Understanding delta

[enjofaes]

Hi @David Harper CFA FRM & @Nicole Seaman,

I'm reviewing chapter 16 of book 4 and came across this example in the instructional video and got confused:

In this example the delta becomes more negative and part of the portfolio should be sold.

However I wrote down in previous part of the video: Negative portfolio delta: to hedge we purchase shares.

In the examples from dynamic hedging by Hull in contrast, a positive delta is reported when writing the option and the market maker then purchases shares to dynamically hedge.

So when exactly do we buy or sell shares. When delta is negative or positive?
My intuition says: we short shares when delta is positive and buy shares when it's negative. But then I don't understand the example above or the signs of delta in the Hull dynamic hedge example.

 

[Torsleno]

If you wrote the put, you might need to inverse the sign of the delta in the right box (short put = long delta). It seems that it is just the delta of the option, you then need to take into account whether you are short or long. You can see in the bullet point that the delta doesn't have a negative sign

 

[David Harper CFA FRM]

Hi @enjofaes the comment by @Torsleno is correct and relevant. Please see my explainer here at https://forum.bionicturtle.com/threads/l1-t4-7-dynamic-delta-hedging.4839/ i.e.,

David's ProTip: I learned from Carol Alexander a useful semantic distinction (not in Hull). Consider a position in 100 call options with per-option delta of 0.6:

• The Percentage Delta is 0.6; this is the unitless first partial derivative, dc/dS
• The Position Delta is 60 because Position Delta = Quantity * Percentage Delta.
• If we are long, we use (+) quantity: Position Delta (long 100 calls) = +100 * 0.6 = +60;
• If we are short , we use (-) quantity: Position Delta (short 100 calls) = -100 * 0.6 = -60
• To neutralize is to get the position Greek to zero

This is robust, for example:

• Selling puts increases position delta because -QTY * -% delta = +position delta; i.e., % delta of puts always negative; % delta of calls is always positive
• Selling calls or puts decreases position gamma because -QTY * +% gamma = - position gamma; i.e., % gamma is always positive for both calls & puts

Just as we use dollar duration (not modified duration) to neutralize duration in the portfolio, we neutralize an option Greek by summing Position Greeks to zero. I often see candidates trying to neutralize with percentage delta directly, but you can't, you need to sum the "Position" Greeks. I hope that's useful! David

So I like to refer to "position Greek". A put option always has a negative percentage (per option) delta but if we write 100 put options and use Q = -100 to signify selling the put contract (and Q = +100 to signify buying the put contract), then it's easy to manage that 100 written put options have a position delta equal to -100 * -D = positive position delta.

The above portfolio insurance approach is a little different, however. It is compared to buying puts. So the (long) portfolio is with $90.0 million and the basic approach is to buy puts. In contrast, this portfolio insurance above does not buy puts, but rather creates a "synthetic option" by dynamically trading in and out of cash at the margin. In this approach, as the portfolio value declines (e.g., from 900 to 880), the percentage (aka, per) put option delta becomes more negative (i.e., approaches toward -1.0 as the put becomes more ITM) but, in this insurance approach, this is just the "imaginary" compared-to put: the more negative per-option put delta implies, in the insurance approach, to sell more of the portfolio (into riskfree cash) but actual put option in involved, it is a "synthetic put option" but it only works if the trading is dynamic. Thanks,

 

[enjofaes]

Thank you very much David, your explainer says it all!