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+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.