Finding a value from normal density funciton -- help me

[axiomlu]

How to find K value from the following equation (1)
a*n(d)-b*N(d)=0 (1)
where
d= ln(c/K)+b (2)
where n(d) and N(d) are normal density function and cumulative normal function.

 

[jvicens]

To find the K value from the given equation (1), we can follow the steps below:

1. Substitute the value of d from equation (2) into equation (1):
a*n(ln(c/K)+b)-b*N(ln(c/K)+b)=0

2. Rearrange the terms in the equation to isolate the K value:
a*n(ln(c/K))+a*b-b*N(ln(c/K))-b=0

3. Use the fact that N(ln(c/K))=1-n(ln(c/K)), which comes from the definition of the cumulative normal function.

4. Substitute this into the equation:
a*n(ln(c/K))+a*b-b*(1-n(ln(c/K)))-b=0

5. Expand the terms:
a*n(ln(c/K))+a*b-b+b*n(ln(c/K))-b*n(ln(c/K))-b=0

6. Rearrange the terms:
(a+b)*n(ln(c/K))+(a-b)*b-b=0

7. Factor out the n(ln(c/K)) term:
n(ln(c/K))*[(a+b)+(a-b)*b-1]=0

8. Use the fact that n(ln(c/K)) is a normal density function and must be non-zero:
(a+b)+(a-b)*b-1=0

9. Simplify the equation:
a+b+a*b-b^2-1=0

10. Rearrange the terms:
a*b-b^2+a+b-1=0

11. Use the quadratic formula to solve for the K value:
K = [-(a+b)+√(a^2+2*a*b+b^2+4)]/2

Therefore, the K value can be found by substituting the values of a and b into the above equation.