1.4.2 Delta

Delta is the first derivative of the option price with respect to spot price, and roughly corresponds to the probability that an option will end up in the money. Delta ranges from 0 to +1 for calls and 0 to -1 for puts, corresponding to an assignment risk between -100 to +100 shares of the underlying. Buying and selling options and the underlying stock such that total portfolio delta is zero eliminates assignment risk and is commonly practiced by market makers.

Example

The SPY is currently trading at $595. A market participant sells 12 SPY calls at the $600 strike expiring in 45 days for $7.00 per contract with a delta of 0.7. The total expected assignment risk is for 70 x 12 = 840 shares of the underlying at expiration, corresponding to a delta dollar value of $499,800. To hedge this risk, the call seller can buy 840 shares of SPY at $595 per share.

Summary of Terms

$V$ = Contract Value

$\Delta$ = Contract Delta

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

\Delta = \frac{\partial V}{\partial S}

\Delta_{calls} = {e^{-q\tau} \over 2}\bigg[1 + erf\bigg({d_+ \over \sqrt2}\bigg)\bigg]

\Delta_{puts} = -{e^{-q\tau} \over 2}\bigg[1 + erf\bigg({-d_+ \over \sqrt2}\bigg)\bigg]

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

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