Asian Call (Control Variates)¶

Background¶

Exotic Options: Asian Call Pricing with Control Variates

This script demonstrates variance reduction via control variates for pricing an arithmetic average Asian call. The European call (with known Black-Scholes price) serves as the control variate, correcting systematic Monte Carlo errors.

Mathematical Framework:

- Control variate estimator:
    V_asian_adj = V_asian_MC + (C_BS - C_MC)
- The correction (C_BS - C_MC) removes the systematic error shared
  between the Asian and European payoff estimates.
- Variance reduction is proportional to Corr(Asian_payoff, Euro_payoff)^2.

Also demonstrates using the geometric average Asian option as a superior control variate (higher correlation with the arithmetic average).

References:

- Glasserman (2003). Monte Carlo Methods in Financial Engineering, Ch. 4.
- Kemna, Vorst (1990). A pricing method for options based upon average values.

Code¶

```python """ Exotic Options: Asian Call Pricing with Control Variates

This script demonstrates variance reduction via control variates for pricing an arithmetic average Asian call. The European call (with known Black-Scholes price) serves as the control variate, correcting systematic Monte Carlo errors.

Mathematical Framework: - Control variate estimator: V_asian_adj = V_asian_MC + (C_BS - C_MC) - The correction (C_BS - C_MC) removes the systematic error shared between the Asian and European payoff estimates. - Variance reduction is proportional to Corr(Asian_payoff, Euro_payoff)^2.

Also demonstrates using the geometric average Asian option as a superior control variate (higher correlation with the arithmetic average).

References: - Glasserman (2003). Monte Carlo Methods in Financial Engineering, Ch. 4. - Kemna, Vorst (1990). A pricing method for options based upon average values. """

import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm

=============================================================================¶

1. Black-Scholes Call Price¶

=============================================================================¶

def black_scholes_call(S, K, T, r, sigma): """Analytical Black-Scholes European call price.""" 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)

=============================================================================¶

2. Standard Asian Call Monte Carlo (baseline)¶

=============================================================================¶

def asian_call_standard(S, K, T, r, sigma, M, N, seed=None): """Standard MC for Asian call.""" if seed is not None: np.random.seed(seed)

dt = T / M
asian_payoff = np.zeros(N)

for i in range(N):
    S_path = [S]
    for _ in range(M):
        z = np.random.normal()
        S_next = S_path[-1] * np.exp(
            (r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * z
        )
        S_path.append(S_next)
    avg_price = np.mean(S_path)
    asian_payoff[i] = np.exp(-r * T) * max(avg_price - K, 0)

price = np.mean(asian_payoff)
se = np.std(asian_payoff) / np.sqrt(N)
return price, se

=============================================================================¶

3. Control Variate: European Call¶

=============================================================================¶

def asian_call_control_variate(S, K, T, r, sigma, M, N, seed=None): """ Asian call with European call as control variate.

Uses the known Black-Scholes price to correct the MC estimate:
    V_asian_adj[i] = V_asian[i] + (C_BS - C_MC_mean)

Parameters
----------
S, K, T, r, sigma : float
    Standard option parameters.
M : int
    Number of time steps per path.
N : int
    Number of simulated paths.

Returns
-------
price : float
    Control-variate-adjusted price.
se : float
    Standard error of the adjusted estimate.
"""
if seed is not None:
    np.random.seed(seed)

dt = T / M
euro_payoff = np.zeros(N)
asian_payoff = np.zeros(N)

for i in range(N):
    S_path = [S]
    for _ in range(M):
        z = np.random.normal()
        S_next = S_path[-1] * np.exp(
            (r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * z
        )
        S_path.append(S_next)

    avg_price = np.mean(S_path)
    euro_final = S_path[-1]

    asian_payoff[i] = np.exp(-r * T) * max(avg_price - K, 0)
    euro_payoff[i] = np.exp(-r * T) * max(euro_final - K, 0)

# Control variate adjustment
euro_mc = np.mean(euro_payoff)
euro_bs = black_scholes_call(S, K, T, r, sigma)
adj_asian = asian_payoff + (euro_bs - euro_mc)

price = np.mean(adj_asian)
se = np.std(adj_asian) / np.sqrt(N)
return price, se

=============================================================================¶

4. Control Variate: Geometric Average Asian Call¶

=============================================================================¶

def geometric_asian_call_exact(S, K, T, r, sigma): """ Exact price of a continuous geometric average Asian call.

Under GBM, the geometric average is lognormal with adjusted parameters:
    sigma_hat = sigma / sqrt(3)
    r_hat = 0.5 * (r - sigma^2/6)
"""
sigma_hat = sigma / np.sqrt(3)
r_hat = 0.5 * (r - sigma**2 / 6)
S_hat = S * np.exp((r_hat - r) * T)

d1 = (np.log(S_hat / K) + (r_hat + 0.5 * sigma_hat**2) * T) / (sigma_hat * np.sqrt(T))
