7 Purposes
13.11 VOLATILITY
In essence, volatility is a measure of the variability (but not the direction) of the price of the underlying instrument, essentially the chances of an option being exercised. It is de- fined as the annualised standard deviation of the natural log of the ratio of two successive prices.
Volatility is a statistical function of the movement of an exchange rate. It measures the speed of movement within an exchange rate band, rather than the width of that band.
Historical volatility is a measure of the standard deviation of the underlying instrument over a past period and is calculated from actual price movements by looking at intraday price changes and comparing this with the average (the standard deviation). The calculation is not affected by the absolute exchange rates, merely the change in price involved. Thus, for example, the starting and finishing points for two separate calculations could be exactly the same but could give two very different levels of volatility depending on how the exchange rate traded in between. Thus, if the market has traded up and down erratically, the reading will be high. If instead it has gradually moved from one point to the other in even steps, then the reading will be lower.
Implied volatilityis the volatility implied in the price of an option, i.e. the volatility that is used to calculate an option price. Implied volatilities rise and fall with market forces and tend to reflect the level of activity anticipated in the future, although supply and demand can at times be dominant factors. In the professional interbank market, two-way volatility prices are traded according to market perception and these volatilities are converted into premium using option models. Implied volatility is the only variable affecting the price of an option that cannot be directly observed in the markets, thus leading to the typical variations in price inherent in any marketplace.
Actual volatilityis the actual volatility that occurs during the life of an option. It is the difference between the actual volatility experienced during delta hedging and the implied volatility used to price an option at the outset, which determines if a trader makes or loses money on that option.
In summary, implied volatility is a timely measure, in that it reflects the market’s perceptions today. On the other hand, historical volatility is a retrospective measure of volatility. This implies that it reflects how volatile the variable has been in the recent past. But it has to be remembered that it is a highly objective measure. Implied volatilities can be biased, especially if they are based upon options that are traded in a market with very little liquidity. Also, historical volatility can be calculated for any variable for which historical data is tracked.
Volatility affects the time value or risk premium of an option, as an increase in volatility increases the time value and thus the price of the option. Likewise, a decrease in volatility lowers the price of the option. For example, consider the position of the writer of an option, whereby, say, a bank sells an option to a client, giving the client the right to purchase dollars and sell Swiss francs in three months’ time. In order to correctly hedge the position, consider what will happen in three months’ time. If the spot is above the strike price of the option, the client will exercise the option and the bank will be obliged to sell dollars and buy francs.
However, if the spot is below the strike price, the client will allow the option to lapse. Hence the bank’s initial hedge for the option will be to purchase a proportion of dollars in the spot market against this potential short dollar position. If the spot subsequently rises, the likelihood of the option being exercised will increase and so the initial hedge will be too small. Therefore, the bank will need to buy some more dollars, which it does at a rate worse than the original rate at which the option was priced, thereby losing money. Conversely, if the spot rate falls, this makes the option less likely to be exercised and the bank will then find itself holding too many dollars and will have to sell them out at a lower price than where they were purchased,
Option Pricing Theories 81
Figure 13.3
Figure 13.4
Figure 13.5
thus losing more money. These losses are called “hedging costs” and each time the spot market moves, the rehedging required will lose the bank money. In essence, the premium received by the writer is effectively the best estimate of these hedging costs over the life of the option.
It is clear to see that as volatility increases, market movement increases and so does the number of times the writer of the option must rehedge. Thus, the hedging costs of the option increases and as volatility rises, the price of the option will rise. Conversely, as volatility decreases, the number of times the writer of the option has to hedge also decreases and so lower hedging costs will be incurred.
It should be noted that longer-term options are more sensitive to volatility. For instance, say a volatility of 10% gives a certain range of outcomes over one time period. See Figure 13.3.
Given more time periods with the same volatility, the range of possible outcomes increases.
See Figure 13.4.
An increase in volatility to 12% will have a larger impact on the range of outcomes in the second case due to the cumulative effect. See Figure 13.5.
The option pricing models make an incorrect assumption regarding the likelihood of exercise of low and high delta options. Pricing models currently in use assume a lognormal distribution of outcomes around the forward outright rate similar to the normal distribution shown in the graph (Figure 13.6). This is an approximation in all respects but it is particularly inaccurate for low and high delta options, where the outcomes are more likely to occur than the distribution would indicate. The volatility trader will, therefore, use a higher implied volatility, which gives a higher option price in order to give a closer estimate of their real value.
Another point to note is that sometimes a volatility quote is wider for low or high delta options. This is because as with gamma, volatility has more effect on 50 delta options than on lower or higher delta options. If the width of a volatility quote for an at-the-money option
Figure 13.6
were maintained for, say, a 10-delta option, the option premium spread would be extremely narrow. For this reason, it is usual practice to widen these quotes to give a more usual price spread. As in all markets, however, it should be noted that supply and demand determine the prices quoted.