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Bond Option Pricing Principles

上海子午线About 5 min

Bond Option Pricing Principles

1. Overview of Bond Options

Bond options grant the holder the right to buy or sell a specific bond at a predetermined price on or before a specified date. In addition to being traded in over-the-counter markets, bond options are often embedded in bond issuances to enhance their appeal to issuers or potential buyers.

Bond options include embedded bond options and European bond options.

1.1 Embedded Bond Options

Callable Bonds are a prime example of embedded bond options. These bonds allow the issuing company to repurchase the bond at a predetermined price at specific future dates. Holders of such bonds effectively sell a call option to the issuer.

  • Strike Price (Call Price): The predetermined price the issuer must pay to the holder.
  • Lock-out Period: Callable bonds typically cannot be called for the first few years after issuance.
  • Decreasing Call Price: For example, a 10-year callable bond may have no call privileges for the first 2 years, a call price of 110 in years 3-4, 107.5 in years 5-6, 106 in years 7-8, and 103 in years 9-10.

The value of the call option is reflected in the bond's quoted yield. Generally, bonds with call features offer higher yields than those without.

Puttable Bonds are another type of embedded option bond, allowing holders to demand early redemption at a predetermined price at specific future dates. Holders of such bonds effectively purchase a put option alongside the bond. Since the put option increases the bond's value to the holder, puttable bonds typically offer lower yields.

Loan and Deposit Instruments often contain embedded bond options. Examples include:

  • A 5-year fixed-rate deposit redeemable without penalty, which includes an American put option.
  • Prepayment privileges on loans and mortgages, which function as call options on bonds.
  • Loan commitments by banks, which act as put options on bonds.

1.2 European Bond Options

A European bond option is a financial derivative that grants the holder the right, but not the obligation, to buy (call option) or sell (put option) the underlying bond at a predetermined price on a specific date. Unlike stock options, bond option pricing must account for the interest rate term structure, coupon payments, and credit risk.

This article focuses on the pricing logic for European bond options.

2. Pricing Methods for European Bond Options

2.1 Black-76 Model

The Black-76 model, a variant of the Black-Scholes model, is specifically designed for pricing options based on forward prices and is the most commonly used model for bond option pricing.

2.1.1 Model Formulas

For a European bond call option:

C=e−rT[FN(d1)−KN(d2)]C = e^{-rT}[FN(d_1) - KN(d_2)]

For a European bond put option:

P=e−rT[KN(−d2)−FN(−d1)]P = e^{-rT}[KN(-d_2) - FN(-d_1)]

where:

d1=ln⁡(F/K)+(σ2/2)TσTd_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}}

d2=d1−σTd_2 = d_1 - \sigma\sqrt{T}

Variable Definitions:

  • F: Forward price of the bond
  • K: Strike price of the option
  • r: Risk-free interest rate
  • T: Time to maturity of the option
  • σ: Volatility of the bond price
  • N(·): Cumulative standard normal distribution function

2.1.2 Implementation Steps

  1. Determine the Forward Price (FBF_B):

FB=B0−IP(0,T)F_B = \frac{B_0 - I}{P(0, T)}

  • B0B_0: Current full price of the bond (clean price + accrued interest)
  • II: Present value of coupons paid during the option's life
  • P(0,T)P(0, T): Risk-free discount factor for maturity TT

  1. Select Appropriate Volatility:

    • Historical volatility: Based on historical bond price data
    • Implied volatility: Derived from market option prices

  2. Determine the Risk-Free Rate:
    Typically, the government bond yield or SHIBOR rate for the same maturity is used.

  3. Calculate the Option Price

2.1.3 Example

Consider a 10-month European call option on a 9.75-year bond with a face value of $1,000 (remaining maturity of 8 years and 11 months at option expiration). Given:

  • Current cash bond price: $960
  • Strike price: $1,000
  • 10-month risk-free rate: 10% (annualized)
  • Forward bond price volatility: 9% (annualized)
  • Bond coupon: 10% (semiannual payments of $50 at 3 and 9 months)

Calculation Steps:

  1. Present value of coupons:

    50e−0.25×0.09+50e−0.75×0.095=$95.4550e^{-0.25 \times 0.09} + 50e^{-0.75 \times 0.095} = \$95.45

  2. Forward bond price:

    FB=(960−95.45)e0.1×0.8333=$939.68F_B = (960 - 95.45)e^{0.1 \times 0.8333} = \$939.68

  3. Option price:
    • If the strike price is the cash price ($1,000), the option price is $9.49.
    • If the strike price is the quoted price (plus accrued interest of $1,008.33), the option price is $7.97.

2.1.4 Applicability

The Black-76 model is particularly suitable for:

  • European bond options
  • Highly liquid bonds
  • Short-term option pricing
  • Over-the-counter market quotations

2.2 Single-Factor Interest Rate Models

Single-factor interest rate models attribute changes in the entire term structure to a single stochastic factor. Common models include Vasicek, CIR, and Hull-White.

