Time Decay of Option Values¶

Background¶

Black Scholes Time Decay

Educational script demonstrating black scholes time decay concepts.

Code¶

```python """ Black Scholes Time Decay

Educational script demonstrating black scholes time decay concepts. """

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black_scholes_TIME_DECAY_OF_OPTION_VALUES.py¶

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

Market parameters¶

if name == "main": r = 0.05 # risk-free rate sigma = 0.2 # volatility T = 1.0 # maturity in years days = np.linspace(0.01, T, 100) # time to maturity

# Create plots for three moneyness cases
fig, ax = plt.subplots(3, 2, figsize=(14, 9))

# Moneyness scenarios
spot_prices = [80, 100, 120]
labels = ["In-the-money", "At-the-money", "Out-of-the-money"]
K = 100  # fixed strike

for i, S in enumerate(spot_prices):

    ax[i, 0].plot(days, bs.bs_call_price(S, K, days, r, sigma), color='blue')
    ax[i, 0].invert_xaxis()
    ax[i, 0].set_title(f"Call Price: {labels[2 - i]}")
    ax[i, 0].set_xlabel("Time to Maturity (Years)")
    ax[i, 0].set_ylabel("Call Value")
    ax[i, 0].grid(True)

    ax[i, 1].plot(days, bs.bs_put_price(S, K, days, r, sigma), color='red')
    ax[i, 1].invert_xaxis()
    ax[i, 1].set_title(f"Put Price: {labels[i]}")
    ax[i, 1].set_xlabel("Time to Maturity (Years)")
    ax[i, 1].set_ylabel("Put Value")
    ax[i, 1].grid(True)

fig.suptitle("European Call and Put Prices vs. Time to Maturity for Different Moneyness Scenarios", fontsize=16)
plt.tight_layout()
plt.show()

```

Exercises¶

Exercise 1. For an ITM call (\(S = 120\), \(K = 100\)), its time value is \(C - \max(S - K, 0)\). Explain why ITM options have less time value than ATM options.

Exercise 2. The put time decay curve differs from the call curve. For a deep ITM put (\(S = 80\), \(K = 100\)), explain whether its time value increases or decreases with time to maturity.

Exercise 3. At expiry (\(T \to 0\)), the call value converges to \(\max(S - K, 0)\) with a kink at \(S = K\). Describe how the time decay curve at \(S = K\) approaches zero as \(T \to 0\).

Exercise 4. For three moneyness levels (\(S = 80, 100, 120\) with \(K = 100\)), rank the call options by their rate of time decay at \(T = 0.5\).