Uncertain Volatility Models for Derivative Pricing
Summary
The document introduces the uncertain volatility model (UVM), which represents volatility as unknown within specified lower and upper bounds, and asks how it compares with stochastic volatility models. Its appeal is a tractable way to incorporate volatility skew and uncertainty into pricing. The cited discussion says that, under bounded stochastic volatility, the UVM gives a higher, worst-case derivative price, and describes implementation through PDE pricing methods.
The answers disagree about how common the model is in practice: one calls it used, while another says it is rarely used. They identify the volatility bounds and the model’s pessimistic worst-case assumption as important limitations. Numerical implementation also has tradeoffs: an explicit Euler scheme may require a restrictive spatial step for stability, while alternative PDE schemes can improve stability and accuracy at added implementation complexity. The document provides qualitative claims and references research, but no market data or comparative pricing results.
Key ideas
- The uncertain volatility model treats volatility as bounded but not known precisely.
- Its worst-case pricing approach can produce more conservative prices than standard Black-Scholes inputs.
- The choice of volatility bounds strongly affects the model’s output.
- PDE implementation is possible, but stability and accuracy depend on the numerical scheme.
- The document reports differing views on how widely practitioners use the model.
Tags
Full text
# Uncertain volatility
# Uncertain volatility
Recently, I have encountered something called "uncertain volatility". Is it a popular concept in QF? Do practitioners use it nowadays? What are its pros and cons compared to e.g more familiar stochastic volatility models?
## Answer by James Spencer-Lavan (score 3, accepted)
https://quant.stackexchange.com/a/33930
It is definitely used in practice:
- It affords a tractable way of pricing in a skew that is easier understood
- Avellaneda proves that a derivative priced under any stochastic volatility process that is bounded by (sigma_min, sigma_max) will produce a cheaper price than under UVM
- It is easily implemented into any PDE pricer at no calc time cost
Edit: just to challenge myself on third bullet point. Given how UVM has two vols, if you are using Euler (i.e. conditionally stable, first-order accuracy) you need to use the larger of the vols for the spatial step size to remain stable. This is not ideal for reasons of accuracy.
Rather than move to ADI (unconditionally stable, second-order) which is not simple to implement, i am a strong pusher of ADE (alternate direction explicit) which achieves the same stability and accuracy ADI but with the code- and computation-complexity of two passes of an Euler discretisation. Look up ADE by Daniel Duffy if of interest.
## Answer by Hernandez Guillaume (score 1)
https://quant.stackexchange.com/a/33925
To learn more about these type of volatility models, I suggest you to have a look on this research paper http://math.cims.nyu.edu/faculty/avellane/UVMfirst.pdf. They provide robust heding of volatility derivatives
## Answer by jherek (score 0)
https://quant.stackexchange.com/a/49511
The model is interesting, but rarely used in practice. One main reason is the choice of the range $\sigma_{min}, \sigma_{max}$. As Martini and Jacquier explain in their article The uncertain volatility model,
> the price corresponds to the worst-case scenario where the Gamma changes signs exactly when the volatility switches regimes. This will hardly happen for real - even if it could.
The pro is to model the uncertainty of the Black-Scholes volatility directly. For exotics, this will give a much more pessimistic price than the regular Black-Scholes model used with the distinct volatilities $\sigma_{min}$ and $\sigma_{max}$. Wilmott shows a concrete example in one of his articles. The con is stated above.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.