Application Itô's Lemma: Forward to Spot process

I am working on the following equation (I want to apply Ito's lemma on it):

and I know that:

and also

and

My problem is that I want the dynamic of F(S,T) without S because I need first to calibrate my model with F(S,T) parameters and then move to S to get a spot price.

When applying ito's lemma, i have a d(ln F(S,T)) with mutliple S, maybe there is a mistake.

Any idea?

Many thanks

• I don't understand: why do you need to calibrate model with $F(S(t),t)$ first to find the spot price? Isn't the current value of $S$ observable? It would help if you tell us what $S$ represents, I suspect $S(t)$ is the instantaneous short rate or instantaneous volatility, something like that, but you need to confirm. – Frido Rolloos Apr 2 '20 at 13:36
• The research papier I am trying to replicate is this one: "Boogert, A., & De Jong, C. (2008). Gas storage valuation using a Monte Carlo method. The journal of derivatives, 15(3), 81-98." ; actually the price S is a spot price and is not observable, that is my problem; i would like to replicate the spot price thanks to the forward price (and for this I need to calibrate One factor Schwartz model). Maybe it is clear now :) – Jul Apr 2 '20 at 14:55
• energyquant.nl/assets/uploads/files/publications/… (pdf of the paper) – Jul Apr 2 '20 at 14:57
• Okay, so I am not a commodity expert, and I browsed through the paper in 30 seconds, but if you can observe futures/forward prices then you should be able to calibrate the model. If options are traded, even better. For now suppose only futures are traded. The unknown parameters are $\kappa$, $\sigma$, which are taken to be constant, and $\alpha^*$. If $\alpha^*$ is constant you need at least three futures/forward prices. If it depends on time then you need to interpolate the futures/forward curve. Does this make sense? Once the parameters are calibrated you can do your Monte Carlo. – Frido Rolloos Apr 2 '20 at 15:04
• Thank you for this complete answer. We consider mu as constant thus we have 3 constant parameters: 𝛼 , 𝜅 , 𝜎. (I dont consider the mu as time varying). My question is more about how to calibrate the model, I would like to implement it on python. – Jul Apr 2 '20 at 15:20