### Zassenhaus formula

Posted:

**July 11th, 2019, 3:20 pm**Ah, what the heck.

So I've been mulling this over for quite some time now: let [$]B[$] be the Black-Scholes generator, and [$]A[$] be SV generator terms, such as the vega, volga, vanna differential operators.

From semi-group theory it is justified to write the formal solution of an SV PDE as [$]C(\tau) = e^{\tau (A + B)} C(0)[$]. Kind of justified as [$] C(0) [$] is not smooth, but let's ignore that for the moment.

We know the Baker-Campbell-Hausdorff formula, but there is another (lesser known) formula called the Zassenhaus formula (see for instance https://arxiv.org/pdf/1702.04681.pdf) which expresses [$] e^{\tau(A+B)} [$] as a product of exponentiated differential operators, in particular one could write [$] e^{\tau(A+B)} = \cdots e^{\tau B} [$]. But [$] e^{\tau B} C(0) = C^{BS} (\tau)[$], which means that if the explicit form of " [$] \cdots [$] " is known then *maybe* a general perturbative solution could be found for any SV model.

Now I think the general form of "[$] \cdots [$]" can be found or is actually known (see link above to ArXiv paper), but I am not sure what the "best" way is to cut-off or Taylor expand the "[$] \cdots [$]" part, i.e. to what order in [$] \tau [$] or other parameters.

Thoughts? My thinking at the moment is that this approach could be too complex, not worth it, but maybe somebody else has some better insights.

So I've been mulling this over for quite some time now: let [$]B[$] be the Black-Scholes generator, and [$]A[$] be SV generator terms, such as the vega, volga, vanna differential operators.

From semi-group theory it is justified to write the formal solution of an SV PDE as [$]C(\tau) = e^{\tau (A + B)} C(0)[$]. Kind of justified as [$] C(0) [$] is not smooth, but let's ignore that for the moment.

We know the Baker-Campbell-Hausdorff formula, but there is another (lesser known) formula called the Zassenhaus formula (see for instance https://arxiv.org/pdf/1702.04681.pdf) which expresses [$] e^{\tau(A+B)} [$] as a product of exponentiated differential operators, in particular one could write [$] e^{\tau(A+B)} = \cdots e^{\tau B} [$]. But [$] e^{\tau B} C(0) = C^{BS} (\tau)[$], which means that if the explicit form of " [$] \cdots [$] " is known then *maybe* a general perturbative solution could be found for any SV model.

Now I think the general form of "[$] \cdots [$]" can be found or is actually known (see link above to ArXiv paper), but I am not sure what the "best" way is to cut-off or Taylor expand the "[$] \cdots [$]" part, i.e. to what order in [$] \tau [$] or other parameters.

Thoughts? My thinking at the moment is that this approach could be too complex, not worth it, but maybe somebody else has some better insights.