Why At-the-Money Put Delta Rises with Implied Volatility

Summary

The document asks why the delta of a standardized, perpetually at-the-money, 30-day put on SPY became less negative during the 2008 financial crisis. The described series uses OptionMetrics data, with standardized option prices derived by linear interpolation across the volatility surface. The explanation is that higher volatility raises the Black–Scholes value of d1 under the stated assumptions of equal spot and strike and equal interest and dividend rates.

For a put, delta is negative; as d1 rises, the normal cumulative probability term in the delta expression falls, making delta less negative. This accounts for the observed direction of change as volatility increased. The explanation is conditional on the simplified assumptions and the Black–Scholes framework. It does not quantify the size of the effect or address changes in rates, dividends, or other market inputs during the crisis.

Key ideas

• For an at-the-money put with equal rates and dividends, higher volatility raises d1.
• The put delta becomes less negative as the volatility-driven d1 rises.
• The example concerns standardized 30-day SPY puts across a historical period.
• The explanation relies on Black–Scholes assumptions and does not isolate other changing market inputs.

Tags

# Delta of a standardized at-the-money 30-day put option


# Delta of a standardized at-the-money 30-day put option












The plot below depicts the delta of a standardized at-the-money 30-day put option on the S&P500 tracker SPY over a 14-year period. This is data from OptionMetrics and standardized prices are calculated using linear interpolation from the volatility surface

My question is: Why does delta increase (i.e. decrease in absolute value) during the 2008 financial crisis?

Link: https://i.sstatic.net/0wtDm.jpg

Delta of a put option over time, whose characteristics are constant. I.e. the underlying option characteristics is modeled so that it is perpetually at the money and 30 days from expiry

## Answer by John (score 1, accepted)

https://quant.stackexchange.com/a/10909

Delta increases as volatility increases.

In particular, the formula for delta of a put is $$\Delta=-exp\left(-qt\right)\Phi\left(-d_{1}\right) $$ with $$d_{1}\equiv\frac{ln\left(S/K\right)+\left(r-q+\frac{\sigma^{2}}{2}\right)t}{\sigma\sqrt{t}} $$ setting $S=K$ and $r=q$ you would get $$d_{1}\equiv\frac{\sigma}{2}\sqrt{t} $$ By the chain rule, an increase in $\sigma$ leads to an increase in $d_{1}$, which leads to a decrease in the $\Phi(-d_{1})$ term but an increase in $\Delta$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.