Summary: Journal of Econometrics 116 (2003) 947
Nonparametric option pricing under shape
Yacine A t-Sahaliaa;
, Je erson Duarteb
aDepartment of Economics, Princeton University and NBER, Princeton, NJ 08544-1021, USA
bDepartment of Finance and Business Economics, University of Washington, 267 MacKenzie Hall,
Box 353200, Seattle, WA 98195-3200, USA
Frequently, economic theory places shape restrictions on functional relationships between eco-
nomic variables. This paper develops a method to constrain the values of the ÿrst and second
derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate
the state price density (SPD), or risk-neutral density, implicit in the market prices of options.
The option pricing function must be monotonic and convex. Simulations demonstrate that non-
parametric estimates can be quite feasible in the small samples relevant for day-to-day option
pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P 500 option
prices, we show that unconstrained nonparametric estimators violate the constraints during more
than half the trading days in 1999, unlike the constrained estimator we propose.
c 2003 Elsevier B.V. All rights reserved.