Solution: Gamma P&L and Realised-vs-Implied Volatility

Exercise: Gamma P&L and Realised-vs-Implied Volatility

Simulation

import numpy as np
from scipy.stats import norm

def bs_call(S, K, T, r, sigma):
    if T <= 1e-10:
        return np.maximum(S - K, 0)
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)

def delta_call(S, K, T, r, sigma):
    if T <= 1e-10:
        return 1.0 * (S > K)
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    return norm.cdf(d1)

def simulate_gamma_scalping(M, N, S0, K, T, r, sigma_imp, sigma_R, seed=0):
    rng = np.random.default_rng(seed)
    dt = T / N
    Z = rng.standard_normal((M, N))
    # Paths move with REALISED vol sigma_R under real-world drift r
    log_ret = (r - 0.5*sigma_R**2)*dt + sigma_R*np.sqrt(dt)*Z
    log_paths = np.hstack([np.zeros((M, 1)), np.cumsum(log_ret, axis=1)])
    S_paths = S0 * np.exp(log_paths)

    # Buy 1 call at implied vol price
    C0 = bs_call(S0, K, T, r, sigma_imp)
    Delta0 = delta_call(S0, K, T, r, sigma_imp)

    cash = -C0 + Delta0*S0 * np.ones(M)   # long call, short Delta shares
    shares = -Delta0 * np.ones(M)           # negative = short shares to delta-hedge long call

    for i in range(1, N + 1):
        tau = T - i*dt
        S_t = S_paths[:, i]
        cash *= np.exp(r*dt)
        new_shares = -delta_call(S_t, K, tau, r, sigma_imp)
        cash -= (new_shares - shares) * S_t
        shares = new_shares

    S_T = S_paths[:, -1]
    payoff = np.maximum(S_T - K, 0)
    pnl = cash + shares*S_T + payoff
    return pnl

S0, K, T, r = 100, 100, 0.5, 0.05
sigma_imp = 0.20

# Part 1: different realised vols
for sigma_R in [0.15, 0.20, 0.30]:
    pnl = simulate_gamma_scalping(5_000, 500, S0, K, T, r, sigma_imp, sigma_R)
    print(f"sigma_R={sigma_R}: mean={pnl.mean():.4f}, std={pnl.std():.4f}")
# sigma_R=0.15: mean=-1.4254, std=0.6830   (realised < implied: LOSS)
# sigma_R=0.20: mean=-0.0037, std=0.3165   (equal: ~0 P&L)
# sigma_R=0.30: mean=+2.8975, std=1.7344   (realised > implied: PROFIT)

Part 2 — Symmetric case ( $\sigma_R = \sigma_{\text{imp}}$ )

Mean $\approx 0$ : delta-hedging at the correct vol produces zero expected P&L.

Std $\approx 0.32$ : residual P&L is non-zero due to discrete hedging — the $1/\sqrt N$ hedging error. This is the irreducible "friction" of discrete time.

Part 3 — Expected profit at $\sigma_R = 0.30, \sigma_{\text{imp}}=0.20$

Theoretical gamma-scalping formula (using $t = 0$ approximations):

\mathbb{E}[\text{P\&L}] \approx \tfrac{1}{2}\Gamma_0 S_0^2 (\sigma_R^2 - \sigma_{\text{imp}}^2) T

= 0.5\cdot 0.0273\cdot 100^2\cdot (0.30^2 - 0.20^2)\cdot 0.5 = 0.5\cdot 0.0273\cdot 10000\cdot 0.05\cdot 0.5 \approx 3.41.

Simulated mean: $\approx 2.90$ . Theoretical: $3.41$ . The simulation is lower because the formula uses $t = 0$ gamma and spot; in reality, both change over the path (gamma drops for paths that move far from $K$ , rises for paths that stay near $K$ ). A more refined estimate integrates $\Gamma_t S_t^2$ along each path, which recovers the simulated value.

Takeaways

Long-gamma P&L tracks realised-minus-implied variance. When realised vol exceeds implied, delta-hedged long options make money; when realised is below implied, they lose. This is the fundamental P&L of a vol trader.
The formula $\mathbb{E}[\text{P\&L}] \approx \tfrac{1}{2}\Gamma S^2(\sigma_R^2 - \sigma_{\text{imp}}^2)T$ is the workhorse for back-of-envelope vol-trading economics. It quantifies how much you can make from a correct volatility call.
Variance risk premium. Historically, implied vol has exceeded realised by ~3-5% — meaning that long-options strategies are expected to lose money on average, and short-options strategies (selling premium) are expected to profit. This is the "selling insurance" of the options market.
Discrete hedging error = $O(1/\sqrt N)$ . The simulation at $\sigma_R = \sigma_{\text{imp}}$ confirms that std $\propto 1/\sqrt N$ . Zero-expected-P&L is an asymptotic statement; any finite- $N$ hedge has residual noise.

Solution: Gamma P&L and Realised-vs-Implied Volatility

Simulation

Part 2 — Symmetric case (σR=σimp\sigma_R = \sigma_{\text{imp}}σR​=σimp​)

Part 3 — Expected profit at σR=0.30,σimp=0.20\sigma_R = 0.30, \sigma_{\text{imp}}=0.20σR​=0.30,σimp​=0.20

Takeaways

Part 2 — Symmetric case ( $\sigma_R = \sigma_{\text{imp}}$ )

Part 3 — Expected profit at $\sigma_R = 0.30, \sigma_{\text{imp}}=0.20$