1.4.8 Speed

Speed is the third derivative of the option price with respect to spot price, the second derivative of delta with respect to price, and the first derivative of gamma with respect to spot price. Incorporating speed into a dynamic delta hedging strategy has diminishing returns compared to using gamma adjustments only, but provides higher hedging accuracy in general and is significant for very large moves of the underlying price or in situations where there is very large gamma (such as a few minutes before expiration on SPX 0dtes).

Summary of Terms

$V$ = Contract Value

$\Delta$ = Contract Delta

$\Gamma$ = Contract Gamma

$U$ = Contract Speed

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

U = \frac{\partial^3 V}{\partial S^3} = \frac{\partial^2 \Delta}{\partial S^2} = = \frac{\partial \Gamma}{\partial S}

U = {-e^{-q\tau} \over S^2\sigma\sqrt{2\pi\tau}}e^{-{1 \over 2}d_+^2}\bigg({d_+ \over \sigma\sqrt{\tau}}+1\bigg)

d_+ = {1 \over \sigma \sqrt{\tau}}\bigg[\ln\bigg({S \over K}\bigg) + \bigg(r + {\sigma^2 \over 2}\bigg)\tau\bigg]

Speed market impact

The figure below shows how speed varies for a $100 strike contract as a function of spot. See the delta and gamma sections for more details on market maker hedging impact. The speed for a contract is positive when out the money and negative when in of the money. This results in an increase in gamma for out of the money contracts as the market approaches expiration, and a decrease for in the money contracts.

speed