# Question regarding the valuation of convertible bond

I would like to ask a question about the valuation of convertible bond using binomial tree.

It is known that the CB can be valued as debt and equity component using risky and risk-free interest rate using binomial tree. Using backward induction, the conversion value (= stock price/conversion price) is assigned as equity component and bond redemption value and/or the coupon is assigned as debt component. The equity and debt values are then discounted from the nodes at the maturity based on decision rule (eg. max(min(roll,call),conv) or min(max(roll,conv),call))

On the other hand, as convertible bond is regarded as straight bond plus the call option on underlying share (conversion option), I attempt to value the convertible bond based on intrinsic value apporach . That is, using backward induction I assign the intrinsic value of conversion option as equity component and redemption amount of CB as debt component. Again, the equity and debt values are then discounted from the nodes at the maturity, based on decision value.

I want to clarify the issue about the use of binomial model to value the convertible bonds based on intrinsic value approach. In order to do so, the following questions are asked,

1. Is the method to value the convertible bond using binomial tree based on intrinsic value of conversion option described correct? If it is incorrect, what things do I omit in the model?

2. Is there any reference that explain the valuation of convertible bond as straight bond and options on underlying share in details, in particular, the option is valued based on intrinsic value of payoff?

I had also posted similar questions in What is the difference between the methods (listed in content) in pricing convertible bond?, you could have a reference.

Here is the second part, I doubt that the intrinsic value approach is correct approach to value the conversion option. Here is my claim,

In the past few days, I tried to use the binomial tree to price the conversion option based on intrinsic value approach and using the decision rule max(min(call,roll),conv), I found that the value of conversion option is smaller than that based on total value approach. Hence, the CB valued is smaller.

I guess the reason is that if the option is exercised, only the intrinsic value is discounted but in fact intrinsic value is not only the part to be discounted by risk-free rate. Hence the option valued is smaller and using intrinsic value approach to value the CB is not correct, is my claim right?