Explaining today's P&L with the greeks

Question

Your option book starts the day with delta $\Delta = +200$ , gamma $\Gamma = +50$ (deltas per $\$ 1 $), vega$ = +$300 $per vol point, and theta$ \Theta = -$120 $/day. During the day the stock rises$ $1.50 $and implied vol falls$ 0.5$ points.

Attribute the day's P&L to each greek. What does a leftover "residual" tell you?

Accepted Answer

The standard first-order-plus-gamma P&L explain (a Taylor expansion of the book's value) is \Delta	ext{P\&L} \approx \underbrace{\Delta\cdot dS}{	ext{delta}} + \underbrace{	frac12\Gamma\,(dS)^2}{	ext{gamma}} + \underbrace{	ext{Vega}\cdot d\sigma}{	ext{vega}} + \underbrace{\Theta\cdot dt}{	ext{theta}}. Compute each piece with dS = +1.50, d\sigma = -0.5, dt = 1 day: - Delta: 200 	imes 1.50 = +\300. - Gamma: 	frac12 	imes 50 	imes (1.50)^2 = 	frac12 	imes 50 	imes 2.25 = +\56.25. - Vega: 300 	imes (-0.5) = -\150. - Theta: -120 	imes 1 = -\120. Total explained: 300 + 56.25 - 150 -…

Explaining today's P&L with the greeks

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