**Delta**

Does anyone remember graph paper? I recall as a 14 year old sitting in a maths class being asked to draw tangents to curves. I never really got the point until I started looking at options. Prior to maturity the relationship between the movement in the underlying price and the option premium is non-linear.

If we look at a call option, the relationship is positive – an increase price leads to an increase in the premium. The magnitude by which the premium moves for a ‘small’ move in price is the delta relationship.

Using an ATM spot option on TESLA with a strike of $1,000, 3 month maturity and implied vol of 60% generates a delta value of 0.5629. This is calculated by a formula not a sheet of graph paper in site! This means that for a ‘unit’ change in the underlying price, the premium will increase by 56.29%.

My option pricing model returns a premium of $120.3391 / share at the initial price of $1,000. If the price increases by $1, delta tells us we should expect the premium to increase by just over 56 cents to $120.902. Using the model, the premium increases by a slightly larger amount to $120.9026. Why is delta not accurate? Think back to the maths class – a tangent works on a linear basis which means in this case it will underpredict the movement in the premium. It cannot pick up the curvature of the premium even for a $1 change. So for complete accuracy you would need to use an even smaller change in the underlying price.

When the option is out of the money (OTM) delta will tend towards 0. For an in the money option, delta will take a value closer to 1 as it is behaving more like the underlying asset. An at the money option has a delta of about 0.5. This is sometimes referred to as a “50 delta” option.

BTW – one class participant many years ago became very angry with me about using an ATM delta that wasn’t exactly 0.5. He argued that “everyone knows” that when the option is ATM there is a 50/50 chance of it being exercised. I have had many heated debates with people over the years on this point but all I can say is that the model is not wrong! Strictly speaking the TESLA example I cite is not truly ATM as since it is a European option the “underlying price” is a forward, which would have a value of 1002.50. Resetting the strike to this level returns a delta of 0.5596, still nowhere near 0.50.

Delta will take a + or – sign depending on how the change in price impacts the premium. If you are long a call option, an increase in price will lead to an increase in the premium – a positive relationship. For a long put option, the relationship is negative as an increase in price will lead to a fall in the option premium. Using the same logic a short call has a negative delta and a short put a positive delta.

We have accepted that delta is a dynamic value but how does delta change?

- As we highlighted, movements in the underlying price will impact the delta value.
- The passage of time will also have an impact. As the option approaches maturity, the relationship between premium and price loses its curvature and starts to resemble the ‘hockey stick’ maturity profile. Delta on OTM options tends towards zero, while ITM options tend towards 1. This can be seen in the image used in part 1 of this blog.
- Changes in implied volatility will also impact the delta value. If implied volatility increases the range of expected values that the underlying will take at maturity will widen. The premium – price relationship line gains more curvature and for OTM and ITM options their delta will tend towards 0.5