Monte Carlo Multi-factor Short Rate Model

The Monte Carlo Multi-factor Short Rate Mode has been used extensively in pricing a variety of interest rate derivative securities. The model assumes that short rates at reset times are lognormally

Monte Carlo Multi-factor Short Rate Model

The Monte Carlo Multi-factor Short Rate Mode has been used extensively in pricing a variety of interest rate derivative securities. The model assumes that short rates at reset times are lognormally distributed.

Let I = [0,T] , where T > 0, be an interval of time and let Dt = T/N, where N is a positive

integer, be an accrual period. We consider a model for N , Dt - period short rates.

Specifically let 0 0 = t < < t = T N 􀀀 , where t i T i = D , be an equally spaced partition of I .

We are interested in the Dt - period short rate at time equal to ti , which we denote by

ri , for i = 1, ... , N .

The FP short rate model derives from an underlying forward rate model. In particular let

f ti , for i = 1,..., N , denote a Dt - period forward rate, beginning at time equal to ti , as

seen at time equal to t . Also let {W_ t I} ti Î , for i = 1, ... , N , denote standard Brownian

motion under a probability measure Q ; furthermore assume that the Brownian motions

{W_ t I} ti Î and {W_ t I} tj Î , for i, j = 1,..., N , have instantaneous constant correlation

equal to _rij . We assume that the process {f t t } ti Î[0, ] , for i = 1, ... , N , satisfies a

stochastic differential equation (SDE) of the form

where _mti and _sti denote piecewise constant drift and volatility terms (i.e., constant over

each subinterval, [t , t ) j-1 j , for j = 1, ... ,i ). We then set r f i ti = (i = 1,..., N ).

Notice that ri is equal to

where f i0 is the Dt - period forward rate beginning time equal to ti , as seen at time equal to zero, Z i is a standard normal random variable, . For consistency, we re-write (2) as the equivalent expression

FP assumes that short rates are lognormally distributed, of the form (3), under the probability measure Q . The model includes as input the initial term structure of forward rates (i.e., f i

0 , for i N = 1,..., ). The volatilities si , for i = 1,..., N , in (3) are also supplied as inputs. Furthermore correlations for Z i and Z i+1 ( i = 1, ... , N ) are input and then combined with the volatilities si ( i = 1, ... , N ), to construct a covariance matrix (under the measure Q ) for the random vector [s s ] 1W1 W NN T ,..., .

Let P( j, i) , for i = 1,..., N and j = 0, ...i , denote the price at time equal to t j of a bond with face value $1 at time equal to ti . Also let {B i N} i | = 0, ... , , where D , denote our money-market, numeraire process under Q . Then the Monte Carlo construction scales the short rate (3), for i = 1,..., N , so that all zero coupon bond prices, as seen at time equal to zero, are repriced, that is, P i E. Furthermore, because this is a short rate model, the martingale (no-arbitrage) condition is then automatically satisfied under Q (where Fi denotes the sigma algebra induced by Wi ,for i = 1,..., N ).

Note that

· all options are of European exercise type,

· by the correlation between adjacent LIBOR rates ri and ri+1 ( i = 1, ... , N -1), we

mean the correlation between the standard normal random variables Z i and Z i+1

described in Section 2, and,

· by the volatility for LIBOR rate ri , we mean the value for the parameter si

In our terminology below, the payoff for a caplet with tenor ti (i = 1,..., N ) is defined

equal to

where ri denotes the spot rate at time equal to ti and X denotes the strike; furthermore the payoff is received at time equal to ti+1 . Caplets were specified based on the following parameters :

· short rate equal to three month LIBOR (see https://finpricing.com/lib/FxForwardCurve.html),

· tenor equal to three months, one year and three years,

· initial forward rate term structure set

· constant at 7%, 7.5% and 8%, and

· linearly upward rising , initially at 7% , with .0025 increments every three

months,

· strike equal to 7%, 8% and 9%,

· LIBOR rate volatility set constant at 10%, 25% and 50%, and

· adjacent LIBOR rate correlation constant in the range of 90% to 99% inclusive.

Caplets were benchmarked using Black’s model, that is, with constant volatility and with

fixed discounting based on the initial term structure of forward rates; FP prices were

based on 10,000 Monte Carlo paths. Numerical test results showed relative differences

between the benchmark and FP model

· not greater than 1% for high (50%) volatility,

· not greater than .1% for low (10%) and medium (25%)volatility.