# How to Benefit from Hedging Inaccurately Priced Options

## Summary

This article describes a hedging strategy whereas the possibility exists to benefit from an inaccurately priced option based on the difference between implied and historical volatilities. Investors also learn, through these volatilities, how to recognize if options are priced above or below their fair value.

The value of a stock and its options are determined by two characteristics, price and volatility. There are two types of volatility: the historical volatility (HV) and the implied volatility (IV).

The HV is the risk of price: it is the dispersion of returns around the average return of a stock. The IV is the volatility of options. Each option has its IV. To simplify, it is a measure of the offer and demand for the option.

Both volatilities fluctuate however, the IV fluctuates much more than the HV. The IV is calculated on every trade whereas the HV is computed using only the daily close price.

Table 1 shows the stock Agrium along with a few of its options. On October 31, AGU closed at \$50.34 and the HV was 45% (A).

The ATM Agrium put December 50 was priced at \$2.90 and the IV of this option was 41.74% (B).

The IV has the property to inflate or deflate option prices. If the IV is high, the option is “fat”, therefore expensive. If the IV is small, the price is “lean” or deflated.

We assume that the price of an option is “fair” when the implied volatility is the same as the HV of the underlying stock. Therefore if the IV is lower, the price is at discount; if the IV is above, it is at premium. We assume also that the fair volatility is the HV and not the IV. Therefore, the option tends to reach the price given by the implied volatility equal to the historical volatility.

Given these premises, in our example the put is under priced. The IV is lower than the HV by 3.26 percentage points (= 45 – 41.74).

Using an option calculator, with an implied volatility of 45% the fair price is \$3.12.

We believe the option will increase to \$3.12. One way to profit from this inaccuracy is to buy puts. However, this strategy is risky, especially if AGU stock increases in value. The best solution is to separate the risk of the stock price from the option inaccuracy by also purchasing AGU shares. How many shares? The option delta is the answer. Delta is the sensitivity of the option price to a change of \$1 in the underlying stock. Because the size of one option contract is 100 shares, delta is equivalent to the number of shares that makes the same gain or loss as one option contract.

The option calculator shows that the delta of this put is – 0.44. This means that the change in the option price is equivalent to 44 shares of AGU, when the stock moves by \$1. If we purchase 10 option contracts, we would need to buy 440 shares of stock.

If the stock price rises by \$1, each put option will decrease in value by \$0. 44 or \$44 per option contract, resulting in a \$440 loss. The Delta shows precisely how many shares of the stock we must hold to offset our exposure to AGU.

 1,000 puts @ \$2.90 \$2, 900 440 shares @ \$50.34/share \$22,149.60 Total outlay \$25,049.60
 Stock price 49 50,.34 51 Put price 3,75 3,12 2,84 Profit (loss) on each put 0,85 0,22 (0,06)
 Stock price 49 50,34 51 Value of 1,000 put options 3 750 3 120 2 840 Value of 440 shares 21 560 22 149,60 22 440 Total 25 310 25 269,60 25 280 Profit ( = Value – Cost from Table 1) 260,40 220,00 230,40

## Conclusion

The profit is small compared to the capital needed to establish this strategy. Volatility estimates are never fully reliable. Both historical and implied volatilities fluctuate over time. Therefore they are like moving targets. Although delta-neutral positions might eliminate exposure to risk from fluctuations in the value of the stock, they still are subject to volatility risk.

## The author

### Charles K. Langford

PhD, Fellow CSI

Charles K. Langford is President of Charles K. Langford, Inc, Portfolio Managers. He teaches portfolio management at School of Management (École des Sciences de la Gestion), University of Québec (Montréal). He is the author of 14 books on portfolio management, derivatives strategies and technical analysis.

Until 2007 he has been vice-president overlay risk management for Visconti Venosta Teaspoon Approach Management, Ltd. Until 1990 he was portfolio manager for Refco Futures (Canada) Ltd.

He has received a Bachelor degree from Université de Montréal, a Master degree and the PhD from McGill University (Montreal); he is Fellow of CSI (Canadian Securities Institute).