# Variable Rate Swap Valuation

Variable Rate Swap Analytics

 

Variable rate swap is a special type of interest rate swap in which one leg of the swap corresponds to fixed rate payments while the other involves fixed rate payments for an initial period of time and a floating rate for the rest. The floating rate on that portion is defined as a minimum of two index rates.

 

The fixed rate leg is similar to a fixed rate bond. The bond price is computed by discounting each coupon descripted at <https://finpricing.com/lib/FiBond.html>

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We treat the two index rates as assets whose values are lognormally distributed random variable. Their pricing procedure uses discount factors retrieved from the EURIBOR curve.

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The present value of the minimum of two assets, *s1* and *s2*, is given by the formula

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&#x20;                  (1)

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where

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·         ,

·         s1 and s2 are the volatilities of *s1* and *s2* respectively,

·         r is the correlation of the two risk factors,

·         *T* is the maturity time,

·          and are the expected values of *s1* and *s2* at maturity (future values),

·         *D(T)* is the discount factor implied by the yield curve.

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It is forward two index rates as retrieved from the yield curves that should be used for  and .

The minimum of two values, *s1* and *s2*, can be expressed as

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*p = min(s1,s2) = s1 – max(s1-s2, 0)*                                          (2)

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If *s1* and *s2* are two risky assets, the present value of *p* at some future time *T* is

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&#x20;                                                        (3)

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where *P(0,T)* is the discount factor implied by the yield curve. Assuming the asset values lognormally distributed, the expectation of *max(s1-s2, 0)* at maturity can be found as the value of the so called “Exchange-One-Asset-for-Another” European option (see <https://finpricing.com/lib/EqBarrier.html>)  and is given by the following equation:

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&#x20;                                          (4)

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where

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·       &#x20;

·         s1 and s2 are the volatilities of *s1* and *s2* respectively,

·         r is the correlation of the two risk factors

·         *T* is the maturity time

·          and are the expected values of *s1* and *s2* at maturity (future values).

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Combined, Eqs. (2), (3) and (4) yield

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&#x20;                        (5)

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