A Risk Neutral Framework For The Pricing Of Credit Derivatives

A Risk Neutral Framework For The Pricing Of Credit Derivatives 1. INTRODUCTION Considerable research effort has gone into Credit Derivatives since the early 1990s. The roots of credit derivatives can be traced back to the notion that the credit risk of a firm can be captured by the credit rating ascribed to it. This premise is also the cornerstone of loan pricing and credit risk management models the world over, including J.P. Morgans CreditMetricsTM. Empirical research enables the predictability of the event of default as well as the Loss in the Event of Default (LIED).

This information is expressed in terms of a transition matrix – a matrix that traces out the probabilities the migration of a firms credit rating. agencies such as Standard & Poor (S&P) provide transition matrices computed from periods of data about bonds – default record and post-default behaviour in the US markets. Lack of adequate data precludes the computation of such matrices in the Indian context, although it is possible to map ratings of Indian rating agencies such as CRISIL onto S&P ratings. 2. TYPES OF CREDIT DERIVATIVES Here is a brief description of some popular types of credit derivatives: 2.1 Credit Default Swaps A credit default swap provides a hedge against default on some payment, such as a bond.

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The counterparty buying credit protection pays the provider a certain amount in return for a guarantee to make good the loss in the event of default. 2.2 Total Return Swaps In this contract, the payer gives a receiver the total return on an asset in return for the returns on a benchmark asset, typically a risk-free asset. The payer has thus eliminated the risk of default in return for a lower but certain risk-free rate of return. 2.3 Credit Spread Derivatives Credit spread derivatives take the form of credit spread options, forwards or swaps. A credit spread call option, for example, is a call option written on the level of the spreads for a given bond. The option, thus increases in value as the spread increases, so that the value of the bond is protected.

3. RISK-NEUTRALITY Hypothesising 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 MIGRATION 4.1 Derivation of 1-period risk neutral transition matrix For 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-1 Empirical Transition Matrix after 1 period Current rating A B C D A 0.80 0.12 0.06 0.02 B 0.10 0.75 0.10 0.05 C 0.05 0.10 0.75 0.10 D 0.00 0.00 0.00 1.00 In 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-2 Prices of Zero coupon Bonds Maturity 2 3 4 A 85.86 74.10 64.59 55.32 B 84.09 71.32 61.30 51.91 C 80.96 66.76 56.06 46.55 In 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 0.12 p(1)AC = p(1)A x 0.06 p(1)AD = p(1)A x 0.02 p(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 = 0.126 p(1)AC = 0.063 p(1)AD = 0.021 The 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-3 Risk neutral probabilities of transition over 1 period after 1 period Current A B C D A 0.790 0.126 0.063 0.021 B 0.110 0.725 0.110 0.055 C 0.058 0.115 0.713 0.115 D 0.000 0.000 0.000 1.000 4.2 Derivation of M-period risk-neutral transition matrix The 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-4a Real world probabilities of transition over 2 periods after 2 periods Current A B C D A 0.655 0.192 0.105 0.048 B 0.160 0.585 0.156 0.100 C 0.088 0.156 0.576 0.181 D 0.000 0.000 0.000 1.000 Table-4b Real world probabilities of transition over 3 periods after 3 periods Current A B C D A 0.548 0.233 0.137 0.081 B 0.194 0.473 0.185 0.148 C 0.114 0.185 0.452 0.248 D 0.000 0.000 0.000 1.000 Table-4c Real world probabilities of transition over 4 periods after 4 periods Current A B C D A 0.469 0.254 0.159 0.118 B 0.212 0.397 0.198 0.194 C 0.133 0.198 0.365 0.305 D 0.000 0.000 0.000 1.000 In order to obtain the 2-period risk-neutral transition probabilities, p(2)Aj , we require the price V(2)A of a two-period zero coupon bond currently rated A.

The expected pay-off in the risk neutral framework is p(2)AA x100 + p(2)AB x100 + p(2)AC x100 + p(2)AD x40. Thus, the following relationship must hold : V(2)A = (5) Since the expected pay-off is calculated using risk-neutral probabilities, it is free from credit risk. Further, since the pay-off is independent of the interest rate, using the 2-period risk free rate for discounting is justified. As in the 1-period case, we assume that the risk-neutral probabilities of transition from state A to other states are proportional to real-world probabilities. The R.H.S of Eqn.(5) reduces to a function of p(2)A.

Thus, the risk neutral transition probabilities p(2)Aj can be obtained. The same methodology may be used to obtain p(2)Bj and p(2)Cj and thus the 2-period risk neutral transition matrix is obtained. Tables 5a-c show the 2,3 and 4-period risk neutral transition matrices respectively. Note that the M-period risk-neutral transition matrix is not the Mth power of the 1-period matrix. Table-5a Risk neutral probabilities of transition over 2 periods after 2 periods Current A B C D A 0.627 0.207 0.113 0.052 B 0.181 0.530 0.176 0.112 C 0.102 0.183 0.503 0.212 D 0.000 0.000 0.000 1.000 Table-5b Risk neutral probabilities of transition over 3 periods after 3 periods Current A B C D A 0.503 0.256 0.151 0.089 B 0.223 0.394 0.213 0.170 C 0.137 0.222 0.343 0.298 D 0.000 0.000 0.000 1.000 Table-5c Risk neutral probabilities of transition over 4 periods after 4 periods Current A B C D A 0.405 0.285 0.178 0.132 B 0.248 0.294 0.231 0.226 C 0.163 0.243 0.219 0.375 D 0.000 0.000 0.000 1.000 Now that the risk neutral transition matrices have been calculated, any credit derivative may be priced. 5.

VALUATION OF CREDIT DERIVATIVES The present model does not consider the credit risk associated with seller of the derivative product. Further, it does not consider the correlation between interest rate changes and credit rating migrations. In the following sections, we illustrate the pricing of two credit derivative products. 5.1 A simple derivative Here again, the valuation of the credit derivative requires only the 2-period risk free zero rate because the pay-off from the derivative is adjusted for credit risk and is independent of the interest rate. 5.2 A multi-period derivative Consider the bond XYZ again.

Now the derivative pays Rs. 100 if the credit rating of the bond changes to C during period 2 or period 3. This derivative can be decomposed into the simple derivative described above and another derivative that pays Rs. 100 in the third period if the rating of the bond is C in the third period but not in the second. For the valuation of this derivative, let us define events Ei and F as under Ej : The rating of the bond changes to i at the end of 2 periods (where i = A or B) F : The rating of the bond changes to C at the end of 3 periods The probability that the derivative yields Rs.

100 in the third period is given by where i = …


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