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