The exam wants you to be familiar with six option Greeks: delta, gamma, vega, theta and rho. Except for gamma (which is a *second *partial derivative), all are** first partial derivatives** with respect to option price (or portfolio value, if we are referring to a portfolio of options): *they are rates of change in option (or portfolio) price with respect to a change in some variable*. The variables are stock price (delta), volatility (vega), time (theta), and the risk-less interest rate (rho).

**Delta **is the change in option price (portfolio value) given a change in the stock price:

**Vega **is the change in option price (portfolio value) given a change in volatility:

**Theta **is the change in option price (portfolio value) given the passage of time:

And**rho **is the change in option price (portfolio value) given a change in the risk-less interest rate:

Gammais a second partial derivative with respect to the option price; thatmakes it the first derivative of the Delta. Remember that Gamma is"related to" Delta. Both are derivatives**with respect to the price **of the underling asset. Delta is the first derivative; Gamma is the second derivative:

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Lognormal Property of Stock PricesPosted by**David Harper** on 23rd August 2006

TheBlack-Scholes option pricing model assumes that a stock moves in aGeometric Brownian motion (GBM). Under GBM, the natural log of thestock price at future time T (ST) is normally distributed:

And the ratio of tomorrow’s stock price divided by today’s stock price (ST/S0) is also normally distributed.**This formula is important:**

In words, the natural log of the wealth ratio (ST/S0) has an expected mean that*almost *reachesthe expected return scaled (multiplied) by time (T) and a standarddeviation scaled (multiplied) by the square root of time. Remember thelognormal/normal distinction:

**returns** are *normally *distributed but stock **prices** are *lognormally *distributed.A periodic stock return can go up or down (e.g., plus or minus 10%) buta stock price cannot fall below zero, nor can the wealth ratio (ST/S0) fall below zero. As such, the stock price/level is captured by a nonzero distribution:

There are two parameters in the normal distribution above: expected value (mean) and standard deviation. The standard deviation*scales with the square root of time*. You know this one by now: volatility scales with the square root of time. The mean looks curious: the expected mean is* depressed by one-half the variance*. The idea is that volatility depresses returns!

Considersix period returns as below. The geometric average (a.k.a., compoundannual growth rate) of this brief series is 105.4% and the arithmeticaverage is 105.8%:

The standard deviation of the population [=STDEVP()] is 8.9% so that the variance is 0.8% [8.9%2].The geometric average (CAGR) is always lower than the arithmeticaverage (unless the series has zero volatility). Notice that the CAGRis lower by one-half (0.4%) the variance.*Volatility depresses the geometric average*.

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And

Gammais a second partial derivative with respect to the option price; thatmakes it the first derivative of the Delta. Remember that Gamma is"related to" Delta. Both are derivatives

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Lognormal Property of Stock PricesPosted by

TheBlack-Scholes option pricing model assumes that a stock moves in aGeometric Brownian motion (GBM). Under GBM, the natural log of thestock price at future time T (ST) is normally distributed:

And the ratio of tomorrow’s stock price divided by today’s stock price (ST/S0) is also normally distributed.

In words, the natural log of the wealth ratio (ST/S0) has an expected mean that

- The stock price and the wealth ratio (ST/S0) are
distributed**lognormally** - The
**natural log of**the stock price [ln(ST)] and the natural log of the wealth ratio [ln(ST/S0)] aredistributed**normally**

There are two parameters in the normal distribution above: expected value (mean) and standard deviation. The standard deviation

Considersix period returns as below. The geometric average (a.k.a., compoundannual growth rate) of this brief series is 105.4% and the arithmeticaverage is 105.8%:

The standard deviation of the population [=STDEVP()] is 8.9% so that the variance is 0.8% [8.9%2].The geometric average (CAGR) is always lower than the arithmeticaverage (unless the series has zero volatility). Notice that the CAGRis lower by one-half (0.4%) the variance.

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