The Curvature
It is evident that the Delta of an option fluctuates in accordance with changes in the underlying.
Examining the blue line which shows the delta of a call option, it is evident that it oscillates between 0 and 1 or inversely from 1 to 0 depending on the situation. The red line representing the delta of a put option displays similar behavior, albeit its value alters between 0 and -1. It is clear from this graph that the delta is ever-changing.
Therefore, one should be prepared to answer the following query:
I’m aware of the alteration in delta, but why should I be concerned by it?
If the variance of delta is significant, what are the chances of predicting its alteration?
Let’s focus on the second query first; I’m sure the response to the initial one will become apparent as we go through this section.
The Gamma provides the rate at which the delta will shift based on movements in the underlying security.
This rate is often shown in deltas either gained or lost per a one-point move in the underlying – showing an increase in delta by gamma sum when rising, and a decrease by this same amount when falling.
For example, consider this –
Nifty Spot = 8500
Strike = 8600
Option type = CE
Moneyness of Option = Slightly OTM
Premium = Rs.30/-
Delta = 0.4
Gamma = 0.003
Change in Spot = 80 points
New Spot price = 8500 + 80 = 8580
New Premium =??
New Delta =??
New moneyness =??
Let’s figure this out –
Change in Premium = Delta * change in spot i.e 0.4 * 80 = 32
New premium = 32 + 30 = 62
Rate of change of delta = 0.003 units for every 1-point change in underlying
Change in delta = Gamma * Change in underlying i.e 0.003*80 = 0.24
New Delta = Old Delta + Change in Delta i.e 0.4 + 0.24 = 0.64
New Moneyness = ATM
When Nifty shifted from 8500 to 8580, the 8600 CE premium rose from Rs.30 to Rs.62, while its Delta went up from 0.4 to 0.64.
The transition from slightly OTM to ATM option is marked by an increase of 80 points, and the delta consequently shifts from 0.4 to near 0.64. It is clearly depicted in the present scenario.
Let us consider the scenario that Nifty rises by 80 points from 8580. Then, what will be the situation with the 8600 CE option?
Old spot = 8580
New spot value = 8580 + 80 = 8660
Old Premium = 62
Old Delta = 0.64
Change in Premium = 0.64 * 80 = 51.2
New Premium = 62 + 51.2 = 113.2
New moneyness = ITM
Therefore, the delta should be higher than 0.5
Change in delta =0.003 * 80 = 0.24
New Delta = 0.64 + 0.24 = 0.88
Proceeding on, if the Nifty drops by 60 points, let’s have a closer look at how 8600 CE performs.
Old spot = 8660
New spot value = 8660 –
60 = 8600
Old Premium = 113.2
Old Delta = 0.88
Change in Premium = 0.88 *(60) = – 52.8
New Premium = 113.2 – 52.8 = 60.4
New moneyness = slightly ITM (hence delta should be higher than 0.5)
Change in delta = 0.003 * (60) = – 0.18
New Delta = 0.88 – 0.18 = 0.7
Take a look at the delta transition and how it adheres to the earlier chapters’ rules. You may be wondering why Gamma stays constant here; indeed, it changes with changes in the underlying. This variation is referred to as “speed”, “Gamma of Gamma”, or “DgammaDspot” – all derivatives of underlying. Unless you’re mathematically inclined or working in investment banking, wherein risks can run up to millions of dollars, discussion of speed is unnecessary.
When a trader has a long position in both Call and Put Option, they are referred to as Long Gamma. On the other hand, should they have a short position on those same options, they would be ‘Short Gamma’.
For instance, let’s say the Gamma of an ATM Put option is 0.005 – if the underlying shifts 15 points, what would you calculate the new Delta as?
I would highly recommend you to spend some time thinking about a potential solution for the issue at hand.
If the ATM Put option we are considering has a Delta of – 0.6, then its Gamma is +0.005; when the underlying moves by 15 points, we will need to evaluate what occurs in both possible directions.
Case 1 – Underlying moves up by 15 points
Delta = – 0.6
Gamma = 0.005
Change in underlying = 15 points
Change in Delta = Gamma * Change in underlying = 0.005 * 15 = 0.075
New Delta = We know the Put option loses delta when the underlying increases, hence – 0.6 + 0.075 = – 0.525
Case 2 – Underlying goes down by 15 points
Delta = – 0.6
Gamma = 0.005
Change in underlying = – 15 points
Change in Delta = Gamma * Change in underlying = 0.005 * – 15 = – 0.075
New Delta = We know the Put option gains delta when the underlying goes down, hence – 0.6 + (-0.075) = – 0.675
So, here’s something to think about – Earlier, we mentioned that the Delta of a Futures agreement is typically 1.
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