FRM 2006-The Option Greeks

发布时间:2010-01-19 共1页

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:
  • The stock price and the wealth ratio (ST/S0) are lognormally distributed
  • The natural log of the stock price [ln(ST)] and the natural log of the wealth ratio [ln(ST/S0)] are normally distributed
Ifa random variable is lognormally distributed (e.g., the stock price),that’s another way of saying the lognormal of the variable (e.g., thenatural log of the stock price) is normally distributed. Notice thatstock 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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