Over the last few posts we have spent a fair amount of time looking at delta as a measure of directional price risk. Arguably, the more difficult concept to grasp is that of gamma. Its formal definition – the rate of change of delta for a given change in the underlying price is perhaps unhelpful. It doesn’t convey why I might be interested in this concept. We can derive a series of alternative definitions that might help us. But before we do that, an example will help set the scene.
Setting up the trade
Suppose we sell 100, 3 month call options on Tesla, struck ATM (say $1,000) using an implied vol of 60%. Premium comes out at $120 / share. The position is delta negative (-0.56, rounded) and so we buy 56 shares at $1k each to create a delta neutral position. Shortly after putting the position on, the price spikes up by 10% to $1,100.
The trader’s initial reaction may well be to assume that they are hedged against this move. The premium has increased to $183, resulting in a mark to market loss of $63 / share, so $6,300 on the option position. However, the shares have increased in value by $100 / share returning a profit of $5,600. However, this indicates a combined loss on the position of $700. This is because the delta on the option has now increased to -0.68 and so it is behaving like a short position of 68 shares. The trader is now faced with a dilemma. Recall the diagram from the previous post – if the price continued to rise the trader will incur greater losses (see previous blog)
Let us say they decide to re-establish delta neutrality at the higher spot price. They will now need to buy an additional 12 shares at this higher price.
Let us now say that the market falls back to its original price. What is the impact? This time the option premium falls by $63 generating a $6,300 profit for the trader. The shares fall in price resulting in a loss of $6,800 (68 shares x $100). Net loss is $500. Again if the price continues to fall, the trader will incur losses and so they may decide to re-establish delta neutrality by selling the additional 12 shares they have just bought.
So what is the overall PnL from the day’s trading? The option premium starts and finishes the day at the same level as we are only analysing a change in price. There is no profit or loss effect. The original delta hedge of 56 shares also has zero profit and loss for the same reason. But buying 12 shares @ $1,100 and then selling them at $1,000 will result in a loss of $1200. This accounts for the two mark to market losses noted earlier ($700 + $500). Note that the loss is wholly attributable to the delta hedging activities.
This is gamma in action, but we will return to the example in the next post to develop the idea further.