Delta is the change in premium per change in the underlying. Technically, the underlying is the forward outright rate, but as the option-pricing model assumes constant interest rates, this is often calculated using spot. For example, if an option has a delta of 25 and spot moved 100 basis points, then the option price gain/loss would be 25 basis points. For this reason, delta is sometimes thought of as representing the ‘spot sensitive’ amount of the option.
An option’s price sensitivity to price changes in the underlying instrument is known as its delta.
Delta can also be thought of as the estimated probability of exercise of the option. As the option-pricing model assumes an outcome profile based around the forward outright rate, an at-the-money option has a delta of 50%. It falls for out-of-the-money options and increases for in-the-money options, but the change is non-linear, in that it changes much faster when the option is close-to-the-money.
An option is said to be delta-hedged if a position has been taken in the underlying in proportion to its delta. For example, if one is short a call option on an underlying with a face value of $1 000 000 and a delta of 0.25, a long position of $250 000 in the underlying will leave one delta-neutral with no exposure to changes in the price of the underlying, but only if these are infinitesimally small.
The delta of an option is altered by changes in the price of the underlying and by its volatility, time to expiry, and interest rates. Hence, the delta hedge must be rebalanced frequently. This is known as delta-neutral hedging.
Gamma
Gamma is the rate of change in an option’s delta for a one-unit change in the underlying.
Gamma is the change in delta per change in the underlying and is important because the option model assumes that delta hedging is performed on a continuous basis. In practice, however, this is not possible as the market gaps and the net amounts requiring further hedging would be too small to make it worth while. The gapping effect that has to be dealt with in hedging an option gives the risk proportional to the gamma of the option.
An option’s gamma is at its greatest when an option is at-the-money and decreases as the price of the underlyingmoves further away from the strike price. Therefore, gamma is U-shaped and is also greater for short-term options than for long-term options.
Volatility
It 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.
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 defined as the annualized standard deviation of the natural log of the ratio of two successive prices.
Historical volatility is a measure of the standard deviation of the underlying instrument over a past period. Implied volatility is the volatility implied in the price of an option.
All things being equal, higher volatility will lead to higher option prices. In traditional Black–Scholes models, volatility is assumed to be constant over the life of an option. Since traders mainly trade volatility, this is clearly unrealistic. New techniques have been developed to cope with volatility’s variability. The best known are stochastic volatility, Arch and Garch.
Actual volatility is 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.
Time Decay (Theta)
Time decay is the effect of time passing on an option’s value.
Theta is the depreciation of the time value element of the premium, that is, it measures the effect on an option’s price of a one-day decrease in the time to expiration. The more the market and strike prices diverge, the less effect theta has on an option’s price.
Obviously, if you are the holder of an option, this effect will diminish the value of the option over time, but if you are the seller (the writer) of the option, the effect will be in your favour, as the option will cost less to purchase. Theta is non-linear, meaning that its value decreases faster the closer the option is to maturity. Positive gamma is generally associated with negative theta, and vice versa.
American versus European
In circumstance where the option enables the purchase of a currency that yields a higher return than the currency that is given up in payment, these early exercise features have value, but in such cases, they are more expensive that their European-style counterparts. Examples where this is the case include currency options in which the call currency interest rate exceeds or is close to the put currency interest rate.
American-style option – an option a purchaser may exercise for early value at any time over the life of the option up to and including its expiration date.
European-style option – an option where the purchaser has the right to exercise only at expiration.
Hence, there is a price difference between the two styles of option, but only sometimes. The difference in price occurs because there is a difference in the interest rates that each currency attracts. With American options, the intrinsic value is priced against the spot or the forward outright price, whichever is the most advantageous. This is because the American option can be exercised for spot value at any time during its life. If the call currency (right to buy) of the option has a higher interest rate than the put currency (right to sell), there will be an advantage in calculating the intrinsic value against spot rather than against the forward outright rate.
Therefore, the main risk of the writer of the American option is that at some point in time, if the option is so far in-the-money that there is negligible time value remaining, the holder may exercise early. This would mean that the writer would incur the differential interest cost of borrowing the higher interest rate currency and lending the lower interest rate currency. If this happens, the option is said to be at logical exercise.
As the American-style option is more flexible, shouldn’t it always be more expensive?
Actually, the American option is not really more flexible than the European option. True, it can be exercised early and therefore the intrinsic value can be realized immediately, but unless the option is at logical exercise, the holder would be better to sell the option back and receive the premium. (Remember, the premium represents the intrinsic value of an option plus time value.) This is true for both American and European options and, in both cases, if the option is not at logical exercise, and the aim is to realize maximum profit, it would be better to sell than to exercise the option.
Examples of cases when it would be better to pay extra premium and buy a more expensive American-style option are:
- In buying an option where the call currency (right to buy) has the higher interest rate and it is expected that the interest rate differential will widen significantly.
- In buying an option where the interest rates are close to each other and it is expected that the call (right to buy) interest rate will move above the put (right to sell) interest rate;
- In buying an out-of-the-money option with interest rates as in (1) or (2), and it is expected that the option will move significantly into the money, then the American-style option is more highly leveraged and will produce higher profits.
Source: A Foreign Exchange Primer
