What is Delta Hedging and why should I do it? Plus, the impact of Path Dependency on Delta Hedging

Level 1 and 2 - Zero to Options Hero - Intermediate Vol Trader (after paywall)

In this post in our Zero to Options Hero series, we will elaborate on the post explaining delta and delta equivalence and explain the concept of delta hedging. Why is delta hedging so important, and why should you consider doing it when trading options?

Delta Hedging

As mentioned in the post about the Black Scholes model, modern options pricing is based on the fundamental assumption that the price of an option is based on the expected costs of replicating the option by trading the underlying asset.

For example, an options market maker who sells a 1-month at-the-money (50% delta) call on TSLA will buy shares of TSLA equal to 50% of the option notional in order to achieve delta equivalence, as previously described. This is also known as the 'initial delta hedge' as it is the first delta hedge that the option seller will make in her attempt to replicate the option via dynamic hedging.¹

Due to the positive gamma of a long call option (which represents negative gamma for the option seller), in order to keep delta equivalence the market maker should continuously adjust the delta hedge amount every time the underlying asset moves. In the Black Scholes model's world, the market maker is trading the underlying asset in continuous time (i.e. every millisecond) with zero transaction costs. In reality, though due to the existence of the bid-offer spread on the underlying asset, the option market maker will delta hedge at certain moments in time, usually based on either:

1. a predetermined time-frequency (i.e. every 5 minutes while the market is open, once a day at market close, etc) or

2. once the underlying asset reaches certain levels away from the current level.

Note that due to the negative gamma of being short the option, in order to delta hedge, the call option seller will need to buy the underlying asset when it moves up and sell the underlying asset when it moves down (otherwise known as stop loss trades (S/L)). If realized volatility is high and the underlying price moves up and down rapidly, the market maker will incur delta hedging losses by buying high and selling low! If the option is fairly priced, then we can expect that the option premium the market maker receives for selling the option is perfectly offset by the delta hedging losses. Intuitively, this would be the case if the realized volatility during the lifetime of the option is equal to the implied volatility that the market maker sold the option at (the implied volatility can be computed using the Black Scholes model based on the option premium at the time of the initial option trade).

Why should I do it?

Although the majority of option traders are looking for a low-cost way to speculate on the direction of the underlying, delta hedging an option can be profitable if the option trader believes/forecasts that the realized volatility of the option through its lifetime is different from the implied volatility that the option is trading at in the market.

Hope this makes sense from a big-picture perspective. Now, we will go into more advanced concepts as to why even if the realized volatility calculated during the lifetime of the option is equal to the implied volatility at trade inception the market maker can end up making a loss at the end of the delta hedging process!

The reason that the market maker can end up with a loss (or a gain!) is due to the concept of path dependency.

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