PnL Decomposition of Options Trading Strategies

Newsletter #59

Backtesting is a core daily task for a quant. Beyond calculating the total cumulative PnL, the goal of backtesting is to analyse where the PnL comes from.

In this article, we focus on backtesting options strategies, where the PnL is typically decomposed into the PnL from each Greek component:

• Delta PnL: The PnL from changes in the underlying asset's price.

• Gamma PnL: The PnL from changes in the option’s delta.

• Vega PnL: The PnL from changes in implied volatility.

• Theta PnL: The PnL from the time decay of the option.

• Unexplained PnL: The difference between the total PnL and the sum of the explained PnL components.

We exclude Rho (the change in option value due to changes in the risk-free rate) as its impact is typically minimal.

Delta PnL

If an option has a delta of 0.5, a $1 increase in the underlying's price would lead to a $0.50 increase in the option's price.

\(\text{Delta PnL} = \text{Delta} \times \Delta S\)

df['delta_pnl'] = df['delta'].shift(1) * df['stock_price'].diff()

Gamma PnL

From the Taylor expansion, the Gamma PnL is computed using the formula below.

As gamma PnL is proportional to the square of the price change, gamma PnL is always positive for long option holders.

\(\text{Gamma PnL} = \frac{1}{2} \times \text{Gamma} \times (\Delta S)^2\)

df['gamma_pnl'] = 0.5 * df['gamma'].shift(1) * df['stock_price'].diff().pow(2)

Vega PnL

If an option has a vega of 0.2, a 1% increase in implied volatility would increase the option's price by $0.20.

\(\text{Vega PnL} \approx \text{Vega} \times \Delta \sigma\)

df['vega_pnl'] = df['vega'].shift(1) * df['implied_vol'].diff()

Theta PnL

If an option has a theta of -0.05, the option's price decreases by $0.05 per day. Long options positions always have negative theta (which balances out the always positive gamma PnL).

\(\text{Theta PnL} \approx \text{Theta} \times \Delta t\)

df['theta_pnl'] = df['theta'].shift(1) * df['date'].diff()

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