The article studies the possibility for analytically approximating the price of a convertible bond within defaultable Markov diffusion models. Since convertible bond pricing requires time-consuming finite difference or tree pricing methods in general, such proxy formulas can help to calibrate model parameters more efficiently. The derivation is based on the idea to “Europeanize” the American conversion option of the holder. Consequently, the quality of the approximations stands and falls with the value of the early conversion premium, and the resulting formulas in general may be viewed as (often quite sharp) lower bounds for the price.
The equity forward with a given maturity is defined as today’s risk-neutral expectation of a stock price at maturity. If arbitrage-free prices for European put and call options with the given maturity are observed for different strike prices, the equity forward can be retrieved from the put-call parity. In particular, it is invariant with respect to different risk-neutral pricing measures which explain observed option prices, i.e. it is a model-free quantity. More precisely, the equity forward is given by the unique root of the (in practice partially) observed difference between calls and puts, viewed as a function in the strike price. In contrast, if only American-style put and call option prices are observed, the lack of a put-call parity makes it more difficult to retrieve the equity forward in an unambiguous way from the observed option data. In particular, the unique root of the difference between calls and puts in general is no longer equal to the equity forward. The present article investigates whether American-style put and call prices also determine the equity forward unambiguously. Unfortunately, this seemingly simple “yes or no”-question appears to be non-trivial and open, and the present investigation is not able to answer it.
Within a traditional Markowitz-setting, it is investigated whether it is possible to improve the performance of a portfolio by adding a macro hedge to it. The result is demonstrated by an application to our fund XAIA Credit Curve Carry on 31 January 2017.