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I am quite new to the topic of financial options. I'm aware of options with analytical solutions (e.g. European options in Black-Scholes and Ornstein-Uhlenbeck models). I read that sometimes (most times?) numerical methods are used for option pricing. I also read that there exist exotic options which simply do not have a closed-form solution and require numerical methods; however, I have not been able to find examples of such options. Could anyone name a few? Cheers.

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American options (on any underlying) is the first that comes to mind. They are often priced using a tree-based algorithm to determine if there is a benefit to early exercise anywhere along the life of the option.

Any options that has non-linearity or trigger conditions (binary, knock-in, etc.) are also candidates for numerical models.

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You can price anything with an analytic formula if you have all of the required parameters.

Let's take european options as an example. We can price these using Black76 if we have the forward price, the appropriate discount factor, expiry, the strike, and the volatility. This last bit - the volatility - needs to have an important consideration: it is the black76 option price implied volatility. If we are pricing using some other model, say Vasicek, then we need a different volatility.

That you're given all of the parameters already is the important point - the model is popular enough that you can often get a vol surface (where the volatility is the black76 vol, assuming a specific conventions around stuff like your calculation of T (i.e. act/act 250 business days per year, etc.)) provided as market data. The vol surface is essentialyl a one to one mapping to the option prices (given all the other params).

Now, if we have some other, more exotic stucture, there's nothing to stop us doing the same thing - we can construct some function that takes in the market data (i.e. numbers observable directly in the market), specifics of the trade, and then some preconstructed surface which maps to price of the instrument.

Does this last example count as analytic? If it does, then anything can be priced analytically...

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