# LIBOR Rate Model

LIBOR Rate Model Analytics

 

LIBOR Rate Model is used for pricing Libor-rate based derivative securities. The model is applied, primarily, to value instruments that settle at a Libor-rate reset point.  In order to value instruments that settle at points *intermediate* to Libor resets, we calculate the numeraire value at the settlement time by interpolating the numeraire at bracketing Libor reset points.

 

Libor rate model is very useful to price callable exotics. Many derivatives have callable features. Callable exotics are among the most challenging derivatives to price. These products are loosely defined by the provision that gives the holder or issuer the right to call the product after a lock-out period (more details at <https://finpricing.com/lib/EqCallable.html>).

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Let  denote a Libor rate that sets at time  for an accrual period .  A European caplet on  is an option with payoff at  of the form

.

Similarly, a floorlet has payoff of the form

.

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Consider a European option on a fixed-for-floating-rate swap specified by

·         forward start time, ,

·         set of future resets, , where ,

·         floating-leg payments of  at where  denotes the spot Libor at  for the accrual period .

A European *payer* swaption has payoff at  of the form

where

is the spot swap rate at time .  A European *receiver* swaption has payoff at  of the form

.

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Consider a set of future Libor reset dates, , where .  Let  denote the forward Libor rate, as seen at time , which sets at time  for the accrual period .     We seek to model Libor rates under the spot Libor measure, which has numeraire process,

,

where  denotes the price at time  of a zero coupon bond with maturity of .&#x20;

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Let  denote the integer, , such that .  Under the spot Libor measure, WM assumes that

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for , where

·           is a vector of uncorrelated, standard Brownian motions,

·          is a time deterministic volatility vector, which we define below.

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We primarily consider interest rate derivatives that depend on the set of Libor rates above, , and that settle at one of the reset times above, .  Consider, for example, a payoff, , at time .  This payoff then has value

.

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A volatility vector is of the form

.

Here

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Furthermore

where

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Here

denotes a Chebyshev polynomial of the first kind.

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In the above, the parameters  and  are determined from calibration.

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