# CMS Spread Option Valuation

CMS Spread Option Analytics

 

A constant maturity swap (CMS) spread option makes payments based on a bounded spread between two index rates (e.g., a GBP CMS rate and a EURO CMS rate).  The GBP CMS rate is calculated from a 15 year swap with semi-annual, upfront payments, while the EURO CMS rate is based on a 15 year swap with annual, upfront payments.

 

Let

·          be a CMS rate fixing time,

·          be a payment time,

·          and  respectively denote the GBP and EURO CMS rates at the fixing time,

·          be a CMS rate spread.

 

Then, at time , the buyer must pay a spread, , bounded below and above by 0% and 10% respectively, multiplied by the accrual period, ;  formally the payoff is given by 

 

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We assume that both the forward GBP and EURO CMS rates follow geometric Brownian motion under their respective -forward measures.  Here respective initial forward CMS rates are calculated.  The forward rates are then convexity adjusted from respective parallel bonds specified using (see <https://finpricing.com/lib/IrCurveIntroduction.html>)

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·         the same payment times as the underlying CMS,

·         coupon equal to the forward CMS rate,

·         initial bond yield equal to the forward CMS rate.

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Let

·          be a GBP swap rate at time ,

·          be a EURO swap rate at time t,

·          be a spread between the two swap rates above.

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Then

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Furthermore, since ,

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The buyer must pay  at time  where  (with )  is an accrual period. Let

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·          denote the domestic  -forward probability measure,

·          denote the domestic -forward probability measure,

·          denote the domestic risk-neutral probability measure.

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Then

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where  is the price at time  of a zero-coupon bond that matures at time  with face value of 1 Euro.

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We assume that the forward CMS rate processes,  and , satisfy, under domestic T-forward measure, the respective SDEs

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,

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where

·          and  are respective constant volatilities,

·          and  are standard Brownian motions with instantaneous correlation, .

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Here  is determined from respective convexity and quanto adjustments, while  is determined from a convexity adjustment.

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Let  satisfy

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Observe that, under domestic -forward probability measure,

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so that  is a martingale under this measure. We assume that the process  satisfies, under domestic T-forward measure, the SDE

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where

·          is a constant volatility,

·          is a standard Brownian motion with respective pairwise correlation to  and ,  and ,

·         .

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Assuming that  satisfies the SDE above, however, the martingale property above does not hold.

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In order to satisfy the martingale condition

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we convexity adjust the initial forward bond yield, .

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Observe that

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where

·          is the spot exchange rate from foreign (GBP) currency into domestic (EURO) currency,

·          is the forward exchange rate, at seen at time , from foreign currency into domestic, at the forward time .

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From the above the correlation, under foreign T-forward measure, between the Brownian motions respectively driving the foreign bond yield and GBP swap rate processes constrains the correlation between the Brownian motions driving the Euro bond yield and forward GBP CMS rate processes under domestic T-forward measure. &#x20;

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To evaluate the expression above, we require the SDE followed by the forward exchange rate process under domestic T-forward measure.  This is derived by changing measure from foreign -forward measure, to domestic risk-neutral measure, to domestic -forward measure.

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