How Delta Hedging Changes Short Straddle Risk and P&L

# Is a volatility forecast essentially a delta forecast in vanilla European options?


# Is a volatility forecast essentially a delta forecast in vanilla European options?












As the title suggests.

I want to understand why delta hedging is done. I'd like to illustrate with an example:

Say you have 7 dte option chain with 15.8% IV ATM straddle on an underlying of spot 100.

Price of straddle 0.8 x 100 x 0.158 x sqrt(7/250) = 2.1151

Now your 7 days volatility forecast says 12% and so you short straddle on that chain.

So for 7 days, the implied move on underlying is 0.8 X 0.12 x sqrt(7/250) = +- 1.606% (forecasted delta).

The range of this implied move on spot is 101.606 and 98.394.

At expiry if the spot expires at 101.606, RV (realised volatility) is equal to forecast and value of straddle is 2.1151 - 1.606 = 0.5091 Similarly, if spot expires at 98.394, RV is equal to forecast and expiry value of straddle is 2.1151-1.606 = 0.5091

So if the forecast is the edge, then why do we need to delta hedge?

What am I missing?

## Answer by Newquant (score 1)

https://quant.stackexchange.com/a/75850

In your specific example, you are trading 7 day volatility, that's fine as it's own trade, however more typical is more frequent delta hedging, or delta hedging when $|\Delta| > \text{threshold}$. Depends on the trader/firm.

In your case, because you aren't trading 1 day volatility, or hourly volatility, you are happy to pick up (and hold) deltas over the life of your trade. So if the market falls, you will get long, and explicitly want the market to return to it's prior level. Delta, generally, is equivalent to edgeless risk, you want the market to go some direction, but the market is noisy and may or may not move in your direction, some don't want to hold that risk over the period. By hedging deltas, traders locally eliminate market risk (ignoring spot-IV correlations).

Consider the toy example where a trader is short a straddle with $\Gamma_{\text{Cash}} = \\\$2mm$. Assuming that this gamma stays constant over the option life and as spot/time changes (a stretch, yes) here are two examples as the market moves down 1% every day for 5 days (for simplicity assume that's 5% down over the week) whilst IV is 8%:

- The trader hedges every day on close:

$P/L = 5 \times \\\$1mm \times (0.005^2 - 0.01^2) = -\\\$375$

- The trader allows deltas to pick up over the week:

$P/L ~= \\\$1mm \times (0.025^2 - 0.05^2) = -\\\$1,875$

Generalising this, the ratio of the less frequent to more frequent hedging over equivalent moves in the underlying (like -1% per day) is:

$(N^2 \times IV^2 - N^2 \times RV^2) / (N \times (IV^2 - RV^2))$

$= N^2 (IV^2 - RV^2) / (N \times (IV^2 - RV^2))$

$= N^2 / N$

$= N $

So when hedging frequency is $(1/N)$-th of the reference frequency (i.e. daily), the ending P/L is $N$ times the magnitude. This is because the local delta-neutral option exposure is to variance, not volatility, so larger moves grow quickly on your P/L. Naturally this cuts both ways, if the ending spot price equals the spot at $t_0$, then you'll collect the full premium, whereas if you'd hedged you'd have added some losses, but that's not the game generally and you'd want to be hedging more frequently to cut market risk.

There's an implicit assumption in your example when not delta hedging that the return process is not independent. Ignoring costs, execution impact, if your volatility forecast is N%, there's no reason to not hedge as often as possible, as this reduces the volatility of your final P/L (proportional $1/\sqrt{\text{frequency}}$), whilst your expected P/L is unchanged. However, if you believe there's a negative autocorrelation in returns (mean-reversion), then you wouldn't want to hedge a short gamma position, since that provides you with the delta you want as the market moves -- positive deltas pick up when the market falls, and vice versa - 4if you hedge these then you won't make as much as you could if the market is actually mean reverting.

So to your point, you are exactly correct that your forecast is your edge, but delta hedging allows you to lock in that edge over each timestep, and through the LLN, if you can lock in your edge over N hedging intervals (effectively a sample of implied - realised variance between $t_0 \to t_0+n$), that is preferable to relying on a single sample of variance between $t_0$ and expiry.

Hope that helps happy to answer any qs. Thx.