Delta-Gamma is a small Python project that compares fast, scenario-based PnL approximations for an at-the-money (ATM) option straddle against exact Black-Scholes repricing.

The goal is to keep the simplicity of a local delta-gamma (second-order Taylor) approximation near the current spot, while preventing the unrealistic quadratic blow-up in large moves by switching to linear tails.

Given a current spot S0 and an ATM straddle (1 call + 1 put, same strike K and maturity T), the script builds a pandas DataFrame across a grid of spot scenarios S:

• pnl_exact_bs
  Exact PnL from repricing the straddle with Black-Scholes at each S and subtracting the initial value at S0.

• pnl_delta_gamma
  Local second-order Taylor approximation about S0: pnl(S) = delta0 * (S-S0) + 0.5 * gamma0 * (S-S0)^2

• pnl_piecewise_clip
  C1 piecewise quadratic/linear proxy:

  • quadratic in the middle (same formula as pnl_delta_gamma)
  • linear in the tails with slopes clipped to the natural straddle delta limits [-1, +1] The linear regions start where the linearized delta hits the bounds: x_L = (delta_L - delta0)/gamma0, x_R = (delta_R - delta0)/gamma0 S_L = S0 + x_L, S_R = S0 + x_R For an ATM straddle, delta_L=-1, delta_R=+1 (and typically delta0 is near 0).

• pnl_piecewise_fit_bs
  Same piecewise quadratic/linear shape, but the join points (S_L, S_R) are chosen so that the piecewise model matches Black-Scholes PnL at the joins while keeping the tail slopes fixed. This usually improves tail accuracy, at the cost of extra setup work (a small 1D root solve per side).

The repository includes a plot generated by the script:

In the plot, dotted vertical lines show the piecewise_clip join points (S_L_clip and S_R_clip). The piecewise_clip curve coincides with pnl_delta_gamma between those lines and becomes linear outside.

• Python 3.9+ (works on newer versions as well)
• numpy
• pandas
• scipy
• matplotlib (optional, for plotting)

Install dependencies:

pip install numpy pandas scipy matplotlib

Run the script (replace with your file name if different):

python xdelta_gamma.py

Edit the parameters near the top of the script to change:

• S0, K, T, sigma
• scenario grid range and spacing
• whether to show the plot

The script prints the key values (V0, delta0, gamma0, join points) and outputs the DataFrame in-memory.

• This is a PnL-vs-spot approximation at a fixed time T and fixed sigma (Black-Scholes). It does not model changes in implied volatility, skew/smile, or time decay across scenarios.
• The piecewise approximations are intended for fast scenario grids and stress testing where a full repricer is too slow.