20.060.02B0.100.750.100.05C0.050.100.750.10D0.000.000.001.00In order to value a credit derivative, we need risk-neutral transition probabilities which may be obtained from the prevailing bond prices (refer Table-2). Let p(M)ij be the risk neutral probability of transition from rating i to rating j over M periods. We shall show how p(1)Aj are to be calculated.If the price of a 1-period zero-coupon bond with a rating A is V(1)A , the following relationship should hold: (1)where r01 = 15% ( the current one period rate)Table-2Prices of Zero coupon BondsMaturity Rating1234A85.8674.1064.5955.32B84.0971.3261.3051.91C80.9666.7656.0646.55In order to estimate p(1)Aj we assume that the risk-neutral probabilities of transition from state A to other states are proportional to empirically observed probabilities. p(1)AB = p(1)A x 0.12p(1)AC = p(1)A x 0.06p(1)AD = p(1)A x 0.02p(1)AA = 1 - p(1)A x 0.2(2)Equations (1) and (2) are used to obtain p(1)A and hence p(1)Aj (3)From eqn.(3), we get p(1)A = 1.05 and hence the risk neutral probabilities p1Aj are p(1)AA = 0.79 p(1)AB = 0.126p(1)AC = 0.063p(1)AD = 0.021The same methodology may be used to obtain p(1)Bj and p(1)Cj and thus the risk neutral transition matrix for 1 period is obtained (refer Table-3).Table-3Risk neutral probabilities of transition over 1 periodRating after 1 periodCurrent RatingABCDA0.7900.1260.0630.021B0.1100.7250.1100.055C0.0580.1150.7130.115D0.0000.0000.0001.0004.2Derivation of M-period risk-neutral transition matrixThe M-period risk-neutral transition matrix requires the M-period real-world transition matrix. If we assume that rating migration in any period is independent of the previous migrations, the M-period real-world transition matrix is given by {T(M)ij } = {T(1)ij }M(4)Tables 4a-c show the real-world transition matrices for 2, 3 and 4 periods respectively. Table-4aReal world probabilities of transition over 2 periodsRating after 2 periodsCurrent RatingABCDA0.6550....
