Exposure Valuations and their Capital Requirements

Abstract

Risk exposures are defined as the instantaneous changes in financial valuations in response to movements in underlying factors. In derivative valuations, the underlying asset's price is the primary determinant. In the absence of movement in an underlying factor the change in valuation must be zero. Hence exposure functions are zero at zero and nonzero exposures cannot be constants. Admitting delta hedging the exposures have zero derivatives at zero as well. They are therefore to first order exposed to quadratic variation. Movements in log prices are modeled using limit laws that have infinite aggregate jump arrival rates. The arrival rates are then not normalizable to a probability. These considerations lead us to value exposures in units of a numeraire exposure that is itself to first order, quadratic variation. The theory of acceptable risks is extended to acceptable exposures with pricing reformulated in nonconstant numeraire exposure 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 then 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. The capital requirements mirror levels of theta induced by jump risk exposures that serve as signals for anticipated increases in implied volatilities.

Keywords: Fundamental Theorem of Asset Pricing, Bilateral Gamma Laws, Measure Distortions, Capital Requirements.

JEL Classification: G10, G11, G12.