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## Option Valuation

Option delta or hedge ratio

The hedge ratio is commonly referred to as option delta. A ratio is a tool that enables us to summarize the overall exposure of a portfolio of options with different exercise prices and maturity periods. The option ratio is the change in option price for a 1$ increase in share price. A call option has a positive hedge ratio and a put option has a negative hedge ratio.

Under the black scale option formula, the hedge ratio for a call option is N (d1) and the hedge ratio for a put is N (d1)-1. Note that N (d) refers to the area under the standard normal curve up to d. Therefore, the call option hedge ratio should be positive and the put option hedge ratio should be negative and have an absolute value less than 1.0.

Implied Volatility

Black scale option valuation assumes volatility is given. We can ask a different question. What is the volatility (or standard deviation) of the option price observed to be consistent with the Blackscale formula? This is the inherent volatility of the stock. Implied volatility is the volatility implied by the option price. An investor can compare actual and implied volatility.

If the actual volatility is greater than the implied volatility, the investor may conclude that the option fair value is greater than the observed value. Hence, she may consider the option as a good investment. You can use an Excel spreadsheet to calculate the black scale option price and implied volatility.

Dividend paying share options

The share price decreases by an amount that reflects the payment of the dividend. Consequently, the price of the call option will decrease and the price of the put option will increase. Share price risk is considered to have a low component and a risky component. The black scale model includes the risky part of the share price. The current value of the dividend (from the ex-dividend date to the present) can be treated as the de-risked component of the share price. Therefore, to value a call option, we must adjust the share price downward for the present value of dividend payments over the life of the option, and then use the black scale model. We also have to adjust for volatility in the case of dividend-paying stocks because in the black scale model it is the volatility of the risky part of the share price. This is usually ignored in practice.

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