Pricing Callable and Putable Bonds Using the Black-Derman-Toy Model

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Abstract

Callable and putable bonds, which embed option-like features, are influenced by both market interest rate fluctuations and the value of the embedded options. Traditional pricing methods struggle to accurately reflect their complexity, while the Black-Derman-Toy (BDT) model provides a flexible interest rate tree-based approach. This article introduces the basic concepts of callable and putable bonds, the theoretical foundations of the BDT model, and explores the practical implementation of pricing these bonds using the BDT model with numerical methods.

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1. Overview of Callable and Putable Bonds

1.1 Definition

Callable and putable bonds are bonds that embed option-like features, granting the issuer or holder the right to redeem or sell the bond early under specified market conditions. Common types include:

• Callable Bonds: The issuer has the right to redeem the bond at a predetermined price on specified dates.
• Putable Bonds: The investor has the right to sell the bond back to the issuer at a predetermined price on specified dates.

1.2 Complexity in Pricing

The value of callable and putable bonds consists of two components:

• Plain Bond Value: The value of the bond without the embedded option.
• Embedded Option Value:
  • For callable bonds: Plain bond value minus the option value (negative for the holder).
  • For putable bonds: Plain bond value plus the option value (positive for the holder).

Thus, pricing these bonds requires considering:

• Market interest rate curves and their volatility.
• Embedded option terms (e.g., call price, put price, exercise dates).

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2. Introduction to the Black-Derman-Toy Model

2.1 Basic Principles

The Black-Derman-Toy (BDT) model is a single-factor interest rate tree model used to describe the stochastic evolution of short-term interest rates. Its core idea is to simulate future interest rate paths using a binomial tree. The short-term rate at each node follows the stochastic process:

$$dr_t = \theta(t) \cdot r_t \cdot dt + \sigma(t) \cdot r_t \cdot dz_t$$

where:

• $r_t$: Short-term interest rate.
• $\theta(t)$: Drift term, reflecting the average market interest rate level.
• $\sigma(t)$: Volatility of the short-term rate.
• $dz_t$: Standard Wiener process.

2.2 Features of the BDT Model

• Single-Factor Model: Assumes the short-term rate is the sole driving factor.
• Lognormal Distribution: Ensures non-negative interest rates.
• Calibration Flexibility: Can be calibrated using market data (e.g., zero-coupon rate curves and volatility curves).

2.3 Building the Interest Rate Tree

The steps to build the BDT interest rate tree are:

1. Calibrate Model Parameters: Use market data to compute the drift term $\theta(t)$ and volatility $\sigma(t)$.
2. Construct the Binomial Tree: Discretize possible interest rate paths into a binomial tree, with each node representing a different interest rate level.
3. Recursive Pricing: Use the no-arbitrage principle to compute bond values, working backward from terminal nodes to the root.

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3. Implementation of Callable and Putable Bond Pricing

3.1 Pricing Approach

Under the BDT framework, the pricing process for callable and putable bonds is as follows:

1. Build the Interest Rate Tree: Generate possible interest rate paths using the BDT model.
2. Price the Plain Bond: Starting from terminal nodes, compute the plain bond value using discount factors.
3. Adjust for Embedded Options:
   • For callable bonds: At each exercise node, compare the bond's holding value with the call price and choose the lower-cost path.
   • For putable bonds: At each exercise node, compare the bond's holding value with the put price and choose the higher-return path.

3.2 Numerical Implementation

The specific steps for pricing callable and putable bonds using the BDT model are:

1. Build the Interest Rate Tree

• Calibrate model parameters $\theta(t)$ and $\sigma(t)$ using market data (e.g., zero-coupon rate curves and volatility curves).
• Construct the interest rate tree using:

  $$r_{i,j} = r_{i-1,j} \cdot e^{\sigma(t) \sqrt{\Delta t}}$$

  where $r_{i,j}$ is the short-term rate at node $(i, j)$.

2. Compute the Plain Bond Value

• Starting from terminal nodes, recursively compute the bond value using discount factors:

  $$V_{i,j} = \frac{1}{1 + r_{i,j} \cdot \Delta t} \left( V_{i+1,j} + V_{i+1,j+1} \right)$$

3. Incorporate Option Terms

• Callable Bonds:
  At each node, adjust the bond value as:

  $$V_{i,j} = \min \left( V_{i,j}, \text{Call Price} \right)$$

• Putable Bonds:
  At each node, adjust the bond value as:

  $$V_{i,j} = \max \left( V_{i,j}, \text{Put Price} \right)$$

4. Obtain the Initial Value

• Starting from the root node (initial short-term rate), compute the current value of the callable or putable bond.

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4. Case Study

4.1 Assumptions

Assume the following market conditions:

• Zero-coupon rate curve: 1-year 2%, 2-year 2.5%, 3-year 3%.
• Volatility: 10%.
• Callable bond terms:
  • Face value: 100.
  • Call price: 102.
  • Maturity: 3 years.
  • Exercise dates: Every 6 months.

4.2 Interest Rate Tree Construction

Build a 3-year binomial interest rate tree using the BDT model, with each time step representing 6 months.

4.3 Recursive Calculation

1. Compute the terminal value of the plain bond.
2. Work backward from terminal nodes to the root, adjusting for the call option terms.

4.4 Results

Through numerical calculations, the theoretical price of the callable bond can be derived. For example, compared to a plain bond, the callable bond's price is typically lower, reflecting the value of the issuer's right to call the bond early.

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5. Advantages and Limitations

5.1 Advantages

• Flexibility: The BDT model can handle different interest rate and volatility curves.
• Broad Applicability: Suitable for various complex bond structures.
• Precise Calibration: Can be calibrated to market data, ensuring pricing accuracy.

5.2 Limitations

• Single-Factor Limitation: Assumes the short-term rate is the only driving factor, ignoring other factors (e.g., credit risk).
• Numerical Complexity: Building the interest rate tree and performing recursive calculations require significant computational resources.

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6. Conclusion

The Black-Derman-Toy model provides a flexible and effective tool for pricing callable and putable bonds. By constructing an interest rate tree and applying the no-arbitrage principle, it accurately captures the impact of market interest rate fluctuations on the value of these bonds. However, in practice, investors should also consider factors such as credit risk and liquidity to comprehensively evaluate the investment value of callable and putable bonds.

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References

1. Black, F., Derman, E., & Toy, W. (1990). A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options. Financial Analysts Journal.
2. Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson Education.
3. Brigo, D., & Mercurio, F. (2006). Interest Rate Models - Theory and Practice. Springer.