# Option Volatility and Pricing [PDF]

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PDF Title : Option Volatility and Pricing 25 Chapters (full) 2nd Edition 588 Pages Sheldon Natenberg 4.6 MB English mheducation.com Available

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### Option Volatility and Pricing – Book

Suppose that a trader sells a realized variance contract at a volatility of 20 (percent), equal to a variance of 202 = 400. If the actual realized volatility over the life of the contract is greater than 20 percent, the trader will lose money; if the actual realized volatility is less than 20 percent, the trader will make money. How can a trader hedge this position? A variance position can be replicated by purchasing a strip of options across all exercise prices.

To create a position with a constant-variance exposure, it is necessary to purchase 1/X 2 (where X is the exercise price) of each option. Then, by dynamically hedging the entire position in order to remain delta neutral throughout the life of the variance swap, the total value of the strategy will exactly match the actually realized variance of the variance contract.

It may seem that a volatility position can be replicated using the same approach. But the fact that volatility is the square root of variance means that if the variance exposure is constant, the volatility exposure cannot be constant. Let’s return to an earlier example where a realized volatility contract with a vega exposure of \$10,000 was purchased at a price of 20.

We can compare the outcomes in two different cases. In the first case, the contract is settled in variance points, with each point having a value equal to the notional vega divided by twice the volatility price: \$10,000/(2 × 20) = \$250. In the second case, the contract is settled in volatility points with each point have a value of \$10,000.

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