One of my favorite follows on #voltwit is @SqueezeMetrics. The account more colloquially known as “the Lemon” has a personal crusade against using implied vol to refer to option prices. Recall, volatility is just the asset’s standard deviation of returns. It’s usually an annualized number. So if the SPX has a 15% volatility that just means you expect the SPX to return +/- 15% about 68% of the time^{1}

“Lemon” prefers using the average expected move, more commonly known as the straddle.

Thus tweeted the Lemon:

*I think the convention of turning the straddle price into an annualized standard deviation is obfuscatory. Straddle gives you the average move that’s priced in. Why complicate that?*

I can see how the distinction between average move (aka the “straddle”) and standard deviation (aka the “vol”) is “obfuscatory”.

So let’s clear it up.

Expect to learn:

- The math relationship between the straddle and the volatility
- How the distinction relates to win rates and expectancy
- Why the spread between the straddle and volatility can vary in turn altering win rates
- My own humble opinion on the matter

**Turning Volatility Into A Straddle and Vice Versa**

A handy formula every novice trader learns is the at-the-money straddle approximation^{2}:

**Straddle = .8Sσ√T**

where S = stock price

σ = implied volatility

T = time to expiry (in years)

Ok, let’s pretend the SPX is $100, there’s 1 year to expiry, and implied volatility is 15%. Plug and chug and we get a straddle value of $12 or 12%. Pretty straightforward.

Straddle/S = .8*σ√T*

If we want to simply speak in annualized terms then we can assume T = 1 and can simplify:

**Straddle as % of Spot = .8 x ***σ*

Which of course means if you know the annualized straddle price as a percent of spot you can go in reverse to get the volatility:

*σ = ***Straddle as % of Spot x 1.25**

**When is this useful?**

Let’s say based on a stock’s past earnings move you see that it usually moves 5% per day. In other words, the earnings day *straddle *should be 5%. Then, you can find the standard deviation:

5% x 1.25 or 6.25%

The standard deviation is a volatility which you can annualize to plug into an options model which will spit out a 5% straddle price.

6.25% x **√**252 = 99.2% vol

Knowing the 1-day implied volatility is useful when you are trying to estimate a term volatility for a longer period that includes the earnings day (topic for another time).

**What’s the practical difference between straddles and volatility?**

Volatility is a number you stick into a model to generate a price for an instrument you actually trade. In this case, a straddle. If you input 15% vol into our above example, you will find that a 1-year straddle will cost you 12% of spot.

If you buy this straddle your return is equal to:

Absolute value of SPX return – 12%

Your worst case scenario is the SPX is unchanged and you lose your entire 12% premium. You are “long volatility” in that you want the SPX to move big one way or another.

So let’s talk about what we really care about — expectancy and win rates.

**Expectancy**

The point of the model is to generate a price that is fair for a given volatility. 12% was the fair theoretical value for a 15% vol asset.

If you pay 12% for the straddle on a 15% vol asset you have **zero expectancy**.

But that’s not the whole story.

**Win Rates**

Expectancy and win rate are not the same. Remember that the most you can lose is 12% but since there is no upper bound on the stock, your win is theoretically infinite. So the expectancy of the straddle is balanced by the odds of it paying off.** You should expect to lose more often than you win for your expectancy to be zero since your wins are larger than your losses.**

So how often do you theoretically win?

A fairly priced straddle quoted as percent of spot costs 80% of the volatility. We know that a 1- standard deviation range encompasses about 68% of a distribution. How about a .8 standard deviation range?

Fire up excel. NORMDIST(.8,0,1,True) for a cumulative distribution function. You get 78.8% which means 21.2% of the time the SPX goes up more than .8 standard deviations. Double that because there are 2 tails and voila…you win about 42% of the time.

**So in Black-Scholes world, if you buy a straddle for correctly priced vol your expectancy is zero, but you expect to lose 58% of the time!**

**Outside Of Black-Scholes World**

The Black Scholes model assumes asset prices follow a lognormal distribution. This leads to compounded or logreturns that are normally distributed. This is the world in which the straddle as percentage of spot is 80% of the annualized volatility.

