Why Default Risk behaves like a Put Option?
June 13, 2006
Every day we talk about CDOs, CDS, correlations, copulas, joint probabilities, and what not to talk default risk. But very few of us try to understand the process of default in an option pricing framework (long live Robert Merton!). Let us consider a very simply and stylized example.
Let us say that a bank enters into a one year fixed for fixed currency swap for $100 million with a counter-party (say, a corporate) whereby it receives interest in US Dollars and pays interest in Euro. Principals are exchanged at the end of one year. That is at the end of the year the bank will receive US$100 mil plus the USD interest and will pay to the counter-party the EUR principal that is dependent on the FX rate at maturity, plus of course, the EUR interest rate locked in today.
| Life of the swap (maturity):
|| 1 year
| Euro Interest rate (one year):
| US Dollar Interest rate (one year):
| Initial EUR-USD FX rate (today):
| Notional USD Principal (today):
|| $100 million
| Notional EUR Principal (today):
|| EUR 78.74 million
So, what is the default risk of this transaction for the USD receiver, i.e. the bank? For that we need to look at what is the net benefit to the USD receiver (the bank) at maturity, i.e. at the end of one year. This net benefit can be thought of the profit from the transaction.
After one year the bank will receive $100 million plus the interest on that which is $5 million. The bank will pay to the counter-party EUR 78.74 million times the FX spot rate after one year plus 3% on EUR 78.74 million times the FX spot rate after one year. This payoff-the net benefit to the bank - can be written as a simply payoff function:
The above is expressed as a maximum function because, if the EUR-USD spot FX rate at maturity is such that the above expression is negative then that would be a negative payoff for the bank which would mean that it is no longer a benefit for the bank. That is, if the above payoff is negative at maturity then the bank would be losing money on the swap transaction and therefore needs to make a net payment to the counter-party. Hence, if there is a loss for the bank, then the issue of default risk is meaningless and there is no default risk for the bank in the underlying transaction.
Rounding off the numbers the above payoff can be expressed as:
However, the above will occur one year in future, so we need to present value the above payoff:
Thus the payoff-or the net benefit-from the swap to the bank is exactly like 81 put options of one year maturity on the EUR-USD FX rate with a strike price of 1.2947. We can very easily calculate the value of the one year Dollar-Euro put at strike 1.2947 using a closed form or numerical solution and estimate what is the quantum of default risk inherent in the swap.
But note that the above calculations will only state the quantum of default risk inherent in the swap transaction for the bank. But what is the probability that such an event will happen? Without knowing that probability we cannot determine the cost of default risk and express it as a percentage of the notional principal. In other words, the above calculation though very crucial is still incomplete. We need the probability of default and then multiply that probability with the quantum (magnitude) of the default risk to get the cost of default risk in basis points.
How do we determine the probability of default? That is truly a million-and these days a billion-dollar question!
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