902 Part VI: Measuring and Trading Volatility 
Each of the risk measures can be derived mathematically by taking the partial 
derivative of the model. However, there is a shortcut approximation that works just 
as well. For example, the formula for gamma is as follows: 
x=ln[ P ]/v-ft+v-ft 
s X (1 + r)t 2 
r - e(-x212) 
- pv ✓ 27tt 
There is a simpler, yet correct, way to arrive at the gamma. The delta is the par­
tial derivative of the Black-Scholes model with respect to stock price - that is, it is 
the amount by which the option's price changes for a change in stock price. The 
gamma is the change in delta for the same change in stock price. Thus, one can 
approximate the gamma by the following steps: 
1. Calculate the delta with p = Current stock price. 
2. Set p = p + 1 and recalculate the delta. 
3. Gamma = delta from step 1 - delta from step 2. 
The same procedure can be used for the other "greeks": 
Vega: 1. Calculate the option price with a particular volatility. 
2. 
3. 
Theta: 1. 
2. 
3. 
Rho: 1. 
2. 
3. 
Calculate another option price with volatility increased by 1 %. 
Vega = difference of the prices in steps 1 and 2. 
Calculate the option price with the current time to expiration. 
Calculate the option price with 1 day less time remaining to expiration. 
Theta = difference of the prices in steps 1 and 2. 
Calculate the option price with the current risk-free interest rate. 
Calculate the option price with the rate increased by 1 % . 
Rho = difference of the prices in steps 1 and 2. 
THE GAMMA OF THE GAMMA 
The discussion of this concept was deferred from earlier sections because it is some­
what difficult to grasp. It is included now for those who may wish to use it at some 
time. Those readers who are not interested in such matters may skip to the next sec­
tion.