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I am only now reading into Mathematical Finance, I understand the derivation of the BS equation with vanilla European options. On the next page of my book it starts to delve into obtaining exact solutions for the BS equation for Euro-options, and the introductory chapter on numerical methods has this to say:

[...] There are many examples (particularly of multi-factor models) where it is not feasible or even not possible to reduce the problem to a constant coefficient diffusion equation; in this case there is little choice but to use finite differences on the BS equation [...]"

There are no links to these models or relevant chapters in the book. I have googled "multi-factor black scholes" and I am not getting anything digestible.


Question:

Could someone take me through an instance of the BS equation using European options that cannot be solved analytically? Possible some references to derivation?

On first thought I thought it was something to do with the type of option (European over American) but it seems as though you can get solutions to American options too.

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This is my first answer here in the StackExchange.

In the standard BS equatation the uncertainty is driven purely by the Brownian shocks and therefore it is a single risk factor model. However, it is possible to extend this model and add another risk factors and you will get a multi-factor model for pricing options. For example, you can do that by introducing uncertainty into the interest rates or volatility process.

One of the widely used multi-factor models is the so-called Heston model, where volatility process is additionally modeled as a stochastic process. See https://en.wikipedia.org/wiki/Heston_model. In the Heston model you can also price option without a finite difference method, but there are more complicated multi-factor models that you would not be able to solve without numerical methods.

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