Comparing Local and Stochastic Local Volatility for Exotic Options
Summary
The document compares local volatility (LV) and stochastic local volatility (SLV) models, focusing on when the added stochastic factor may matter. LV can fit the current volatility skew, while stochastic volatility adds a representation of how volatility may evolve over time. Because path-dependent exotic prices depend on market dynamics beyond today’s vanilla option quotes, the discussion suggests that SLV may be more suitable for products such as barriers. For vanilla options, the added model complexity may offer less benefit.
It proposes a diagnostic: price the target product with a calibrated stochastic volatility model, derive its vanilla implied volatility surface, construct an LV model from that surface using the Dupire method, and compare the two product prices. A substantial difference suggests that LV alone may be inadequate and motivates investigating SLV. This is an indication rather than proof of real-market mispricing: the stochastic model calibration is described as rough, and building a calibrated SLV model is acknowledged to be difficult.
Key ideas
- Local volatility can match the current smile but may not capture future volatility dynamics.
- Stochastic volatility adds volatility dynamics that can affect path-dependent exotic option values.
- Compare stochastic volatility and local volatility prices for the target product as a practical diagnostic.
- A large difference in the diagnostic prices suggests that local volatility alone may be insufficient.
- The proposed comparison depends on model calibration and does not guarantee that SLV is necessary.
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Full text
# Is Local Stochastic Vol needed in order to price barrier options?
# Is Local Stochastic Vol needed in order to price barrier options?
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!
## Answer by Mild_Thornberry (score 6)
https://quant.stackexchange.com/a/49859
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
## Answer by Antoine Conze (score 5)
https://quant.stackexchange.com/a/49860
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.