Greeks use case: Delta hedging in action

Derivatives are called as such because they “derive” their price from the underlying asset(s). The latter is a central component for pricing and risk management purposes. Let’s look at how the Delta impacts the price of an option by looking at a use case.

Delta and Gamma are typically used on an aggregated basis in practical contexts, which is straightforward since the figures are simply added together

Delta is technically the first derivative of the pricing formula (of the derivative) with respect to the price of the underlying: it measures the change in the price, of a forward or an option, for example, given any (small) change in the underlying’s price.

That allows us to determine how much of the underlying asset the derivative currently (just now) represents. To momentarily hedge exposure, we simply need to take the opposite position in the underlying. That creates a Delta-neutral (combined) position.

Because that will change, for larger changes in the underlying a new Delta becomes relevant: Gamma shows by how much Delta would change as the underlying moves (relevant for options, of course). So, the two risk measures are to be looked at as a pair.

Delta and Gamma are typically used on an aggregated basis in practical contexts, which is straightforward since the figures are simply added together. Of course, these values can be positive or negative depending on the derivative exposure. That is why it is essential to know whether the position is long or short!

Currently, stock exchanges have experienced a sharp increase in volatility (in the order of a doubling since the beginning of the year) due to US fiscal policy regarding global tariffs, either already applied or potentially to be applied - with the level of impact varying significantly depending on the country or region.

On 9 April 2025 when the US President postponed to a larger extent the tariffs during the day, it triggered a sharp intraday stock rally: The S&P 500 index ended the day at 5,456.90 when the previous session had closed at 4,982.77.

The VIX, the forward-looking volatility, also suffered a strong move but in the opposite direction from 52.33% on the previous day (the highest closing year-to-date up to the 17 of April 2025) to 33.62%.

Considering a hypothetical European long call option on 8 April with:

• Spot: 4,982.77
• Strike: 5,000, one-month expiration
• Risk free: 4.36%
• Dividend yield: 1.27%
• and Volatility: 52.33%

Its price would have been 297.64 (Forward: 4,995.62), its Delta: 0.527236 (expected as the Strike is close to the Forward) and its Gamma: 0.000528.

At the end of the next day, using the spot and volatility values of 9 April but not changing the option’s expiration we had a different picture.

^{The table below shows the Delta-hedging in action - simulation considering large index moves}

Day	Spot	Long Call price	Delta (at closing)	Position in index after action	Action performed
08-Apr	4,982.77	297.64	0.527236	-0.52724	Sell 0.527236 index at 4,982.77
09-Apr	5,456.90	602.73 (same Volatility & Expiration)	0.748233	-0.74823	Sell further 0.220997 at 5,456.90
10-Apr	5,268.05	468.73	0.668699	-0.6687	Buy 0.079534 at 5,268.05
11-Apr	5,363.36	534.48	0.710474	-0.71047	Sell 0.041775 at 5,363.36

_{Source: Evolids Finance}

Its price would have been 517.51, its Delta: 0.834595 (an expected increase, as the option is now well In-The-Money) and its Gamma: 0.000467. That’s a significant move!

Changing only Spot, the pricing model will typically consider that volatility is (known and) not changing, we get: its price of 602.73, its Delta: 0.748233 (expected as the option is now well In-The-Money) and its Gamma: 0.000386.

The approximative change in option value with constant volatility should have been:

0.527236 (Delta) * 474.13 (change in Spot) + 0.5 * 0.000528 (Gamma) * 474.13^2 (change in Spot ^2) = 249.98 (Delta effect) + 59.35 (Gamma effect) = 309.33.

The true change is: 602.73-297.64=305.09.

Pretty precise here… But we should not forget that volatility did change a lot.

The next article will look at Vega and the impact of volatility on options embedded in structured products.

_{Image: KanawatTH/Adobe Stock}

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_{DISCLAIMER: This content is not intended as a solicitation or an offer; it is provided solely for informational purposes to professional investors. The information presented herein has been prepared with great care; however, errors may still occur.}