Sensitivity Analysis¶

Background¶

Black Scholes Sensitivity

Educational script demonstrating black scholes sensitivity concepts.

Code¶

```python """ Black Scholes Sensitivity

Educational script demonstrating black scholes sensitivity concepts. """

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

black_scholes_SENSITIVITY_ANALYSIS.py¶

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

import black_scholes as bs import matplotlib.pyplot as plt import numpy as np

def main(): # Model parameters S0 = 100 # Current stock price K = 105 # Strike price T = 0.25 # 3 months to expiration r = 0.05 # 5% risk-free rate sigma = 0.2 # 20% volatility q = 0.02 # 2% dividend yield mu = 0.1

# =============================================================================
# SENSITIVITY ANALYSIS
# =============================================================================
print("Analyzing option price sensitivity to parameters...")

# Volatility sensitivity
vol_range = np.linspace(0.1, 0.4, 10)
call_prices_vol = []

for vol in vol_range:
    bs_temp = bs.BlackScholes(S0, K, T, r, vol, q)
    call_temp, _ = bs_temp.price_analytical()
    call_prices_vol.append(call_temp)

plt.figure(figsize=(12, 4))

plt.subplot(1, 2, 1)
plt.plot(vol_range*100, call_prices_vol, 'b-', linewidth=2, marker='o')
plt.axvline(sigma*100, color='red', linestyle='--', alpha=0.7, 
            label=f'Current σ = {sigma*100:.1f}%')
plt.xlabel('Volatility (%)')
plt.ylabel('Call Price ($)')
plt.title('Call Price vs Volatility')
plt.grid(True, alpha=0.3)
plt.legend()

# Stock price sensitivity
S_range = np.linspace(80, 120, 20)
call_prices_S = []

for S in S_range:
    bs_temp = bs.BlackScholes(S, K, T, r, sigma, q)
    call_temp, _ = bs_temp.price_analytical()
    call_prices_S.append(call_temp)

plt.subplot(1, 2, 2)
plt.plot(S_range, call_prices_S, 'g-', linewidth=2, marker='o')
plt.axvline(S0, color='red', linestyle='--', alpha=0.7, 
            label=f'Current S₀ = ${S0}')
plt.axhline(K, color='orange', linestyle=':', alpha=0.7, 
            label=f'Strike K = ${K}')
plt.xlabel('Stock Price ($)')
plt.ylabel('Call Price ($)')
plt.title('Call Price vs Stock Price')
plt.grid(True, alpha=0.3)
plt.legend()

plt.tight_layout()
plt.show()

if name == "main": main() ```

Exercises¶

Exercise 1. Plot (conceptually) the call price as a function of volatility \(\sigma \in [0.1, 0.4]\) for an ATM option. Is the relationship linear, convex, or concave?

Exercise 2. For \(S_0 = 100\), \(K = 105\), \(T = 0.25\), \(r = 0.05\), \(\sigma = 0.2\), \(q = 0.02\), compute the call price sensitivity to a 1% increase in volatility (from 20% to 21%).

Exercise 3. Explain why the call price is convex in the underlying price \(S\) (i.e., gamma is positive) and the financial implication for delta hedging.

Exercise 4. Compare the sensitivity of call and put prices to the risk-free rate \(r\). Which has positive rho and which has negative rho? Explain the economic intuition.