VII.3. The Greeks of options. Why are options gaining or losing their value so quickly?

To understand what happens to options at the time of sharp price movements and over time, we need to get acquainted with the so–called “Greeks” of options – the parameters of the price calculation model that affect the value of the option. The ability to read the “Greeks” allows you to work more subtly with options, pulling the maximum out of them.

To begin with, let’s look at the graphs of these same Greeks for example in Fig. 136. Under the green line “Portfolio-1” we will see in the table a number of values “delta”, “gamma”, “vega”, “tetta”.

Рис.137. Значения «греков» опциона CALL.

Fig.138. Delta (D) options.

Fig.139. Gamma (g) options.

Fig.140. Vega (u) options.

Fig.141. Theta (q)options.

“Greeks” are the parameters of an option that describe the sensitivity of its price to such variables as the price of the underlying asset, the strike price, the number of days before expiration, and volatility. Figure 138-141 shows the charts of these “Greeks” for the option from our example in Figure 136. Let’s analyze what we see on these charts and what to pay attention to. It is clear that the blue line is the Greek value at the time of expiration, and the red line is the current value of this parameter. To find the Greek value for the current strike price of 122530, we are simply looking for the intersection of the vertical strike price line with the desired graph.

Delta is the coefficient of transition of the change in the price of the underlying asset into an option premium. The delta of calls varies from 0 to 1, and the delta of puts varies from 0 to -1. If the delta is equal to 1, then a change in the futures price by 1000 pp will bring a change in the option premium by a similar value: 1000 * 1 = 1000. And if the delta is 0.5, then 1000 pp of movement on the underlying asset will shift the option by only 500 pp: 1000 * 0.5 = 500. The delta is also called the probability coefficient of the option going into the money. That is, when an option is in the money, its delta is 0.5 (-0.5 for puts). An option on money is a little bit in the money, then a little out of the money. He’s in a pretty precarious position. In fact, it is decided whether the market will go towards a money option or not. And the probability of this movement is 0.5: either yes or no. If the option is already deep in the money, then when the futures move by 1000 or 500 pp, these points turn into an option, since even if there is no liquidity on the strike, we can simply execute it. And the probability of an option going out in money into money at the time of expiration is very high (of course, a sharp reverse movement is possible, which can leave the option “overboard”, but it is unlikely). Options outside of money have almost no delta. Their delta is approximately equal to 0. Therefore, the delta of options outside of money is very small and continues to fall as far as possible, approaching 0. I.e., according to Figures 137 and 138, we see that our delta = 0.75, i.e. the option is in our money and if the strike is at least 117500, then the red curve it will connect with the blue one, and the delta will become equal to 1 by the time of expiration.

Vega shows how the value of the option will change when the volatility changes. The increase in volatility increases the value of the option, since the desired profit can be earned faster on volatile instruments. To a greater extent, it affects the time value of the option, rather than the base one. Money options have the biggest vega, since they have the maximum time value. Options in money have a minimal vega, since the effect of volatility on the base price is small – options in money are affected by changes in the price of the underlying asset and delta. Out-of-money options also have a minimal vega. 140 we see that we have a large Vega, because the option is on the money.

Theta (time decay) shows how the option price will change per day. Since options inevitably move in time to the date of their expiration, their value is constantly reduced by the amount of time decay. Theta has an effect only on the time value. The basic cost of theta cannot be reduced, since in American options there is a possibility of execution before the expiration date. Theta has the maximum values for options on money that have the maximum time value. For options outside the money and on the money, theta is reduced. Theta is a nonlinear parameter. Thus, one day of holding a long-term expiration option from the point of view of temporary decay will be cheaper. In Fig.141, we see that 1 day of holding our option for the current day costs 211.75 rubles. If we buy and sell an option during a 1-day trading session, then we do not pay tet.

Mathematically, delta is the 1st derivative of the option price relative to the price of the underlying asset. Gamma is the 2nd derivative relative to the asset price.

Gamma indicates a change in the delta when the price of the underlying asset moves, i.e. from a mathematical point of view, gamma represents the derivative of the delta of the option relative to the price of the underlying asset. The delta (the coefficient of the slope of the tangent to the option price chart) is not a constant value, but depends on the asset price.

Options traders compare delta with duration (duration of the flow) when considering the relationship between the bond price and interest rates, and gamma with convection (transfer).

Gamma (or convection, using bond terminology) measures the rate of change in the delta of an option. Gamma indicates the degree of convexity (roundness) of the option price chart relative to the underlying asset at a certain point where the asset price is located. The larger the convexity of the graph, the faster the delta changes.

The purchase of call and put options leads to a long (or positive) range, and the sale of options leads to a negative gamma position.

The gamma of the on-the-money call option increases as it approaches expiration, while the gamma of the in-the-money and off-the-money call option decreases. Due to the small amount of time until the expiration, the probability of non-execution of out-of-money and in-money options increases, the deltas of which approach zero and one (or minus 1 for the put option), respectively. To fundamentally change the situation, a significant movement in the price of the underlying asset is necessary. On the other hand, the fate of the on-the-money option is characterized by greater uncertainty when it expires after three days, since a small fluctuation in the price of the underlying asset can greatly change the delta of the option, either to zero or to one (or minus 1)

This phenomenon of an increasing gamut for on-the-money options, both call and put, is of critical importance for a trader who has a short option position. The higher the gamma value, the more unstable the delta is. This leads to more frequent delta hedging, and, consequently, higher losses from portfolio rebalancing.

A short option position forces a trader to hedge the delta at a loss by buying and selling the underlying asset. As the stock price rises, the seller of call options must buy more and more shares in order to reduce the delta to zero. Then, if the stock price falls, the trader needs to sell the purchased shares at lower prices so that the delta is zero again. On the other hand, the option holder (call or put) will always hedge the delta with a profit due to the positive gamma. This means that the portfolio delta will increase when the asset price rises, and decrease when the asset price falls. Our option is on the money, so in Fig.139 we see that the gamma has a small value for us. Another Greek is RO (r).It is commonly used in specialized option trading packages. Reflects the sensitivity of the theoretical value of the option to changes in interest rates. The RH indicator is less important compared to the Greeks delta, theta, Vega, gamma. Options that are deep in the money have the highest ROS, as they require the most costs. The longer the time to expiration, the greater the RO. Stock options have much more RO than futures options. Due to the relatively small significance of RO, it is usually not taken into account when analyzing option strategies.

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