Black-Scholes Model



There are three basic types of information that are relevant to the default probability of a publicly traded firm: financial statements, market prices of the firm's debt and equity, and subjective appraisals of the firm's prospects and risk.Prices,by their nature, are inherently forward looking.In determining market prices, investors use,amongst many other things,subjective appraisals of the firm's prospects and risk, financial statements and other market prices.This information is combined using their own analysis and synthesis and results in their willingness to buy and sell the debt and equity securities of the firm. Market prices are the result of the combined willingness of many investors to buy and sell and thus prices embody the synthesized views and forecasts of many investors.

Merton's general derivative pricing model was the genesis for understanding the link between the market value of the firm's assets and the market value of its equity.We use the Black-Scholes option pricing model as a special case of Merton's model,to illustrate some of the technical details of estimating EDF values.Here, I work the calculation of an EDF value using the BS option-pricing model.

Equity has a residual claim on the assets after all the obligations have been met.It also has limited liability.A call option on the underlying assets has a claim on the assets after meeting the strike price of the option.In this case the strike of the call option is equal to the book value of the firm's libilities. If the value of the assets is insufficient to meet the liabilities of the firm then the shareholders, holders of the call option will not exercise their option and leave the firm to its creditors.

We exploit the option nature of equity to derive the market value and volatility of the firm's underlying assets implied by the equity's market value.In particular, we solve backwards from the option price volatility for the imlplied asset value and asset volatility.

The Black-Scholes formula gives the value of equity today as:


The risk-neutral probability that the company will default on the debt is N(-d2).To calculate this, we require V and . Neither of these are directly observable.However if the company is publicly traded we can observe E.This means that the above equation provides one condition that must be satisfied by V and . We can also estimate . From Ito's lemma,


This provides another equation that must be satisfied by V and . The above 2 equations can be solved simultaneously for V and .

Due to the arbitrary complexity of solving the above two equations for a large number of firms , I use the book-value-implied asset volatility instead of the above equity-implied asset volatility given by Ito's lemma.The book-value-implied asset volatility is as follows:


Recall also,that the distance to default is the number of standard deviations that the firm is away from default and thus in the BS world is given by d2.

Estimating equity/asset volatility from Historical Data

Volatility is measured as the standard deviation of the returned provided by the equity/asset in one year when the return is expressed using continuous compounding. Using the lognormal property of stock/equity values volatility is calculated as follows:
  • Compute S(t)/S(t-1), where S(t) is quarter t equity value
  • Compute u(t)= ln[S(t)/S(t-1)]
  • Find the Standard Deviation of u(t).This estimates the quarterly volatility
  • If the volatility is v for one unit of time the volatility for t units of time is vt1/2.Thus we go from quarterly to annual volatility by multiplying by 2.


    • The only problem in implementing remains in calculating the cumulative normal distribution function,N. A polynomial approximation that gives six-decimal-place accuracy is as follows(See M.Abramowitz and I.Stegun, 'Handbook of Mathematical Functions,Dover Publications,New York,1972):

    Results

    Click here to see the probability of default for the S&P 500 publicly traded companies.
    Click here for the code.