Pricing the edge when realized beats implied

Question

You are long an option and delta-hedge it. The option is priced at implied volatility $\sigma_{\text{impl}} = 20\%$ , but your forecast is that the stock will realize $\sigma_{\text{real}} = 30\%$ . The stock is at $S = 100$ and the option's gamma is roughly $\Gamma = 0.05$ .

What is the expected P&L of the hedged position, per day and annualized?

Accepted Answer

The expected P&L of a delta-hedged option over a short interval, after theta and gamma are combined, is E[d\Pi] \approx 	frac{1}{2}\,\Gamma\,S^2\big(\sigma{	ext{real}}^2 - \sigma{	ext{impl}}^2\big)\,dt. You earn the gap between the variance the stock actually delivers and the variance you paid for. Plug in \Gamma = 0.05, S = 100, \sigma{	ext{real}} = 0.30, \sigma{	ext{impl}} = 0.20: 	frac{1}{2}(0.05)(10{,}000)\big(0.30^2 - 0.20^2\big) = 	frac{1}{2}(0.05)(10{,}000)(0.09 - 0.04) = 	frac{1}{2}(0.05)(10{,}000)(0.05). Working it through: 0.05 	imes 10{,}000 = 500, then 500 	imes 0.05 = 2…

Pricing the edge when realized beats implied

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