Abstract
Different from diversification of stocks, there are two strategies to diversify portfolios consisting of options: one is to combine options on single underlying stocks, and the other one is to buy an option based on the index of these stocks. In this paper we analyse which diversification strategy is optimal for classical rational investors with constant relative risk aversion. We employ the Black–Scholes model and the stochastic volatility model of Heston for generating the processes of underlying stocks as well as pricing the derivatives. The results are developed first for options and then extended to some important classes of structured financial products: capital protected notes, discount certificates and bonus certificates. We find that investors’ choices on the two diversification strategies differ noticeably, but in general for convex payoffs index options are preferable, whereas for concave payoffs a portfolio of single stock options has usually higher utility.
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Notes
Empirical studies typically find a premium of overpricing of about 0.5% for simple products on large markets, but average values are as high as 8% in other situations. Some single products can have even larger mispricing.
See Branger and Schlag (2004) for example.
Since our aim is the comparison of derivatives on different underlying stocks ceteris paribus, we do not search for the optimal fraction of the wealth in derivatives nor the exposure of the optimal portfolio to different risk factors for the retail investor.
This corresponds to the uncapped capital protection (1100) according to the SVSP Swiss Derivative Map.
Sometimes called Low Exercise Price Option (LEPO), basically the underlying without dividend payments.
This allows a comparison with Hens and Rieger (2014).
The parameter properties of the Heston model have been estimated by a large number of studies, and the estimated parameters may differ from paper to paper. Our chosen parameters are in the generally agreed region as basically in line with Liu and Pan (2003).
See Branger and Schlag (2004). The analytical solution would not exist even if each stock of the index followed a geometric Brownian motion and all the stocks were independent.
Summary of results is available upon request.
The predefined level refers to the nominal protection level for the capital protected note, the limited profit potential (cap) for the discount certificate, and the conditional protection level for the bonus certificate if the barrier is not breached.
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Yuan, S., Rieger, M.O. Diversification with options and structured products. Rev Deriv Res 24, 55–77 (2021). https://doi.org/10.1007/s11147-020-09169-x
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DOI: https://doi.org/10.1007/s11147-020-09169-x