# Convertible option

### Conversion feature

The number of shares for which the bond can be exchanged is given by the conversion ratio, which is usually stated at bond issuance. While the investor holds the bond and does not exercise their option to convert, they will receive periodic payments at the stated coupon level. In essence, a convertible bond can be thought of as a regular bond with an embedded equity call option.

A convertible bond, also called a convertible note, is similar to a regular corporate bond with one exception: It can be exchanged for shares of stock. In addition, you can convert the bond to a specified number of underlying stock shares. The price of convertible debt stems from its value as a bond and its value if converted to stock. Typically, the conversion price on newly issued convertible bonds is well above the current stock price. Interpreting Conversion Premium If you put the conversion premium on a percentage basis, you can compare it with those of similar bonds.

Convertible bonds can also be issued with other embedded features such as time varying issuer calls and investor puts. As the underlying convertible option price increases, the bond will behave more like an option.

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The main advantage to the issuer of a convertible bond is that the bond will usually be issued with a lower coupon rate than the equivalent non-convertible bond. This is because the holder of the convertible bond is compensated with the right to convert the bond into stock of the issuing company.

If conversion does occur, dilution may also occur if new shares must be issued. The valuation of a convertible bond is made more difficult due to the underlying characteristics.

### Convertible bond

When pricing, one must consider the underlying bond and equity details. For example, the equity price, maturity, coupon, volatility and spread must all be considered. FINCAD currently has two methods for the pricing of convertible bonds: the first uses a trinomial tree and the second method, introduced in v10, uses a partial differential equation method, specifically the Crank-Nicolson finite-difference method.

Tsiveriotis and C. The main idea behind this model is that a convertible bond consists of two components, an equity component and a debt component, and these convertible option have different default risks. The equity component has no default risk since the issuer can always issue its own stock.

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- Additional features[ edit ] Any convertible bond structure, on top of its type, would bear a certain range of additional features as defined in its issuance prospectus: Conversion price: The nominal price per share at which conversion takes place, this number is fixed at the issuance but could be adjusted under some circumstance described in the issuance prospectus e.

Thus the equity component should be discounted at the risk-free rate. Coupon and principal payments, and any put provisions depend on the issuer's timely access to the convertible option cash and thus introduce credit risk.

Thus the debt component should be discounted at the risk-free rate plus a credit spread. To split the convertible bond CB up into its components, a new hypothetical security is defined, called the "cash-only part of the convertible bond" COCB.

This hypothetical security is defined as follows: "The holder of a COCB is entitled to all cash flows, and no equity flows, that an optimally behaving holder of the corresponding convertible bond would receive.

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This results in a system of two coupled Black-Scholes equations for the value of the convertible bond: Equation 1 Equation 2 where U is the value of the convertible bond, V is the value of COCB, S is the underlying stock price, r is the risk free rate, q is the dividend yield, and rc is the credit spread. The above system of coupled partial differential equations must be solved simultaneously.

Looking at Equation 2, it appears at first that it is independent of Equation 1. However, the two equations are coupled due to the fact that this convertible option a free boundary problem. The free boundary comes from the convertible option to exercise at times other than maturity, and becomes an unknown in the problem that must be solved for at each time step.

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It is because of this common free boundary that Equations 1 and 2 are coupled and must therefore be solved simultaneously. The stock price is modeled using a trinomial tree, and the value of the convertible bond is calculated at the final nodes based on any conversion options help at that time and then rolled back through the tree. This method is equivalent to a fully implicit finite difference scheme.

Finite Difference Method In this implementation, the system of coupled partial differential is convertible option using a finite difference method, specifically the Crank-Nicholson scheme. This scheme is unconditionally stable and has better convergence properties than either the fully explicit or fully implicit schemes.

### Convertibles and preferreds

As part of the valuation process, a grid of time-steps is setup from the value date to the maturity date. The implementation also ensures that a time-step falls on every cash flow date, conversion dates, call and put dates and all discrete dividend dates.

The key advantages that a user would see are in convergence to a solution and accuracy. Convergence - The Crank-Nicolson method is unconditionally stable, having no relevant restriction on the number of time steps required to converge.

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The greater the number of additional features, the greater the number of how to create make money required to ensure that all features are picked up by the trinomial tree. Accuracy - the accuracy of the Crank-Nicolson method is better than our current tree implementation.

Unless otherwise stated, both implementations can handle the feature listed below: Time varying conversion ratio Callable European, Bermudan, or American Putable European, Bermudan, or American Cap on Conversion Price.