Exposure valuations and their capital requirements

Abstract

Risk exposures are defined to be the change in financial valuation induced by a movement in an underlying factor. For derivative valuations, the underlying asset's price is the primary determining factor. When there is no move in the underlying factor there is also no change in valuation and therefore exposure functions are zero at zero. With delta hedging the exposures may also be taken to have zero derivatives at zero. As a consequence the first order of exposure is quadratic variation. These considerations lead us to value exposures in units of quadratic variation as the numeraire. The theory of acceptable risks is then extended to acceptable exposures with pricing reformulated in quadratic variation units. Explicit computations are made using the hyperbolic cosine for the numeraire exposure and the bilateral gamma law for the motion of the underlying asset price. Capital requirements for risk exposures are defined using measure distortions to develop a conservative exposure valuation. Computations illustrate applications to options on ten underlying assets over a six year period. The capital requirements increase with moneyness and volatility and decrease with maturity with a premium for puts over calls.