# Pricing of convertible bonds

I'm trying to evaluate a convertible bond using the structural approach : the price of convertible bond is an option (call) on the firm value. We suppose that the firm value is equal to the sum of the debt (in this case, convertible bond) and number of stocks multiplied by their market price : $$V(t) = B(t) + NS(t)$$ The authors of the book where I found this method explain that, at the maturity, the price of the convertible is equal to $B(T) = max[min(V(T), D), \kappa V(T)]$ where $D$ is redemption price and $\kappa$ is the inverse of dilution coefficient. Thus, it the bond price at $t=0$ can be calculated using binomial tree method.

There is one thing that I don't inderstand, the CB price is equal to the option on firm value, but the firm value depends on CB price, too. How can I estimate the firm value $V(T)$ ?

• Can you please cite which book are you referring to? – RandomGuy Jul 18 '16 at 17:33
• Finance de marché : instruments de base, produits dérivés, portefeuilles et risques. Author : Roland Portait. It is in French. – Carl Zeiss Jul 18 '16 at 17:37

In classical structural modeling, we have that the firm asset value $A(t)$ is the fundamental stochastic variable, following an SDE like the one presented by @Dom

$$\frac{dA}{A} = \mu_A d\,t + \sigma_A d\,W$$

The stock price is then itself an option on the asset value at some horizon $T_H$,

$$S_t = S(A(t), \mu_A, \sigma_A, T_H)$$

and the convertible bond price, being an option on $S$, can then be viewed as a compound option, with a payoff in the $S$ options. In particular at bond maturity time $T \leq T_H$

$$CB(T) = \max( \rho \tilde{S}(A(T), \mu_A, \sigma_A, T), F)$$

where $\rho$ is conversion ration, $F$ is the final coupon plus face value and $\tilde{S}$ is diluted stock value.

Thus, to do pricing we run a binomial tree backwards on $A$ and $S$, from $T_H$ to $T$, find bond prices on the nodes at $T$, and then continue to backwardate from $T$ to time zero. (American exercise adds only minor complexity to the calculations.)

Note that we typically also have a bankruptcy barrier condition, namely that there exists some $L$ such that if any $A(t)<L$ then the stock value drops to zero and the bond value drops to some bankruptcy recovery value.

I think the point of this approach is to model the firm value $V(t)$ using some appropriate probability distribution, then deduce the dustribution of the CB price. Thus the CB price depends on the firm value, but not vice versa.

You can copy the Merton model which assumes that the firm value evolves according to

$dV = \mu V dt + \sigma V dW$

So you can simulate from $t=0$ to $T$ to get $V(T)$ in one step using.

$V(T) = V(0) \exp \left( (\mu-\sigma^2/2) T + \sigma W_T \right)$

You have to get $V(0)$.

I am not sure I like this model. It will be hard to make it fit the market and the payoff at expiry is not handled correctly as the terminal stock price is a call option on the future firm value, e.g. it should be zero if $V(T)<D$.

And there are more problems too. Do not use it for anything serious.