d2 = d1 - sigma_hat * np.sqrt(T)

return np.exp(-r * T) * (S_hat * np.exp(r_hat * T) * norm.cdf(d1) - K * norm.cdf(d2))

def asian_call_geometric_cv(S, K, T, r, sigma, M, N, seed=None): """ Asian call with geometric average Asian call as control variate.

The geometric average is more highly correlated with the arithmetic
average than the European terminal price, providing superior variance
reduction.
"""
if seed is not None:
    np.random.seed(seed)

dt = T / M
arith_payoff = np.zeros(N)
geom_payoff = np.zeros(N)

for i in range(N):
    S_path = [S]
    for _ in range(M):
        z = np.random.normal()
        S_next = S_path[-1] * np.exp(
            (r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * z
        )
        S_path.append(S_next)

    arith_avg = np.mean(S_path)
    geom_avg = np.exp(np.mean(np.log(S_path)))

    arith_payoff[i] = np.exp(-r * T) * max(arith_avg - K, 0)
    geom_payoff[i] = np.exp(-r * T) * max(geom_avg - K, 0)

# Control variate adjustment
geom_mc = np.mean(geom_payoff)
geom_exact = geometric_asian_call_exact(S, K, T, r, sigma)
adj_arith = arith_payoff + (geom_exact - geom_mc)

price = np.mean(adj_arith)
se = np.std(adj_arith) / np.sqrt(N)
return price, se

=============================================================================¶

5. Main: Comparison¶

=============================================================================¶

if name == "main": # Parameters S = 100 K = 100 T = 1 r = 0.05 sigma = 0.2 M = 252 N = 50000

print("=" * 70)
print("ASIAN CALL PRICING: Control Variate Methods")
print("=" * 70)
print(f"Parameters: S={S}, K={K}, T={T}, r={r}, sigma={sigma}")
print(f"Paths: N={N}, Steps: M={M}")
print()

# --- Run all three methods ---
std_price, std_se = asian_call_standard(S, K, T, r, sigma, M, N, seed=42)
cv_euro_price, cv_euro_se = asian_call_control_variate(
    S, K, T, r, sigma, M, N, seed=42
)
cv_geom_price, cv_geom_se = asian_call_geometric_cv(
    S, K, T, r, sigma, M, N, seed=42
)

# Reference values
bs_price = black_scholes_call(S, K, T, r, sigma)
geom_price = geometric_asian_call_exact(S, K, T, r, sigma)

print(f"Reference: BS Call = {bs_price:.4f}, Geometric Asian = {geom_price:.4f}")
print()

print(f"{'Method':<30} {'Price':>10} {'Std Err':>10} {'Var Ratio':>10}")
print("-" * 62)
print(f"{'Standard MC':<30} {std_price:10.4f} {std_se:10.4f} {'1.00':>10}")
print(f"{'CV: European Call':<30} {cv_euro_price:10.4f} {cv_euro_se:10.4f}"
      f" {std_se**2/cv_euro_se**2:10.2f}")
print(f"{'CV: Geometric Asian':<30} {cv_geom_price:10.4f} {cv_geom_se:10.4f}"
      f" {std_se**2/cv_geom_se**2:10.2f}")

# --- Convergence comparison ---
N_values = [500, 1000, 2000, 5000, 10000, 25000, 50000]
results = {'Standard': [], 'CV Euro': [], 'CV Geom': []}

for n in N_values:
    _, se1 = asian_call_standard(S, K, T, r, sigma, M, n, seed=42)
    _, se2 = asian_call_control_variate(S, K, T, r, sigma, M, n, seed=42)
    _, se3 = asian_call_geometric_cv(S, K, T, r, sigma, M, n, seed=42)
    results['Standard'].append(se1)
    results['CV Euro'].append(se2)
    results['CV Geom'].append(se3)

# --- Plots ---
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Panel 1: Standard error comparison
axes[0].plot(N_values, results['Standard'], 'ro-', markersize=5, label='Standard MC')
axes[0].plot(N_values, results['CV Euro'], 'go-', markersize=5, label='CV: European')
axes[0].plot(N_values, results['CV Geom'], 'bo-', markersize=5, label='CV: Geometric')
axes[0].set_xlabel('Number of Simulations (N)')
axes[0].set_ylabel('Standard Error')
axes[0].set_title('Standard Error Comparison')
axes[0].set_xscale('log')
axes[0].set_yscale('log')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Panel 2: Variance reduction ratios
ratios_euro = [s**2 / e**2 for s, e in zip(results['Standard'], results['CV Euro'])]
ratios_geom = [s**2 / g**2 for s, g in zip(results['Standard'], results['CV Geom'])]

x = np.arange(len(N_values))
width = 0.35
axes[1].bar(x - width/2, ratios_euro, width, label='CV: European',
            color='green', alpha=0.7, edgecolor='black')
axes[1].bar(x + width/2, ratios_geom, width, label='CV: Geometric',
            color='blue', alpha=0.7, edgecolor='black')
axes[1].set_xticks(x)
axes[1].set_xticklabels([str(n) for n in N_values], rotation=45)
axes[1].set_xlabel('Number of Simulations (N)')
axes[1].set_ylabel('Variance Reduction Ratio')
axes[1].set_title('Variance Reduction: Euro vs Geometric CV')
axes[1].axhline(y=1, color='r', linestyle='--', alpha=0.5)
axes[1].legend()
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('asian_call_control_variate.png', dpi=150, bbox_inches='tight')
plt.show()

print("\nPlot saved to asian_call_control_variate.png")

```

Exercises¶

Exercise 1. Explain the control variate technique for Asian option pricing. Why is the European call a natural control variate?

Exercise 2. Why is the geometric average Asian option an even better control variate than the European call?

Exercise 3. Derive the optimal control variate coefficient \(c^*\) and the resulting variance reduction formula.

Exercise 4. Can multiple control variates be combined? Describe how to use both the European call and geometric Asian as controls simultaneously.