Uncertain volatility and the risk-free synthesis of derivatives

To price contingent claims in a multidimensional frictionless security market it is sufficient that the volatility of the security process is a known function of price and time. In this note we introduce optimal and risk-free strategies for intermediaries in such markets to meet their obligations when the volatility is unknown, and is only assumed to lie in some convex region depending on the prices of the underlying securities and time. Our approach is underpinned by the theory of totally non-linear parabolic partial differential equations (Krylov and Safanov, 1979; Wang, 1992) and the non-stochastic approach to Ito's formation first introduced by Follmer (1981a,b). In these more general conditions of unknown volatility, the optimal risk-free trading strategy will, necessarily, produce an unpredictable surplus over the minimum assets required at any time to meet the liabilities. This surplus, which could be released to the intermediary or to the client, is not required to meet the contingent claim. One sees that the effect of unknown volatility is the creation of a 'with profits' policy, where a premium is paid at the beginning, the contingent claim is collected at the terminal time, but that in addition an unpredictable surplus available as well. The risk-free initial premium required to meet the contingent claim is given by the solution to the Dirichlet problem for a totally non-linear parabolic equation of the Pucci-Bellman type. The existence of a risk-free strategy starting with this minimum sum is dependent upon theorems ensuring the regularity of the solution and upon a non-probabilistic understanding of Ito's change of variable formulae. To illustrate the ideas we give a very simple example of a one-dimensional barrier option where the maximum Black-Scholes price of the option over different fixed values for the volatility lying in an interval always underestimates the risk-free 'price' under the assumption that the volatility can vary within the same interval. This paper puts together rather standard mathematical ideas. However, the author hopes that the overall result is more than the sum of its parts. The ability to hedge under conditions of uncertain volatility seems to be of considerable practical importance. In addition it would be interesting if these ideas explained some features in the design of existing contracts.