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convertible bond pricing


Convertible bond (CB) pricing functions have to be correctly implemented and tested before assessing their impact on regulatory capital charges. This page is a short summary of our technical paper on convertible bonds and introduces a basic pricing model based on a binomial tree covering CBs with e.g. callable and putable features. Moreover, we extend the model using more sophisticated credit risk models and investigate sensitivity towards various parameters.

 

The Process of the Underlying Share

 

The main stochastic factor of our pricing model is the price of the underlying equity which follows a Geometric Brownian Motion (GBM). First we build our equity tree to discretize the GBM process based on the work of Cox, Ross and Rubinstein (1979) and the tree is given by the following three equations:
\begin{align}
u &= e^{\sigma\sqrt{\Delta t}}, \\
d &= e^{-\sigma\sqrt{\Delta t}}, \\
q(t) &= \frac{e^{(f(t)-\delta)\Delta t}-d}{u-d},
\end{align}
where $u$ and $d$ are the “up” and “down” movements of the equity tree, $q(t)$ is the risk-neutral
probability, $\Delta t$ is the time step of the tree, $\delta$ is the continuously compounded dividend
yield, $f(t)$ denotes the forward interest rate between $t$ and $t + \Delta t$ and $\sigma$ is the
volatility of the underlying stock.

convertible-bond-pricing-tree

 

Convertible Bond Pricing Tree

 

Our convertible bond pricing model is based on the equity tree and uses a backward iteration on the binomial tree. The calculation starts at the last level of the tree where the formula of the nodes is is given by

\begin{equation}
V=\max \bigg (N \cdot R+N \cdot \frac{c}{f_{c}}, S \cdot C_{r} \bigg ),
\end{equation}

where $N$ is the face value, $R$ is the redemption in percent of the face value,
$c$ is the annual coupon payment expressed as a percent of the face value,
$f_{c}$ is the coupon frequency and $C_{r}$ is the conversion ratio.

The algorithm continues by calculating the continuation value of the convertible bond which is
given by

\begin{equation}
H=\exp(-f(t)\Delta t)(q(t)V_{u}+(1-q(t))V_{d})+c_{USD},
\end{equation}

where $V_{u}$ and $V_{d}$ denote the “up” and “down” values of the CB at the next node and
$c_{USD}$ is the amount of the coupon payment in the currency of the CB.

At each node before the maturity the model calculates the value of the convertible bond as

\begin{equation}
V=\max(H,S \cdot C_{r}).
\end{equation}

The algorithm continues until it reaches the first node of the convertible bond pricing tree.

 

convertible bond pricing tree

Including Credit Risk into the Model

We extend our model using three different implementations of credit risk.

The first implementation applies a constant credit spread. Let $r_{b}(t)$ be the risky forward rate between
$t$ and $t + \Delta t$ given by

\begin{equation*}
r_{b}(t) = f(t) + CS,
\end{equation*}

where $CS$ is the constant credit spread. Using this risky forward rate the calculation of the continuation
value can be written as

\begin{equation*}
H=\exp(-r_{b}(t)\Delta t)(q(t)V_{u}+(1-q(t))V_{d})+c_{USD}.
\end{equation*}

The second approach uses the so-called credit-adjusted discount rate (Goldman Sachs (1994))
where the the discount rate lies between the risk-free rate and the risky rate ($r_{b}(t)=f(t)+CS$).
Let $p_{conv}$ be the probability that the CB will be converted into stocks in the future and $1-p_{conv}$
the probability that it will behave like a corporate bond. Moreover, let $y(t)$ be the credit-adjusted
discount rate at time $t$ defined as

\begin{equation}\label{credit_adj_disc_rate}
y(t)=p_{conv}f(t)+(1-p_{conv})r_{b}(t).
\end{equation}

This calculation introduces two new binomial trees in the pricing model: the first tree models the
conversion probabilities while the second tree models the credit-adjusted discount rates.
The calculation of the tree of the conversion probabilities is the following:

  • The value of $p_{conv}$ at every node is equal to $1$ if the CB is converted into stocks at the node
    (even if it is a result of forced conversion) and the value of $p_{conv}$ at every node is equal to $0$ if
    the CB is redeemed or put at the node.
  • If the CB is not put, redeemed or converted at the current node,
    the calculation of $p_{conv}$ corresponds to\begin{equation*}
    p_{conv}=q(t) p_{conv}^{up} + (1-q(t)) p_{conv}^{down},
    \end{equation*}where $p_{conv}^{up}$ and $p_{conv}^{down}$ are the conversion probabilities
    and $q(t)$ is the risk-neutral probability at time $t$.

