Change in price of underlying impact on delta gamma and vega

Question

I am working my way through Natenberg's book as well as the accompanying workbook, and there is a question I cannot figure out (p86).

• Futures price = 149.65
• time to August expiration = 8 weeks
• annual volatility = 24.20%

You have the following position:

• -32 August 140 puts
• +30 August 160 calls
• -15 August futures contracts

with the options having these risk sensitivities

option	delta	gamma	theta	vega
Aug 140 put	-22.6	2.12	-.0381	.176
Aug 160 call	25.5	2.26	-.0407	.188

I correctly calculated that the greeks for my position were

delta	gamma	theta	vega
-11.8	-0.04	+0.0018	0.008

The question is

What will happen to your delta, gamma, and vega position if the futures price rises while all other market conditions remain unchanged?

The answers say that the delta, gamma, and vega would all become positive. I don't see how, with rising prices and negative gamma, even if its very small, the delta becomes positive. Why does the gamma and the vega become positive as well?

Would somebody please explain to me why this is the case?

Thank you in advance!

Answer 1

Without having done the maths myself - as you drift away from the short put strike towards the long call strike, your gamma also starts becoming more positive. This is because the future drifts away from the short put strike of 140 to being closer to the 160 long calls (atmf options habe more gamma than otm/itm options). This also naturally means that you start accruing deltas, as you drift.

The small -ve gamma that you calculate is only locally correct, gamma also changes with the level of the underlying with respect to the strikes of the portfolio. The vega becoming (more) positive has a similar rationale.