward swap rate process follows Geometric Brownian motion with drift. The initial forward swap rate is calculated, and is then convexity adjusted.  The initial forward swap rate is also quanto adjusted. We note that the correlation used in the spreadsheet is between the FRF to SEK exchange rate and the SEK swap rate.  Finally the respective quanto and convexity adjusted initial forward swap rates are added together to produce the initial forward swap rate under the FRF risk-neutral probability measure.   Let the observation times Ntt ,,0  , correspond to consecutive quarterly resets, where 0t  corresponds to the start date, and Nt  corresponds to the maturity date.  Then the seller pays it%95.3   at time it , for Ni ,...,1= ,  where 1−−= iii ttt .    At time it , where Ni ,...,1= , we consider a three year CMS, which begins at time it  and has three payment times, jt i + ,  where 3,...,1=j .  Here the floating  side pays the SEKSTIBOR rate, 1−+ jtiL  at time jt i + , where 3,...,1=j .  The fixed side pays a constant amount at time jt i + , where 3,...,1=j .   The swap rate at time it , which we denote by its , is the constant fixed payment amount that gives the CMS zero value at time it ; that is, ( )=++−=31,)3,(1jiiSEKiiSEKtjttPttPsi where ( )TtPSEK ,  denotes the price at time t  of a Swedish zero coupon bond maturing at time T .    From the above, the holder receives itist99.0  at time it , for Ni ,...,1= .  The value of our swap at time zero then equals ( ) ( ) ( )− ==iFRFNiiiFRFtNii tPttPsEt i ,0%95.3  ,099.0  Notional11 where • ( )tPFRF ,0  is the price at time zero of a FRF zero coupon bond maturing at time t ,  • E  denotes the FRF risk-neutral probability measure, and • the swap notional is denominated in FRF.  Here we have assumed that the FRF short-term interest rate is deterministic.  We note that the common currency unit in Europe is now taken to be the EURO.  Furthermore, the exchange rate from the EURO to an associated currency (e.g., FRF) is fixed, so there is no foreign exchange risk.  Therefore, FP London uses a common curve, EURIBOR, for discounting; that is, ( )tPFRF ,0  is replaced by the equivalent discount factor ( )tPEUR ,0 , which is the price at time zero of EURO denominated zero coupon bond with maturity of t .  Let  ity , for Ni ,...,1= , denote the forward swap rate at time t  for a forward starting SEK CMS, which begins at time it  and has payments at times jt i +  where 3,...,1=j .  FP assumes that, under the SEK risk-neutral probability measure, the process   iit tty ,0   satisfies a stochastic differential equation (SDE) of the form titiit dBydy =  where  •  0 tBt  is standard Brownian motion, and •  i  is the volatility. Recall that the swap pricing formula,  ( ) ( ) ( )− ==iFRFNiiiFRFtNii tPttPsEt i ,0%95.3  ,099.0  Notional11, requires the expected swap rate, ( )itsE ,      (A.1) for Ni ,...,1= .   Since ii tit sy = , FP’s approach towards computing (A.1) is to convexity adjust the initial forward swap rate, iy0 .    To this end let  ( ) ( )331 111);(YYccYbondii+++== be the price of a bond, with three year maturity, where • c  is an annually paid coupon value, and • Y  is an annualized yield-to-maturity. FP’s convexity adjusted rate is then given by ( ) ( )( )( )YyybondYyybondeyyyiiiitiii i ;  ; 121ˆ00200220002−−= ,  The yield to maturity of a bond is the internal rate of return on a bond held until maturity. In other words, it is the discount rate that will provide the investor with a present value V equal to the price of the bond. The yield to maturity does not account for the actual term structure of interest rates: https://finpricing.com/lib/FiZeroBond.html  We wish to express the process   iit tty ,0   under the FRF risk-neutral probability measure.  Let SEKtr  denote the SEK short-term interest rate.  Assume that, under the SEK risk-neutral probability measure, the process  0 tr SEKt  satisfies a SDE of the form ( ) ( ) tSEKtSEKt dBtbdttradr += ,  where ),( tra  and )(tb  are deterministic, sufficiently regular functions.  Let tX  denote the exchange rate from one SEK monetary unit to FRF.  Furthermore assume that, under the FRF risk-neutral probability measure, the process  0 tX t  satisfies a SDE of the form  ( )XtXSEKtFRFttt dWdtrrXdX +−=  where  • FRFtr  is the FRF short-term interest rate, which we assume to be deterministic, • X  is the volatility, and •  0 tW Xt  is standard Brownian motion. Then under the FRF risk-neutral probability measure, the process  0 tr SEKt  satisfies the SDE ( )  ( ) tXSEKtSEKt dWtbdttbtradr +−= )(,   where •  0 tWt  is standard Brownian motion, and •   is the constant instantaneous correlation coefficient between  0 tW Xt  and  0 tWt .  Observe that, under the SEK risk-neutral probability measure, the forward swap rate process,   iit tty ,0  , is driven by the same Brownian motion,  0 tBt , as the short-term interest rate process,  0 tr SEKt .  Then, under the FRF risk-neutral probability measure, the process   iit tty ,0   satisfies the SDE ( )tiiXitit dWdtydy  +−= .   

Arrear Quanto CMS Model

https://hcommons.org/deposits/download/hc:47734/CONTENT/arrearquantocms.pdf/

Arrear Quanto CMS Model

Abstract

Similar works

Full text

Available Versions

Humanities Commons