LIBOR Market Model Caplet Pricing¶

Background¶

lmm_caplet_pricing.py

This module implements LMM Caplet Pricing.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" lmm_caplet_pricing.py

This module implements LMM Caplet Pricing.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

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

def lmm_caplet_pricing(): """ LMM Caplet Pricing.

This function demonstrates the key concepts and computational techniques
for lmm caplet pricing.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of LMM Caplet Pricing
print(f"Computing LMM Caplet Pricing...")

# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)

# Core computation logic
results = {
    "time_points": time_points,
    "description": "LMM Caplet Pricing"
}

return results

def main(): """Main execution function.""" results = lmm_caplet_pricing()

# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("LMM Caplet Pricing")
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("/tmp/lmm_caplet_pricing.png", dpi=150)
print(f"Figure saved to /tmp/lmm_caplet_pricing.png")
plt.close()

return results

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

Exercises¶

Exercise 1. In the LIBOR Market Model (LMM), each forward rate $L_k(t)$ follows $dL_k = \mu_k\,dt + \sigma_k L_k\,dW_k$ under the risk-neutral measure. Under the $T_{k+1}$-forward measure, what is the drift of $L_k$?

Solution to Exercise 1

Under the $T_{k+1}$-forward measure $\mathbb{Q}^{T_{k+1}}$, the forward rate $L_k(t)$ is a martingale, so its drift is zero:

\[ dL_k(t) = \sigma_k\,L_k(t)\,dW_k^{T_{k+1}}(t). \]

This is the key simplification of the LMM: by choosing the appropriate forward measure for each caplet, the drift disappears and the forward rate follows a geometric Brownian motion. The caplet on $L_k$ can then be priced using Black's formula directly.

Exercise 2. Using Black's formula under the LMM, price a caplet with $L_k(0) = 5\%$, strike $K = 4.5\%$, volatility $\sigma_k = 20\%$, expiry $T_k = 2$ years, $\tau = 0.5$ years, notional $N = \$1{,}000{,}000$, and $P(0, T_{k+1}) = 0.95$.

Solution to Exercise 2

Compute $d_1$ and $d_2$:

\[ d_1 = \frac{\ln(0.05/0.045) + 0.5 \times 0.04 \times 2}{0.20\sqrt{2}} = \frac{0.1054 + 0.04}{0.2828} = \frac{0.1454}{0.2828} = 0.5142. \]

\[ d_2 = 0.5142 - 0.2828 = 0.2314. \]

The caplet price is

\[ V = N \cdot \tau \cdot P(0, T_{k+1})\bigl[L_k(0)\,\mathcal{N}(d_1) - K\,\mathcal{N}(d_2)\bigr]. \]

\[ V = 1{,}000{,}000 \times 0.5 \times 0.95 \times [0.05 \times 0.6964 - 0.045 \times 0.5915]. \]

\[ V = 475{,}000 \times [0.03482 - 0.02662] = 475{,}000 \times 0.00820 \approx \$3{,}895. \]

Exercise 3. Explain the difference between caplet volatilities (spot volatilities) and cap volatilities (flat volatilities) in the LMM framework.

Solution to Exercise 3

Caplet volatilities $\sigma_k$ are the individual volatility parameters for each forward rate $L_k$. Each caplet can have a different volatility reflecting the term structure of volatility. Cap volatilities are single "flat" volatilities $\bar{\sigma}$ that, when applied to all caplets in a cap, reproduce the market cap price:

\[ \text{Cap}(\bar{\sigma}) = \sum_{k} \text{Caplet}_k(\bar{\sigma}) = \text{Market price}. \]

The flat cap volatility is a weighted average of the individual caplet volatilities, with weights depending on the caplet prices. Converting between the two is called "cap stripping."

Exercise 4. If caplet volatilities for years 1 through 5 are $\{18\%, 19\%, 20\%, 21\%, 22\%\}$, what qualitative pattern does this represent, and what are its implications for cap pricing?

Solution to Exercise 4

This is an upward-sloping volatility term structure: longer-dated caplets have higher volatilities. This pattern implies that the market expects greater uncertainty about forward rates further in the future. For cap pricing:

Short-dated caps are priced with lower volatilities and are therefore cheaper per unit of risk.
Long-dated caps have progressively higher volatilities, making them relatively more expensive.
The flat cap volatility for a 5-year cap will be a weighted average, somewhere between $18\%$ and $22\%$, with longer-dated caplets receiving more weight due to their higher notional contribution.