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# Rules of Thumb

Skew Lambda

### Average move vs 1 SD Move

• Doc: Probability Density Function (PDF) of a normal distribution follows 1/2pi* some stuff. To get log normal you have to transform but pi remains.
• So if you know what the sigma is, you can find the average move using the PDF:
• $\color{white} Avg = \sigma * \underbrace{\sqrt{2/\pi}}_{\text{.798}}$

• Related, ATM Straddle in dollars is spot price * .8 * vol * sqrt(252/days of straddle)

### Delta vs ITM Probability

In Black Scholes: The term for delta is N(d1). The term for the probability of finishing in the money is N(d2). What’s the relationship between d2 and d1?

$D2 = D1 - \sigma\sqrt{t}$ The math defines the relationship we figured out intuitively:

The higher the volatility the more delta and probability will diverge!

Delta and probability are only similar when an option is near expiration or when it’s vol is “low”.”

### Butterflies, Straddles and 50d Options

• Even in a continuous distribution, the higher the volatility, the more positively skewed the distribution, the further OTM the 50d call strike lives.
• The cheapest straddle will occur at the median outcome or the ATM strike (should be 50d strikes?)
• The most expensive butterfly will have its “body” near the theoretical mode. This makes sense since a butterfly which is just a spread of 2 vertical spreads is a pure bet on the distribution. If you chart the price of all the butterflies equidistantly across strikes you will have drawn the probability density function implied by the options market

### Variance Drain

• The higher the variance, the lower the median and mode! The distribution gets “squished to the left” as the probability the stock declines increases in exchange for a longer right tail like we saw during the dotcom days.
• The median expected stock price is S – .5 * variance.
• The mode is S – 1.5*variance.

### Variance Equations

Variance Equation Cheat Sheet

## Vol Drag

Geo Return = Arithmetic Return - 1/2 * Vol squared

# Index Correlation

• The average cross-correlation of stocks in an index can be approximated by the ratio of index variance to average weighted stock variance.
• Assume index vol 17, avg stock vol 33
• Using our estimates (.17^2) / (.33^2) = .27 which is in the ballpark of where long term average SP500 index correlations have realized
• Although option folks know how spikey that number can be, especially on short measures

# Monte Carlo Simulation

If X and Y are correlated (retunrs) and you want to simulate with Monte Carlo, say.

Given X(t) and Y(t), pick 2 independent standard normal variables W1 & W2

X(t+dt) = X(t)*(1 + (r – repoX))*dt) + X(t)*volX*W1*sqrt(dt)

Y(t + dt) = Y(t)*( 1 + (r – repoY))*dt) + Y(t)*volY*(W1*rho + W2*sqrt(1 – rho^2))*sqrt(dt)

This is the most simple way. Don’t worry about the exponentials.

# Probability Density Functions

• The
Rebalancing Portfolio of Long Volmacrocephalopod (@macrocephalopod) Tweeted: I thought about power perpetuals a bit more and I really like the design. If price of the underlying goes S to S’ and back to S, price of the power perp also round-trips (unlike a leveraged token, which has a lower price in this scenario because it rebalances to higher leverage)Ben Eofert Hedge Ratioben eiffert volgaSelect content from the quant and vol communityThink Like a Market Maker — Understanding Implied Volatility | by Harel Jacobson | Mar, 2022 | MediumNotes on The Laws Of TradingMoontower on Gamma - Party at the MoontowerWeighting dispersionThe Gamma Of Levered ETFs - Party at the MoontowerVariance and VIX FormulasMoontower Quant CodexInvestment Blogs I Read