A Fourier-based approach for valuing bonds with embedded options

Abstract

Due to their attractive characteristics, convertible and callable bonds became a more important class
of fixed-income products within the financial market. Therefore, the need for fair and accurate
pricing of convertible and callable bonds increases. Where the convertible option can be considered
as a right for the bondholder, the callable option is a right assigned to the bond issuer. Moreover, due
to the hybrid nature of a convertible bond, it both contains characteristics of fixed income and equity
products. As a consequence, both the risk of default and the possible equity profits need to be taken
into account when valuing convertible bonds. Furthermore, as many convertible bonds also include
a call option, an extra early exercise feature due to this call option needs to be taken into account. This
early exercise option further increases the complexity of the valuation problem, as a possible buyback
of the bond before maturity needs to be considered in the value of the bond. Computing a fair and
accurate price for convertible and callable bonds, therefore gives rise to a complex valuation problem
which leads to the need for further research. 
In this thesis, a structural default model is used to value callable convertible bonds. Contrary to reduced-form models, structural default models are characterized by the rationale behind the default event.
Due to the better rationale and new insights into possible better market fits, structural default models
have gained new interest in academic research. Recent research is conducted on finding
new numerical techniques used for approximating the prices of complex financial products. This
thesis discussed different valuation methods as proposed by Longstaff and Schwartz, Lord et al. and
Oosterlee and Grzelak applied to the valuation of callable convertible bonds. In particular, a convolution-based method (CONV) and a cosine-based (COS) method are discussed. Whereas the Monte Carlo
methods and the CONV method are already used in different articles, the COS method was not yet
applied to the callable convertible bond valuation. As the COS method has been proven to be
an efficient algorithm for approximating the values of financial derivatives, this thesis uses the COS
method to compare its convergence to seek more insights into the convergence of the CONV method.
Monte Carlo methods are used to verify the obtained approximations. 
For the problem considered under constant interest rates, this thesis shows that the COS method can
be derived and applied. For small values of the grid size, the COS method showed to converge much
faster than the CONV method. For larger values of the grid size, the CONV method showed to catch
up with the COS method to become almost equally accurate. The results also showed that, contrary to
the COS method, the CONV method was robust under the choice of the hyper-parameter concerning
the integration grid. For the COS method, it was shown that a bad choice of the hyper-parameter could
lead to a bad approximation. Although the two-dimensional CONV method shows to converge to the
value obtained under Monte Carlo simulation, the results are still off for grid sizes of intermediate size.
Results seem to indicate that the amount of grid points is not sufficient for the proposed integration
interval and that therefore more grid points are needed. A greater amount of grid points, however, will
also indicate a requirement for a greater amount of resources which may not always be available. 
From the results of the zero-coupon case, the COS method showed a more rapid convergence towards the approximations obtained by the Monte Carlo methods than the approximations obtained
using the CONV method. Furthermore, when taking the hyper-parameters into account, the CONV
method showed less robust features than the COS method in the two-dimensional case. Only for a
small range of the hyper-parameters convergence is obtained for the CONV method. On the other
hand, the COS method clearly shows convergence for a much wider range of hyper-parameters.