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I am trying to workout how to calculate the implied volatility of the overnight movement from market close to open. That is, the volatility of the from the closing price $S_t$ to the opening price $S_{t+1}$.

If we want an estimate of current volatility from close to close, we can use ATM options that are close to expiry, then turn the annualized vol into a daily vol by:

$$\sigma_{daily} = \frac{\sigma_{annual}}{\sqrt{252}}$$

But since the time interval is affectively 1 tick:

$$\lim_{t\to \infty} \sigma_{overnight} = \frac{\sigma_{annual}}{\sqrt{t}} = 0$$

Which obviously has no basis in reality when the stock does jump up/down quite a bit. Rather than using historical measure, how can we use options to get the implied overnight move?

I am basically trying to get the market’s estimate of overnight volatility

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The implied variance you infer at time $t$ for an option with expiry $T$ is the integral of the instantaneous variances: $$(T-t)\hat\sigma^2(t,T) = \int_t^T\sigma_u^2 du$$ so your request is a bit impossible without extra assumptions (eg. overnight variance is $x\%$ of total day variance).

However, in some cases, you may have an option with the same underlier but in a different region which could help you "complete" your term structure.

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The limit you created is misleading because:

$lim_{t→∞}σ_{daily}$ is defined w.r.t a sequence of values and is not the same as $σ_{daily}$, which is a single value and NOT 0 in a squeezed time interval. For example, if the stock is B.M., the vol in time interval $dt$ is $sqrt(dt)$, which is a part of a sequence which converges to 0 as you squish $dt$ but it is itself NOT 0.

There are no options that trade at close and expire at open, so you cannot directly infer what the market thinks about implied vol overnight.

One way is, you will need an estimate of forward vol from open to close tomorrow and subtract it from current implied vol from close today to close tomorrow to get an estimate. However these aren't very very reliable because short term options aren't very sensitive to vol.

One hack is to scale the normal implied volatility by (Historical_overnight_vol)/(Historical_overall_vol).

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Implied volatilities by definition is sqrt integrated variance till expiry. There is no way to tell any particular section of time apart.

You can make historical estimates that’s it. Sometimes if there’s an overnight event (like CPI), you need to price it accordingly too. Or if you can find a paired stock/index that trades whole day, you can calculate some proxy vol.

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