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I recently come across Merton's model to estimate the default probability and recovery rate of the company. Here is the inputs

Market value of equity =  4,242,509,661
Debt to be paid =  3,397,334,000
equity Volatility = 34%
Risk-free rate = 0.38%
Time to maturity = 2.29

I simply follow the way that Chapter 20 in John Hull 6th edition suggested, in which excel "solver" is used to find the total market value of the asset and its volatility. However, the result is strange and I cannot get positive expected loss and recovery rate greater than 1.

For your information, I initialize

V =  7,500,000,000 
sig_V = 10%

But the solution becomes

V =  7,500,000,000 
sig_V = 19%
Expected loss = 0.0046807
Recovery rate = 8.2132 > 1

If I initialize

V =  7,619,759,237 
sig_V = 10%

Then, the solution becomes,

V =  7,619,759,237 
sig_V = 19%
Default probability = 0.00304
Recovery rate = 0.04853

Did anyone come across the Merton's model before? Can anyone explain what's wrong of the model or what I did? Thanks.

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    $\begingroup$ I've played a bit with the basic version on Merton's model, that is, the one without any kind of stochastic volatility and exotic options' adjustments to simulate the mess following an haircut of issuer's debt. I've often seen gradient-based optimisation algorithms to fail, i.e. to produce inconsistent results. I would suggest you to give a look at nleqslv, which can solve non linear equation systems according to the way Hull himself suggest in his book. Excel solver is not the best way to deal with such a problem. $\endgroup$ – Lisa Ann Oct 7 '13 at 7:55
  • $\begingroup$ How do you compute the recovery rate? If V is the value of the underlying assets than the recovery rate should be monotone in V so your results make no sense. Consider using a more stable method such as those implemented here. $\endgroup$ – Benjamin Christoffersen Apr 29 '18 at 12:36

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