A convertible bond (or preferred share) is a hybrid security, part debt and part equity. Its valuation is derived from both the level of interest rates and the price of the underlying equity. Several convertible bond pricing approaches are available to value these complex hybrid securities such as Binomial Tree, Partial Differential Equation and Monte Carlo simulation. One of the earliest approaches was the Binomial Tree model originally developed by Goldman Sachs [1,2] and this model allows for an efficient implementation with high accuracy. The Binomial Tree model is flexible enough to support the implementation of bespoke exotic features such as redemption and conversion by the issuer, lockout periods, conversion and retraction by the share owner etc.

In this post, we will summarize the key steps in valuing a convertible bond using the Binomial Tree approach. Detailed description of the method and examples are provided in references [1,2].

Generally, the value of a convertible bond with embedded features depends on:

- The underlying common stock price
- Volatility of the common stock
- Dividend yield on the common stock
- The risk free interest rate
- The credit worthiness of the preferred share issuer

## Convertible bond pricing using a binomial tree

Within the binomial tree framework, the common stock price at each node is described as

where *S _{0}* is the stock price at the valuation date;

*u*and

*d*are the up and down jump magnitudes. The superscript

*j*refers to the time step and

*i*to the jump. The up and down moves are calculated as

and

where is the stock volatility, and is the time step.

The risk neutral probability of the up move, *u*, is

and the probability the down move is *1-p*

After building a binomial tree for the common stock price, the convertible bond price is then determined by starting at the end of the stock price tree where the payoff is known with certainty and going backward until the time zero (valuation date). At each node, *Pj,i* the value of the convertible is

where *m* denotes the conversion ratio.

If the bond is callable, the payoff at each node is

The payoff of a putable bond is

Here *C* and *P* are the call and put values respectively; *r* denotes the risk-free rate.

The above equations are the key algorithms in the binomial tree approach. However, there are several considerations that should be addressed due to the complexities of the derivative features

- Credit spreads (credit risk) of the issuers which usually are not constant.
- Interest rates can be stochastic.
- Discount rate
*ri,j*depends on the conversion probability at each node. This is due to the fact that when the common share price is well below the strike, the preferred share behaves like a corporate bond and hence we need to discount with a risky curve. If the share is well above the strike then the preferred behaves like a common stock and the riskless curve need to be used. - The notice period: the issuer tends to call the bond if the stock price is far enough above the conversion price such that a move below it is unlikely during the notice period. For most accurate results, the valuation would require a call adjustment factor. This factor is empirical and its value could be determined by calibration to stock historical data.

## Convertible bond pricing using Partial Differential Equation (PDE) approach

An alternative way of pricing a convertible bond is to use the PDE method developed by K. Tsiveriotis and C. Fernandes [3]. In this method, we have to solve a system of coupled equations,

where *U *denotes the convertible bond’s price,

* V *is the value of the cash-only part,

* S* is the stock price,

* r* is the risk-free interest rate,

* q *denotes the dividend yield,

* r _{c} *is the credit spread.

The above 2 equations can be solved simultaneously by using the Finite Difference Method in which the partial derivatives are approximated by finite differences.

## Convertible bond pricing when there is a contingent convertible feature

*A contingent convertible bond is defined with two elements: the trigger and the conversion rate. While the trigger is the pre-specified event causing the conversion process, the conversion rate is the actual rate at which debt is swapped for equity. The trigger, which can be bank specific, systemic, or dual, has to be defined in a way ensuring automatic and inviolable conversion. A possibility of a dynamic sequence exists—conversion occurs at different pre-specified thresholds of the trigger event. Since the trigger can be subject to accounting or market manipulation, a commonly used measure has been the market’s measure of bank’s solvency. The design of the trigger and the conversion rate are critical in the instrument’s effectiveness. **Read more*

From a quantitative point of view, modeling such a contingent conversion feature is far from trivial. Usually, the 20/30 contingent conversion can be modeled somewhat explicitly on the PDE lattice by introducing an extra state variable which tracks whether the condition was satisfied in the previous monitoring quarter. The soft call feature is often more material as it is in the interest of the issuer to call the bond as soon as the value equals the call price (plus accrued interest).

Readers who would like to learn more about the mathematical aspects of pricing a contingent convertible bond can start with the following articles:

Valuing convertible bonds with 20-of-30 soft call provision

## Convertible bond pricing example

The binomial convertible bond pricing approach presented in the previous section can be implemented in scripting languages such as VBA or Matlab. Here we present a simple example of pricing a convertible bond in Excel. We are going to value a hypothetical convertible bond. The specifics of the hypothetical convertible bond are as follows

INPUTS | |

Stock price | 100 |

Volatility | 0.25 |

Risk Free Rate | 0.02 |

Risky Rate (risk free+credit spread) | 0.08 |

Coupon | 0.06 |

Maturity (in years) | 5 |

Conversion ratio | 10 |

Using the formula provided in the previous section, we calculated the up and down moves and the probability of the up move. The results are

MODEL PARAMETERS | |

dt | 1.00 |

u | 1.28 |

d | 0.78 |

Probability Up | 0.48 |

Once the tree parameters are calculated, we next build the tree and then work backward from the end nodes in order to obtain the convertible bond’s price at time zero. As the final result, the bond price is $1319 (per $1000 notional)

Note that this is a simplified example. In real life, convertible bonds are usually more complex. They often include features such as call, put, contingency conversion options. The call and put options can be implemented using the formula given in the previous section.

A sample Excel workbook for pricing a convertible bond can be downloaded by following the instruction below.

**References**

[1] *Valuing Convertible Bonds as Derivatives, *Quantitative Strategies Research Notes, Goldman Sachs, November 1994.

[2] *Pricing Convertible Bonds, *Kevin B. Connolly, Wiley, 1998.

[3] Tsiveriotis, K. and Fernandes, C. (1998), *Valuing Convertible Bonds with Credit Risk, *Journal of Fixed Income, 8(2): pp. 95–102.

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