Monday, January 4, 2010


Option valuation

Option’s delta or Hedge ratio

We have earlier explained the concept of the option’s delta. The hedge ratio is commonly called the option’s delta. The hedge ratio is a tool that enables us to summarize the overall exposure of portfolios of options with various exercise prices and maturity periods. An option’s hedge ratio is the change in the option price for a 1$ increase in the share price. A call option has a positive hedge ratio and a put option has a negative hedge ratio.

Under the black scholes option formula, the hedge ratio of a call option is N (d1) and the hedge ratio for a put is N (d1)-1. Recall that N (d) stands for the area under the standard normal curve up to d. Therefore, the call option hedge ratio must be positive and the put option hedge ratio is negative and of smaller absolute value than 1.0.

Implied Volatility

The black scholes option valuation assumes that the volatility is given. We can ask a different question. What is the volatility (or standard deviation) for the observed option price to be consistent with the black scholes formula? This is implied volatility of the stock. Implied volatility is the volatility that the option price implies. An investor can compare the actual and implied volatility.

If the actual volatility is higher than the implied volatility, the investor may conclude that the option’s fair price is more than the observed price. Hence, she may consider option as potentially a good investment. You can use the excel spreadsheet to calculate the black scholes option price and implied volatility's.

Dividend paying share option

The share prices go down by an amount reflecting the payment of dividend. As a consequence, the value of a call option will decrease and the value of a put option will increase. The share price is assumed to have a risk less component and a risky component. The black scholes model includes the risky component of the share price. The present value of dividends (from ex-dividend dates to present) can be treated as the risk-less component of the share price.

Thus, for valuing a call option, we should adjust downwards the share price for the present value of the dividend payments during the life of the option, and then use the black scholes model. We also need to adjust the volatility in case of a dividend-paying share since in the black scholes model it is the volatility of the risky part of the share price. This is generally ignored in practice.

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