Investments, tenth edition

 Figure 21.13 

 Value of implicit put option on a loan guarantee as 

a percentage of the face value of debt  (Debt maturity  5  1 year; 

Standard deviation of value of firm  5  40%; Risk free rate  5  6%)  

bod61671_ch21_722-769.indd   753

bod61671_ch21_722-769.indd   753

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P A R T   V I

  Options, Futures, and Other Derivatives

beta and standard deviation of 

the underlying asset. 

 

Therefore, when compil-

ing the “input list” for creating 

an efficient portfolio, we may 

wish to think of equity as an 

implicit call option and com-

pute its elasticity with respect 

to the total value of the firm. 

For example, if the covariance 

of the firm’s  assets  with other 

securities is stable, then we 

can use elasticity to find the 

covariance of the firm’s  equity

with those securities. This will 

allow us to calculate beta and 

standard deviation. 

 Unfortunately, elasticity can 

itself be a moving target. As 

the firm gets weaker, its elas-

ticity will increase, potentially 

very quickly.  Figure 21.14  uses 

the Black-Scholes model to plot call option elasticity as a function of the value of the 

underlying stock. Notice that as the option goes out of the money (the stock price falls 

below 100), elasticity increases rapidly and without limit. Similarly, as the firm gets closer 

to insolvency (the value of firm assets falls below the face value of debt), equity elasticity 

shoots up, and even small changes in financial condition can lead to major changes in risk. 

Elasticity is far more stable (and closer to 1) when the firm is healthy, i.e., the implicit call 

option is deep in the money. Similarly, equity risk characteristics will be far more stable for 

healthy firms than for precarious ones.