Description of Tree Model for the Valuation of a Convertible Bond with Credit Risk

Source

Adapted and extended from Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. P. 646f

Valuation of convertible bonds (CBs) in a binomial tree model

Credit risk plays an important role in the valuation of convertible bonds. One approach to valuing convertible bonds is to construct a tree in the usual way to represent the behavior of the company's stock price. The life of the tree should be set equal to the life of the convertible bond. The value of the convertible at the final nodes of the tree can be calculated based on any conversion options that the holder has at that time. We then roll back trough the tree. At nodes where the terms of the instrument allow conversion we test whether conversion is optimal. If the company has the option to call the bond, we test whether it is optimal for it to do so and in turn re-test whether holders should convert if the bond is called. This is equivalent to setting the value at a node equal to

Max[Min(Q₁,Q₂),Q₃]

where Q₁ is the value given by the rollback (before possible conversion or call), Q₂ is the call price, Q₃ is value if holder elects to convert.

Discount rate when discounting in the tree

If the bond remains a bond, i.e. is not converted, it is appropriate to use the credit risk adjusted discount rate, i.e. the risk free rate plus a credit spread. On the other hand, if conversion is certain (conversion value >> redemption price) we then use the risk-free rate. The CB value is then correctly calculated as the value of the equity underlying the bond.

Unfortunately in practice we are uncertain whether the bond will eventually be converted. In an approach formalized by Tsiverotis and Fernandes (1998) we therefore arrange the calculations so that the value of the bond at each node is divided into two components:

1. "Equity" component - arising from situations where a CB will be converted.
>Apply risk-free rate when rolling back in tree.

2. "Debt" component - arising from situations where CB is redeemed at redemption price.
>Apply credit risk adjusted rate when rolling back in tree.

Handling of Coupon

If the CB pays coupons (C_i) and most CBs do, the present value of the coupon(s) to be paid during the subsequent time step is added to the debt value component is added at the particular time step. This illustrated in the following figure. Discounting of coupons is obviously at the rate including credit spread.

T

Final Maturity

T- 4Dt

T- Dt

Dividends

The model handles continuously paid dividends (rate q) only. For more a more accurate model one would have to treat dividends similar to the method above. Note also that a change in q will affect p (probability of an up-movement) in the tree.

VBA Code

The Excel file you download will have its VBA module protected. You may find the code of this function further down.

A simplified numerical example

The Excel workbook also contains a simplified three-step model implemented as a spreadsheet-only solution. This model does not allow the setting of CB coupons, an initial non-conversion period or a put option by the CB holder. These features are only handled by the VBA version.