Break-even volatility for delta hedge portfolio

After simulating practical and theoretical PnL of a delta hedged portfolio on some data from the SPX500 under 0.15 management Vol I want to find the Vol which gives me an accumulated PnL of 0.

Initially I tried using the formula squared-vol = (-2 * theta) / (sigma * S^2) but that just returns whatever management Vol I calculate the portfolio under.

Can someone tell me what I'm missing?

• at which point does the implied implied vol come in for which you purchase the option ? If that is the same as the vol that you run your simulation at, then the sum of hedging pnl and options pnl should always work out to zero – ZRH Mar 14 '19 at 7:35
• I'm not entirely sure exactly what you're asking but the average vol over the period for the SPX close is about 0.158 but we're just assuming a management Vol of 0.15 and using that for BS price and greeks calculation. Currently the PnL is about +3% on the delta hedged portfolio, -70% if on just a call portfolio and -0.5% on the underlying (SPX), I want to calculate the management vol at which the PnL for the hedge portfolio is 0%, does that make sense? – DasBoot Mar 14 '19 at 8:28
• So basically you are asking what vol should you have used in the past to come up with zero PnL ? – ZRH Mar 14 '19 at 18:47
• How did you do your simulation of the underlying (which appears to be the SPX)? Did you use an actual historical path of the SPX? – Frido Rolloos Mar 15 '19 at 6:10