I 'he problem with buying volatility low is that one may experience even lower volatility and that selling volatility high one may experience even higher volatility.
Unless the low is extremely low (or in the case of Japanese warrants, zero) or the high is extremely high there is no way of guaranteeing a profit. There are occasions when it is possible to simultaneously buy volatility low and sell volatility high on the same instrument. In such situations, whatever the actual volatility ;i profit results. Consider the following as an extreme example. A i vitiiin stock is trading at $100. There exist two. one-year call options. v\ith strike prices of $100 and $110 priced at $5.98 and $5.04 respec-n\cl\. These prices imply volatilities of 15% and 22% respectively. I IKTC is clearly something wrong here.
Both options are exercisable •nto the common underlying stock and so both should imply the same
volatility. If this situation were to exist in reality a trader would construct a portfolio long of the (cheap) $100 strike option and short of the (expensive) $110 strike option in an attempt to capitalise on the difference in the two prices. The portfolio would be the vertical call spread or combination #1 discussed in Chapter 7. The combination has long exposure and would be hedged with short stock. The rationale behind this strategy is that whatever the true volatility of the stock, buying at 15% and selling at 22% will result in an arbitrage profit. This profit may materialise in one of two ways. One way would be that the market-place suddenly adjusted the option prices so that both have the same (new) implied volatility. For example, if the market adjusted both options to have implied volatilities of 20%
the prices would be $7.97 and $4.29 respectively and a profit of 100 X (7.97 - 5.98 + 5.04 - 4.29) = $274 would result. Another way may be that the position is run to expiry with continual dynamic hedging. Whatever the final volatility, the profit (in theory) should be and often will be similar, but because hedging costs are not zero, the final profit will usually be smaller. However, even in this situation a profit is not always guaranteed. It is possible to think of a number of very contrived stock price paths that result in a loss. The simplest one is that of the stock price staying completely fixed at $100 whereupon both options would expire worthless resulting in a loss of 100 x (5.98 -5.04) = $96.
In the example above the simultaneous buying of volatility at 15% and selling at 22% would most probably result in a profit, but what if the implied volatilities were nearer at say 15 and 17% respectively? Here, we most certainly could not guarantee a profit. In most exchange traded markets it is not uncommon to find options in the same expiry cycles with different strike prices having differing implied volatilities. Initially many practitioners observing such differences put on hedged positions such as the one described above. In the early stages of exchanged traded options markets when the anomalies were large, many of these positions produced profits but some produced losses. Over time, the magnitude of the anomalies have reduced, but they are still there. Some practitioners say that the model is wrong and that implied volatility is not a valid measure of value. Others say that the assumption of a lognormal distribution is not valid and research in this area is ongoing. However, some academics have shown that the anomalies can be explained by relaxing the constant volatility assumption. Almost everyone agrees that volatility varies and that in some
markets is related to the price of the underlying. When stock prices fall (rise) there is often an increase (decrease) in volatility. It has been shown that incorporating these and other aspects into the Black and Scholes model would be consistent with different strike prices having different implied volatilities. So the model can still be used but one needs to input different volatilities. Most market participants follow this practice.
The way in which the implied volatilities vary across strike prices depends on the market and market conditions. Stock options typically have higher volatilities at lower strikes and lower volatilities at higher strikes. The standard reasoning behind this type of volatility profile is that in a falling market everyone needs out-of-the-money puts for insurance and will pay a higher price for the lower strike options.
Also equity fund managers round the world are long billions of dollars worth of stock and like writing (selling) out-of-the-money call options against their holdings as a way of generating extra income. It is believed that this large scale selling of options has the effect of lowering the implied volatilities of higher strikes. Figure 8.2 shows a graph of such volatilities versus strike prices and this is referred to as a volatility profile or volatility skew. Practitioners refer to this particular profile as a volatility smirk.
Other markets such as options on commodities and commodity futures sometimes have a reverse volatility smirk profile. This profile that places higher volatility on higher strikes and lower volatility on lower strikes is sometimes explained by a combination of government intervention and the risk of shortages.
In certain commodity markets
Figure 8.2 Volatility profiles
such as soyabeans there is an unspoken belief that the government will always support the farmer and not let the price fall too low. If the government is going to support a market then there may be no need to worry about a large price fall.
Believing this, some speculators are tempted to sell aggressively puts with the result that prices and hence implied volatilities fall. At the other extreme, there are from time to time unforeseen factors that will result in supply shortages and since, in theory, there is no upper limit on the price of a commodity, individuals are prepared to pay up for short cover insurance. The history of commodity markets is littered with examples of extremely violent up moves and it is easy to understand the demand for higher strike price options.
Some markets, such as interest rate options, display what is known as a volatility smile. This is where the at-the-money option has a low volatility and either side the volatility is higher. One explanation for this is that individuals have a propensity to sell at-the-money options and buy out-of-the-money options. A popular strategy known as the Butterfly involves selling two at-the-money options, hedged with the purchase of one out-of-the-money option and one in-the-money option. This and other similar strategies are low risk and result in maximum pay-off if the options expire with the underlying exactly at the middle strike price.
Whatever the reason, options with different strike prices often have different implied volatilities and this should be incorporated in any risk-monitoring exercise.
The software supplied with this book gives the user the ability to input a different volatility for each strike price. Figure 8.3 shows one aspect of the effects of using different volatility profiles with the complex combination #4 discussed in Chapter 7.