Markets, Profits, Capital, Leverage and Return

31 Pages Posted: 20 Sep 2010  

Peter Carr

New York University (NYU) - Courant Institute of Mathematical Sciences

Juan Jose Vicente Alvarez

Morgan Stanley

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

Date Written: August 18, 2010

Abstract

The theory of two price markets of Cherny and Madan (2010) yields closed forms for bid and ask prices. Defining profits as the difference between the mid quote and the risk neutral expectation and capital as difference between the ask and the bid price one obtains precise expressions for these entities and thereby also returns. 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 Merton Scholes model, followed by variance swaps and call options for variance gamma underliers. It is argued that markets should economize capital and furthermore the maximization of expected utility may involve an uneconomic use of capital. We further observe that stock positions should be revised downwards from zero delta in left skewed markets in response to the target gamma when minimizing capital committments.

Keywords: Acceptable Risks, Bid and Ask Prices, Variance Gamma Model, Variance Swaps

JEL Classification: G10, G12, G13

Suggested Citation

Carr, Peter and Vicente Alvarez, Juan Jose and Madan, Dilip B., Markets, Profits, Capital, Leverage and Return (August 18, 2010). Available at SSRN: https://ssrn.com/abstract=1679503 or http://dx.doi.org/10.2139/ssrn.1679503

Peter P. Carr

New York University (NYU) - Courant Institute of Mathematical Sciences ( email )

251 Mercer Street
New York, NY 10012
United States

Juan Jose Vicente Alvarez

Morgan Stanley ( email )

1585 Broadway
New York, NY 10036
United States

Dilip B. Madan (Contact Author)

University of Maryland - Robert H. Smith School of Business ( email )

College Park, MD 20742-1815
United States
301-405-2127 (Phone)
301-314-9157 (Fax)

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