Given that an option value is determined by five variables — spot price, strike price, time to expiry, risk-free rate, and volatility — it is interesting to understand the interaction between these variables. This week, we discuss the impact of change in volatility on the delta of an option.
Volatility matters
Delta is the change in the option price for a one-point change in the underlying. If an option delta is 0.51, then a one-point change in the underlying price will change the option price by approximately 0.51 points. Conceptually, delta can be interpreted in several ways. One such interpretation, though with less practical application for creating option strategies, is that the delta is approximately the probability of an option expiring in-the-money (ITM). This interpretation, however, provides us with an intuitive understanding of why at-the-money (ATM) option has a delta close to 0.50. This strike has a 50-50 chance of expiring ITM or OTM (out-of-the-money).
Interestingly, the delta of an option changes with change in volatility. This is captured by a second-order Greek called vanna. Suffice it to know that vanna is greatest for OTM and ITM options. That is, when you increase volatility, the delta of the OTM and ITM options change the most. The delta of an ATM option does not change when you increase volatility! In other words, the vanna of an ATM option is approximately zero.
Put differently, when you increase volatility, both the ITM and the OTM option tend towards ATM. That means, the delta of an ITM decreases and the delta of an OTM option increases when you increase volatility. This argument is easier to understand using the probability interpretation. When you increase volatility, the risk of an ITM option becoming ATM increases. Therefore, the probability of this option expiring ITM decreases. On the other hand, an increase in volatility increases the chances that an OTM option will expire ITM. Therefore, its delta increases. Note that the probability of an ATM expiring ITM remains close to 0.50. Hence, its delta does not change.
Traders typically prefer ATM options when they expect volatility to explode (increase). This is because ATM options increase the most when volatility increases. How is this possible if the delta does not change? The answer lies in an option’s vega — the change in option price for a one percentage point change in (implied) volatility. ATM option has the highest vega.
Optional reading
The above discussion shows that an option price can change without a significant change in its delta. This understanding is important when you bet on volatility explosion, especially when you are close to option expiry. This could happen when a company-specific or macro-level event is scheduled to happen at or near option expiry, but the outcome of the event is uncertain. Because of option’s asymmetric payoff, this uncertainty could lead to greater demand for options on the run-up to the event without a marked increase in the underlying price.
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