Appendix
- Numeric implementation for PDE
In this section, we describe the numerical method used to solve discrete
forms of (23) and (28). Let and define backward time as . The equations
(23) and (28) can be rewritten as
(A1)
(A2)
The equations (A1) and (A2) can be approximated using Crank-Nicolson
rule. We discretize the x to be equally spaced as a grid of nodes
0 ~ M . At the maturity, and are determined
according to (29) and (30). At any time i+1, the boundary conditions are
(A3)
(A4)
Then, we conduct the backward induction. The procedure is as follows.
For i = penultimateTime to currentTime
// determine accrual interest and call/put prices
// determine boundary nodes
// use the PSOR (Projected Successive Over Relaxation) method to obtain
the continuation value of the bond component and the continuation value
of the equity component , applying the constraints (31).
EndFor
The value at node[0][y] is the convertible bond price where the
equity price at node[0][y] is equal to the current market stock
price.
- Binomial tree algorithm
A binomial tree method is equivalent to an explicit difference scheme.
Suppose that the stock price S will either move up to the valueuS with probability or down to the value dS with
probability . As the binomial tree is a discrete approximation to the
continuous distribution of equation (16), the expectation and variance
of the discrete distribution should be equal to those of the continuous
distribution. This method is commonly referred to as the moment matching
technique.
To match the expectation, we have
(B1)
or
(B2)
where
(B3)
where q is the dividend.
To match the variance, we get
(B4)
or
(B5)
Solving equations (B2) and (B5) according to the usual tree-symmetry
condition: u = 1/d , we obtain
(B6)
(B7)
(B8)
There are many ways to approximate equations (B7) and (B8). The most
well-known one is the Cox, Ross, and Rubinstein (1979) type
approximation that is up to order accuracy and is given by
(B9)
(B10)
Equations (B6), (B9) and (B10) specify the binomial risky tree
parameters that are used to map the continuous stock price dynamics into
the lattice representation.
Suppose that there is a convertible bond. Let us construct a trading
strategy to hold units of the risky stock and units of the risky bond.
At time the convertible bond value is where is the bond component and is
the stock component; the stock value is ; and the bond value is . At
time , the bond value becomes where is the risky rate of the bond; the
stock value becomes either or ; and the convertible value has two
possible outcomes: or corresponding to either an up movement or a down
movement in the stock price. The discounted portfolio should replicate
the discounted convertible bond11Unlike the risk-free tree, the
risky tree tries to match the discounted value of the replicating
portfolio to the discounted value of the convertible bond in order to
catch credit risk properly., which yields
(B11)
(B12)
Solving for yields
(B13)
(B14)
For a self-financing portfolio, the initial wealth needed to finance
this strategy (sometimes called the manufacturing cost of the contingent
claim) is
(B15)
where is defined in (B6).
We split equation (B15) into an equity equation and a bond equation, and
get
(B16)
(B17)
Equations (B16) and (B17) tell us that the fair price of an equity
component or a bond component is equal to the expected value of its
future payoffs discounted by the associated risky rate. The expected
value is calculated using the corresponding values from the latter two
nodes (up or down) weighted by the transition probabilities.
- A comparison of results
Let us now briefly turn to a comparison with previous works. We use a
simple example described in Table C1, which is similar to the example
used in Tsiveriotis and Fernandes (1998) and Ayache, et al. (2003). We
assume that the interest rate, the bond spread, and the volatility are
flat. The hazard rate is . As the call and put prices are quoted using
the clean prices, we need to convert them to the dirty prices as
(C1)
where the accrued interest is give by
(C2)
where C denotes the coupon, denotes the accrual factor or day count
fraction for period (,) where , denotes the start time of the accrual
period, and denotes the end time of the accrual period. The numerical
results are shown in Table C2, from which we can see that our model
generates lower results than AVF and TF models.