Why do we need delta and gamma cash?

Because traders think in percentage movements rather than absolute movements.

1. Delta Cash

Assuming there is no convexity from the option position

\[\delta_{cash} = \delta . S_t\]

The profit for 1% move in spot is:

\[\delta_{cash/1 \%} = \frac{\delta . S_t}{100}\]

Assuming a trader shorts 100,000 1 year call options on VIC with price of 25 with delta of 0.5.

The cash delta is equal to:

\[-0.5 \times 100,000 \times 25 = 1,250,000\]

The 1% cash delta in this case is equal to:

\[\delta_{cash/1 \%} = \frac{-0.5\times100,000\times25}{100} = -12,500\]

So when VIC stock increases by 1%, the trader loses -12,500. To fully hedge the position, he needs to buy the number of stock as below:

\[\frac{-1,250,000}{25} = 50,000\]

2. Gamma Cash

Assuming now there is some convexity in the option position.

The gamma is equal to \(\gamma\). Gamma cash is defined as the change in delta cash for a 1% move in the underlying. We have the formula for gamma cash:

\[\gamma_{cash} = \frac{\gamma S_t^2}{100}\]

Note that the new delta cash after 1% move from spot is defined as:

\[(\delta + \frac{\gamma S_t}{100})S_t\]

Continue with the example above, now assuming we have 1% gamma cash of 50,000 from the short option position (remember that when we short option, we short gamma, so the sign will be negative). When price move up by 1%, the delta cash after the increase will be:

\[\delta S_t + \frac{\gamma S_t^2}{100} = -0.5\times 25 \times 100,000 - 50,000 = -1,300,000\]

Assuming the trader wants to delta-hedge the option position after 1% move, he will need to long total:

\[\frac{-1,300,000}{25}= 52,000\]

To maintain the portfolio delta-neutral, in this case, the trader needs to buy extra 2,000 shares.