Loanable funds versus money creation in banking: a benchmark result

We establish a benchmark result for the relationship between the loanable-funds and the money-creation approach to banking. In particular, we show that both processes yield the same allocations when there is no uncertainty. In such cases, using the much simpler loanable-funds approach as a shortcut does not imply any loss of generality. When there is aggregate risk with complete contracts and complete markets, we indicate that a restricted equivalence result holds.

Pricing Product Options and Using Them to Complete Markets for Functions of Two Underlying Asset Prices

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The first perspective uses a geometric two-dimensional Brownian motion model. The second inverts two-dimensional characteristic functions. The third uses a bootstrapped physical measure to propose a risk charge minimizing hedge using options on the two underlying assets. The options are priced at the cost of the hedge plus the risk charge.

Sharing Asymmetric Tail Risk: Smoothing, Asset Prices and Terms of Trade

Crises and tail events have asymmetric effects across borders, raising the value of arrangements improving insurance of macroeconomic risk. Using a two-country DSGE model, we provide an analytical and quantitative analysis of the channels through which countries gain from sharing (tail) risk. Riskier countries gain in smoother consumption but lose in relative wealth and average consumption. Safer countries benefit from higher wealth and better average terms of trade. Calibrated using the empirical distribution of moments of GDP-growth across countries, the model suggests non-negligible quantitative effects. We offer an algorithm for the correct solution of the equilibrium using DSGE models under complete markets, at higher order of approximation.

Dynamically complete markets under Brownian motion

This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

Competitively-Issued Convertible Bank Notes in a Theory of Finance: Earl Thompson Meets Fischer Black

In this paper, I show the validity of and the relationship between two previously unrelated claims in monetary theory. The first claim, made by Earl Thompson, is that privately-issued bank notes pay a positive rate of return in a competitive equilibrium. The second claim, made by Fischer Black, is that it is possible to have a gold standard in which the gold reserves of the central bank are near zero. I show that both of these claims are correct under the assumption of complete markets and perfect commitment. The link between these claims is the Black-Scholes equation applied to convertible bank notes. In commodity-based monetary systems, bank notes are perpetual American options. I extend the model to consider the implications of a lack of commitment on the part of the bank and incomplete markets. I show that both arguments break down when banks lack commitment to redemption or markets are incomplete. I conclude with implications for macroeconomic theory.

Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete‐time markets

We examine the connection between discrete‐time models of financial markets and the celebrated Black–Scholes–Merton (BSM) continuous‐time model in which “markets are complete.” Suppose that (a) the probability law of a sequence of discrete‐time models converges to the law of the BSM model and (b) the largest possible one‐period step in the discrete‐time models converges to zero. We prove that, under these assumptions, every bounded and continuous contingent claim can be asymptotically synthesized, controlling for the risks taken in a manner that implies, for instance, that an expected‐utility‐maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete‐time economies as she can at the continuous‐time limit. Hence, in economically significant ways, many discrete‐time models with frequent trading resemble the complete‐markets model of BSM.