Delta-Hedging Simulation and Rebalancing Frequency Analysis

📊 Description

Monte Carlo simulation of a delta-hedging strategy on a European Call under the Black-Scholes model. The project studies the impact of the rebalancing frequency (daily, weekly, monthly, quarterly) on the hedging error, and compares with an unhedged strategy.

🎯 Objectives

• Quantify the replication error as a function of the rebalancing frequency
• Illustrate the convergence toward a perfect hedge as the time step decreases
• Compare the P&L distribution with and without delta-hedging

📐 Mathematical Model

Underlying dynamics

Under the risk-neutral measure, the price follows a geometric Brownian motion:

$$dS_t = r S_t , dt + \sigma S_t , dW_t$$

Trajectories are simulated in log-price using the Euler scheme:

$$\ln S_{t_{k+1}} = \ln S_{t_k} + \left(r - \tfrac{1}{2}\sigma^2\right) h + \sigma \sqrt{h} , Z_k, \qquad Z_k \sim \mathcal{N}(0,1)$$

Hedging portfolio

At each rebalancing date, the stock position is adjusted to the Black-Scholes delta:

$$\Delta_t = \Phi(d_1), \qquad d_1 = \frac{\ln(S_t / K) + (r + \tfrac{1}{2}\sigma^2)\tau}{\sigma\sqrt{\tau}}$$

The discounted hedging P&L is given by:

$$\text{PnL}_T = \sum_{k=0}^{n-1} \Delta_{t_k} \left( S_{t_{k+1}} - S_{t_k} e^{rh} \right) e^{r(T - t_{k+1})}$$

The terminal portfolio value reads:

$$V_T = C_0 , e^{rT} + \text{PnL}_T - \max(S_T - K, 0)$$

where $C_0$ is the initial call price. A perfect hedge yields $V_T = 0$.

Unhedged strategy

$$V_T^{\text{no hedge}} = C_0 , e^{rT} - \max(S_T - K, 0)$$

🔧 Parameters

Parameter	Value
$S_0$	100
$K$	100 (ATM)
$r$	5%
$\sigma$	20%
$T$	1 year
Simulations	16,384

Tested rebalancing frequencies

• Daily: $h = 1/252$
• Weekly: $h = 1/52$
• Monthly: $h = 1/12$
• Quarterly: $h = 1/4$

📈 Results

The script generates:

• Histograms of the terminal portfolio value for each rebalancing frequency and for the unhedged strategy
• Descriptive statistics (mean, standard deviation, quantiles) allowing the hedging error to be quantified

🚀 Usage

python delta_hedging.py

📦 Dependencies

pip install numpy pandas matplotlib scipy

👨‍💻 Author

Alexandre R. - Université Paris Cité