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I'm trying to understand when it is appropriate to use stochastic local volatility models rather than local volatility ones.

More precisely, for which products is it appropriate to introduce a stochastic multiplier $e^{u(t)}$ on top of the local one, e.g.

$$ \frac{\mathrm{d}S(t)}{S(t)} = r(t)\mathrm{d}t + e^{u(t)} \sigma(t, S(t))\mathrm{d}W(t) $$

where $u$ follows, say, some Orstein-Uhleneck process. Are there cases when $u\equiv 0$ leads to wrong prices? Any examples/references would be really appreciated!

Thanks!

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  • $\begingroup$ Going by the title of the question, for barrier options the standard-ish approach at least in FX is mixture of local and stochastic volatility. Local does not do the job because for example the smile dynamics (sticky strike vs sticky money-ness etc) are not great. $\endgroup$ – Magic is in the chain Nov 22 at 17:55
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Local volatility models capture skew today but not dynamics tomorrow. Stochastic vol captures dynamics tomorrow but not necessarily skew today (how well does your calibrated vol surface match observation?). To answer your question: if you're pricing exotic options that are path dependent, stochastic local vol is more accurate. If you're pricing vanilla options, it's just an added layer of complexity.

Check out some of this source material. A few excerpts:

"Stochastic volatility in a local volatility context permits the exact calibration of vanilla options while at the same time addressing the exposure of financial contracts to the rate of mean reversion in volatility and the volatility of volatility."

https://staff.fnwi.uva.nl/p.j.c.spreij/winterschool/16RenMadanQian.pdf

"Barrier prices and path dependent options are in general not determined by vanilla market quotes. They also depend on the dynamics of the market."

https://www.scribd.com/document/405131295/Bloomberg-Stochastic-Local-Volatility

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You can compute the SV - LV price difference and see if it is substantial or not. This is easily done and will give you an indication of whether your product can be safely priced with LV only.

  • start with a pure SV model: choose $\sigma(t,S)=1$ and do a rough calibration of the parameters that drive $u_t$, to historical data for instance
  • Price your product with this model. You get the SV price
  • Also price vanillas for all $K$ and $T$, compute the corresponding implied vols. This generates the SV recomputed implied volatility surface $\Sigma(K,T)$
  • Now build the pure LV model on this volatility surface, $u_t=0$ and $\sigma(t,S)$ obtained from the Dupire formula applied to $\Sigma(K,T)$ and price your product with this model. You get the LV price

If the SV - LV price difference is subtantial for your product, this is an indication that in real life, with an LSV and a LV both calibrated to the current market smile, the LSV and the LV price will be subtantially different. In this case you have to build an LSV model, not a so simple task.

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