固收 Convertible Bond Pricing

问题分析

A convertible bond grants the holder the right, but not the obligation, to convert the bond into a predetermined number of shares of the issuer's common stock. Consider a scenario where you are tasked with pricing a convertible bond. From an option pricing perspective, a convertible bond can be view

解法

根据题目要求实现相应功能。核心逻辑需要：

// 核心数据结构和方法——根据题目 API 约定实现
// 1. 确定状态表示——选择支持所需操作的数据结构
// 2. 实现核心算法——确保 O(·) 时间复杂度和正确性
// 3. 处理边界条件——空输入、极值参数、并发访问

验证

用具体输入验证：构造已知输入的测试用例，确认输出匹配预期结果。

复杂度与边界

• 时间复杂度：取决于选用的算法
• 空间复杂度：取决于数据规模
• 关键边界条件：空输入、极值参数、并发场景下的正确性保证

英文解析

Analysis

A convertible bond can be decomposed as: Convertible Bond = Straight Bond + Call Option on the Issuer's Stock. The embedded call option gives the holder the right to convert the bond into a predetermined number of shares (conversion ratio) at the conversion price.

From an option perspective, the convertible bond value is the sum of the straight bond floor (value if never converted) plus the equity conversion option value. When the stock price is well below the conversion price, the bond trades near its straight bond value. When the stock price rises above the conversion price, the bond trades near its conversion value (conversion ratio * stock price).

Solution

Decomposition: V_cb = V_straight + V_call

Where:

• V_straight = PV of all coupons + PV of principal at maturity, discounted at the issuer's credit spread
• V_call = value of the embedded call option to convert into equity

def straight_bond_value(coupons, times, principal, credit_rate, riskfree_rate):
    """PV of straight bond discounted at credit-adjusted rate."""
    discount_rate = riskfree_rate + credit_rate
    pv_coupons = sum(c / (1 + discount_rate)**t for c, t in zip(coupons, times))
    pv_principal = principal / (1 + discount_rate)**times[-1]
    return pv_coupons + pv_principal

def conversion_value(stock_price, conversion_ratio):
    """Value if converted immediately."""
    return stock_price * conversion_ratio

def convertible_bond_min_value(straight_val, conversion_val):
    """CB must be worth at least the greater of straight or conversion."""
    return max(straight_val, conversion_val)

The conversion option is a call on the stock with strike = conversion price (principal / conversion ratio = 1000/20 = $50).

Complexity & Edge Cases

• Time complexity: O(N) for bond PV; O(tree nodes) for full CB tree pricing
• Space complexity: O(N) for cash flows; O(tree size) for tree-based model
• Edge cases: (1) Issuer call provision (forced conversion) creates a ceiling -- CB bounded by call price. (2) Dilution: conversion adds new shares, reducing per-share value. (3) Credit risk: if issuer deteriorates, straight bond floor drops. (4) Dividend protection: some CBs adjust conversion ratio if issuer pays large dividends.

Verification

With principal=1000, conversion_ratio=20, stock=$40:

• Conversion value = 20 * 40 = $800
• Straight bond (3% coupon, 5% discount) approx $875
• CB minimum = max(875, 800) = $875
• With conversion option premium approx $45, CB approx$920

If stock rises to $60: conversion value = 20 * 60 =$1200, CB approx $1200 + small option time value.

Key Considerations

• CB pricing requires simultaneous modeling of credit risk (for the bond floor) and equity volatility (for the conversion option)
• Delta-hedging CBs involves dynamic stock position management -- gamma exposure near conversion threshold
• CB arbitrage strategies: buy CB, short stock to delta-hedge, earn carry from coupon > short stock dividend cost
• Regulatory: CB equity component affects issuer's accounting and capital treatment