Algorithm 2 Estimation of joint state price density of a portfolio
Input: Single asset option price
, 
and 
and 
Output: 
and
, 
and 
such that the Gaussian mixture distribution minimizes the square loss 
and has correlation matrix 
Method:
1. for 
do
2. Apply Algorithm 1 and get the optimal Gaussian mixture density estimate 
to minimize 
3. end for
4. Chosse 
5. for 
with 
for all 
do
6. Construct 
7. Construct 
8. end for
9. Choose 
such that 
has diagonal elements 
and the right hand side of (7) is 
10. Return
, 
, 
and