• Rules of Thumb
• Average move vs 1 SD Move
• Delta vs ITM Probability
• Butterflies, Straddles and 50d Options
• Variance Drain
• Variance Equations
• Vol Drag
• Index Correlation
• Vol Regime Evidence
• Monte Carlo Simulation
• Probability Density Functions

Rules of Thumb

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

Interesting Observations About 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

Vol Drag

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

Vol Regime Evidence

Monte Carlo Simulation

Here, for your notes -Doc

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