In that world, you lose when you buy a fairly priced straddle 58% of the time. Of course fairly priced means your expectancy is zero. What happens if we change the distribution?

I’m going to borrow an example of a binary distribution from my election straddle post:

- 90% of the time the SPX goes up 5.55%
- 10% of the time the SPX goes down 50%
Expected move size = 90% x 5.55% + 10% x 50% = 10%

Expected move is the same as a straddle. The straddle is worth 10% of spot. **Your expectancy from owning it is 0.**

If this was Black-Scholes world, we would say the volatility is 1.25 x 10% = 12.5% (not annualized). But this is not Black Scholes world. This is a binary distribution not a lognormal one. What is the standard deviation of this binary asset?

We can compute the standard deviation just as we do it for coin tosses or dice throwing.

*σ= √(.9 x .0555*^{2}* + .1 x .50*^{2}*)*

*σ = 16.7% *(again, not annualized so we can compare)

Note that your straddle is 10% but your volatility is 16.7%. That ratio is not the 80% we saw in the lognormal world, but instead it is 60%.

Note you cannot repeat the earlier process to find the win rate. You can’t just NORMDIST(.6,0,1,True) because the distribution of returns is not normal. Luckily, with a binary distribution our win rate is easy to see. In this example, if you pay 10% for the straddle you lose 90% of the time.

** Even if you paid 6% for the straddle you still lose 90% of the time. However if you bought the straddle that ‘cheap’, your expectancy will be massively positive! **

**My Own Humble Opinion**

When there is a short time to expiration, arbitrarily let’s say a few weeks, my mind’s intuition might latch on to a straddle price. I might think in terms of expected move as one does for earnings in getting a feel for what is the right price. But on longer time frames I prefer to think of implied vol because I am going to be dynamically hedging. Measures of realized vol can be readily compared with implied vol.

If I look at a straddle price for a long period of time, say 1 year, I might fall into a trap thinking “20%? That just sounds high.” I’d rather just compare the implied vol which would be 25% (remember 1.25 x straddle), to realized vol since I am interested in the expectancy of the trades, not the win-rate.

There are all kinds of house of mirrors when looking at vols and straddles and thinking about winning percentages. As Lemon says, it’s “obfuscatory”. Everyone should do what works for them.

If you tend to be long vol, be aware having more losing months than winning months might be completely normal. It’s baked into the math. And the more skewed the distribution, the worse your batting average will be.

But in the long run it’s your slugging percentage that matters.

**Recap**

- Straddles as a percent of spot are 80% of the volatility (all annualized)
- Straddles tell you the average move.
- Fair straddles have zero expectancy.
- You lose more often when you win when you are long a straddle.
- Your win sizes are larger than your losses.
- Skewed distributions change the relationship between win rates and expectancy. They also change the relationship between straddle prices and standard deviations.

- This assumes returns follow a normal distribution which we know is not true. There’s fat tails. Outside of derivative traders, most people think there’s some positive mean return made of a risk free rate + risk premium. This would shift the mean and draw the boundaries of the 1 standard deviation range around that. So if the mean return of the SPX was 10% then the 1 standard deviation range would extend from -5% to +25%.
- Technically, this approximation is for the at-the-forward strike not the ATM strike. The ATF strike is the strike where the call and put are equal because it is the carry-adjusted spot price. For a complete derivation see https://brilliant.org/wiki/straddle-approximation-formula/

Incredible post as always.

Could I point out one correction (I might be mistaken):

With regard to the stock that has an expected move of 5% on earnings day,

The O/N vol should be approx 125%

S.D. of 6.25% * SQRT (365) (instead of * SQRT(252)) since we quote all options with calendar day convention.

Do let me know if you agree.

I generally defer to specifying the model when talking about it, so it’s annoying but just caveating what model you use when you make the assumption is best. I used calendar models but adjusted the vols they spat out for analytics purposes.

By habit i would typically presume a vol was trading day not calendar day (as an approximation it’s closer to the actual day count I would use to clean model outputs)