d is protected.3.RISK-NEUTRALITYHypothesising the existence of a risk-neutral world is extremely useful in the pricing of instruments whose value is derived from a stochastic process. In the real world, the present price is less than the expected net present value of the likely outcomes in future. Thus, for example, if the price of a commodity can become either Rs. 100 or Rs. 200 in the next period, its value today (ignoring time value of money) will be less than Rs. 150 (say it is Rs. 140). This is because the uncertainty associated with future values tends to depress current values. This follows from a risk-averse mindset of the real world. However, this poses severe problems in the pricing and valuation of such instruments, because the time-tested expected value criterion fails to hold good.It therefore becomes necessary to risk-neutral values of the underlying factor that causes the uncertainty in prices, for instance, interest rate in case of bond prices. These risk-neutral values can be used in the expected present value context, because the probabilities of the possible outcomes (Rs. 100 and Rs. 200 in our above example) are suitably adjusted so as to yield the observed prices. Thus, the risk-neutral probabilities that the price in the next period would be Rs. 100 and Rs. 200 would be 0.6 and 0.4 respectively so as to yield a current price of Rs. 140.4.RISK-NEUTRAL PROBABILITIES OF CREDIT RATING MIGRATION4.1Derivation of 1-period risk neutral transition matrixFor the purpose of illustration, let us say there are four credit ratings - A, B, C and D (default). It is also assumed that the empirically observed one-period transition matrix is as shown in Table -1 and that the amount recovered in case of default is 40% of the face value of the bond. Further, the zero yields for the first three periods are assumed to be 15%, 14.35% and 13.58% respectively.Table-1Empirical Transition MatrixRating after 1 periodCurrent ratingABCDA0.800.1...
