Covered warrants: the greeks – Part Two

The price of a covered warrant is calculated using software programmes based on complex mathematical models, the most widespread being the one developed by Black & Scholes.

We might summarise by saying that the mathematical function relating to price is dependent upon 5 market factors:

1.  The price of the underlying asset;
2.  The volatility of the underlying asset;
3.  The time remaining until expiry;
4.  The so-called free-risk interest rate (namely the market rate paid by debtors with zero default risk);
5.  Expected dividends.

All other conditions remaining the same, a change in any one of the factors listed above is sufficient to modify the theoretical value of a covered warrant.  The following table shows the (positive or negative) effect produced by an increase in the aforesaid market variables on the theoretical value of a covered warrant according to whether it is a call or a put.

Effect on the warrant’s theoretical value of a rise (­­) in the level of the market variables

The table helps us to understand the change sign, and hence for example to foresee a rise in the price of a call covered warrant if there is an increase in the price of the underlying asset.

But how much will it rise? We are told this by the so-called Greeks, namely the sensitivity indicators of a covered warrant’s price with respect to the parameters used to determine it:

• Delta (Δ): this measures, with other market factors remaining the same, how much the price of a covered warrant varies given a one unit change in the price of the underlying.  The Delta is a number between 0 and 1 for calls and between 0 and -1 for puts and essentially measures the probability that the covered warrant will end up “in the money” on expiry.  Consequently a covered warrant where the price of the underlying asset is very near to the strike price (so-called “at the money”) has a Delta very close to 0.5, one that is “deep out of the money” has a delta close to 0, one that is “deep in the money” has a delta close to 1;
• Gamma (Γ): the Delta’s accuracy as an indicator is inversely proportional to the size of the change in the underlying. If the changes are substantial it is helpful to also use the so-called Gamma, which measures the change in the Delta parameter due to a change in the price of the underlying. The Gamma basically represents the Delta of the Delta;
• Theta (Θ): a covered warrant, whether call or put, loses value simply due to the passage of time, even without a movement in the price of the asset with which it is associated.  In fact, as expiry approaches, the probability that the price of a certain underlying will move as necessary to reach the hoped-for result decreases.  The concept is the same as in the following example: it is more likely that a share will achieve a 20% performance in a year than in a week.  The daily change in the value of a covered warrant due to the simple passage of time, other market factors remaining the same, is therefore measured by the so-called Theta;
• Vega (Ñ): this measures, other market factors remaining the same, how much the price of a covered warrant changes given a one percentage point change in the value of the underlying’s volatility.  The Vega chart is bell-shaped, with the summit corresponding to a value of the underlying close to strike price (“at the money”).  This parameter is always positive, but assumes a value close to zero if the covered warrant is “deep in the money” or “deep out of the money”;
• Rho (R): this measures the sensitivity of a covered warrant’s price to a change in interest rates and hence in the cost of funding.  In terms of absolute value this parameter is much lower than the Delta and the Vega;
• Phi (F): this measures the change in a covered warrant’s price with respect to a change in expectations as regards the dividends to be paid by the underlying during the life of the covered warrant.  The same consideration applies, with regard to the parameter’s value, as previously made for the Rho.

It is important to emphasise that all these parameters are dynamic, in the sense that they provide an accurate measurement of the change in price of a covered warrant only insofar as all other conditions remain the same. Hence where there is a simultaneous change in both the price and the volatility of the underlying, the effects in terms of Delta and Vega take into account both changes.

Covered Warrant definitions

• At the Money: definition of a covered warrant where the current value of the underlying is exactly equal to the strike price;
• Call: right to purchase at a certain price;
• Deep in the money: definition of a call covered warrant where the current value of the underlying is much higher than the strike price, and put covered warrant where the current value of the underlying is much lower than the strike price;
• Deep out of the money: definition of a call covered warrant where the value of the underlying is much lower than the strike price, and put covered warrant where the value of the underlying is much higher than the strike price;
• In the money: definition of a call covered warrant where the current value of the underlying is higher than the strike price, and put covered warrant where the current value of the underlying is lower than the strike price;
• Out of the money: definition of a call covered warrant where the value of the underlying is lower than the strike price, and put covered warrant where the value of the underlying is higher than the strike price;
• Put: right to sell at a certain price;
• Strike Price: price at which the right to purchase (call) or sell (put) can be exercised;
• Volatility: This is a statistical indicator that measures the magnitude of the underlying asset’s price fluctuations around its average value.