The third method uses a reduced form credit risk model. Let $\lambda$ be the so-called default intensity
and let us define the default as the first arrival time ($\tau$) of a Poisson process with a constant
$\lambda$ parameter.

convertible bond pricing with default
The pricing model assumes that in case of default the stock price will drop to zero.
The left hand side of the figure shows one branch of the stock price tree used in this
modification of the model. The calculation of the risk neutral probability of the stock tree changes
and it is given by

\begin{equation*}
q(t)=\frac{e^{(f(t)+ \lambda – \delta)\Delta t}-d}{u-d}.
\end{equation*}

The right hand side of the figure shows the CB valuation tree: there is a positive probability of default
in the tree with the value of the CB dropping to the recovery value (denoted by $V_{R}$). The
recovery value is the cash amount that a bond holder gets in case of the CB’s default. If expressed
as a percentage of the face value, this is typically referred to as recovery rate ($RR$). The model
uses a more advanced method to calculate the recovery value:

\begin{equation}\label{present_value_recovery}
V_{R}=RR \cdot (N \exp(-r(T-t)).
\end{equation}

Due to the default intensity the calculation of the continuation value is changing as

\begin{equation}
H=\underbrace{\exp(-(f(t)+\lambda)\Delta t)(q(t)V_{u}+(1-q(t))V_{d})}_{a)} +\underbrace{\exp(-\lambda \Delta t)c_{USD}}_{b)} + \underbrace{\exp(-f(t) \Delta t)(1-\exp(- \lambda \Delta t))V_{R}}_{c)}.
\label{cont_value_intensity}
\end{equation}

For a better interpretation, this equation can be decomposed into three parts a), b) and c).
Part a) and b) represent the tree in case of survival (hence the $\exp(-\lambda \Delta t)$ part),
while part c) represents the default event (hence the $1-\exp(-\lambda \Delta t)$ part).
Part a) is the present value of the expected value of the up and down values ($V_{u}$ and $V_{d}$) of
the CB on the next level of the tree in case of survival, while part b) is the present value of expected
coupons. Part c) represents the present value of the recovery value in case of default.

 

Special Features of Convertible Bonds

The market for convertible bonds is innovative and therefore a variety of specifications on the rights
and obligations of issuers and investors can be found. Our pricing model implements two important
features of convertible bonds which are the following: callability and putability (for the implementation
of other exotic features see our convertible bond pricing technical document).

The calculation algorithm changes if we take into account the special features.
The formula of the nodes including callability and putability of the convertible bond is given by

\[ V(t) = \left\{ \begin{array}{ll}
\max(H,S \cdot C_{r}) & \textrm{if } t \not \in \Omega_{Call} \textrm{ and } t \not \in \Omega_{Put},\\
\max(\min(H,K_{c}^{USD}+c_{USD}),S \cdot C_{r}) & \textrm{if } t \in \Omega_{Call} \textrm{ and } t \not \in \Omega_{Put},\\
\max(H,K_{p}^{USD}+c_{USD},S \cdot C_{r}) & \textrm{if } t \not \in \Omega_{Call} \textrm{ and } t \in \Omega_{Put},\\
\max(\min(H,K_{c}^{USD}+c_{USD}),K_{p}^{USD}+c_{USD},S \cdot C_{r}) & \textrm{if } t \in \Omega_{Call} \textrm{ and } t \in \Omega_{Put},
\end{array} \right. \]

where $K_{c}^{USD}$ and $K_{p}^{USD}$ are the call and put price, respectively, values in the currency of
the convertible bond and $\Omega_{Call}$ and $\Omega_{Put}$ are the periods where callability and
putability apply.

 

Sensitivity Analysis of the Convertible Bond Pricing Model

 

Investigating model sensitivities is particularly important considering the Solvency II and Swiss Solvency Test use cases of the pricing model.

The figure shows the sensitivity analysis of three sample CBs with different maturities to the parity, the volatility of the underlying equity, the changes in the risk-free interest rate and credit spread. As the chart shows the price of the CB is an increasing function of the parity and the volatility due to the conversion option. However, the increase in interest rate or credit spread decrease the value of the CB.

(For further details about the model and the sensitivity analysis see our convertible bond pricing technical document).

sensitivity analysis convertible bond pricing model