Why is this delta-hedging/P&L example on a variance swap call correct?

Question

I'm looking into this article about var swaps: http://sbossu.com/docs/VarSwaps.pdf and not sure how to correctly interpret Exhibit 2.1.1.

"In this example an option trader sold a 1-year call struck at 110% of the initial price on a notional of 10,000,000 for an implied volatility of 30%, and delta-heged his position daily. The realized volatility was 27.50%, yet his final trading P&L is down $150k. Furthermore, we can see (Figure a) ....."

Is the trading p&l meant to be the delta-hedging p&l? It looks like it, but

• how come p&l is raising steadily even when stock price is rising? the trader should be losing money on the delta hedging because he is short gamma?
• why does it start from zero ? When trader has sold the option, he should have bought the stock and trading p&l at that point was negative?

Answer 1

To answer your questions:

Is the trading p&l meant to be the delta-hedging p&l? Yes, in his example it concerns delta hedged pnl.

how come p&l is raising steadily even when stock price is rising? the trader should be losing money on the delta hedging because he is short gamma? He is short gamma but long theta. He is initially making money because as per his chart (the bottom one) initially when the stock is rising the realized vol is below 30%. His gamma pnl would equal his theta pnl if realized would equal 30%. However initially the realized vol is less than 30% therefore he makes more on his theta than he loses on his gamma.

This follow from the black scholes PDE. $\theta + 0.5 \sigma^2 S^2 \gamma + r S \delta - rV = 0$ (perhaps a bit sloppy. you can derive this much nicer with r>0) but take r = 0. Then you will see immediately that $\theta = - 0.5 \sigma^2 \gamma$

why does it start from zero ? When trader has sold the option, he should have bought the stock and trading p&l at that point was negative?

If at time 0 the value of any asset is say S, and you buy it at S and immediately you look at your pnl then you bought at S and you can sell at S. Therefore your theoretical pnl is 0.