I am trying to repeat calculations from Hull(options futures and other derivatives) chapter "Using Equity Prices to Estimate Default Probabilities". I want to solve system of 2 equations:
\begin{cases} E_0 = V_0 N(d_1) - L e^{-rt}N(d_2) \\ \sigma_E E_0 = N(d_1) \sigma_V V_0 \end{cases} to find $V_0, \sigma_V$, where you recognise Merton model. We assume that other variables are given, hence should be easy to solve numerically system of 2 equations with 2 unknowns. Using matlab fsolve function, even with feeding analytical Jacobian for different data inputs sometimes returns me:
No solutions found. or Equation solved, fsolve stalled.
This is even worse if I use Black Cox model, as the 1st equation(barrier option) and 2nd equation(because of $\frac{\partial E }{\partial V}$ term) is getting huge.
Could somebody give some hint how to solve this system?
Could somebody give tips or ideas how to choose good initial conditions for fsolve? ( Finally, I eager to be able to solve Black Cox model)
Update: I have been thinking to search for reference how to work with ill-posed problems, or trying to identify if condition number is good, but I can not do the last as the system is non linear( I can not represent it as a matrix).