This paper shows how to pick quantities of options to maximize two common risk‑reward scores when the underlying stock returns are skewed and have heavy tails. The authors work with a skew‑elliptical t distribution (a skewed version of the Student’s t) to model the asset returns. They derive explicit formulas for the option holdings that maximize the Sharpe ratio and a return‑to‑Value‑at‑Risk (R‑VaR) ratio under that return model.

Options are contracts that give the right, but not the obligation, to buy or sell a stock. Because option payoffs are nonlinear, the distribution of an options portfolio is hard to write down exactly. The authors use a common shortcut called the delta‑gamma approximation, a second‑order Taylor expansion, to approximate the portfolio profit and loss. They then combine that approximation with the skew‑t return model and a Cornish‑Fisher expansion for Value‑at‑Risk to obtain tractable expressions for the portfolio mean and variance. From those, they give closed‑form formulas for the optimal option weights. In words, the Sharpe‑maximizing weights are an explicit function of the portfolio sensitivity matrix Q, the expected change u, and the vector of option values v; a similar closed form holds for the R‑VaR maximizer but with an extra constant that depends on the VaR tail parameter.

The paper includes numerical experiments on portfolios of at‑the‑money options written on five stocks (DIS, XOM, PFE, MO and INTC). The experiments show that the allocations that maximize the Sharpe ratio differ from those that maximize the R‑VaR ratio. For example, in one call‑option case the allocation to MO differed by about 9% between the two objectives. They also report that solving these ratio problems directly tends to produce highly leveraged portfolios, which may be impractical for real investors.

Because the analytic optima can be extreme, the authors also solve constrained versions with box limits on position sizes. Imposing these limits makes the Sharpe‑ and R‑VaR‑optimized portfolios much more similar in practice. The paper therefore provides both closed‑form guidance and a practical route to safe implementations.