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Numerical Solution of Black-Scholes Equation (KMV Model)

Salman Zaffar
Posted 1 year ago

Hi All, I am trying to numerically solve Black Scholes Equation as it is used in Credit Risk Analysis through KMV method. Enclosed please find the snapshot of the two nonlinear equations that are to be solved simultaneously. Is there anything wrong with the Mathematica code? Mathematica code is given below 

r=10.35;

T=1.0;

Debt=100000.0;

d1=(Log[VA/EqVaVo[[1,1]]]+(r+sigA^2/2)*T)/(sigA*Sqrt[T]);

d2= (Log[VA/EqVaVo[[1,1]]]+(r-sigA^2/2)*T)/(sigA*Sqrt[T]);

F=CDF[d1];

G=CDF[d2];

Equation1=VA*F-Debt*Exp[-r*T]*G;

Equation2=EqVaVo[[1,1]]*EqVaVo[[1,2]]-VA*F*sigA;

NSolve[{Equation1 == 0, Equation2 == 0}, {VA,sigA}]

The result I get is

KMV Equations

Result of NSolve
POSTED BY: Salman Zaffar
5 Replies
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This has syntax errors, check the documentation for PDF.

pdf = PDF[NormalDistribution[0, 1], [Xi]]
POSTED BY: Rohit Namjoshi
Updating Name
Updating Name
Posted 1 year ago

Hi, I changed the code a bit as follows. But its still not getting numerically solved. please help

EqVaVo=Import["..../HDEqVaVo.csv"];

(* EqVaVo[[1,1]]=48885.7 and  EqVaVo[[1,2]]8.752*^-6 *)

r=0.1035;

T=1.0;

Debt=37100.0;

d1=(Log[VA/EqVaVo[[1,1]]]+(r+sigA^2/2)*T)/(sigA*Sqrt[T]);

d2=(Log[VA/EqVaVo[[1,1]]]+(r-sigA^2/2)*T)/(sigA*Sqrt[T]);

pdf = PDF[NormalDistribution[0, 1], \[Xi]];

F=Integrate[pdf, {\[Xi], -Infinity, d1}]

G=Integrate[pdf, {\[Xi], -Infinity, d2}]

NSolve[{VA*F==Debt*Exp[-r*T]*G,EqVaVo[[1,1]]*EqVaVo[[1,2]]== VA*F*sigA}, {VA,sigA}]
POSTED BY: Updating Name

You have syntax error in CDF !.

From documentation of CDF we have:

CDF[dist,x] gives the cumulative distribution function for the distribution dist evaluated at x.

You must define at first ProbabilityDistribution for example:

 d= ProbabilityDistribution[Sqrt[2]/Pi/(1 + x^4), {x, -Infinity, Infinity}];(* and then *)
 CDF[d, x]


 (*(\[Pi] - ArcTan[1 - Sqrt[2] x] + ArcTan[1 + Sqrt[2] x] + 
  ArcTanh[(Sqrt[2] x)/(1 + x^2)])/(2 \[Pi])*)

In your case:

 CDF[ProbabilityDistribution[(10 (207/2 + sigA^2/2) + Log[(10 VA)/488857])/(
            Sqrt[10] sigA), {sigA, -1, 1}], SigA]

or:

  CDF[ProbabilityDistribution[(
    10 (207/2 + sigA^2/2) + Log[(10 VA)/488857])/(
    Sqrt[10] sigA), {VA, -1, 1}], VA]
POSTED BY: Mariusz Iwaniuk
Salman Zaffar
Salman Zaffar
Posted 1 year ago 
EqVaVo=Import[".../HDEqVaVo.csv"];

 (*Values of EqVaVo[[1,1]]  & EqVaVo[[1,2]]   as read from HDEqVaVo.csv file*)

EqVaVo[[1,1]]=48885.7

EqVaVo[[1,2]]=8.752*^-6

r=10.35; T=1.0; Debt=100000.0;

d1=(Log[VA/EqVaVo[[1,1]]]+(r+sigA^2/2)*T)/(sigA*Sqrt[T]);

d2= (Log[VA/EqVaVo[[1,1]]]+(r-sigA^2/2)*T)/(sigA*Sqrt[T]);

F=CDF[d1];

G=CDF[d2];

Equation1=VA*F-Debt*Exp[-r*T]*G;

Equation2=EqVaVo[[1,1]]*EqVaVo[[1,2]]-VA*F*sigA;

NSolve[{Equation1 == 0, Equation2 == 0}, {VA,sigA}]
POSTED BY: Salman Zaffar 
 EqVaVo[[1, 1]] (* and *)EqVaVo[[1, 2]]

not defined ? paste a full code?
POSTED BY: Mariusz Iwaniuk
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