Total P&L of a vol-arb trade over the option's life

Question

You buy an at-the-money option at an implied vol of $18\%$ and delta-hedge it to expiry. Over its $60$ -day life, the stock ends up realizing $24\%$ volatility. The option's vega is about $\$ 0.30$ per vol point (per share).

Roughly what does the delta-hedged trade earn, and what's the exact form of that P&L?

Accepted Answer

Quick estimate (the vega rule). To first order, a delta-hedged option bought at implied vol \sigmai and held while the stock realizes \sigmar earns approximately 	ext{P\&L} \approx 	ext{Vega} 	imes (\sigmar - \sigmai) = 0.30 	imes (24 - 18) = 0.30 	imes 6 = \1.80 	ext{ per share.} You bought cheap volatility (18) and the world delivered expensive volatility (24); the vega times the 6-point gap is your edge. Over a 10{,}000-share option position that's \approx\18{,}000. The exact form. Rigorously, the total delta-hedged P&L is the running gamma-vs-theta accumulation over the life of the …

Total P&L of a vol-arb trade over the option's life

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