Pricing Very Short-Dated Options Around Events and Hedging Limits

# How to price very short dated options?


# How to price very short dated options?












I was wondering if there is any industry standard in pricing very short dated options, from say 6h options down to 5 minute options.

My thinking is that as time to expiry gets shorter and shorter, the stock price should resemble more and more a GBM, so a naive B&S should do the job.

With the rise of 0DTE options I'm curious if any practitioner has insights behind pricing those. Also what about binary options exchanges, with like 5-10 minute options. Are those even priced as options?

## Answer by dm63 (score 6)

https://quant.stackexchange.com/a/76135

In interest rate options you can observe the behavior of listed options on bond futures on the last day before expiration. What I’ve noticed:

(A) the most important consideration is whether there are any important events prior to expiration (earnings release, unemployment number, etc). If so, a lot of the implied volatility will be concentrated around this exact time. Thus the BS implied vol is very far from constant. In fact it would just collapse as soon as the event has passed.

(B) as time gets shorter, the bid-offer in volatility terms widens out due to the impracticability of delta hedging over such a short period.

## Answer by fes (score 3)

https://quant.stackexchange.com/a/76137

While I am also not an expert on such options, here are some points to add to the other answers:

- In the question it is suggested that returns over very short horizons would become GBM-like. This is wrong. If anything longer horizon returns should rather look more like generated by such a process. Short horizon returns do not follow a log-normal distribution.

- To add to @dm63 great point about the importance of events, intraday volatility can also be subject to different types of intraday patterns. For e.g. in many markets volatility follows a U-shaped pattern, being highest near the open and close. In this case a very short maturity option should be more valuable near the close than around noon.

## Answer by Arshdeep (score 2)

https://quant.stackexchange.com/a/76133

While I'm not experienced, my guess is in 5 minutes you can't delta hedge enough times to be able to be anywhere close to risk free i.e. you can't expect realized vols to equal the implied vols in such a short period.

So in my mind as long as no arbitrage conditions are satisfied I would price based on supply/demand/market speculation rather than replication. One solution is to raise the implied vol quite high to compensate for the now unhedgeable PnL variance. You can raise it such that say the 10th pecentile of your PnL distribution is greater than some tolerance level.

Edit: Bit of formalism to address the comment:

The delta hedge done once leaks a PnL:

$Gamma*(vol_{implied}-vol_{realized})$

But if we delta hedge multiple times over the life of the option, the net PnL is:

$∑_{i=1}^NGamma*(vol_{implied}-vol_{realized})$

which is sum sort of a weighted average of the difference between implied vol (fixed) and realized vols (random variables). If chosen correctly, sum of realized vols converges to the implied vol in a central limit theorem style (ignoring for once the complexity of gamma weights, and vol autocorrelation).

In this case PnL variance due to discrete hedging is essentially the "standard error" of the convergence procedure. So the more times you delta hedge the better convergence you can get.

In this instance you possibly delta hedge only once so the PnL variance is massive. "n" of the central limit theorem is small. So you can't really expect any meaningful hedging. Also therefore idiosyncratic moves in the stock (which otherwise average out over multiple delta hedges) now completely control your PnL in line with dm63's first comment.

Not to mention that gamma of the option is also quite large close to expiry. Therefore my comment that the product has to be priced with how much "comfort" you have over the PnL variability.

## Answer by Frido (score 1)

https://quant.stackexchange.com/a/76136

I cannot say / do not know if this is industry standard, but how I would go about it:

You know that if $T_1 < T_2$ then $C(S_t, K, T_1) < C(S_t, K, T_2)$ where I have assumed rate and dividend yield are zero for simplicity (but you can easily relax this assumption).

Let $T_1 = 1$ day and $T_2 = 1$ month. Suppose you have an estimate for $IV_1$, and $IV_2$ can be observed in the market. Then find the number $N$ of options $C(S_t, K, T_2)$ to trade such that $$ C^{BS}(S_t, K, T_1; IV_1(t)) = N C^{BS}(S_t, K, T_2; IV_2(t)) $$ So this breaks the problem down into two components:

- The pricing problem which is to estimate $IV_1(t)$, where potential jumps and/or events will play an important role.

- The easier hedging problem which is to find $N$.

At $t=T_1$ you should have $(S_{T_1} - K)_+ \approx N C^{BS}(S_{T_1}, K, T_2; IV_2(T_1))$. Since you've basically hedged an option with an option you do not need to worry about delta-gamma hedging because the option $C^{BS}(S_{T_1}, K, T_2; IV_2(T_1))$ can still be traded quite easily.