2.2.1 Vasicek Model

Short-rate dynamics:

drt=a(b−rt)dt+σdWtdr_t = a(b - r_t)dt + σdW_t

Bond option pricing formula:

C=LP(0,s)N(h)−KP(0,T)N(h−σP)C = LP(0,s)N(h) - KP(0,T)N(h-σ_P)

h=1σPln⁡LP(0,s)P(0,T)K+σP2h = \frac{1}{σ_P}\ln\frac{LP(0,s)}{P(0,T)K} + \frac{σ_P}{2}

σP=σa[1−e−a(s−T)]1−e−2aT2aσ_P = \frac{σ}{a}[1-e^{-a(s-T)}]\sqrt{\frac{1-e^{-2aT}}{2a}}

2.2.2 Hull-White Model

An extension of the Vasicek model with time-dependent parameters:

drt=[θ(t)−art]dt+σdWtdr_t = [θ(t) - a r_t]dt + σdW_t

2.2.3 Coupon Bond Option Pricing

The "decomposition method" converts coupon bond option pricing into a portfolio of zero-coupon bond options:

  1. Calculate the critical rate r* where the coupon bond price equals the strike price.
  2. Decompose the coupon bond into zero-coupon bonds.
  3. Price each zero-coupon bond option (strike price corresponds to r*).
  4. Sum to obtain the coupon bond option price.

2.2.4 Implementation Challenges

  1. Complex parameter calibration (mean reversion speed a, volatility σ, etc.)
  2. Requires a complete term structure model.
  3. Computationally intensive.

2.3 Key Parameter Selection

2.3.1 Forward Price Determination

Bond forward prices must account for:

  • Spot price (use full price, not clean price)
  • Cost of carry (repo financing cost)
  • Coupon income
  • Accrued interest

Exact formula:

FB=B0−IP(0,T)F_B = \frac{B_0 - I}{P(0, T)}

2.3.2 Interest Rate Selection

  1. Risk-Free Rate:

    • Government bond yield for the same maturity
    • SHIBOR/SHIBOR3M
    • Swap rate

  2. Discount Rate:

    • Should match the option's maturity
    • Consider the risk-neutral measure

2.3.3 Volatility Selection

  1. Historical Volatility:

    • Based on historical bond price data
    • Typically a 1-year rolling volatility

  2. Implied Volatility:

    • Derived from market option prices
    • Requires liquid option markets

  3. Model Volatility:

    • σ_P in single-factor models
    • Requires parameter calibration

2.4 Model Comparison and Selection

Comparison DimensionBlack-76 ModelSingle-Factor Interest Rate Model
ComplexityLowHigh
Data RequirementsMinimal (F, σ, r)Extensive (model calibration)
Computational EfficiencyHighLow
AccuracyBetter for short-term optionsMore accurate for long-term options
Applicable ProductsEuropean optionsAmerican/complex options
Market PracticeCommon in OTC marketsAcademic/complex derivatives
Parameter StabilityVolatility needs frequent updatesRelatively stable parameters

Recommendations:

  1. For standard European bond options, prefer the Black-76 model.
  2. For bonds with embedded options or complex structures, consider single/multi-factor models.
  3. In China, Black-76 is more practical (better data availability).
  4. For American options, use binomial trees or Monte Carlo simulations.

3. Additional Considerations

3.1 Credit Risk Adjustments

For non-government bonds (e.g., corporate bonds), pricing must account for credit risk:

  • Adjust discount rates using credit spreads.
  • Incorporate default probabilities (using CDS spreads).
  • Adjust volatility (corporate bonds typically have higher volatility than government bonds).

3.2 Liquidity Impact

Challenges in China's bond option market:

  • Wide bid-ask spreads necessitate liquidity premiums.
  • Limited historical data complicates volatility estimation.
  • May require referencing similar bonds.

3.3 Tax Considerations

Tax treatment varies by bond type:

  • Government bond interest is tax-exempt.
  • Policy bank bonds (e.g., China Development Bank) are taxable.
  • Pricing must account for after-tax returns.

4. Pricing Methodology in the Mathema System

Bond option pricing is complex and requires selecting appropriate methods based on product features and market conditions. In China, our current approach includes:

  1. Using the Black-76 model for European bond options due to its simplicity.
  2. Key parameter selection:
    • Use ChinaBond full prices as underlying prices.
    • Government bond yields for the same maturity as risk-free rates.
    • Combine historical and implied volatility (if available).
  3. Credit bond options require credit spread adjustments.
  4. Long-term or complex options may use single-factor interest rate models.
  5. Continuous validation of model outputs against market quotations.
Pricing, Valuation and Risk Management
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