What is the Numerical Relationships between Price, Delta and Gamma

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We all know that words can have various meanings, it is the same case with a delta in options trading. Delta is most commonly defined as the change of premium in relationship with a 1-point change in the underlying.

Let’s take an example. 

If an option trader has selected a delta of 0.87, then for a 1-point move in the underlying, the option premium would increase by 87 cents. However, if the stock moves 2 points, do you think that the premium will increase by the delta of 0.87 or will it increase more than that? 

The answer lies in the Greek component known as gamma.

Black-Scholes-Merton Model

Over the years, many methods of options pricing have been analysed. This is because it is a complex process to value options. So far, the most popular model to be used is Black-Scholes-Merton model, which is based on the theory that markets are arbitrage free. It also assumes that the price of the underlying asset is characterized by a Geometric Brownian motion. 

For finding the option price, we  first find the expected value  of  the  price  of  the  underlying  asset  on  the expiration date. Since the price is a random variable one possible way of finding its expected value  is by simulation. This  model can  be adapted  to  pricing  almost any  type  of option. 

Delta Vs Gamma

The first option Greek, delta is a derivative of option price and is thus known as the first order derivative. Gamma is the derivative of Delta, and hence is known as the second order derivative. It is the rate of change in Delta for every 1 point move in the price of the underlying. 

Delta measures the directional risk of an option, whereas, Gamma measures the changes in that directional risk. Gamma helps interpret the potential movement in Delta that could be caused by changes in the underlying price. Depending on the value of Gamma, a trader can assess the changes in his position risk. Gamma can be expressed as the following equation: 

This equation can be also be expressed as:

Change in Delta = Gamma * Change in the underlying price

Please note that here,
Change in the underlying price = New underlying price - Old underlying price

From this equation, we can also calculate the new value of Delta with the following equation:

New Delta = Old Delta + Change in Delta

Time for an Example!

To understand this further, let’s take an example. Let us assume that a stock is currently trading at ₹100 and that we are looking at an ATM call option having a strike price of ₹100. Let us also assume that at present, this call option’s Delta is 0.5, Gamma is 0.02, and Option price is ₹6.

Let’s assume that the stock price rises to ₹105 in a few days. Using this, we can calculate the new option price as ₹6 (Old Option price) + 0.5 (Delta) * ₹5 (Change in the underlying price). This turns out to be ₹8.5. 

Given that the underlying price has now changed and that the option has moved ITM, the Delta will also change. Now, let’s calculate the new delta! 

Delta can be calculated as 0.5 (Old Delta) + 0.02 (Gamma) * 5 (Change in the underlying price). This turns out to be 0.6. So, the new Delta for a 5 point rise in the underlying is 0.6. Notice that as the option has now moved ITM, its Delta has moved above 0.5.

On the other hand, can you guess what would have happened if the stock price had declined to ₹95 instead? In that case, the option price would have fallen to ₹3.5. Moreover, the change in the underlying price would lead to a change in the value of the Delta as well. This change can be calculated as 0.02 * (-5), which is -0.1. Hence, the new Delta would be 0.5 + (-0.1) = 0.4. Notice that as the option has now moved OTM, its Delta has declined below 0.5.

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Delta Ratio

Delta ratio is defined as the percentage change in the option premium for each dollar change in the underlying. 

Let’s take an example: 

If you have a call option for ABC Ltd with a strike price of INR 300, and the stock price moves from INR 300 to INR 301, it will cause the option premium to increase by a certain amount — let's say it increases by INR 0.50. Then the option will have a positive delta of 50%, because the option premium increased INR 0.50 for an increase of INR 1 in the stock price. Remember, delta is often denoted by a whole number, so if an option has a 50% delta, then it will often be denoted as 50 delta. For a put option, data will always be negative.

Delta and Probability

Delta is also used as a proxy for the probability that a call will expire in the money. A stock with a delta of 85% will have 85% chance of finishing in the money. Delta can only serve as a proxy for the probability because both delta and the probability that a call will go or stay in the money increase as the option goes further into the money.

Delta is not a direct measure of the probability. Most of the value of a call will depend on the intrinsic value, which is the amount that the underlying price exceeds the strike price of the call. If the underlying asset increases by INR 1, then a call would have to increase by nearly the same amount; otherwise, arbitrageurs could sell a stock short and buy the call to make a riskless profit.

This is why, on the last trading day, delta would have to be 100% for an in-the-money call; nonetheless, there is still a high probability that the option can go out of the money in the remaining time, especially if volatility is high, as it often is on the last trading day of the option, so the probability that the call will remain in the money is much less than 100%.

All about Gamma

Gamma is the change in delta for each unit change in the price of the underlying. Delta increases as the time to expiration of the option decreases, and as its intrinsic value increases. Thus, in the above example, as the intrinsic value of the puts increased and time to expiration decreased, the delta of the puts decreased to almost -1, where each INR 1 drop in the price of the stock increased the price of each share of the puts by INR1. The only way you would lose with this strategy is if the stock didn't do much of anything until expiration — then you would lose the premiums that you paid for the puts, but at least your loss was limited to the INR 400 dollars plus commissions.

Wrapping up

The relationship between Delta and Gamma

  • Gamma changes in predictable ways. As an option goes more into the money, delta will increase until it tracks the underlying dollar for dollar; however, delta can never be greater than 1 or less than -1. When delta is close to 1 or -1, then gamma is near zero, because delta doesn't change much with the price of the underlying. 
  • Gamma and delta are greatest when an option is at the money — when the strike price is equal to the price of the underlying. The change in delta is greatest for options at the money, and decreases as the option goes more into the money or out of the money. 
  • Both gamma and delta tend to zero as the option moves further out of the money.

A Quick Recap

  • The most popular model for analysing price options is Black-Scholes-Merton model.
  • The first option Greek, delta is a derivative of option price and is thus known as the first order derivative. 
  • Gamma is the derivative of Delta, and hence is known as the second order derivative. 
  • It is the rate of change in Delta for every 1 point move in the price of the underlying. 
  • Delta ratio is defined as the percentage change in the option premium for each dollar change in the underlying. 
  • Delta is also used as a proxy for the probability that a call will expire in the money.
  • Delta is not a direct measure of probability. 
  • Gamma is the change in delta for each unit change in the price of the underlying.
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