On a late afternoon in autumn last year two credit risk officers in one of the top four consulting houses in Hong Kong asked a rather theoretical question. Can HSBC every default on its debt? We all know the answer and even they knew the answer: of course not. Not in a million years!

Their question was rather motivated by a pressing problem in hand. They were advising a large bank in Korea about its credit risk management models and non-performing loans portfolio and the key problem faced by their client was: is it possible to forecast default by a firm based on information from the equity markets? And a bigger question: can a blue chip financial institution which holds a huge portfolio of non-performing debt as its assets itself default on its obligations? And an interesting question: if this bank/financial institution is traded publicly on the stock market then are the buyers and sellers of the bank's stock be sufficiently informed to sense the trouble ahead or whether the actions of the buyers and sellers of stock have a reverse causal effect on the assets of the firm thereby accelerating default.

These are indeed very interesting and complex questions, and regardless of the volumes of research devoted to this subject by the experts around the world, no one knows for sure how exactly default happens. And more importantly, no one knows for sure what exactly triggers default.

But anyway, these questions ultimately led our two consultant friends to ask the very theoretical question that is it ever possible for HSBC to default on its obligation? Or, put in a different way, is there anyway, that the actions of a large number of investors buying and selling HSBC stock (or for that matter the stock of any financial or non-financial institution) cause it to default on its obligation?

Moreover, after the collapse of Enron, this line of reasoning was very much in vogue amongst the rather astute rating analysts and risk managers. These days, there is a strong school of thought that believes that equity markets are perhaps a far better indicator of default than the debt markets.

The problem can be formulated in Merton's original framework. Merton's model of a firm's equity as a call option on the assets of the firm looks as default as a put option. Further, in Merton's framework it is assumed that there is a ˇ§down-and-outˇ¨ random barrier beyond which the default occurs. As long as the assets of the firm, and thereby the value of the firm, are within that random default barrier the firm will not default.

This problem actually has a simple intuitive explanation using the stopping time (first passage time) of a Brownian motion. And a neat closed form solution can be found. But there is a problem. We will come to that in a minute.

Assuming that the assets of the firm follow a geometric Brownian motion, with asset returns following a log-normal distribution, the can say that:

If L is a global recovery of a firm's liabilities and D is the debt (per share basis) then in Merton's framework LD forms the down-and-out default barrier.

Default occurs when V_{t} < LD

The problem is that in this framework the default boundary itself has a volatility (percentage volatility), which is assumed to be constant in the model. In fact, JP Morgan and other large institutions have deduced the value of this percentage volatility ¡V called lambda ¡V using S & P's database as well as their own proprietary portfolio management database. Generally, speaking lambda is assumed to be 0.3 (30% volatility). Also, the mean of the recovery rate (global recovery) L is estimated to remain constant at 0.5.

Thus using the stopping time model for the geometric Brownian motion of the firm we can calculate the probability of not defaulting (i.e. the value of the firm will not hit the default barrier) as:

The formula is a modified form of the famous first passage time formula for a geometric Brownian motion. The only modifications have been that the process is assumed to be random without any drift and the default barrier itself has been modeled as a random walk with a percentage volatility lambda.

The asset volatility is estimated from the equity volatility using boundary conditions and observable parameters.

B_{T} is a square root function of the asset volatility and the default barrier volatility and k is a function of the default barrier, the value of assets at time t=0 and the barrier volatility.

Now turning to the HSBC problem, as of November 15^{th}, 2004 the value of HSBC shares was $80.20, the debt per share was $110.74 and the asset volatility, as estimated from the stock volatility and the share value and the value of the firm, was 8.87%. This gave us a value for **B(T) = 0.3128** and **k = 2.679.**

If we assume lambda = 0.3 and L= 0.5 then the probability that HSBC will default in one year's time is 0.002646 or 0.26%. This is what the rating agency will also tell you that the probability of default for HSBC, or a similar financial institution like HSBC, is extremely small.

But note that the above was deduced in an exact closed form solution and we didn't have to resort to any historical analysis of transition matrices or default; or even use some kind of option-pricing framework to estimate this. But rather the first passage time problem of a Brownian motion is able to completely explain default of a firm.

But, suppose due to extraordinary events and circumstances completely unrelated to HSBC, the stock volatility increases by 500 basis points (a very big increase perhaps, signifying a general market turmoil) and this causes the otherwise constant lambda (the default barrier volatility) to increase by a 100 basis points to 0.31. Note that though historically speaking, lambda is assumed to be constant, but there is nothing sanctimonious about this fact. Lambda can, and perhaps will change, though this change is far more complex to model and understand.

If this is now factored in the model then the probability of default for HSBC in the next one year increases significantly to 0.46%, i.e. it becomes almost twice more likely to default than otherwise predicted under normal circumstances.

Hence, the most important factor in understanding equity induced models of default is the impact of stock volatility on the default barrier volatility. Currently, to the best of our knowledge, no model clearly explains this phenomenon.

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