Journal of Risk

Markets, profits, capital, leverage and return

Peter Carr, Dilip B. Madan, Juan Jose Vicente Alvarez


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.

Sorry, our subscription options are not loading right now

Please try again later. Get in touch with our customer services team if this issue persists.

New to View our subscription options

You need to sign in to use this feature. If you don’t have a account, please register for a trial.

Sign in
You are currently on corporate access.

To use this feature you will need an individual account. If you have one already please sign in.

Sign in.

Alternatively you can request an individual account here