Greetings Jonathan -

I have been testing your function a little more and I wanted to verify with you that I am seeing the correct behavior. Please let me know if I am seeing the correct behavior. I pulled the Options Data for US Equity ticker XOM. I setup a calibration and arrived at the following:

Current Variance: +0.071
Theta: +0.066
Rho: -0.999
Kappa: +5.382
Sigma: +0.500

I then priced a selection of strikes assuming Maturity of .5 Years. After pricing, I backed out the Black Scholes Implied. This is a picture verifying my results. I think it looks correct. (Rho is nearly -1).

After this I tried to run another test. I simulated a set of log-normal fwd prices .25 years away. I assumed in each Price Scenario, 2500 of them using Monte Carlo, that I buy the current ATM strike. In this case the price is currently 82.80, so the Strike in each case is 83. For each Fwd Price, I used your model to back out the Option Price. Again, I backed out the implied volatility. I show the graph below:

However on this graph, I want to verify what may be correct- but feels counter-intuitive: here we are seeing forward implied volatility which is lower for forward spot prices. Is this correct? This runs counter to my expectation. I tried to rationalize why this would be and decided to post here to get your feedback. Please let me know your thoughts.

with certain option montages, the calculated value for the alphas is ending up NaN. this in turn is causing strange optimization behavior. most of the time the optimizer does not crash, however the warnings are continuously presented. on occasion the calibration routine halts completely. i have attempted a numerical analysis on the data but have had no luck so far.

the following code + workspace zip file will replicate the warning:

xoptP = lsqnonlin(@(x) Heston1993KahlJaeckelLordRev3(PC+1, S,K,T,t,r,q,x(1),x(2),x(3),x(4),x(5)) - MarketPrice(:,2),...
startparameters, ... %start values
... % v0,theta,rho,kappa,sigma
[ eps eps -1+eps eps eps ], ... % lower bound for parameter vector
[ 100 100 +1-eps 100 100 ], ... % upper bound for parameter vector
options);

optimizer status:

                                     Norm of      First-order 

Iteration Func-count f(x) step optimality
0 6 836.793 1.1e+05
1 12 183.198 0.0511948 3.27e+04
2 18 19.5157 0.0481355 3.69e+03
3 24 13.8879 0.218201 2.43e+03
4 30 13.8879 0.418154 2.43e+03
5 36 8.22993 0.104539 434
6 42 4.2232 0.209077 124
7 48 1.37073 0.311223 53.4
8 54 0.827855 0.3201 41
9 60 0.53483 0.268784 27
10 66 0.278322 0.255432 10.2
Exiting fzero: aborting search for an interval containing a sign change
because NaN or Inf function value encountered during search.
(Function value at 1.71 is -Inf.)
Check function or try again with a different starting value.
349 warning(message('MATLAB:integral:NonFiniteValue'));

alpha_warnings.zip