We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σminand σmax. These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the 'pricing' volatility is selected dynamically from the two extreme values, σmin, σmax, according to the convexity of the value-function. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model captures the importance of diversification in managing derivatives positions. It can be used systematically to construct efficient hedges using other derivatives in conjunction with the underlying asset.hedging, volatility risk,
We present an algorithm for hedging option portfolios and custom-tailored derivative securities, which uses options to manage volatility risk. The algorithm uses a volatility band to model heteroskedasticity and a non- linear partial differential equation to evaluate worst-case volatility scenarios for any given forward liability structure. This equation gives sub-additive portfolio prices and hence provides a natural ordering of prefer- ences in terms of hedging with options. The second element of the algorithm consists of a portfolio optim- ization taking into account the prices of options available in the market. Several examples are discussed, including possible applications to market-making in equity and foreign-exchange derivatives.Uncertain volatility, dynamic hedging, hedging with options,
We i n troduce a new class of strategies for hedging derivative securities in the presence of transaction costs assuming lognormal continuous-time prices for the underlying asset. We do not assume necessarily that the payo is convex as in Leland 11] or that transaction costs are small compared to the price changes between portfolio adjustments, as in Hoggard, Whalley and Wilmott 8]. The type of hedging strategy to be used depends on the value of the Leland number A = q 2 k p t , where k is the round-trip transaction cost, is the volatility of the underlying asset, and tis the time-lag between transactions. If A < 1 it is possible to implement modi ed Black-Scholes delta-hedging strategies, but not otherwise. We propose new hedging strategies that can be used with A 1 to control e ectively hedging risk and transaction costs. These strategies are associated with the solution of a nonlinear obstacle problem for a di usion equation with volatility A = p 1 + A. In these strategies, there are periods in which rehedging takes place after each interval t and other periods in which a static strategy is required. The solution to the obstacle problem is simple to calculate, and closed-form solutions exist for many problems of practical interest.
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