The Influence of Volatility on the Value of Convertible Bonds (Der Einfluss der Volatilität auf den Wert der Wandelobligation)

By Marco Frenkel, Corporate Finance at the Swiss Banking Institute of the University of Zurich , Prof. Dr. R. Volkart, 4 October 2005

This thesis attempts to provide a condensed overview of the prevalent valuation models for convertible bonds and does this in the form of an empirical study.

Executive Summary

Objective

The objective of this thesis is to provide a condensed overview of the prevalent valuation models for convertible bonds and subsequently to carry out an empirical study on the valuation of convertibles. It examines the question of whether volatility can symbolically serve as a sort of 'coil spring': When the price of the underlying stock falls, is the drop in the price of the convertible cushioned due to the rising implied volatility? When the price of the underlying stock recovers, is the increase in the value of the convertible lessened because of the simultaneous drop in implied volatility? The thesis reviews these hypotheses and compares the findings with the results obtained when employing historical volatilities.

Conclusions

A convertible bond is a complex, hybrid financial instrument that is very difficult to value correctly: Since in the case of a convertible bond – in contrast to a warrant bond – the conversion right and the bond are inseparably linked, these two components cannot be valued separately from each other. Furthermore, the valuation is complicated by the various options on the part of the investors (e.g. option to demand early repayment of the bond) and the issuer (e.g. option to call the bond early). In practice, there are naturally numerous such options, which differ depending on the type of convertible but nevertheless significantly influence its value. It is therefore decisive for a valuation model for convertibles to exhibit broad flexibility so that it is capable of taking into account the various options.

In view of these considerations, highly flexible binomial and multinomial models have gained acceptance in recent years for the valuation of convertibles: Since any number of conditions can be examined in one node of a binomial tree, it is possible to consider all of the options related to a convertible bond. With multinomial models, in addition to the share price trend underlying the convertible bond, it is also possible to model a second variable as a stochastic process (in practice generally the interest rate trend, but in rare cases also the exchange rates in the case of cross-currency convertible bonds). Nevertheless, the latest scientific publications on the valuation of convertibles show that the future likely lies in simulation-based valuations methods (Monte Carlo simulations) because they will enable even more realistic modeling of the stochastic processes for the share price and interest rates.

Given that convertible bonds can have very long maturities of up to 30 years, volatility plays a decisive role in the valuation of convertibles. And there is a positive correlation between the value of a convertible bond and the volatility of the underlying stock: A rising volatility increases the value of the convertible because this raises the probability of a positive share price trend and thus a possible conversion of the bond. However, the volatility influences the price trend of a convertible in a second manner, which is not taken into account in the prevalent valuation models for convertible bonds: Namely – as this paper has identified – there is an inverse correlation between the share price trend and the implied volatility of the convertible, so that rising share prices lead to lower implied volatility, and vice versa. As a result, the volatility has a dampening effect on the price trend of a convertible bond: When share prices rise, the price increase of the convertible is moderated by the fallen volatility, whereas when share prices fall, the volatility has a stabilizing effect. By means of an empirical study based on three selected convertible bond indices for the three regions of Europe, the USA and Japan, the dampening effect of volatility was also confirmed for the index prices observable in the markets.