help required for valuation of custom structure

[Abraham Robinson-3]

Hi,

 

I would like to develop a workable  two-factor trinomial tree for valuation of a custom structure- a Convertible Bond with Embedded Double-Barrier Swaption-Bermudan style.

 

We are ready to pay a fair price to anybody who can deliver us the workable software that can be run on Windows using Excel.

 

Here's the mathematical formula of our custom structure:

 

This writeup assumes the reader has knowledge of the binomial tree model, as well as the Black-Derman-Toy (BDT) model.

 

We will price our structure with a two-factor model which uses a trinomial tree. The concept of the trinomial tree is very similar to a binomial tree, except that each node turns into 3 nodes instead of 2:

 

 

 

                        O

           

O               O

 

                        O

 

This is not the same as a Hull-White trinomial tree, because the middle node is not necessarily equal to “no change” in the underlying asset. Instead, we have two quantities, the stock price S and the short-term interest rate Y, which have values in each node. So instead of a tree with one number in each node representing the price of the underlying asset, we have a tree with two numbers in each node, representing the price of the stock and the short rate. If we label the top node A, the middle node B, and the bottom node C, we say that there are three possible moves in each: RS(A), RS(B) and RS(C) for the returns for the stock, and RY(A), RY(B) and RY(C) for the changes in Y.

 

By making Rn(z) multiplicative, meaning the value after a move of m is P * m, this results in a recombining trinomial tree, where 1 node becomes 3, 3 become 6,  6 become 10, and so on. It is recombining because a move of type A followed by a move of type B is S * RS(A) * RS(B)  which is the same as the price from a move of B followed by  move of A: S * RS(A) * RS(B). This obviously holds true for both S and Y for all combinations of moves (A then B, A then C, etc.).

 

The first step is to generate the returns (RS(A), RS(B), etc.) that are consistent with the input volatilities and correlation between the stock and short rate.  This allows us to build the trinomial tree of stock prices and short-term rates.

 

The second step is to adjust the short-term rates with a drift term, exactly as in the Black-Derman-Toy model, to make the interest rate tree be arbitrage-free.

 

Given the coupon on the non-convertible bond (call the bond value NCB), we can now follow the standard BDT model to calculate NCB in each node: in the terminal nodes it is par plus the last coupon, in the nodes before that it is the weighted average discounted value of NCB in the three nodes which emanate from that node, and in all the nodes before that it is also the weighted average discounted value of NCB in the three nodes which emanate from that node.

 

We now have a tree with a stock price S and a non-convertible bond price NCB in each node.

 

For each time period in the tree, we can also denote the amount of cash that must be “given back” upon exercise as C(t) where t is the time the bond has been outstanding.

 

In each node, we can calculate the intrinsic value of the conversion option as:

 

S-(NCB+C(t))

 

Where S and NCB are specific to the node, and C(t) is the same in all nodes for  given time period.

 

We work backwards through the tree again, calculating the option value in each node as follows:

 

Intrinsic value = Max(S-(NCB+C(t)),0)

 

when S is within the allowable price range for that period, and 0 otherwise (the allowable price range can be fixed or a function of time, i.e. based on an annualized rate of return). If the payoff is based on the appreciation of the stock, then replace S with S-S0 in the formula, with S0 representing the starting price of the stock.

 

Value of option = Max(intrinsic value, replicating value)

 

when exercise is allowed (in case there is a lockout period), and replicating value otherwise.

 

Replicating value is the weighted average discounted value of the option in the three nodes that emanate from the node in which the value is being calculated.

 

The value of the convertible bond is then NCB in the root node of the tree, plus the value of the option in the root node of the tree.

 

 

Regards,

 

Abraham Robinson

 

 

 

---

Make free worldwide PC-to-PC calls. Try the new Yahoo! Canada Messenger with Voice