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# High level lessons

A refresher on what a batch is

A set of 500 simulations in which we:
1. Buy a 1-year ATM call for 25% implied vol. The call’s initial value is \$9.95 and has a vega of \$.396
1. This implies our daily breakeven move is 1.5%
1. Hedge the delta daily
1. Each batch is seeded with an expected` realized volatility` that governs the stock diffusion process. Any single 1-year run is a sample that can differ from the batch `realized vol`
In this section, instead of zooming in a particular run we will look examine what happens in a batch in aggregate. The format will be:
1. Description of notable stats
1. Observations from charts
1. Summary table to compare batches

### Description of notable stats

Let’s explore the batch of runs for 25% realized volatility.
For the 500 runs:
• Mean realized volatility is 24.77% with a standard deviation of 1.09%
• Therefore the realized underperformed the IV on average by .23% or less than a quarter vol point.
• The average P/L as a percentage of the initial option price of \$9.95 over the 500 runs is -.5%. This makes sense since the RV slightly underperformed the IV.
• The initial vega is \$.396. You would expect to make about \$.40 per option for every 1 vol point of outperformance. Since we lost -.5% of our premium on average we can convert this to the statement: We lost .12 vol points on average with a standard deviation of 1.38 vol points.
This forms the basis of a simple, handy chart.

### Observations from charts

We can see how our P/L varies in vol points as a function of `RV-IV` in vol points.
• When `RV-IV` is positive, the RV outperformed the 25% IV and vice versa

• The black line is a fabricated slope of 1 line. It’s a useful visual reference because it’s the line you’d get if your P/L in vol points simply comes down to “I win or lose vol points in direct proportion to the spread between RV and IV.”
• The red line is the regression line from the sample data. The R-square is for the red line.

Based on this batch of runs, it seems that when IV and RV are close together that the expected p/l in vol points will mirror the spread RV vs IV.

### Summary table to compare batches

Note: Sharpe, P/L and Win% are from the perspective of the winning side (so if RV < IV it’s the seller’s POV)

Thoughts
• St dev of of RV-IV spread is larger if the RV is simply larger, but it’s close to a constant proportion of the RV level
• The goodness of fit between the P/L in vol points vs outperformance decays as RV and IV diverge
• Across the board, the standard deviation of the p/l in vol points is wider than the standard deviation of the RV-IV spread
• If you buy an option for about 5% less than what it’s worth (so if you bought 25% IV when the realized ended up being 26.25%), the Sharpe ratio would be about 1. In this example, the 1-year option priced with 25% vol is worth \$9.95 so you would need to buy it for about \$9.50 or a bit more than a vol point of edge to have a Sharpe 1 trade.
• The Sharpe of selling an option for 5 vol points of edge (25% IV vs 20% RV) vol seems to be about 50% higher than buying an option for 5 vol points of edge (25% IV vs 30% RV). I’m not sure what’s going on there but I’d hazard a guess based on a possible asymmetry — if the RV is very low, the stock probably doesn’t travel far so you maintain short gamma for a longer time as the stock stays nearer to the strike. Remember your hedged profits are a function of the daily over/underperformance of the implied move times the gamma. If you buy an option for much less than it’s RV, it’s possible that the high RV is generated by a large move that takes you far from the strike but then you are left without any gamma to scalp over the remaining option life. The results are congruent with this guess — the higher the realized vol, the less average gamma you have.
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Real World Caveat
If the realized vol is high, the implied vol will tend to follow it. For OTM options, the higher IV will narrow the distance between the spot and strike in standard deviations. This will “push” the option close to ATM, thus increasing its gamma.
There’s no advice here. It’s just a reminder that you need to think dynamically about how the greeks would change, in turn, altering your hedge ratios as your gamma and delta change.
• When you mark IVs to market, you surrender control of your hedge ratios.
• If you mark your IV to model, you are ignoring any wisdom-of-crowds IV information from the surface.
You just have to understand the trade-offs and how the biases of each method interact with your particular strategy.