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zequant
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Joined: December 5th, 2019, 7:50 pm

### SABR for Equity modelling

Hi there,

dumb question: do people actually use SABR for equity modelling? If so, what do they do with the drift of the asset? (e.g. (r-q) for the prop div)
Clearly you could use the standard (driftless) SABR process for the forward price of the equity, but then the spot process depends on the forward expiry, which seems nonsensical. Other transformations that kill the drift seem to lead to explicit time dependence in the backbone (the F^beta in the classic SABR model), which is also no good. Guess you could postulate that the spot process satisfies something like

dS = (r-q)*S*dt + sigma*S*B(S*exp(-(r-q)t))*dW

where B(x) is the backbone, and then change var to F(t)=S(t)*exp(-(r-q)t), but this seems a little artificial.

I'm a bit puzzled because I thought I heard of people using SABR for equities back in the day.

Thanks.

Cuchulainn
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### Re: SABR for Equity modelling

I thought SABR was more to compute iv? @Alan is the quant-in-residence expert on SABR

A recent thesis on SABR meets ML by Dalvir Singh Mantara might be relevant.

https://www.datasim.nl/blogs/26/msc-the ... al-finance

JohnLeM
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Joined: September 16th, 2008, 7:15 pm

### Re: SABR for Equity modelling

And a nice, second order accurate, SABR simulation can be found here. Note however that it is a martingale process used for rates modeling.
Regarding your question, extensions to equity that I know are written as
dS = (r-q)*S*dt + sigma*S^\beta dW^1
But Alan is surely a better expert than me.

Cuchulainn
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### Re: SABR for Equity modelling

@JohnLeM is the quant_NOT_always_in_residence expert on SABR

JohnLeM
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### Re: SABR for Equity modelling

@JohnLeM is the quant_NOT_always_in_residence expert on SABR
I do declare myself not very competent on SABR. I just know how to model it accurately By the way, I did not write the most important part, that is an log-normal process with return to average
d \sigma = \kappa(\theta - \alpha) dt + \nu \sigma dW^2

Alan
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### Re: SABR for Equity modelling

I have a big chapter of SABR in "Option Valuation under Stochastic Volatility II". My interest is largely in solving it exactly as a problem in mathematical finance.

For equities, it's kind of a silly model, but I have a few comments about that in the book chapter -- excerpt attached.

For those with more interest, if you sign in to amazon, do a "Look inside", and search for: SABR, you'll get the flavor of the chapter.
Attachments
Lewis.Volii.SABRexcerpt.pdf

zequant
Topic Author
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Joined: December 5th, 2019, 7:50 pm

### Re: SABR for Equity modelling

Sorry, I have just gotten back to this. Thanks for the responses. I realise I wasn't very clear in the original post. My question is regarding the analytic IV formulas and the semi-analytic approaches like Antonov et al. These rely on certain "nice" properties of the dynamics, and in particular it seems to me they rely on the underlying "S" being driftless. If this is correct, it would make these analytical approaches unuseable for equities, which would seem to defeat the point of using SABR, hence my question. As I said above, one could always model the equity forward as a driftless SABR process, and thus retain the tractability, but this would imply crazy dynamics for the spot, so doesn't seem a good way forward either.

Alan
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### Re: SABR for Equity modelling

Let's take the lognormal SABR model as an example.
So, let's put in a drift, as that seems to be your issue:
$dS_t = r S_t dt + \sigma_t S_t dB_t$,  $d\sigma_t = \sigma_t dW_t$, $dB_t dW_t = \rho \, dt$.

Now suppose I gave you three terms of the small-T power series expansion for the BS implied volatility for an option with strike $K$:

$\sigma^{imp}_T = a_0(X,\sigma_0) + a_1(X,\sigma_0) T + a_2(X,\sigma_0) T^2 + O(T^3)$,

where $X = \log K/S_0$, and I suppress the dependencies on $(\rho,r)$.

What would you do with it?

zequant
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Joined: December 5th, 2019, 7:50 pm

### Re: SABR for Equity modelling

Well, ideally I'd like it for general LV and general drift, i.e.
dS/S = mu * dt + sigma * C(S) * dW
What would I do with it? I might use it as a fast approximation near the money, or I might use it in the context of a large simulation (like for counterparty risk) as a fast approximate conditional option pricer, or I may use it in combination with some ad-hoc arb removal methodology to mark options, or...
but mainly I just wanted to know if the expansion techniques work with drift, as I haven't seen this. I am aware btw of double expansions in expiry/moneyness using PDE techniques (for even more general LSV models), but that is a somewhat different thing.

Alan
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### Re: SABR for Equity modelling

Yes, I was going to suggest you work on the double expansion for what you need. That can be automated to very high order. In general, the "exact" terms $a_i(X,\sigma_0)$ are very difficult to find analytically, while in contrast the double expansion terms can be found analytically with a good computer algebra program like Mathematica.

I will add that, in the abstract, the exact expansion certainly 'works' in the sense that the power series exists. It is a general consequence of the so-called heat kernel expansion for n-dimensional diffusions. But, as I said, actually getting an analytic expression, beyond say $O(T)$, can be very difficult. Just look at what Paulot had to do to get the SABR case done exactly through $O(T^2)$ and you'll see the difficulties. Likely some numerics with geodesics would get you the terms $a_i(X,\sigma_0)$, numerically. But, if you're going to go that route, you might as well just solve the full evolution PDE numerically.