Valuation of portfolios under uncertain volatility: Black-Scholes-Barenblatt equations and the static hedging

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The famous Black-Scholes (BS) model used in the option pricing theory contains two parameters - a volatility and an interest rate. Both parameters should be determined before the price evaluation procedure starts. Usually one use the historical data to guess the value of these parameters. For short lifetime options the interest rate can be estimated in proper way, but the volatility estimation is, as well in this case, more demanding. It turns out that the volatility should be considered as a function of the asset prices and time to make the valuation self consistent. One of the approaches to this problem is the method of uncertain volatility and the static hedging. In this case the envelopes for the maximal and minimal estimated option price will be introduced. The envelopes will be described by the Black - Scholes - Barenblatt (BSB) equations. The existence of the upper and lower bounds for the option price makes it possible to develop the worse and the best cases scenario for the given portfolio. These estimations will be financially relevant if the upper and lower envelopes lie relatively narrow to each other. One...