The Pricing of a Convertible Bonds with a Jump-to-Default Model simply explained

Convertible bonds are hybrid financial instruments that blend the features of debt and equity. They include an embedded option that allows the bondholder to convert the bond into a predefined number of shares of the issuing company. This conversion is governed by two main parameters: the conversion ratio and the conversion price.

- Embedded Option: Provides the right to convert the bond into stock.
- Conversion Ratio ('CR'): Dictates the number of shares obtainable per bond upon conversion.
- Conversion Price: Sets the effective price per share for conversion.

At maturity, assuming the option isn't exercised early, converting a bond with a conversion ratio 'CR' and a given conversion price is equivalent to owning 'CR' call options on the underlying stock, each with a strike price equal to the conversion price. The hazard rate, integral to credit risk analysis, plays a significant role in the valuation of convertible bonds.

The Jump-to-Default Approach with Hazard Rate is modeled with an SDE providing the Stock Price Dynamics:

dS_t = μS_tdt + σS_tdW_t + J_tdN_t, where each term represents the stock price, drift rate, volatility, Wiener process, jump size due to default, and Poisson process for default events.

The hazard rate (λ) quantifies the issuer's default risk, influencing the likelihood of a jump (default) event in the SDE.

The value of the convertible bond at maturity can be expressed as: Value at Maturity = max(Face Value of Bond, CR * max(S_T - Conversion Price, 0)).

Future payoffs are discounted using a risk-adjusted rate, factoring in the hazard rate and default risk.

Monte Carlo simulations are used to estimate the convertible bond's price, factoring in various asset price paths and jump-to-default events.

Callability[^1] and Putability[^2] are also Important features that can significantly affect the bond’s valuation.

[^1]: Callability is a feature that allows the bond issuer to repurchase and retire the bond before the scheduled maturity date, usually at a predetermined price.

[^2]: Putability gives bondholders the right to sell the bond back to the issuer at certain times before maturity, typically at face value, offering protection against adverse conditions.

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