Markets, profits, capital, leverage and return

Cherny and Madan’s theory of two-price markets yields closed forms for bid and ask prices. By defining profits as the difference between the mid quote and the risk-neutral expectation, and capital as the difference between the ask and bid prices, we obtain precise expressions for profit, capital and, hence, return. New expressions are developed for the bid and ask prices in terms of the sensitivity of the inverse distribution function to the quantile level. The latter turns out to be a measure of risk exposure at the quantile level. The theory is illustrated on unhedged exposures in the Black-Scholes-Merton model, followed by variance swaps and call options for variance gamma underliers. It is argued that markets should economize capital and, furthermore, that the maximization of expected utility may involve an uneconomic use of capital. We further observe that stock positions should be revised downward from zero delta in left-skewed markets in response to the target gamma when minimizing capital commitments.