# 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?

• Not a full answer, but for EQ Options people tend to assume Sticky-Strike (see e.g. deltaquants.com/volatility-sticky-strike-vs-sticky-delta), which means that the implied vol of your particular option remaims the same (irrespective of spot/moneyness moving). This has an impact on your spot delta. Now if you e.g. want to simulate simple parallel vol scenarios you could look at historic ATM vol moves (obviously adjust them for the sticky-strike assumption; note if you assumed sticky-delta you'd not have to adjust them). – Phil-ZXX Jun 18 '18 at 22:41
• Thanks for your helpful comments. Can you please describe your last sentence in brackets in more detail? Let's suppose that today S=105 and K=100 for a call option yielding impl. vol of 10%. In a sticky strike world this impl vol would not change as spot moves. When generating a new impl vol level based on historical ATM moves, how should I adjust for sticky strike rule in order to derive a new simulated vol? – user31190 Jun 19 '18 at 18:17

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 $$\sigma_{sim_t}(K,T) = \sigma_{today}(K,T) + s_t(T)$$

• Thanks a lot. This really helps. However, let me address the following question: – user31190 Jun 25 '18 at 18:33
• Thanks a lot. This really helps. However, let me address the following question: In your difference function s_t (K,T), you presented three scenarios from ideal, incorrect up to the correct one. What is the issue with the ideal solution? I mean at the end we are picking individual points on historical vol surfaces in all three formulas, right? One issue with the ideal solution for me would be that underlying spot 250 days ago may have traded at completely different levels than now. In this case we may not have meaningful/quoted impl. vols for our current absolute strike K, right? – user31190 Jun 25 '18 at 18:42
• You are right, this approach (or any of them) only works if you have meaningful vol surface data for more than just one strike, etc. But in any case, you are not querying vols at today's spot level (today), you'd be querying vols as of $t-1$ at the next day's level $t$. So I wouldn't expect spot to have moved massively by then. – Phil-ZXX Jun 26 '18 at 7:53
• @user31190 would appreciate it if you could formally accept the answer then. :) – Phil-ZXX Jun 28 '18 at 19:45
• I don't think that your approach ensures that the simulated implied volatility surface is arbitrage free or even that the simulated implied volatilities are positive since the shock might be negative and larger in magnitude than the today's base implied vol level. – Confounded May 30 '19 at 18:12