Generating Option VaR Scenarios with Sticky-Strike Volatility
Summary
The document outlines a historical-scenario approach for estimating one-year value at risk on a single-name equity option. It focuses on how to pair changes in the underlying spot price with changes in implied volatility when repricing the option, and uses sticky strike as the simplifying assumption: volatility at a fixed absolute strike and maturity remains attached to that strike as spot moves.
Because a full history of volatility changes for every strike and maturity is impractical, the answer suggests using parallel volatility moves by maturity, approximated with at-the-money volatility. Under sticky strike, the relevant historical move compares volatility at the current day’s spot level across adjacent dates, rather than subtracting at-the-money readings tied to different spot levels. That move can then be added to today’s volatility to form a scenario. This is a simplified proxy: it does not describe a complete volatility-surface model, discuss calibration, or compare sticky strike with alternatives such as sticky delta. The scenario choice and historical data remain material limits.
Key ideas
- Sticky strike assumes implied volatility at a fixed absolute strike and maturity does not shift with spot.
- A complete history of volatility changes across all strikes and maturities is difficult to maintain.
- Parallel volatility shifts by maturity can be approximated using at-the-money volatility.
- To isolate a sticky-strike volatility move, compare adjacent dates at the same spot reference level.
- Historical scenario results depend on the simplifying assumption and the selected data window.
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Full text
# Volatility scenario generation for value-at-risk
# Volatility scenario generation for value-at-risk
I have the following problem: For a single name plain vanilla equity option calculate 1y VaR for given confidence level.
Is there any state-of-the-art or current market practice known on how to generate new impl. vol scenarios based on historical returns? Obviously there is an interplay with underlying spot levels as new spot scenarios would imply different moneyness levels. In order to revaluate todays option with historical data, what would be your estimates for simulated spot and vol levels in BS Formula?
Can you recommend some literature on this?
## Answer by Phil-ZXX (score 2, accepted)
https://quant.stackexchange.com/a/40418
Let us denote the implied vol on day $t$ for absolute strike $K$ and maturity tenor $T$ as $$\sigma_t(K,T)$$ If $S_t$ denotes the spot value on day $t$ then $\sigma_t(S_t,T)$ is referred to as At-The-Money (ATM) vol. (Note: I'll ignore things like ATMF here)
If we assume sticky-strike (i.e. any option's implied vol doesn't move in absolute strike-terms when spot moves), then for any fixed strike $K$ and maturity tenor $T$ a scenario move could be $$s_t(K,T) = \sigma_t(K,T) - \sigma_{t-1}(K,T)$$ and if $\sigma_{today}(K,T)$ is today's value then the simulated vol scenario value could be $$\sigma_{sim_t}(K,T) = \sigma_{today}(K,T) + s_t(K,T)$$
Now, the problem is that you cannot record moves $s_t(K,T)$ for all possible $K,T$. I mean, where does it end? So one example of a simpler approach is to look at parallel vol moves (one per maturity), which we could proxy via the ATM vol.
Since we are assuming sticky-strike we cannot use $$\text{ATMVol}_{t} - \text{ATMVol}_{t-1} = \sigma_t(S_t,T) - \sigma_{t-1}(S_{t-1},T)$$ as a valid vol move, because in general $S_t\ne S_{t-1}$. The correct move to look at is $$s_t(T) = \sigma_t(S_t,T) - \sigma_{t-1}(S_{t},T)$$ Note the subtle difference in subscripts: $S_{t-1}\to S_{t}$. Then a simulated vol scenario could be $$\sigma_{sim_t}(K,T) = \sigma_{today}(K,T) + s_t(T)$$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.