Uncertainty and stochastic theories on European options valuation and their delta and vega risks

Uncertainty finance presents alternative models for derivative valuation relevant to markets willing to consider subjective information or expert criterium in their operation. This paper proposes a methodology based on experimental data for comparing the prices and the delta and vega risks for European options via the uncertain Liu stock model with respect to the stochastic Black and Scholes model. To this end, the Greek letters delta and vega are estimated from the uncertain approach, and then two comparison criteria based on order relations and matrix norm metrics are established. An extensive numerical experiment shows the measurements and comparisons from a set of parameters that incorporate different market parameters and expert views. The empirical results are summarised in a data-driven set of approximate Facts. They suggest that the inclusion of subjective information determines surfaces of prices and risks that move away from the stochastic benchmark according to well-identifiable parametric regions, which could imply, through uncertainty, new projections in the performance and cost of hedging. In the end, two delta hedging examples under uncertainty are discussed, showing satisfactory results of hedging, performance and cost