Thank you, Michael, for your comment. You have actually raised a valid point - I should have made this explicit in the article. I am taking this as a recommendation for my future work, and will try to make it clear from the very beginning.

Thank you for your valuable contribution to this discussion!

Igor

Hello Michael

I will try to be clearer to eliminate any confusion:

1) FX options - 1st currency measure This is the 'standard' FX option pricing formula for USD/EUR - i.e. option on buying €1 for K dollars and this is implemented through usdOpt formula. This is standard lognormal ito process with mean = F[0]Exp[(r-q)T] and volatility = sigma*Sqrt[T]. This is essentially the same B/S formula as the equity option on the stock paying continuous dividend q.

When I talk about the options pricing consistency, the LHS of the equality refers to the above FX option - i.e. usdOpt formula

2) The Ito triplet you refer to in your comment is quite different - essentially unrelated to the above. Therefore it cannot have any run-on effect on usdOpt formula above. This uses standard Ito process ipUSD - different from other processes derived delow.

The entire second section deals with derivation of a probability measure when the investor position changes from being USD to the one governed by EUR. Although I use the same notation, the process for the inverted FX rate is different and it results in a different Ito SDE where both location and scale parameters are different. I started with zero drift for the second derivation of FX process from practical point of view, knowing that the process is martingale and hence the drift is zero. I could have started with any drift - say mu and then drop it the final SDE formulation.

The whole point of section 2 is to show that the Ito SDE for (1/F) is indeed different and requires different probability measure.

The point of the article is to prove that investor's stance (USD or EUR) cannot lead to arbitrage when the option to buy €1 for K USD is priced. LHS of the consistency equation is the usdOpt, the RHS is the eurOpt. To prove they are consistent we need to show the premiums are the same. usdNum - eurNum shows this is the case.

So, any correctness or not should be just based on this relationship - i.e. comparison of standard (well-known) FX option against the same option evaluation from the second currency measure. The rest are just partial steps to come to the final result.

Hope this clarifies my point.

Igor

Hello Michael Thank you again for your comments and observations. 1) I start from zero-drift to show how the real drift is actually obtained. In FX market this is usually a function of domestic and foreign rate. This is described both from USD and EUR investor points of view

2) Consistency equation you refer to is merely a description of non-arbitrage condition for FX options when investor position changes. LHS of the equation is implemented through the usdOpt formula (call option on USD/EUR) whereas the RHS is calculated in eurOpt which is the option on the inverted FX rate. RHS is particular was done this way to highlight the case we were dealing with the inverted rate.

To assess the consistency, pls look at both option premium formulas - usdOpt and eurOpt.. As I have shown, the premiums are the same , hence no arbitrage exists when investor position changes. This shows formulas are consistent.

Hope this answers your queries. Igor