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]

Interactive Chart

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