Delta hedging… Spot up or down: who cares?

Have you ever been sure that the spot price is going to be volatile, but unsure of the general direction? In this article we look at strategies that profit from the moves in spot regardless of direction.

In order to build such a strategy, we will need to understand the Delta of the Option and the concept of Delta hedging. The Delta is the price sensitivity of the option with respect to the spot value. This can be understood in the following way: If we have an option with a Delta of 50% then if spot moves up 1 pip the price of the option moves up by 0.5 pips (a 50% move of spot). Likewise, if the Delta is -30% then for spot to move up of 1 pip the option looses 0.3 pips in value.

Note that the Delta of a long spot position is 100%, i.e. the profit and loss moves 1 pip for every 1 pip move in the spot. Therefore, the exposure in the option towards the spot price move can be removed by taking the opposite position in spot. In the case of a 50% Delta option this is done by shorting 50% of the notional amount of the option in the spot. This is called Delta hedging.

However, the Delta of the option changes as spot moves. This is the result of Gamma or the curvature of the option price. A Delta hedged option thus has Gamma exposure and we will now see how this can work to our advantage. Looking at a 1 month EURUSD call strike 1.5000 with spot at 1.4800 and implied volatility of 12%. The price is 120 pips and the Delta is 35%. If we buy 1,000,000 notional we are thus required to sell 350,000 in spot in order to Delta hedge the option. The initial value of the portfolio is thus going to be USD 12,000, the premium of the option. In Table 1 we show what happens to the portfolio value if the spot moves to different levels.

Note that the spot position matches the losses in the option and vice versa. Actually even more so, we get a positive P/L no matter where the spot moves. We are thus long volatility. We simply want spot to move we do not care where it goes. Furthermore, assume that spot goes to 1.4600 we can Delta hedge again, thereby locking in the profit of USD 1,250. Due to the gamma exposure, the Delta at 1.4600 has now fallen to 22%, and since we were short 350,000 we can buy back 130,000 in order to have a short position of 220,000 in spot. The result is shown in Table 2.

Should the spot go back up to 1.4800 we make money again and if it moves further down we still make money. This continuous Delta hedging is called dynamic Delta hedging, and the strategy of earning money by going long an option and dynamically Delta hedging is called Gamma scalping.

So we make money no matter where spot goes, but what is the catch? There is no free lunch so let us see what happens to the situation in Table 1 if we let one day pass. This is shown in Table 3 and Figure 1.

We start losing money if spot stays at the same level. The effect comes from the time decay of the option, i.e. how much value does the option lose from one day to the next.  Being long Gamma/Volatility means having time decay. Making money from being long an option and dynamically Delta hedging requires the spot to be volatile. The condition is that the actual volatility needs to be greater than the implied volatility of the option.

Conversely, if you go short an option and dynamically Delta hedge (you are short volatility), the actual volatility needs to be less than the implied volatility in order to make money.

The following chart displays an example of Profit and Loss resulting from a long option position, and shows the time decay graphically.