Analysis of Asset Allocation

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Options: Implied volatility
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As we have seen, the volatility of an asset is a measure of the variability of its returns (see Black and Scholes model). Traditionally, the volatility is measured on past prices of the assets. But for investors who have a high regard for the wisdom of the market, the ideal estimate of volatility would come from pooling those who are trading in the market.

We have seen in the Black and Scholes formula that the fair option value is a function of the security price, the time to expiration, the exercise price of the option, the riskless interest rate, and the security's price volatility. Given the values of the other parameters and the market price of the option, we can solve for a unique volatility that will lead to that market option price. If the market price of the option is taken to be the correct price, the volatility we back out of the formula will be the estimate of the correct volatility.

Presented another way, since in the aggregate the market takes the market price to be correct, the volatility implied by the market price is the market's opinion of what the volatility should be.

The implied volatility is the value that solves the equation (see Black and Scholes):

d1 = [ln(X / K) + (S2 / 2) t] / [S (t)0.5]

d2 = d1 - S (t)0.5

C = [X N(d1) - K N(d2) e-rt

P = K N(-d2) e-rt - X N(-d1)

In the next topic, we will study the different options' strategies.

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